751.24 Retaining Walls: Difference between revisions

From Engineering_Policy_Guide
Jump to navigation Jump to search
EPGsysop (talk | contribs)
Per BR, guidance for bridge deck drain, approach drainage, roadway drainage or collected drain water discharging into a MSE wall backfill area was revised
→‎751.24.2.1 Design: updated per RR3960
 
(33 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{|style="padding: 0.3em; margin-left:5px; margin-right:2px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#ffddcc" width="220px" align="right"
<div style="float: right; margin-left: 30px; margin-bottom: 30px;">__TOC__</div>
|-
|[[media:751.24 LFD Retaining Walls Sept 2011.pdf|'''Printable Version of September 2011 LFD Retaining Walls Info''']]
|-
|align="left"|EPG 751.24 LFD Retaining Walls presents the very latest information, but this pdf file may be helpful for those wanting to easily print the LFD seismic information as it was in September 2011.
|}


==751.24.1 General==
==751.24.1 General==
{|style="padding: 0.3em; margin-left:6px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
{| style="padding: 0.3em; margin-left:6px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
|-
|-
|'''Additional Information'''
| '''Additional Information'''
|-
|-
|AASHTO 5.1
| LRFD 11
|}
|}
Retaining wall shall be designed to withstand lateral earth and water pressures, including any live and dead load surcharge, the self weight of the wall, temperature and shrinkage effect, live load and collision forces, and earthquake loads in accordance with the general principles of AASHTO Section 5 and the general principles specified in this article.
For understanding the equivalency of seismic design category (SDC) and seismic zone for LRFD, see [[751.9_Bridge_Seismic_Design#751.9.1.1_Applicability_of_Guidelines_and_Seismic_Design_Philosophy|EPG 751.9.1.1]]  and [https://epg.modot.org/forms/general_files/BR/Bridge_Seismic_Design_Flowchart.pdf Bridge Seismic Design Flowchart].
 
Retaining wall shall be designed to withstand lateral earth and water pressures, including any live and dead load surcharge, the self weight of the wall, temperature and shrinkage effect, live load and collision forces, and earthquake loads in accordance with the general principles of LRFD Section 11 and the general principles specified in this article.
 
Seismic analysis provisions shall not be ignored for walls that support another structure (i.e. support abutment fill or building) in SDC B or C (seismic zone 2 or 3). No-seismic-analysis provisions may be considered for walls that do not support another structure (i.e. most of District walls) in SDC B or C (seismic zone 2 or 3) in accordance with LRFD 11.5.4.2 and Geotech report. Seismic analysis provisions shall not be ignored for walls in SDC D (seismic zone 4).


===751.24.1.1 Wall Type Selection===
===751.24.1.1 Wall Type Selection===
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
{| style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
|-
|-
|'''Additional Information'''
| '''Additional Information'''
|-
|-
|AASHTO 5.2.1
| LRFD 11
|}
|}
Selection of wall type shall be based on an assessment of the magnitude and direction of loading, depth to suitable foundation support, potential for earthquake loading, presence of deleterious environmental factors, wall site cross-sectional geometry, proximity of physical constraints, tolerable and differential settlement, facing appearance and ease and cost of construction.
Selection of wall type shall be based on an assessment of the magnitude and direction of loading, depth to suitable foundation support, potential for earthquake loading, presence of deleterious environmental factors, wall site cross-sectional geometry, proximity of physical constraints, tolerable and differential settlement, facing appearance and ease and cost of construction.
Line 27: Line 26:
* Mechanically Stabilized Earth Retaining Walls
* Mechanically Stabilized Earth Retaining Walls
* Cast-In-Place Concrete Cantilever Retaining Walls
* Cast-In-Place Concrete Cantilever Retaining Walls
:* Cantilever Walls on Spread Footings
:Cantilever Walls on Spread Footings
:* Cantilever Wall on Pile Footings
:Cantilever Wall on Pile Footings
:* L-Shaped Walls on Spread Footings
:L-Shaped Walls on Spread Footings


'''Mechanically Stabilized Earth (MSE) Retaining Walls'''
'''Mechanically Stabilized Earth (MSE) Retaining Walls'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
|-
|-
|'''Additional Information'''
| '''Additional Information'''
|-
|-
|AASHTO 5.2.1.4 & 5.8
|LRFD 11.10,</br>FHWA-NHI-10-024 and 025
|}
|}
MSE retaining walls use precast block or panel like facing elements combined with either metallic or geosynthetic tensile reinforcements in the soil mass. MSE walls are preferred over cast-in-place walls because they are usually more economical. Other advantages include a wide variety of design styles, ease and speed of installation, and their ability to accommodate total and differential settlements. Wall design heights upwards of 80 ft. are technically feasible (FHFW-SA-96-071). MSE walls may be used to retain fill for end bents of bridge structures.
MSE retaining walls use precast block or panel like facing elements combined with either metallic or geosynthetic tensile reinforcements in the soil mass. MSE walls are preferred over cast-in-place walls because they are usually more economical. Other advantages include a wide variety of design styles, ease and speed of installation, and their ability to accommodate total and differential settlements. Wall design heights upwards of 80 ft. are technically feasible (FHFW-SA-96-071). MSE walls may be used to retain fill for end bents of bridge structures.


Situations exist where the use of MSE walls is either limited or not recommended. Some obstacles such as drop inlets, sign truss pedestals or footings, and fence posts may be placed within the soil reinforcement area, however, these obstacles increase the difficulty and expense of providing sufficient soil reinforcement for stability. Box culverts and highway drainage pipes may run through MSE walls, but it is preferable not to run the pipes close to or parallel to the walls. Utilities other than highway drainage should not be constructed within the soil reinforcement area. Be cautious when using MSE walls in a flood plain. A flood could cause scouring around the reinforcement and seepage of the backfill material. Soil reinforcements should not be used where exposure to ground water contaminated by acid mine drainage or other industrial pollutants as indicated by a low pH and high chlorides and sulfates exist. Galvanized metallic reinforcements shall not be used where stray electrical ground currents could occur as would be present near an electrical substation.  
Situations exist where the use of MSE walls is either limited or not recommended. Some obstacles such as drop inlets, sign truss pedestals or footings, and fence posts may be placed within the soil reinforcement area, however, these obstacles increase the difficulty and expense of providing sufficient soil reinforcement for stability. Box culverts and highway drainage pipes may run through MSE walls, but it is preferable not to run the pipes close to or parallel to the walls. Utilities other than highway drainage should not be constructed within the soil reinforcement area. Be cautious when using MSE walls in a floodplain. A flood could cause scouring around the reinforcement and seepage of the backfill material. Soil reinforcements should not be used where exposure to ground water contaminated by acid mine drainage or other industrial pollutants as indicated by a low pH and high chlorides and sulfates exist. Galvanized metallic reinforcements shall not be used where stray electrical ground currents could occur as would be present near an electrical substation.  


Sufficient right of way is required to install the soil reinforcement which extends into the backfill area at least 8 ft., 70 % of the wall height or as per design requirements, whichever is greater. For more information regarding soil reinforcement length and excavation limits see [[751.6 General Quantities#751.6.2.17 Excavation|EPG 751.6.2.17 Excavation]]. Finally, barrier curbs constructed over or in line with the front face of the wall shall have adequate room provided laterally between the back of the wall facing and the curb or slab so that load is not directly transmitted to the top wall facing units.  
Sufficient right of way is required to install the soil reinforcement which extends into the backfill area at least 8 feet, 70 percent of the wall height or as per design requirements set forth in [[751.6 General Quantities#751.6.2.17 Excavation|EPG 751.6.2.17 Excavation]], whichever is greater. For more information regarding soil reinforcement length, excavation limits and Minimum Embedment Depth of MSEW, see [[751.6 General Quantities#751.6.2.17 Excavation|EPG 751.6.2.17 Excavation]].
 
Finally, barrier curbs constructed over or in line with the front face of the wall shall have adequate room provided laterally between the back of the wall facing and the curb or slab so that load is not directly transmitted to the top of MSE wall or facing units.
----
<div style="text-align: center; margin-left: auto; margin-right: auto;">
<gallery mode=packed widths=200px heights=200px>
File:751.24.1.1_barrier_top_MSE_wall-01.jpg| '''Barrier at Top of MSE Wall'''
File:blank.jpg|
File:751.24.1.1_barrier_front_MSE_wall-01.jpg| '''Barrier in Front of MSE Wall'''
</gallery>
</div>
----


'''Concrete Cantilever Wall on Spread Footing'''
'''Concrete Cantilever Wall on Spread Footing'''
Line 48: Line 59:
Concrete cantilever walls derive their capacity through combinations of dead weight and structural resistance. These walls are constructed of reinforced concrete.
Concrete cantilever walls derive their capacity through combinations of dead weight and structural resistance. These walls are constructed of reinforced concrete.


Concrete cantilever walls are used when MSE walls are not a viable option. Cantilever walls can reduce the rock cut required and can also provide solutions when there are right of way restrictions. Concrete walls also provide better structural capacity when barrier curbs on top of the walls are required.
Concrete cantilever walls are used when MSE walls are not a viable option. Cantilever walls can reduce the rock cut required and can also provide solutions when there are right of way restrictions. Concrete walls also provide better structural capacity when barrier or railing on top of the walls are required.


Counterforts are used on rare occasions. Sign-board type retaining walls are a special case of counterfort retaining walls. They are used where the soil conditions are such that the footings must be placed well below the finished ground line. For these situations the wall is discontinued 12 in. below the ground line or below the frost line. Counterforts may also be a cost-savings option when the wall height approaches 20 ft. (''Foundation Analysis and Design'' by Joseph E. Bowles, 4th ed., 1988). However, other factors such as poor soil conditions, slope of the retained soil, wall length and uniformity in wall height should also be considered before using counterforts.
Counterforts are used on rare occasions. Sign-board type retaining walls are a special case of counterfort retaining walls. They are used where the soil conditions are such that the footings must be placed well below the finished ground line. For these situations the wall is discontinued 12 in. below the ground line or below the frost line. Counterforts may also be a cost-savings option when the wall height approaches 20 ft. (''Foundation Analysis and Design'' by Joseph E. Bowles, 4th ed., 1988). However, other factors such as poor soil conditions, slope of the retained soil, wall length and uniformity in wall height should also be considered before using counterforts.
Line 54: Line 65:
'''Concrete Cantilever Wall on Pile Footing'''
'''Concrete Cantilever Wall on Pile Footing'''


Concrete cantilever walls on pile footings are used when the soil conditions do not permit the use of spread footings. These walls are also used when an end bent requires wings longer than 22 feet. In these cases a stub wing is left attached to the end bent and the rest of the wing is detached to become a retaining wall.
Concrete cantilever walls on pile footings are used when the soil conditions do not permit the use of spread footings. These walls are also used when an end bent requires wings longer than 22 feet for seismic category A or 17 ft. for seismic category B, C or D. In these cases a stub wing is left attached to the end bent and the rest of the wing is detached to become a retaining wall as shown in [[751.35_Concrete_Pile_Cap_Integral_End_Bents#751.35.3.5_Wing_and_Detached_Wing_Walls|751.35.3.5 Wing and Detached Wing Walls]].


'''Concrete L-Shaped Retaining Wall on Spread Footings'''
'''Concrete L-Shaped Retaining Wall on Spread Footings'''
Line 63: Line 74:


===751.24.1.2 Loads===
===751.24.1.2 Loads===
Conventional retaining walls: Loads shall be determined in accordance with LRFD 3 and 11.6.
MSE retaining walls: Loads shall be determined in accordance with LRFD 3 and 11.10.
Note: For guidance, follow the [[751.40_LFD_Widening_and_Repair#751.40.8.15_Cast-In-Place_Concrete_Retaining_Walls|751.40.8.15 Cast -In-Place Concrete Retaining Walls]] and modify guidance of ASD as necessary to meet LRFD requirements until this section is modified for LRFD.


'''Dead Loads'''
'''Dead Loads'''
Line 68: Line 84:
Dead loads shall be determined from the unit weights in [[751.2 Loads#751.2.1.1 Dead Load |EPG 751.2.1.1 Dead Load]].
Dead loads shall be determined from the unit weights in [[751.2 Loads#751.2.1.1 Dead Load |EPG 751.2.1.1 Dead Load]].


'''Equivalent Fluid Pressure (Earth Pressures)'''
==751.24.2 Mechanically Stabilized Earth (MSE) Walls==
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
 
|-
===751.24.2.1 Design===
|'''Additional Information'''
 
|-
Designs of Mechanically Stabilized Earth (MSE) walls shall be completed by consultants or contractors in accordance with Section 11.10 of LRFD specifications, FHWA-NHI-10-024 and FHWA-NHI-10-025 for LRFD. [https://www.modot.org/bridge-pre-qualified-products-list Bridge Pre-qualified Products List (BPPL)] provided on MoDOT's web page and in Sharepoint contains a listing of facing unit manufacturers, soil reinforcement suppliers, and wall system suppliers which have been approved for use. See [http://www.modot.org/business/standards_and_specs/SpecbookEPG.pdf#page=11 Sec 720] and [http://www.modot.org/business/standards_and_specs/SpecbookEPG.pdf#page=14 Sec 1010] for additional information. The Geotechnical Section is responsible for checking global stability of permanent MSE wall systems, which should be reported in the Foundation Investigation Geotechnical Report. For MSE wall preliminary information, see [[751.1_Preliminary_Design#751.1.4.3_MSE_Walls|EPG 751.1.4.3 MSE Walls]]. For design requirements of MSE wall systems and temporary shoring (including temporary MSE walls), see [[:Category:720_Mechanically_Stabilized_Earth_Wall_Systems#720.2_Design_Requirements|EPG 720 Mechanically Stabilized Earth Wall Systems]]. For staged bridge construction, see [[751.1_Preliminary_Design#751.1.2.11_Staged_Construction|EPG 751.1.2.11 Staged Construction]].
|AASHTO 3.20.1
|}


For determining equivalent earth pressures for Group Loadings I through VI the Rankine Formula for Active Earth Pressure shall be used.
For seismic design requirements, see [https://epg.modot.org/forms/general_files/BR/Bridge_Seismic_Design_Flowchart.pdf Bridge Seismic Design Flowchart]. References for consultants and contractors include Section 11.10 of LRFD, FHWA-NHI-10-024 and FHWA-NHI-10-025.


Rankine Formula: <math>P_a = \frac{1}{2}C_a\gamma_sH^2</math> where:
'''Design Life'''  
:''C<sub>a</sub>'' = <math>\cos\delta\Bigg[\frac{\cos\delta - \sqrt{\cos^2\delta - \cos^2\phi}}{\cos\delta + \sqrt{\cos^2\delta - \cos^2\phi}}\Bigg]</math> = coefficient of active earth pressure


:''P<sub>a</sub>'' = equivalent active earth pressure
* 75 year minimum for permanent walls (if retained foundation require 100 year than consider 100 year minimum design life for wall).


:''H'' = height of the soil face at the vertical plane of interest
'''Global stability:'''


:<math>\boldsymbol{\gamma_s}</math> = unit weight of soil
Global stability will be performed by Geotechnical Section or their agent.


:<math>\boldsymbol{\delta}</math>= slope of fill in degrees
'''MSE wall contractor/designer responsibility:'''


:<math>\boldsymbol{\phi}</math> = angle of internal friction of soil in degrees
MSE wall contractor/designer shall perform following analysis in their design for all applicable limit states.
[[image:751.24.1.2.jpg|center|485px]]


'''Example'''
:* External Stability
::* Limiting Eccentricity
::* Sliding
::* Factored Bearing Pressure/Stress ≤ Factored Bearing Resistance
:* Internal Stability
::* Tensile Resistance of Reinforcement
::* Pullout Resistance of Reinforcement
::* Structural Resistance of Face Elements
::* Structural Resistance of Face Element Connections
:* Compound Stability
:: Capacity/Demand ratio (CDR) for bearing capacity shall be ≥ 1.0
:: <math>Bearing\ Capacity\ (CDR)  = \frac{Factored\ Bearing\ Resistance}{Maximum\ Factored\ Bearing\ Stress} \ge 1.0</math>
:: Strength Limit States:
:: Factored bearing resistance = Nominal bearing resistance from Geotech report X Minimum Resistance factor (0.65, Geotech report)  LRFD Table 11.5.7-1 


Given:
:: Extreme Event I Limit State:
:: Factored bearing resistance = Nominal bearing resistance from Geotech report X Resistance factor
:: Resistance factor = 0.9  LRFD 11.8.6.1


:''δ'' = 3:1 (H:V) slope
:: Factored bearing stress shall be computed using a uniform base pressure distribution over an effective width of footing determined in accordance with the provisions of LRFD 10.6.3.1 and 10.6.3.2, 11.10.5.4  and Figure 11.6.3.2-1 for foundation supported on soil or rock.


:''ϕ'' = 25°
:: B’ = L – 2e


:''γ<sub>s</sub>'' = 0.120 kcf
:: Where,
::: L = Soil reinforcement length (For modular block use B in lieu of L as per LRFD 11.10.2-1)
::: B’ = effective width of footing
::: e = eccentricity
::: Note: When the value of eccentricity e is negative then B´ = L.  


:''H'' = 10 ft
::Capacity/Demand ratio (CDR) for overturning shall be ≥ 1.0
::<math>Overtuning\ (CDR)  = \frac{Total\ Factored\ Resisting\ Moment}{Total\ Factored\ Driving\ Moment} \ge 1.0</math>


''δ'' = arctan<math>\Big[\frac{1}{3}\Big]</math> = 18.4°
::Capacity/Demand ratio (CDR) for eccentricity shall be ≥ 1.0
::<math>Eccentricity\ (CDR)  = \frac{e_{Limit}}{e_{design}} \ge 1.0</math>


''C<sub>a</sub>'' = <math>\cos (18.4^\circ)\Bigg[\frac{\cos(18.4^\circ) - \sqrt{\cos^2(18.4^\circ) - \cos^2(25^\circ)}}{\cos(18.4^\circ) + \sqrt{\cos^2(18.4^\circ) - \cos^2(25^\circ)}}\Bigg]</math> = 0.515
::Capacity/Demand ratio (CDR) for sliding shall be ≥ 1.0 &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.10.5.3 & 10.6.3.4
::<math>Sliding\ (CDR) = \frac{Total\ Factored\ Sliding\ Resistance}{Total\ Factored\ Active\ Force} \ge 1.0</math>


''P<sub>a</sub>'' = (1/2)(0.515)(0.120 kips/ft<sup>3</sup>)(10 ft)<sup>2</sup> = 3.090 kips per foot of wall length
::Capacity/Demand ratio (CDR) for internal stability shall be ≥ 1.0


The ''ϕ'' angle shall be determined by the Materials Division from soil tests. If the ''ϕ'' angle cannot be provided by the [http://wwwi/intranet/cm/default.htm Construction and Materials Division] a ''ϕ'' angle of 27 degrees shall be used.
::Eccentricity, (e) Limit for Strength Limit State: &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.6.3.3 & C11.10.5.4
::: For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, L or (e ≤ 0.33L).


Drainage shall be provided to relieve water pressure from behind all cast-in-place concrete retaining walls. If adequate drainage can not be provided then walls shall be designed to resist the maximum anticipated water pressure.
::Eccentricity, (e) Limit for Extreme Event I (Seismic): &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.6.5.1
:::For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, L or (e ≤ 0.33L) for  γ<sub>EQ</sub> = 0.0 and middle eight-tenths of the base width, L or (e ≤ 0.40L) for  γ<sub>EQ</sub> = 1.0.  For γ<sub>EQ</sub>  between 0.0 and 1.0, interpolate e value linearly between 0.33L and 0.40L. For γ<sub>EQ</sub>  refer to LRFD 3.4.


'''Surcharge Due to Point, Line and Strip Loads'''
:::Note: Seismic design shall be performed for γ<sub>EQ</sub> = 0.5


Surcharge due to point and line loads on the soil being retained shall be included as dead load surcharge. The effect of these loads on the wall may be calculated using Figure 5.5.2B from AASHTO.
::Eccentricity, (e) Limit for Extreme Event II:
:::For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle eight-tenths of the base width, L or (e ≤ 0.40L).


Surcharge due to strip loads on the soil being retained shall be included as a dead load surcharge load. The following procedure as described in ''Principles of Foundation Engineering'' by Braja M. Das (1995) shall be applied to calculate these loads when strip loads are applicable. An example of this application is when a retaining wall is used in front of an abutment so that the wall is retaining the soil from behind the abutment as a strip load on the soil being retained by the wall.
'''General Guidelines'''


[[image:751.24.1.2 retaining.jpg|center|255px|thumb|<center>'''Retaining Wall in front of an Abutment'''</center>]]
* Drycast modular block wall (DMBW-MSE) systems are limited to a 10 ft. height in one lift.


The portion of soil that is in the active wedge must be determined because the surcharge pressure only affects the wall if it acts on the active wedge. The actual failure surface in the backfill for the active state can be represented by ABC shown in the figure below. An approximation to the failure surface based on Rankine's active state is shown by dashed line AD. This approximation is slightly unconservative because it neglects friction at the pseudo-wall to soil interface.
* Wetcast modular block wall (WMBW-MSE) systems are limited to a 15 ft. height in one lift.


The following variables are shown in the figure below:
* For Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems, top cap units shall be used and shall be permanently attached by means of a resin anchor system.


:''β'' = slope of the active failure plane in degrees
* For precast modular panel wall (PMPW-MSE) systems, capstone may be substituted for coping and either shall be permanently attached to wall by panel dowels.
:''δ'' = slope of fill in degrees
:''H'' = height of the pseudo-wall (fom the bottom of the footing).
:''L<sub>1</sub>'' = distance from back of stem to back of footing heel
:''L<sub>2</sub>'' = distance from footing heel to intersection of failure plane with ground surface


[[image:751.24.1.2 wedges.jpg|center|575px|thumb|<center>'''Determination of Active Wedges'''</center>]]
* For precast modular panel wall (PMPW-MSE) systems, form liners are required to produce all panels. Using form liner to produce panel facing is more cost effective than producing flat panels. Standard form liners are specified on the [https://www.modot.org/mse-wall-msew MSE Wall Standard Drawings]. Be specific regarding names, types and colors of staining, and names and types of form liner.


In order to determine ''β'', the following equation which has been derived from Rankine's active earth pressure theory must be solved by iteration:
* MSE walls shall not be used where exposure to acid water may occur such as in areas of coal mining.


:<math>\tan (-\beta) + \frac{1}{\tan (\beta - \phi)} - \frac{1}{\tan (\beta - \delta)} + \frac{1}{\tan (90^\circ + \phi + \delta - \beta)} = 0</math>
* MSE walls shall not be used where scour is a problem.


:''ϕ'' = angle of internal friction of soil in degrees
* MSE walls with metallic soil reinforcement shall not be used where stray electrical ground currents may occur as would be present near electrical substations.


A good estimate for the first iteration is to let ''β'' = 45° + (ϕ/2). In lieu of iterating the above equation a conservative estimate for ''β'' is 45°. Once β has been established, an estimate of L<sub>1</sub> is needed to determine L<sub>2</sub>. From the geometry of the variables shown in the above figure:
* No utilities shall be allowed in the reinforced earth if future access to the utilities would require that the reinforcement layers be cut, or if there is a potential for material, which can cause degradation of the soil reinforcement, to leak out of the utilities into the wall backfill, with the exception of storm water drainage.


:<math>L_2 = H\frac{\cos\delta\cos\beta}{\sin(\beta - \delta)}</math>
* All vertical objects shall have at least 4’-6” clear space between back of the wall facing and object for select granular backfill compaction and soil reinforcement skew limit requirements. For piles, see pipe pile spacers guidance.


The resultant pressure due to the strip load surcharge and its location are then determined. The following variables are shown in the figure below:
* The interior angle between two MSE walls should be greater than 70°. However, if unavoidable, then place [[751.50_Standard_Detailing_Notes#J._MSE_Wall_Notes_.28Notes_for_Bridge_Standard_Drawings.29|EPG 751.50 J1.41 note]] on the design plans.


:''q'' = load per unit area
* Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems may be battered up to 1.5 in. per foot. Modular blocks are also known as “segmental blocks”.
:''P<sub>s</sub>'' = resultant pressure on wall due only to surcharge earth pressure
:<math>\bar{z}</math> = location of P<sub>s</sub> measured from the bottom of the footing
:''L<sub>3</sub>'' = distance from back of stem to where surcharge pressure begins


[[image:751.24.1.2 surcharge.jpg|center|625px|thumb|<center>'''Surcharge Pressure on Retaining Wall'''</center>]]
* The friction angle used for the computation of horizontal forces within the reinforced soil shall be greater than or equal to 34°.


From the figure:
* For epoxy coated reinforcement requirements, see [[751.5 Structural Detailing Guidelines#751.5.9.2.2 Epoxy Coated Reinforcement Requirements|EPG 751.5.9.2.2 Epoxy Coated Reinforcement Requirements]].


:P<sub>s</sub> = <math>\frac{q}{90}\big[H(\theta_2 - \theta_1)\big]</math> where
* All concrete except facing panels or units shall be CLASS B or B-1.


:<math>\theta_1 = \arctan\Big[\frac{L_3}{H}\Big] \ and \ \theta_2 = \arctan\Big[\frac{L_2}{H}\Big]</math>
* The friction angle of the soil to be retained by the reinforced earth shall be listed on the plans as well as the friction angle for the foundation material the wall is to rest on.


:<math>\bar{z} = \frac{H^2(\theta_2 - \theta_1) - (R - Q) + 57.03L_4H}{2H(\theta_2 - \theta_1)}</math> where
* The following requirement shall be considered (from 2009_FHWA-NHI-10-024 MSE wall 132042.pdf, page 200-201) when seismic design is required:  
:* For seismic design category, SDC C or D (Zones 3 or 4), facing connections in modular block faced walls (MBW) shall use shear resisting devices (shear keys, pin, etc.) between the MBW units and soil reinforcement, and shall not be fully dependent on frictional resistance between the soil reinforcement and facing blocks. For connections partially dependent on friction between the facing blocks and the soil reinforcement, the nominal long-term connection strength T<sub>ac</sub>, should be reduced to 80 percent of its static value.


:<math>R = (L_2)^2(90^\circ - \theta_2) \ and \ Q = (L_3)^2(90^\circ - \theta_1)</math>
* Seismic design category and acceleration coefficients shall be listed on the plans for categories B, C and D. If a seismic analysis is required that shall also be noted on the plans. See [[751.50_Standard_Detailing_Notes#A._General_Notes|EPG 751.50 A1.1 note]].


When applicable, P<sub>s</sub> is applied to the wall in addition to other earth pressures. The wall is then designed as usual.
* Plans note ([[751.50_Standard_Detailing_Notes#J._MSE_Wall_Notes_.28Notes_for_Bridge_Standard_Drawings.29|EPG 751.50 J1.1]]) is required to clearly identify the responsibilities of the wall designer.


'''Live Load Surcharge'''
* Do not use Drycast modular block wall (DMBW-MSE) systems in the following locations:
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 3.20.3 & 5.5.2
|}
Live load surcharge pressure of not less than two feet of earth shall be applied to the structure when highway traffic can come within a horizontal distance equal to one-half of the wall height, measured from the plane where earth pressure is applied.


[[image:751.24.1.2 live load1.jpg|center|475px]]
::* Within the splash zone from snow removal operations (assumed to be 15 feet from the edge of the shoulder).


[[image:751.24.1.2 live load surcharge.jpg|center|575px|thumb|<Center>'''Live Load Surcharge'''</center>]]
::* Where the blocks will be continuously wetted, such as around sources of water.


:''P<sub>LLS</sub>'' = (2 ft.) ''γ<sub>s</sub> C<sub>a</sub> H'' = pressure due to live load surcharge only
::* Where blocks will be located behind barrier or other obstacles that will trap salt-laden snow from removal operations.


:''γ<sub>s</sub>'' = unit weight of soil (Note: AASHTO 5.5.2 specifies a minimum of 125 pcf for live load surcharge, MoDOT policy allows 120 pcf as given from the unit weights in [[751.2 Loads#751.2.1.1 Dead Load |EPG 751.2.1.1 Dead Load]].)
* Do not use Drycast modular block wall (DMBW-MSE) systems or Wetcast modular block wall (WMBW-MSE) systems in the following locations:
:''C<sub>a</sub>'' = coefficient of active earth pressure


:''H'' = height of the soil face at the vertical plane of interest
::* For structurally critical applications, such as containing necessary fill around structures.


The vertical live load surcharge pressure should only be considered when checking footing bearing pressures, when designing footing reinforcement, and when collision loads are present.
::* In tiered wall systems.
 
* For locations where Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems are not desirable, consider coloring agents and/or architectural forms using precast modular panel wall (PMPW-MSE) systems for aesthetic installations.


'''Live Load Wheel Lines'''
* For slab drain location near MSE Wall, see [[751.10 General Superstructure#General Requirements for Location and Spacing of Slab Drains|EPG 751.10.3.1 Drain Type, Alignment and Spacing]] and [[751.10 General Superstructure#751.10.3.3 General Requirements for Location of Slab Drains|EPG 751.10.3.3 General Requirements for Location of Slab Drains]].


Live load wheel lines shall be applied to the footing when the footing is used as a riding or parking surface.
* Roadway runoff should be directed away from running along face of MSE walls used as wing walls on bridge structures.
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 3.24.5.1.1 & 5.5.6.1
|}


Distribute a LL<sub>WL</sub> equal to 16 kips as a strip load on the footing in the following manner.
* Drainage:


:*Gutter type should be selected at the core team meeting.


:P = LL<sub>WL</sub>/E
:* When gutter is required without fencing, use Type A or Type B gutter (for detail, see [https://www.modot.org/media/16880 Std. Plan 609.00]).


::where E = 0.8X + 3.75
:* When gutter is required with fencing, use Modified Type A or Modified Type B gutter (for detail, see [https://www.modot.org/media/16871 Std. Plan 607.11]).


::::X = distance in ft. from the load to the front face of the wall
:* When fencing is required without gutter, place in tube and grout behind the MSE wall (for detail, see [https://www.modot.org/bridge-standard-drawings MSE Wall Standard Drawings - MSEW], Fence Post Connection Behind MSE Wall (without gutter).
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 3.24.2 & 3.30
|}


The wheel lines shall move 1 ft. from the barrier curb or wall to 1 ft. from the toe of the footing.
:* Lower backfill longitudinal drainage pipes behind all MSE walls shall be two-6” (Min.) diameter perforated PVC or PE pipe (See Sec 1013) unless larger sizes are required by design which shall be the responsibility of the District Design Division. Show drainage pipe size on plans. Outlet screens and cleanouts should be detailed for any drain pipe (shown on MoDOT MSE wall plans or roadway plans). Lateral non-perforated drain pipes (below leveling pad) are permitted by Standard Specifications and shall be sized by the District Design Division if necessary. Lateral outlet drain pipe sloped at 2% minimum.


[[image:751.24.1.2 wheel.jpg|center|350px]]
::* Identify on MSE wall plans or roadway plans drainage pipe point of entry, point of outlet (daylighting), 2% min. drainage slopes in between points to ensure positive flow and additional longitudinal drainage pipes if required to accommodate ground slope changes and lateral drainage pipes if required by design.


'''Collision Forces'''
::* Adjustment in the vertical alignment of the longitudinal drainage pipes from that depicted on the MSE wall standard drawings may be necessary to ensure positive flow out of the drainage system.
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
::* Identify on MSE wall plans or roadway plans the outlet ends of pipes which shall be located to prevent clogging or backflow into the drainage system. Outlet screens and cleanouts should be detailed for any drain pipe.
|'''Additional Information'''
|-
|AASHTO Figure 2.7.4B
|}


Collision forces shall be applied to a wall that can be hit by traffic. Apply a point load of 10 kips to the wall at a point 3 ft. above the finished ground line.
:* For more information on drainage, see [[#Drainage at MSE Walls|Drainage at MSE Walls]].


[[image:751.24.1.2 collision section.jpg|center|450px|thumb|<center>'''Section'''</center>]]
'''MSE Wall Construction: Pipe Pile Spacers Guidance'''


Distribute the force to the wall in the following manner:
For bridges not longer than 200 feet, pipe pile spacers or pile jackets shall be used at pile locations behind mechanically stabilized earth walls at end bents. Corrugated pipe pile spacers are required when the wall is built prior to driving the piles to protect the wall reinforcement when driving pile for the bridge substructure at end bents(s). Pile spacers or pile jackets may be used when the piles are driven before the wall is built. Pipe pile spacers shall have an inside diameter greater than that of the pile and large enough to avoid damage to the pipe when driving the pile. Use [[751.50 Standard Detailing Notes#E1. Excavation and Fill|EPG 751.50 Standard Detailing Note E1.2a]] on bridge plans.


:Force per ft of wall = (10 kips)/2L
For bridges longer than 200 feet, pipe pile spacers are required and the pile spacer shall be oversized to mitigate the effects of bridge thermal movements on the MSE wall. For HP12, HP14, CIP 14” and CIP 16” piles provide 24-inch inside diameter of pile spacer for bridge movement. Minimum pile spacing shall be 5 feet to allow room for compaction of the soil layers. Use [[751.50 Standard Detailing Notes#E1. Excavation and Fill|EPG 751.50 Standard Detailing Note E1.2b]] on bridge plans.


[[image:751.24.1.2 collision profile.jpg|center|350px|thumb|<center>'''Profile'''</center>]]
The bottom of the pipe pile spacers shall be placed 5 ft. min. below the bottom of the MSE wall leveling pad. The pipe shall be filled with sand or other approved material after the pile is placed and before driving. Pipe pile spacers shall be accurately located and capped for future pile construction.  


When considering collision loads, a 25% overstress is allowed for bearing pressures and a factor of safety of 1.2 shall be used for sliding and overturning.
Alternatively, for bridges shorter than or equal to 200 feet, the contractor shall be given the option of driving the piles before construction of the mechanically stabilized earth wall and placing the soil reinforcement and backfill material around the piling. In lieu of pipe pile spacers contractor may place pile jackets on the portion of the piles that will be in the MSE soil reinforced zone prior to placing the select granular backfill material and soil reinforcement. The contractor shall adequately support the piling to ensure that proper pile alignment is maintained during the wall construction. The contractor’s plan for bracing the pile shall be submitted to the engineer for review.  


'''Wind and Temperature Forces'''
Piling shall be designed for downdrag (DD) loads due to either method. Oversized pipe pile spacers with sand placed after driving or pile jacket may be considered to mitigate some of the effects of downdrag (DD) loads. Sizing of pipe pile spacers shall account for pile size, thermal movements of the bridge, pile placement plan, and vertical and horizontal placement tolerances.


These forces shall be disregarded except for special cases, consult the Structural Project Manager.
When rock is anticipated within the 5 feet zone below the MSE wall leveling pad, prebore into rock and prebore holes shall be sufficiently wide to allow for a minimum 10 feet embedment of pile and pipe pile spacer. When top of rock is anticipated within the 5 to 10 feet zone below the MSE wall leveling pad, prebore into rock to achieve a minimum embedment (pile only) of 10 feet below the bottom of leveling pad. Otherwise, the pipe pile spacer requires a minimum 5 feet embedment below the levelling pad. Consideration shall also be given to oversizing the prebore holes in rock to allow for temperature movements at integral end bents.  


When walls are longer than 84 ft., an expansion joint shall be provided.  
For bridges not longer than 200 feet, the minimum clearance from the back face of MSE walls to the front face of the end bent beam, also referred to as setback, shall be 4 ft. 6 in. (typ.) unless larger than 18-inch pipe pile spacer required. The 4 ft. 6 in. dimension is based on the use of 18-inch inside diameter pipe pile spacers & FHWA-NHI-10-24, Figure 5-17C, which will help ensure that soil reinforcement is not skewed more than 15° for nut and bolt reinforcement connections. Similarly, the minimum setback shall be determined when larger diameter pile spacers are required. For bridges longer than 200 feet, the minimum setback shall  be 5 ft. 6 in. based on the use of 24-inch inside diameter of pipe pile spacers. Other types of connections may require different methods for splaying. In the event that the minimum setback cannot be used, the following guidance for pipe pile spacers clearance shall be used: pipe pile spacers shall be placed 18 in. clear min. from the back face of MSE wall panels; 12 in. minimum clearance is required between pipe pile spacers and leveling pad and 18 in. minimum clearance is required between leveling pad and pile.


Contraction joint spacing shall not exceed 28 feet.
'''MSE Wall Plan and Geometrics'''


'''Seismic Loads'''
* A plan view shall be drawn showing a baseline or centerline, roadway stations and wall offsets. The plan shall contain enough information to properly locate the wall. The ultimate right of way shall also be shown, unless it is of a significant distance from the wall and will have no effect on the wall design or construction.


Retaining walls in Seismic Performance Category A (SPC A) and SPC B that are located adjacent to roadways may be designed in accordance with AASHTO specifications for SPC A. Retaining walls in SPC B which are located under a bridge abutment or in a location where failure of the wall may affect the structural integrity of a bridge shall be designed to AASHTO specifications for SPC B. All retaining walls located in SPC C and SPC D shall be designed in accordance to
* Stations and offsets shall be established between one construction baseline or roadway centerline and a wall control line (baseline). Some wall designs may contain a slight batter, while others are vertical. A wall control line shall be set at the front face of the wall, either along the top or at the base of the wall, whichever is critical to the proposed improvements. For battered walls, in order to allow for batter adjustments of the stepped level pad or variation of the top of the wall, the wall control line (baseline) is to be shown at a fixed elevation. For battered walls, the offset location and elevation of control line shall be indicated. All horizontal breaks in the wall shall be given station-offset points, and walls with curvature shall indicate the station-offsets to the PC and PT of the wall, and the radius, on the plans.  
AASHTO specifications for the corresponding SPC.


In seismic category B, C and D determine equivalent fluid pressure from Mononobe-Okabe static method.
* Any obstacles which could possibly interfere with the soil reinforcement shall be shown. Drainage structures, lighting, or truss pedestals and footings, etc. are to be shown, with station offset to centerline of the obstacle, with obstacle size. Skew angles are shown to indicate the angle between a wall and a pipe or box which runs through the wall.  
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|1992 AASHTO Div. IA Eqns. C6-3 and C6-4
|}
''P<sub>AE</sub>'' = equivalent active earth pressure during an earthquake


''P<sub>AE</sub>'' = 0.5 γ<sub>s</sub>H<sup>2</sup>(1 - k<sub>v</sub>)K<sub>AE</sub> where
* Elevations at the top and bottom of the wall shall be shown at 25 ft. intervals and at any break points in the wall.


''K<sub>AE</sub>'' = seismic active pressure coefficient
* Curve data and/or offsets shall be shown at all changes in horizontal alignment. If battered wall systems are used on curved structures, show offsets at 10 ft. (max.) intervals from the baseline.


:<math>K_{AE} = \frac{\cos^2(\phi - \theta - \beta)}{\cos\theta\cos^2\beta\cos(\delta + \beta + \theta)\Big\{1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \theta - i)}{\cos(\delta + \beta + \theta)\cos(i - \beta)}}\Big\}^2}</math>
* Details of any architectural finishes (formliners, concrete coloring, etc.).


* Details of threaded rod connecting the top cap block.


''γ<sub>s</sub>'' = unit weight of soil
* Estimated quantities, total sq. ft. of mechanically stabilized earth systems.


{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
* Proposed grade and theoretical top of leveling pad elevation shall be shown in constant slope. Slope line shall be adjusted per project. Top of wall or coping elevation and stationing shall be shown in the developed elevation per project. If leveling pad is anticipated to encounter rock, then contact the Geotechnical Section for leveling pad minimum embedment requirements.
|-
|'''Additional Information'''
|-
|AASHTO 5.2.2.3 & Div. IA 6.4.3
|}
''k<sub>v</sub>'' = vertical acceleration coefficient


''k<sub>h</sub>'' = horizontal acceleration coefficient which is equal to 0.5A for all walls,
'''MSE Wall Cross Sections'''
:::but 1.5A for walls with battered piles where
:::A = seismic acceleration coefficient


The following variables are shown in the figure below:
* A typical wall section for general information is shown.


''ϕ'' = angle of internal friction of soil
* Additional sections are drawn for any special criteria. The front face of the wall is drawn vertical, regardless of the wall type.


''θ'' = <math>\arctan\ \Big(\frac{k_h}{1 - k_v}\Big)</math>
* Any fencing and barrier or railing are shown.


''β'' = slope of soil face
* Barrier if needed are shown on the cross section. Barriers are attached to the roadway or shoulder pavement, not to the MSE wall. Standard barriers are placed along wall faces when traffic has access to the front face of the wall over shoulders of paved areas.


''δ'' = angle of friction between soil and wall in degrees
<div id="Drainage at MSE Walls"></div>
'''Drainage at MSE Walls'''


''i'' = backfill slope angle in degrees
*'''Drainage Before MSE Wall'''


''H'' = distance from the bottom of the part of the wall to which the pressure is applied to the top of the fill at the location where the earth pressure is to be found.
:Drainage is not allowed to be discharged within 10 ft. from front of MSE wall in order to protect wall embedment, prevent erosion and foundation undermining, and maintain soil strength and stability.


[[image:751.24.1.2 active soil.jpg|center|450px|thumb|<center>'''Active Soil Wedge'''</center>]]
*'''Drainage Behind MSE Wall'''


<div id="Group Loads"></div>
::'''Internal (Subsurface) Drainage'''
'''Group Loads'''


For SPC A and B (if wall does not support an abutment), apply AASHTO Group I Loads only. Bearing capacity, stability and sliding shall be calculated using working stress loads. Reinforced concrete design shall be calculated using load factor design loads.
::Groundwater and infiltrating surface waters are drained from behind the MSE wall through joints between the face panels or blocks (i.e. wall joints) and two-6 in. (min.) diameter pipes located at the base of the wall and at the basal interface between the reinforced backfill and the retained backfill.
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO Table 3.22.1A
|}


AASHTO Group I Load Factors for Load Factor Design of concrete:
::Excessive subsurface draining can lead to increased risk of backfill erosion/washout through the wall joints and erosion at the bottom of walls and at wall terminal ends. Excessive water build-up caused by inadequate drainage at the bottom of the wall can lead to decreased soil strength and wall instability. Bridge underdrainage (vertical drains at end bents and at approach slabs) can exacerbate the problem.
''γ'' = 1.3


''β<sub>D</sub>'' = 1.0 for concrete weight
::Subsurface drainage pipes should be designed and sized appropriately to carry anticipated groundwater, incidental surface run-off that is not collected otherwise including possible effects of drainage created by an unexpected rupture of any roadway drainage conveyance or storage as an example.


''β<sub>D</sub>'' = 1.0 for flexural member
::'''External (Surface) Drainage'''


''β<sub>E</sub>'' = 1.3 for lateral earth pressure for retaining walls
::External drainage considerations deal with collecting water that could flow externally over and/or around the wall surface taxing the internal drainage and/or creating external erosion issues. It can also infiltrate the reinforced and retained backfill areas behind the MSE wall.  


''β<sub>E</sub>'' = 1.0 for vertical earth pressure
::Diverting water flow away from the reinforced soil structure is important. Roadway drainage should be collected in accordance with roadway drainage guidelines and bridge deck drainage should be collected similarly.


''β<sub>LL</sub>'' = 1.67 for live load wheel lines
*'''Guidance'''


''β<sub>LL</sub>'' = 1.67 for collision forces
:ALL MSE WALLS


{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
:1. Appropriate measures to prevent surface water infiltration into MSE wall backfill should be included in the design and detail layout for all MSE walls and shown on the roadway plans.  
|-
|'''Additional Information'''
|-
|AASHTO 5.14.2
|}


''β<sub>E</sub>'' = 1.67 for vertical earth pressure resulting from live load surcharge
:2. Gutters behind MSE walls are required for flat or positive sloping backfills to prevent concentrated infiltration behind the wall facing regardless of when top of backfill is paved or unpaved. This avoids pocket erosion behind facing and protection of nearest-surface wall connections which are vulnerable to corrosion and deterioration. Drainage swales lined with concrete, paved or precast gutter can be used to collect and discharge surface water to an eventual point away from the wall. If rock is used, use impermeable geotextile under rock and align top of gutter to bottom of rock to drain. (For negative sloping backfills away from top of wall, use of gutters is not required.)


''β<sub>E</sub>'' = 1.3 for horizontal earth pressure resulting from live load surcharge
:District Design Division shall verify the size of the two-6 in. (min.) diameter lower perforated MSE wall drain pipes and where piping will daylight at ends of MSE wall or increase the diameters accordingly.  This should be part of the preliminary design of the MSE wall. (This shall include when lateral pipes are required and where lateral drain pipes will daylight/discharge).
:BRIDGE ABUTMENTS WITH MSE WALLS


For SPC B (if wall supports an abutment), C, and D apply AASHTO Group I Loads and seismic loads in accordance with AASHTO Division IA - Seismic Design Specifications.
:Areas of concern: bridge deck drainage, approach slab drainage, approach roadway drainage, bridge underdrainage:  vertical drains at end bents and approach slab underdrainage, showing drainage details on the roadway and MSE wall plans


{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
:3. Bridge slab drain design shall be in accordance with [[751.10 General Superstructure#751.10.3 Bridge Deck Drainage - Slab Drains |EPG 751.10.3 Bridge Deck Drainage – Slab Drains]] unless as modified below.
|-
|'''Additional Information'''
|-
|AASHTO Div. IA 4.7.3
|}


When seismic loads are considered, load factor for all loads = 1.0.
:4. Coordination is required between the Bridge Division and District Design Division on drainage design and details to be shown on the MSE wall and roadway plans.  


==751.24.2 Mechanically Stabilized Earth (MSE) Walls==
:5. Bridge deck, approach slab and roadway drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.  
::*(Recommended) Use of a major bridge approach slab and approach pavement is ideal because bridge deck, approach slab and roadway drainage are directed using curbs and collected in drain basins for discharge that protect MSE wall backfill. For bridges not on a major roadway, consideration should be given to requiring a concrete bridge approach slab and pavement incorporating these same design elements (asphalt is permeable).


===751.24.2.1 Design===
::*(Less Recommended) Use of conduit and gutters:


Designs of Mechanically Stabilized Earth (MSE) walls are completed by consultants or contractors in accordance with Section 5 of the AASHTO Specifications. MoDOT Internet site contains a listing of facing unit manufacturers, soil reinforcement suppliers, and wall system suppliers which have been approved for use. See [http://www.modot.org/business/standards_and_specs/SpecbookEPG.pdf#page=11 Sec 720] and [http://www.modot.org/business/standards_and_specs/SpecbookEPG.pdf#page=14 Sec 1010] for additional information. The [http://sharepoint/systemdelivery/CM/geotechnical/default.aspx Geotechnical Section] is responsible for checking global stability, which should be reported on the Foundation Investigation Geotechnical Report. For MSE wall preliminary information, see [http://epg.modot.mo.gov/index.php?title=751.1_Preliminary_Design#751.1.4.3_MSE_Walls EPG 751.1.4.3 MSE Walls].
:::* Conduit: Drain away from bridge and bury conduit daylighting to natural ground or roadway drainage ditch at an eventual point beyond the limits of the wall. Use expansion fittings to allow for bridge movement and consider placing conduit to front of MSE wall and discharging more than 10 feet from front of wall or using lower drain pipes to intercept slab drainage conduit running through backfill.


'''General Guidelines'''
:::* Conduit and Gutters: Drain away from bridge using conduit and 90° elbow (or 45° bend) for smoothly directing drainage flow into gutters and that may be attached to inside of gutters to continue along downward sloping gutters along back of MSE wall to discharge to sewer or to natural drainage system, or to eventual point beyond the limits of the wall.  Allow for independent bridge and wall movements by using expansion fittings where needed. See [[751.10 General Superstructure#751.10.3.1 Type, Alignment and Spacing|EPG 751.10.3.1 Type, Alignment and Spacing]] and [[751.10 General Superstructure#751.10.3.3 General Requirements for Location of Slab Drains|EPG 751.10.3.3 General Requirements for Location of Slab Drains]].


* Small block walls are limited to a 10 ft. height in one lift.
:6. Vertical drains at end bents and approach slab underdrainage should be intercepted to drain away from bridge end and MSE wall.


* For small block walls, top cap units shall be used and shall be permanently attached by means of a resin anchor system.
:7. Discharging deck drainage using many slab drains would seem to reduce the volume of bridge end drainage over MSE walls.


* For large block walls, capstone may be substituted for coping and either shall be permanently attached to wall by panel dowels.
:8. Drain flumes at bridge abutments with MSE walls do not reduce infiltration at MSE wall backfill areas and are not recommended.


* For large block walls, form liners are required to produce all panels. Using form liner to produce panel facing is more cost effective than producing flat panels. Standard form liners are specified on the [http://www.modot.org/business/consultant_resources/bridgestandards.htm Bridge Standard Drawings - MSE Wall MSEW]. Be specific regarding names, types and colors of staining, and names and types of form liner.
:DISTRICT DESIGN DIVISION MSE WALLS


* MSE walls shall not be used where exposure to acid water may occur such as in areas of coal mining.
:Areas of concern: roadway or pavement drainage, MSE wall drainage, showing drainage details on the roadway and MSE wall plans.


* MSE walls shall not be used where scour is a problem.
:9. For long MSE walls, where lower perforated drain pipe slope become excessive, non-perforated lateral drain pipes, permitted by Standard Specifications, shall be designed to intercept them and go underneath the concrete leveling pad with a 2% minimum slope. Lateral drain pipes shall daylight/discharge at least 10 ft. from front of MSE wall. Screens should be installed and maintained on drain pipe outlets.


* MSE walls with metallic soil reinforcement shall not be used where stray electrical ground currents may occur as would be present near electrical substations.
:10. Roadway and pavement drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.  


* No utilities shall be allowed in the reinforced earth if future access to the utilities would require that the reinforcement layers be cut, or if there is a potential for material, which can cause degradation of the soil reinforcement, to leak out of the utilities into the wall backfill, with the exception of storm water drainage.
:11. For district design MSE walls, use roadway or pavement drainage collection pipes to transport and discharge to an eventual point outside the limits of the wall.


* The interior angle between two walls should be greater than 70°. However, if unavoidable, then place [[751.50 Standard Detailing Notes#J1.31|EPG 751.50 J1.31 note]] on the plans.
:Example: Showing drain pipe details on the MSE wall plans.


* Small block walls may be battered up to 1.5 in. per foot.
<gallery mode=packed widths=300px heights=300px>
File:751.24.2.1_elev_drain_pipe-01.png| <big>'''ELEVATION SHOWING DRAIN PIPE'''</big>
File:751.24.2.1_elev_drain_pipe_alt-01.png| <big>'''Alternate option'''</big>
</gallery>
<gallery mode=packed widths=400px heights=400px>
File:751.24.2.1_sec_A-A-02.png| <big>'''Section A-A'''</big>
</gallery>
{| style="text-align: left; margin-left: auto; margin-right: auto;"
|
Notes:</br>
(1) To be designed by District Design Division.</br>
(2) To be designed by District Design Division if needed. Provide non-perforated lateral drain pipe under leveling pad at 2% minimum slope. (Show on plans).</br>
(3) Discharge to drainage system or daylight screened outlet at least 10 feet away from end of wall (typ.). (Skew in the direction of flow as appropriate).</br>
(4) Discharge to drainage system or daylight screened outlet at least 10 feet away from front face of wall (typ.). (Skew in the direction of flow as appropriate).</br>
(5) Minimum backfill cover = Max(15”, 1.5 x diameter of drain pipe).</br>
|}


* The friction angle used for the computation of horizontal forces within the reinforced soil shall be greater than or equal to 34°.
===751.24.2.2 Excavation===


* For epoxy coated reinforcement requirements, see [[751.5 Structural Detailing Guidelines#751.5.9.2.2 Epoxy Coated Reinforcement Requirements|EPG 751.5.9.2.2 Epoxy Coated Reinforcement Requirements]].
For estimating excavation and minimum soil reinforcement length, see [[751.6 General Quantities#751.6.2.17 Excavation|EPG 751.6.2.17 Excavation]].


* All concrete except facing panels or units shall be CLASS B or B-1.  
For division responsibilities for preparing MSE wall plans, computing excavation class, quantities and locations, see [[:Category:747 Bridge Reports and Layouts#747.2.6.2 Mechanically Stabilized Earth (MSE) Wall Systems|EPG 747.2.6.2 Mechanically Stabilized Earth (MSE) Wall Systems]].


* The friction angle of the soil to be retained by the reinforced earth shall be listed on the plans as well as the friction angle for the foundation material the wall is to rest on.
===751.24.2.3 Details===
<center>
{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
|+
| style="background:#BEBEBE" width="300" |'''[https://www.modot.org/bridge-standard-drawings Bridge Standard Drawings]'''
|-
|align="center"| MSE Wall - MSEW
|}
</center>


* Seismic performance category and acceleration coefficient shall be listed on the plans.
<gallery mode=packed widths=625px heights=625px>
File:751.24.2.3_mse_wall-01.png | <big>'''Fig. 751.24.2.3.1 MSE Wall Developed Elevation and Plan'''</big>
</gallery>
{| style="text-align: left; margin-left: auto; margin-right: auto;"
|
(1) Minimum embedment = maximum (2 feet; or embedment based on Geotechnical Report and global stability requirements;</br>or FHWA-NH1-10-0124, Table 2-2); or as per Geotechnical Report if rock is known to exist from Geotechnical Report.
|}


* Factors of Safety for MSE walls shall be 2.0 for overturning, 1.5 for sliding, 2.0 for ultimate bearing capacity and 1.5 for pullout resistance.
'''Drycast Modular Block Wall Systems and Wetcast Modular Block Wall Systems'''


* Factors of Safety for seismic design shall be 1.5 for overturning and 1.1 for sliding.
Battered mechanically stabilized earth wall systems may be used unless the design layout specifically calls for a vertical wall (precast modular panel wall systems shall not be battered and drycast modular block wall systems or wetcast modular block wall systems may be built vertical). If a battered MSE wall system is allowed, then [[751.50_Standard_Detailing_Notes#J1._General|EPG 751.50 J1.19]] note shall be placed on the design plans:


* Do not use small block walls in the following locations:
For battered walls, note on the plans whether the horizontal offset from the baseline is fixed at the top or bottom of the wall. Horizontal offset and corresponding vertical elevation shall be noted on plans.


::* Within the splash zone from snow removal operations (assumed to be 15 ft. from the edge of the shoulder).
<gallery mode=packed widths=400px heights=400px>
File:751.24.2.3_typ_dmbw-mse-01.png | <big>'''Fig. 751.24.2.3.2 Typical Section Through Generic Drycast Modular Block Wall (DMBW-MSE) System or Wetcast Modular Block Wall (WMBW-MSE) System'''</big>
</gallery>
{| style="text-align: left; margin-left: auto; margin-right: auto;"
|
<nowiki>*</nowiki> The maximum vertical spacing of reinforcement should be limited to two times the block depth or 32 in., whichever is less.</br>For large modular block (block height > 16 in.), maximum vertical spacing of reinforcement equal to the block height.  
|}


::* Where the blocks will be continuously wetted, such as around sources of water.
'''Fencing (See [https://www.modot.org/bridge-standard-drawings Bridge Standard Drawing] for details)'''


::* Where blocks will be located behind barrier curbs or other obstacles that will trap salt-laden snow from removal operations.
Fencing may be installed on the Modified Type A or Modified Type B Gutter or behind the MSE Wall.


::* For structurally critical applications, such as containing necessary fill around structures.
For Modified Type A and Modified Type B Gutter and Fence Post Connection details, see [https://www.modot.org/media/16871 Standard Plan 607.11].


::* In tiered wall systems.
==751.24.3 Cast-In-Place Concrete Retaining Walls==


* For locations where small block walls are not desirable, consider coloring agents and/or architectural forms using large block walls for aesthetic installations.
===751.24.3.1 Unit Stresses===
<div id="For slab drain location near MSE Wall,"></div>


* For slab drain location near MSE Wall, see [[751.10 General Superstructure#General Requirements for Location and Spacing of Slab Drains|EPG 751.10.3.1 Drain Type, Alignment and Spacing]] and [[751.10 General Superstructure#751.10.3.3 General Requirements for Location of Slab Drains|EPG 751.10.3.3 General Requirements for Location of Slab Drains]].
'''Concrete'''
Concrete for retaining walls shall be Class B Concrete (f'c = 3000 psi) unless the footing is used as a riding surface in which case Class B-1 Concrete (f'c = 4000 psi) shall be used.


* Roadway runoff should be directed away from running along face of MSE walls used as wing walls on bridge structures.
'''Reinforcing Steel'''


* Drainage:
Reinforcing Steel shall be Grade 60 (fy = 60,000 psi).


:*Gutter type should be selected at the core team meeting.
'''Pile Footing'''


:* When gutter is required without fencing, use Type A or Type B gutter (for detail, see [http://www.modot.mo.gov/business/standards_and_specs/documents/60900.pdf Std. Plan 609.00]).
For steel piling material requirements, see the unit stresses in [[751.50 Standard Detailing Notes#A1. Design Specifications, Loadings & Unit Stresses and Standard Plans|EPG 751.50 A1.3 note]].


:* When gutter is required with fencing, use Modified Type A or Modified Type B gutter (for detail, see [http://www.modot.mo.gov/business/standards_and_specs/documents/60711.pdf Std. Plan 607.11]).
'''Spread Footing'''


:* When fencing is required without gutter, place in tube and grout behind the MSE wall (for detail, see Fig. 751.24.2.1.7, Fence Post Connection Behind MSE Wall (without gutter).
For foundation material capacity, see Foundation Investigation Geotechnical Report.


:* Lower backfill longitudinal drainage pipes behind all MSE walls shall be  two-6” (Min.) diameter perforated PVC or PE pipe (See Sec 1013) unless larger sizes are required by design which shall be the responsibility of the District Design Division. Show drainage pipe size on plans. Outlet screens and cleanouts should be detailed for any drain pipe (shown on MoDOT MSE wall plans or roadway plans). Lateral non-perforated drain pipes (below leveling pad) are permitted by Standard Specifications and shall be sized by the District Design Division if necessary. Lateral outlet drain pipe sloped at 2% minimum.
===751.24.3.2 Design===


::* Identify on MSE wall plans or roadway plans drainage pipe point of entry, point of outlet (daylighting), 2% min. drainage slopes in between points to ensure positive flowand additional longitudinal drainage pipes if required to accommodate ground slope changes and lateral drainage pipes if required by design.
Note: For design concepts and guidance, follow the design process ([[751.40_LFD_Widening_and_Repair#751.40.8.15_Cast-In-Place_Concrete_Retaining_Walls|EPG 751.40.8.15]]) and modify design/details of ASD as necessary to meet LRFD requirements until [https://epgtest.modot.org/index.php/751.24_Retaining_Walls EPG 751.24] is updated for LRFD.


::* Adjustment in the vertical alignment of the longitudinal drainage pipes from that depicted on the MSE wall standard drawings may be necessary to ensure positive flow out of the drainage system.
Capacity/Demand ratio (CDR) for bearing capacity shall be ≥ 1.0
: <math>Bearing\ Capacity\ (CDR) = \frac{Factored\ Bearing\ Resistance}{Maximum\ Factored\ Bearing\ Stress} \ge 1.0</math>
::* Identify on MSE wall plans or roadway plans, the outlet ends of pipes that shall be located to avoid clogging or flow into the drainage system. Outlet screens and cleanouts should be detailed for any drain pipe.
: Strength Limit States:
: Factored bearing resistance = Nominal bearing resistance from Geotech report X
: Minimum Resistance factor (0.55, Geotech report) &nbsp;&nbsp;&nbsp;&nbsp; LRFD Table 11.5.7


:* For more information on drainage, see [[#Drainage at MSE Walls|Drainage at MSE Walls]].
: Extreme Event I and II Limit State:
: Factored bearing resistance = Nominal bearing resistance from Geotech report X Resistance factor
: Resistance factor = 0.8 &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.5.8


<div id="MSE Wall Construction:"></div>
: When wall is supported by soil:
'''MSE Wall Construction: Pipe Pile Spacers Guidance'''
: Factored bearing stress per LRFD eq. 11.6.3.2-1


Pipe pile spacers shall be used at pile locations behind mechanically stabilized earth walls to protect the wall reinforcement when driving pile for the bridge substructure at end bents(s). Pipe pile spacers shall have an inside diameter greater than that of the pile and large enough to avoid damage to the pipe when driving the pile. The bottom of the  pipe pile spacers shall be placed 5 ft. min. below the bottom of the MSE wall leveling pad. The pipe shall be filled with sand or other approved material after the pile is placed and before driving. Pipe pile spacers shall be accurately located and capped for future pile construction.  
: When wall is supported by a rock foundation:
: Factored bearing stress per LRFD eq. 11.6.3.2-2 and 11.6.3.2-3


Alternatively, the contractor shall be given the option of driving the piles before construction of the mechanically stabilized earth wall and placing the soil reinforcement and backfill material around the piling. The contractor shall adequately support the piling to ensure that proper pile alignment is maintained during the wall construction. The contractor’s plan for bracing the pile shall be submitted to the engineer for review.  
: Note: When the value of eccentricity e is negative then ''use e = 0''.  


Piling shall be designed for downdrag (DD) loads due to either method. Oversized  pipe pile spacers with sand placed after driving may be considered to mitigate some of the effects of downdrag (DD) loads. Sizing of  pipe pile spacers shall account for pile size, thermal movements of the bridge, pile placement plan, and vertical and horizontal placement tolerances.  
Capacity/Demand ratio (CDR) for overturning shall be ≥ 1.0
: <math>Overtuning\ (CDR) = \frac{Total\ Factored\ Resisting\ Moment}{Total\ Factored\ Driving\ Moment} \ge 1.0</math>


The minimum clearance from the back face of MSE walls to the front face of the end bent beam shall be 3 ft. 9 in. (typ.). The 3 ft. 9 in. dimension is based on the use of 18 in.  Pipe pile spacers& FHWA-NHI-10-24, Figure 5-17C, which will help ensure that soil reinforcement is not skewed more than 15° for nut and bolt reinforcement connections. Other types of connections may require different methods for splaying. In the event that the 3 ft. 9 in. dimension or setback cannot be used, the following guidance for  pipe pile spacers clearance shall be used: pipe pile spacers shall be placed 18 in. clear min. from the back face of MSE wall panels; 12 in. minimum clearance is required between  pipe pile spacers and leveling pad and 18 in. minimum clearance is required between leveling pad and pile.  
Capacity/Demand ratio (CDR) for eccentricity shall be ≥ 1.0
: <math>Eccentricity\ (CDR) = \frac{e_{Limit}}{e_{design}} \ge 1.0</math>


'''MSE Wall Plan and Geometrics'''
Capacity/Demand ratio (CDR) for sliding shall be ≥ 1.0
: <math>Sliding\ (CDR) = \frac{Total\ Factored\ Sliding\ Resistance}{Total\ Factored\ Active\ Force} \ge 1.0</math>


* A plan view shall be drawn showing a baseline or centerline, roadway stations and wall offsets. The plan shall contain enough information to properly locate the wall. The ultimate right of way shall also be shown, unless it is of a significant distance from the wall and will have no effect on the wall design or construction.
: Sliding shall be checked in accordance with LRFD 11.6.3.6 and 10.6.3.4


* Stations and offsets shall be established between one construction baseline or roadway centerline and a wall control line (baseline). Some wall designs may contain a slight batter, while others are vertical. A wall control line shall be set at the front face of the wall, either along the top or at the base of the wall, whichever is critical to the proposed improvements. For battered walls, in order to allow for batter adjustments of the stepped level pad or variation of the top of the wall, the wall control line (baseline) is to be shown at a fixed elevation. For battered walls, the offset location and elevation of control line shall be indicated. All horizontal breaks in the wall shall be given station-offset points, and walls with curvature shall indicate the station-offsets to the PC and PT of the wall, and the radius, on the plans.  
Eccentricity, (e) Limit for Strength Limit State: &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.6.3.3
:* For foundations supported on soil, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, B or (e ≤ 0.33B).
:* For foundations supported on rock, the location of the resultant of the reaction forces shall be within the middle nine-tenths of the base width, B or (e ≤ 0.45B).


* Any obstacles which could possibly interfere with the soil reinforcement shall be shown. Drainage structures, lighting, or truss pedestals and footings, etc. are to be shown, with station offset to centerline of the obstacle, with obstacle size. Skew angles are shown to indicate the angle between a wall and a pipe or box which runs through the wall.  
Eccentricity, (e) Limit for Extreme Event I (Seismic): &nbsp;&nbsp;&nbsp;&nbsp; LRFD 11.6.5.1
:* For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, B or (e ≤ 0.33B) for  γ<sub>EQ</sub> = 0.0 and middle eight-tenths of the base width, B or (e ≤ 0.40B) for  γ<sub>EQ</sub> = 1.0.  For γ<sub>EQ</sub> between 0.0 and 1.0, interpolate e value linearly between 0.33B and 0.40B. For γ<sub>EQ</sub> refer to LRFD 3.4.


* Elevations at the top and bottom of the wall shall be shown at 25 ft. intervals and at any break points in the wall.
:Note: Seismic design shall be performed for γ<sub>EQ</sub> = 0.5


* Curve data and/or offsets shall be shown at all changes in horizontal alignment. If battered wall systems are used on curved structures, show offsets at 10 ft. (max.) intervals from the baseline.
Eccentricity, (e) Limit for Extreme Event II:
:* For foundations supported on soil or/and rock, the location of the resultant of the reaction forces shall be within the middle eight-tenths of the base width, B or (e ≤ 0.40B).


* Details of any architectural finishes (formliners, concrete coloring, etc.).
For epoxy coated reinforcement requirements, see [[751.5 Structural Detailing Guidelines#751.5.9.2.2 Epoxy Coated Reinforcement Requirements|EPG 751.5.9.2.2 Epoxy Coated Reinforcement Requirements]].


* Details of threaded rod connecting the top cap block.
If the height of the wall or fill is a variable dimension, then base the structural design of the wall, toe, and heel on the high quarter point between expansion joints.


* Estimated quantities, total sq. ft. of mechanically stabilized earth systems.
[[image:751.24.3.2.jpg|center|600px|thumb|<center>'''Fig. 751.24.3.2'''</center>]]


* Proposed grade and theoretical top of leveling pad elevation shall be shown in constant slope. Slope line shall be adjusted per project. Top of wall or coping elevation and stationing shall be shown in the developed elevation per project. If leveling pad is anticipated to encounter rock, then contact the Geotechnical Section for leveling pad minimum embedment requirements.


'''MSE Wall Cross Sections'''


* A typical wall section for general information is shown.


* Additional sections are drawn for any special criteria. The front face of the wall is drawn vertical, regardless of the wall type.


* Any fencing and barrier curb are shown.


* Barriers if needed are shown on the cross section. Concrete barriers are attached to the roadway or shoulder pavement, not to the MSE wall. Standard Type B barrier curbs are placed along wall faces when traffic has access to the front face of the wall over shoulders of paved areas.
<!-- moved
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


<div id="Drainage at MSE Walls"></div>
'''Drainage at MSE Walls'''


:'''Drainage Before MSE Wall'''
====751.24.3.2.1 Spread Footings====


:Drainage is not allowed to be discharged within 10 ft. from front of MSE wall in order to protect wall embedment, prevent erosion and foundation undermining, and maintain soil strength and stability.
'''Location of Resultant'''


*'''Drainage Behind MSE Wall'''
The resultant of the footing pressure must be within the section of the footing specified in the following table.


::'''Internal (Subsurface) Drainage'''
{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
|+
! style="background:#BEBEBE" |When Retaining Wall is Built on: !! style="background:#BEBEBE"|AASHTO Group Loads I-VI !! style="background:#BEBEBE"|For Seismic Loads
|-
|  align="center" |Soil<sup>a</sup> || align="center"|Middle 1/3||  align="center"|Middle 1/2 <sup>b</sup>
|-
|  align="center"|Rock<sup>c</sup> || align="center"|Middle 1/2||align="center"|Middle 2/3
|-
|colspan="3"|<sup>'''a'''</sup> Soil is defined as clay, clay and boulders, cemented gravel, soft shale, etc. with allowable bearing values less than 6 tons/sq. ft.
|-
|colspan="3"|<sup>'''b'''</sup> MoDOT is more conservative than AASHTO in this requirement.
|-
|colspan="3"|<sup>'''c'''</sup> Rock is defined as rock or hard shale with allowable bearing values of 6 tons/sq. ft. or more.
|}


::Groundwater and infiltrating surface waters are drained from behind the MSE wall through joints between the face panels or blocks (i.e. wall joints) and two-6 in. (min.) diameter pipes located at the base of the wall and at the basal interface between the reinforced backfill and the retained backfill.
Note: The location of the resultant is not critical when considering collision loads.


::Excessive subsurface draining can lead to increased risk of backfill erosion/washout through the wall joints and erosion at the bottom of walls and at wall terminal ends. Excessive water build-up caused by inadequate drainage at the bottom of the wall can lead to decreased soil strength and wall instability. Bridge underdrainage (vertical drains at end bents and at approach slabs) can exacerbate the problem.
'''Factor of Safety Against Overturning'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


::Subsurface drainage pipes should be designed and sized appropriately to carry anticipated groundwater, incidental surface run-off that is not collected otherwise including possible effects of drainage created by an unexpected rupture of any roadway drainage conveyance or storage as an example.
AASHTO Group Loads I - VI:
* F.S. for overturning ≥ 2.0 for footings on soil.
* F.S. for overturning ≥ 1.5 for footings on rock.


::'''External (Surface) Drainage'''
For seismic loading, F.S. for overturning may be reduced to 75% of the value for AASHTO Group Loads I - VI. For seismic loading:
* F.S. for overturning ≥ (0.75)(2.0) = 1.5 for footings on soil.
* F.S. for overturning ≥ (0.75)(1.5) = 1.125 for footings on rock.


::External drainage considerations deal with collecting water that could flow externally over and/or around the wall surface taxing the internal drainage and/or creating external erosion issues. It can also infiltrate the reinforced and retained backfill areas behind the MSE wall.  
For collision forces:
* F.S. for overturning ≥ 1.2.


::Diverting water flow away from the reinforced soil structure is important. Roadway drainage should be collected in accordance with roadway drainage guidelines and bridge deck drainage should be collected similarly.
'''Factor of Safety Against Sliding'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


:'''Guidance'''
Only spread footings on soil need be checked for sliding because spread footings on rock or shale are embedded into the rock.
* F.S. for sliding ≥ 1.5 for AASHTO Group Loads I - VI.
* F.S. for sliding ≥ (0.75)(1.5) = 1.125 for seismic loads.
* F.S. for sliding ≥ 1.2 for collision forces.


:ALL MSE WALLS
The resistance to sliding may be increased by:
* adding a shear key that projects into the soil below the footing.
* widening the footing to increase the weight and therefore increase the frictional resistance to sliding.


:1. Appropriate measures to prevent surface water infiltration into MSE wall backfill should be included in the design and detail layout for all MSE walls and shown on the roadway plans.
'''Passive Resistance of Soil to Lateral Load'''


:2. Gutters behind MSE walls are required for flat or positive sloping backfills to prevent concentrated infiltration behind the wall facing regardless of when top of backfill is paved or unpaved. This avoids pocket erosion behind facing and protection of nearest-surface wall connections which are vulnerable to corrosion and deterioration. Drainage swales lined with concrete, paved or precast gutter can be used to collect and discharge surface water to an eventual point away from the wall. If rock is used, use impermeable geotextile under rock and align top of gutter to bottom of rock to drain. (For negative sloping backfills away from top of wall, use of gutters is not required.)
The Rankine formula for passive pressure can be used to determine the passive resistance of soil to the lateral force on the wall. This passive pressure is developed at shear keys in retaining walls and at end abutments.


:District Design Division shall verify the size of the two-6 in. (min.) diameter lower perforated MSE wall drain pipes and where piping will daylight at ends of MSE wall or increase the diameters accordingly.  This should be part of the preliminary design of the MSE wall. (This shall include when lateral pipes are required and where lateral drain pipes will daylight/discharge).
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
:BRIDGE ABUTMENTS WITH MSE WALLS
|'''Additional Information'''
|-
|AASHTO 5.5.5A
|}


:Areas of concern: bridge deck drainage, approach slab drainage, approach roadway drainage, bridge underdrainage:  vertical drains at end bents and approach slab underdrainage, showing drainage details on the roadway and MSE wall plans
The passive pressure against the front face of the wall and the footing of a retaining wall is loosely compacted and should be neglected when considering sliding.


:3. Bridge slab drain design shall be in accordance with [[751.10 General Superstructure#751.10.3 Bridge Deck Drainage - Slab Drains |EPG 751.10.3 Bridge Deck Drainage – Slab Drains]] unless as modified below.
Rankine Formula: <math>P_p = \frac{1}{2}C_p\gamma_s[H^2-H_1^2]</math> where thefollowing variables are defined in the figure below
:
:''C<sub>p</sub>'' = <math>\tan \big( 45^\circ + \frac{\phi}{2}\big)</math>


:4. Coordination is required between the Bridge Division and District Design Division on drainage design and details to be shown on the MSE wall and roadway plans.
:''y<sub>1</sub> = <math>\frac{H_1y_2^2 + \frac{2}{3}y_2^3}{H^2 - H_1^2}</math>


:5. Bridge deck, approach slab and roadway drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.
:''P<sub>p</sub>'' = passive force at shear key in pounds per foot of wall length
::*(Recommended) Use of a major bridge approach slab and approach pavement is ideal because bridge deck, approach slab and roadway drainage are directed using curbs and collected in drain basins for discharge that protect MSE wall backfill. For bridges not on a major roadway, consideration should be given to requiring a concrete bridge approach slab and pavement incorporating these same design elements (asphalt is permeable).


::*(Less Recommended) Use of conduit and gutters:
:''C<sub>p</sub>'' = coefficient of passive earth pressure


:::* Conduit: Drain away from bridge and bury conduit daylighting to natural ground or roadway drainage ditch at an eventual point beyond the limits of the wall.  Use expansion fittings to allow for bridge movement. (can consider placing conduit to front of MSE wall and discharging more than 10 feet from front of wall or using lower drain pipes to intercept slab drainage conduit running through backfill.
:<math>\boldsymbol{\gamma_s}</math> = unit weight of soil


:::* Conduit and Gutters: Drain away from bridge using conduit and 90° elbow (or 45° bend) for smoothly directing drainage flow into gutters and that may be attached to inside of gutters to continue along downward sloping gutters along back of MSE wall to discharge to sewer or to natural drainage system, or to eventual point beyond the limits of the wall.  Allow for independent bridge and wall movements by using expansion fittings where needed. See [[751.10 General Superstructure#751.10.3.1 Type, Alignment and Spacing|EPG 751.10.3.1 Type, Alignment and Spacing]] and [[751.10 General Superstructure#751.10.3.3 General Requirements for Location of Slab Drains|EPG 751.10.3.3 General Requirements for Location of Slab Drains]].
:''H'' = height of the front face fill less than 1 ft. min. for erosion


:6. Vertical drains at end bents and approach slab underdrainage should be intercepted to drain away from bridge end and MSE wall.
:''H<sub>1</sub>'' = H minus depth of shear key


:7. Discharging deck drainage using many slab drains would seem to reduce the volume of bridge end drainage over MSE walls.
:''y<sub>1</sub>'' = location of ''P<sub>p</sub>'' from bottom of footing


:8. Drain flumes at bridge abutments with MSE walls do not reduce infiltration at MSE wall backfill areas and are not recommended.
:<math>\boldsymbol{\phi}</math> = angle of internal friction of soil


:DISTRICT DESIGN DIVISION MSE WALLS
[[image:751.24.3.2.1 passive.jpg|center|500px]]
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.2
|}
The resistance due to passive pressure in front of the shear key shall be neglected unless the key extends below the depth of frost penetration.
{|style="padding: 0.3em; margin-right:7px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="left"
|-
|'''Additional Information'''
|-
|[http://sp/sites/cm/Pages/default.aspx MoDOT Materials Division]
|}


:Areas of concern: roadway or pavement drainage, MSE wall drainage, showing drainage details on the roadway and MSE wall plans.
Frost line is set at 36 in. at the north border of Missouri and at 18 in. at the south border.


:9. For long MSE walls, where lower perforated drain pipe slope become excessive, non-perforated lateral drain pipes, permitted by Standard Specifications, shall be designed to intercept them and go underneath the concrete leveling pad with a 2% minimum slope. Lateral drain pipes shall daylight/discharge at least 10 ft. from front of MSE wall. Screens should be installed and maintained on drain pipe outlets.
'''Passive Pressure During Seismic Loading'''


:10. Roadway and pavement drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.  
During an earthquake, the passive resistance of soil to lateral loads is slightly decreased. The Mononobe-Okabe static method is used to determine the equivalent fluid pressure.


:11. For district design MSE walls, use roadway or pavement drainage collection pipes to transport and discharge to an eventual point outside the limits of the wall.
:''P<sub>PE</sub>'' = equivalent passive earth pressure during an earthquake
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|1992 AASHTO Div. IA Eqns. C6-5 and C6-6
|}
:<math>P_{PE} = \frac{1}{2}\gamma_sH^2(1 - k_v)K_{PE}</math> where:


Example: Showing drain pipe details on the MSE wall plans.
:''K<sub>PE</sub>'' = seismic passive pressure coefficient


[[image:751.24.2.1.jpg|center|825px]]
:<math>K_{PE} = \frac{\cos^2(\phi - \theta - \beta)}{\cos\theta\cos^2\beta\cos(\delta + \beta + \theta)\Bigg[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \theta - i)}{\cos(\delta + \beta + \theta)\cos(i - \beta)}}\Bigg]^2}</math>


::<math>\boldsymbol{\gamma}_s</math> = unit weight of soil


[[image:751.24.2.1 alternate.jpg|center|600px]]
:''H'' = height of soil at the location where the earth pressure is to be found


:''k<sub>V</sub>'' = vertical acceleration coefficient


[[image:751.24.2.1 section AA.jpg|center|700px]]
:<math>\boldsymbol{\phi}</math> = angle of internal friction of soil


:<math>\boldsymbol{\theta} =  arctan \big[\frac{k_h}{1 - k_V}\big]</math>


::Notes:
:''k<sub>H</sub>'' = horizontal acceleration coefficient
::'''*''' To be designed by District Design Division.


::'''**''' To be designed by District Design Division if needed. Provide non-perforated lateral drain pipe under leveling pad at 2% minimum slope. (Show on plans).
:<math>\boldsymbol{\beta}</math> = slope of soil face in degrees


::'''***''' Discharge to drainage system or daylight screened outlet at least 10 feet away from end of wall (typ.). (Skew in the direction of flow as appropriate).
:''i'' = backfill slope angle in degrees


::'''****''' Discharge to drainage system or daylight screened outlet at least 10 feet away from front face of wall (typ.). (Skew in the direction of flow as appropriate).
:<math>\boldsymbol{\delta}</math> = angle of friction between soil and wall


===751.24.2.2 Excavation===
'''Special Soil Conditions'''


For estimating excavation, see [[751.6 General Quantities#751.6.2.17 Excavation|EPG 751.6.2.17 Excavation]].
Due to creep, some soft clay soils have no passive resistance under a continuing load. Removal of undesirable material and replacement with suitable material such as sand or crushed stone is necessary in such cases. Generally, this condition is indicated by a void ratio above 0.9, an angle of internal friction (<math>\boldsymbol{\phi}</math>) less than 22°, or a soil shear less than 0.8 ksf. Soil shear is determined from a standard penetration test.


For division responsibilities for preparing MSE wall plans, computing excavation class, quantities and locations, see [[:Category:747 Bridge Reports and Layouts#747.1.6.2 Mechanically Stabilized Earth (MSE) Wall Systems|EPG 747.1.6.2 Mechanically Stabilized Earth (MSE) Wall Systems]].  
:Soil Shear <math>\Big(\frac{k}{ft^2}\Big) = \frac{blows \ per\ 12\ in.}{10}</math>


===751.24.2.3 Details===
'''Friction'''


<center>
In the absence of tests, the total shearing resistance to lateral loads between the footing and a soil that derives most of its strength from internal friction may be taken as the normal force times a coefficient of friction. If the plane at
{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
which frictional resistance is evaluated is not below the frost line then this resistance must be neglected.
|+
 
| style="background:#BEBEBE" width="300" |'''[http://www.modot.org/business/consultant_resources/bridgestandards.htm Bridge Standard Drawings]'''
[[image:751.24.3.2.1 friction 2016.jpg|center|450px|thumb|<center>'''When A Shear Key Is Not Used'''</center>]]
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|-
|align="center"|[http://www.modot.org/business/standard_drawings2/mse_wall_new_title_block.htm MSE Wall]
|AASHTO 5.5.2B
|}
|}


</center>
Sliding is resisted by the friction force developed at the interface between the soil and the concrete footing along the failure plane. The coefficient of friction for soil against concrete can be taken from the table below. If soil data
is not readily available or is inconsistent, the friction factor (f) can be taken as


[[image:751.24.2.2.jpg|center|825px|thumb|<center>'''Fig. 751.24.2.2.1 MSE Wall Developed Elevation and Plan'''</center>]]
: ''f'' =<math>tan \Big(\frac{2\phi}{3}\Big)</math> where <math>\boldsymbol{\phi}</math> is the angle of internal friction of the soil (''Civil Engineering Reference Manual'' by Michael R. Lindeburg, 6th ed., 1992).


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
|+
!style="background:#BEBEBE" colspan="2"|Coefficient of Friction Values for Soil Against Concrete
|-
! style="background:#BEBEBE" |Soil Type<sup>a</sup> !! style="background:#BEBEBE"|Coefficient of Friction
|-
|  align="center" |coarse-grained soil without silt || align="center"|0.55
|-
|  align="center"|coarse-grained soil with silt  || align="center"|0.45
|-
|align="center"|silt (only)||  align="center"|0.35
|-
|align="center"|clay||  align="center"|0.30<sup>b</sup>
|-
|colspan="2"|<sup>'''a'''</sup> It is not necessary to check rock or shale for sliding due to embedment.
|-
|colspan="2"|<sup>'''b'''</sup> Caution should be used with soils with <math>\boldsymbol{\phi}</math> < 22° or soil shear < 0.8 k/sq.ft. (soft clay soils). Removal and replacement of such soil with suitable material should be considered.
|}


'''Battered Small Block Walls'''
[[image:751.24.3.2.1 soil and soil.jpg|center|450px|thumb|<center>'''When A Shear Key Is Used'''</center>]]


Battered mechanically stabilized earth wall systems may be used unless the design layout specifically calls for a vertical wall (large block walls shall not be battered and small block walls may be built vertical). If a battered MSE wall system is allowed, then the following note shall be placed on the design plans:
When a shear key is used, the failure plane is located at the bottom of the shear key in the front half of the footing. The friction force resisting sliding in front of the shear key is provided at the interface between the stationary layer of soil and the moving layer of soil, thus the friction angle is the internal angle of friction of the soil (soil against soil). The friction force resisting sliding on the rest of the footing is of that between the concrete and soil. Theoretically
the bearing pressure distribution should be used to determine how much normal load exists on each surface, however it is reasonable to assume a constant distribution. Thus the normal load to each surface can be divided out between the two surfaces based on the fractional length of each and the total frictional force will be the sum of the normal load on each surface
multiplied by the corresponding friction factor.


:"The top and bottom of wall elevations are given for a vertical wall. If a battered small block wall system is used, the height of the wall shall be adjusted as necessary to fit the ground slope. If fence is built on an extended gutter, then the height of the wall shall be adjusted further."
'''Bearing Pressure'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 4.4.7.1.2 & 4.4.8.1.3
|}


For battered walls, note on the plans whether the horizontal offset from the baseline is fixed at the top or bottom of the wall. Horizontal offset and corresponding vertical elevation shall be noted on plans.
:'''Group Loads I - VI'''


[[image:751.24.2.2 battered.jpg|center|700px|thumb|<center>'''Fig. 751.24.2.2.2 Typical Section Through Generic Small Block Wall'''</center>]]
:The bearing capacity failure factor of safety for Group Loads I - VI must be greater than or equal to 3.0. This factor of safety is figured into the allowable bearing pressure given on the "Design Layout Sheet".


'''Fencing (See [http://www.modot.org/business/standard_drawings2/mse_wall_new_title_block.htm Standard Drawing] for details)'''
:The bearing pressure on the supporting soil shall not be greater than the allowable bearing pressure given on the "Design Layout Sheet".


Fencing may be installed on the Modified Type A or Modified Type B Gutter or behind the MSE Wall.
:'''Seismic Loads'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO Div. IA 6.3.1(B) and AASHTO 5.5.6.2
|}


For Modified Type A and Modified Type B Gutter and Fence Post Connection details, see [http://www.modot.mo.gov/business/standards_and_specs/documents/60711.pdf Standard Plan 607.11].
:When seismic loads are considered, AASHTO allows the ultimate bearing capacity to be used. The ultimate capacity of the foundation soil can be conservatively estimated as 2.0 times the allowable bearing pressure given on the "Design Layout".


==751.24.3 Cast-In-Place Concrete Retaining Walls==
:'''Stem Design'''
:The vertical stem (the wall portion) of a cantilever retaining wall shall be designed as a cantilever supported at the base.


===751.24.3.1 Unit Stresses===
:'''Footing Design'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.1
|}


'''Concrete'''
::'''Toe'''
Concrete for retaining walls shall be Class B Concrete (f'c = 3000 psi) unless the footing is used as a riding surface in which case Class B-1 Concrete (f'c = 4000 psi) shall be used.


'''Reinforcing Steel'''
::The toe of the base slab of a cantilever wall shall be designed as a cantilever supported by the wall. The critical section for bending moments shall be taken at the front face of the stem. The critical section for shear shall be taken at a distance d (d = effective depth) from the front face of the stem.


Reinforcing Steel shall be Grade 60 (fy = 60,000 psi).
::'''Heel'''


'''Pile Footing'''
::The rear projection (heel) of the base slab shall be designed to support the entire weight of the superimposed materials, unless a more exact method is used. The heel shall be designed as a cantilever supported by the wall. The critical section for bending moments and shear shall be taken at the back face of the stem.


For piling capacities, see the unit stresses in [[751.50 Standard Detailing Notes#A1. Design Specifications, Loadings & Unit Stresses |EPG 751.50 Standard Detailing Notes]].
:'''Shear Key Design'''


'''Spread Footing'''
:The shear key shall be designed as a cantilever supported at the bottom of the footing.
 
====751.24.3.2.2 Pile Footings====


For foundation material capacity, see the Unit Stresses Section of the Bridge Manual and the Design Layout Sheet.
Footings shall be cast on piles when specified on the "Design Layout Sheet". If the horizontal force against the retaining wall cannot otherwise be resisted, some of the piles shall be driven on a batter.


===751.24.3.2 Design===
:'''Pile Arrangement'''
For epoxy coated reinforcement requirements, see [[751.5 Structural Detailing Guidelines#751.5.9.2.2 Epoxy Coated Reinforcement Requirements|EPG 751.5.9.2.2 Epoxy Coated Reinforcement Requirements]].


If the height of the wall or fill is a variable dimension, then base the structural design of the wall, toe, and heel on the high quarter point between expansion joints.
:For retaining walls subject to moderate horizontal loads (walls 15 to 20 ft. tall), the following layout is suggested.


[[image:751.24.3.2.jpg|center|600px|thumb|<center>'''Fig. 751.24.3.2'''</center>]]
[[image:751.24.3.2.2 batter piles.jpg|center|300px|thumb|<center>'''Section'''</center>]]
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


====751.24.3.2.1 Spread Footings====
[[image:751.24.3.2.2 plan 2016.jpg|center|450px|thumb|<center>'''Plan'''</center>]]


'''Location of Resultant'''
:For higher walls and more extreme conditions of loading, it may be necessary to:


The resultant of the footing pressure must be within the section of the footing specified in the following table.
:* use the same number of piles along all rows


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
:* use three rows of piles
|+
! style="background:#BEBEBE" |When Retaining Wall is Built on: !! style="background:#BEBEBE"|AASHTO Group Loads I-VI !! style="background:#BEBEBE"|For Seismic Loads
|-
|  align="center" |Soil<sup>a</sup> || align="center"|Middle 1/3||  align="center"|Middle 1/2 <sup>b</sup>
|-
|  align="center"|Rock<sup>c</sup> || align="center"|Middle 1/2||align="center"|Middle 2/3
|-
|colspan="3"|<sup>'''a'''</sup> Soil is defined as clay, clay and boulders, cemented gravel, soft shale, etc. with allowable bearing values less than 6 tons/sq. ft.
|-
|colspan="3"|<sup>'''b'''</sup> MoDOT is more conservative than AASHTO in this requirement.
|-
|colspan="3"|<sup>'''c'''</sup> Rock is defined as rock or hard shale with allowable bearing values of 6 tons/sq. ft. or more.
|}


Note: The location of the resultant is not critical when considering collision loads.
:* provide batter piles in more than one row


'''Factor of Safety Against Overturning'''
::'''Loading Combinations for Stability and Bearing'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


AASHTO Group Loads I - VI:
::The following table gives the loading combinations to be checked for stability and pile loads. These abbreviations are used in the table:
* F.S. for overturning ≥ 2.0 for footings on soil.
* F.S. for overturning ≥ 1.5 for footings on rock.


For seismic loading, F.S. for overturning may be reduced to 75% of the value for AASHTO Group Loads I - VI. For seismic loading:
:::DL = dead load weight of the wall elements
* F.S. for overturning ≥ (0.75)(2.0) = 1.5 for footings on soil.
* F.S. for overturning ≥ (0.75)(1.5) = 1.125 for footings on rock.


For collision forces:
:::SUR = two feet of live load surcharge
* F.S. for overturning ≥ 1.2.


'''Factor of Safety Against Sliding'''
:::E = earth weight
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.5
|}


Only spread footings on soil need be checked for sliding because spread footings on rock or shale are embedded into the rock.
:::EP = equivalent fluid earth pressure
* F.S. for sliding ≥ 1.5 for AASHTO Group Loads I - VI.
* F.S. for sliding ≥ (0.75)(1.5) = 1.125 for seismic loads.
* F.S. for sliding ≥ 1.2 for collision forces.


The resistance to sliding may be increased by:
:::COL = collision force
* adding a shear key that projects into the soil below the footing.
* widening the footing to increase the weight and therefore increase the frictional resistance to sliding.


'''Passive Resistance of Soil to Lateral Load'''
:::EQ = earthquake inertial force of failure wedge


The Rankine formula for passive pressure can be used to determine the passive resistance of soil to the lateral force on the wall. This passive pressure is developed at shear keys in retaining walls and at end abutments.
{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
 
|+
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"  
!style="background:#BEBEBE" rowspan="2"|Loading Case !!style="background:#BEBEBE" rowspan="2"|Vertical Loads !!style="background:#BEBEBE" rowspan="2"|Horizontal Loads !!style="background:#BEBEBE" rowspan="2"|Overturning Factor of Safety !!style="background:#BEBEBE" colspan="2"|Sliding Factor of Safety
|-
!style="background:#BEBEBE" |Battered Toe Piles !!style="background:#BEBEBE" |Vertical Toe Piles
|-
|align="center"|I<sup>a</sup>||align="center"| DL+SUR+E ||align="center"|EP+SUR||align="center"| 1.5||align="center"| 1.5||align="center"|2.0
|-
|-
|'''Additional Information'''
|align="center"|II||align="center"| DL+SUR+E ||align="center"|EP+SUR+COL||align="center"| 1.2|| align="center"|1.2||align="center"| 1.2
|-
|-
|AASHTO 5.5.5A
|align="center"|III||align="center"| DL+E||align="center"| EP||align="center"| 1.5||align="center"| 1.5||align="center"| 2.0
|}
|-
 
|align="center"|IV<sup>b</sup>||align="center"| DL+E ||align="center"|None||align="center"| -||align="center"| -||align="center"| -
The passive pressure against the front face of the wall and the footing of a retaining wall is loosely compacted and should be neglected when considering sliding.
|-
|align="center"|V<sup>c</sup>||align="center"| DL+E||align="center"| EP+EQ||align="center"| 1.125||align="center"| 1.125||align="center"| 1.5
|-
|colspan="6"|<sup>'''a'''</sup> Load Case I should be checked with and without the vertical surcharge.
|-
|colspan="6"|<sup>'''b'''</sup> A 25% overstress is allowed on the heel pile in Load Case IV.
|-
|colspan="6"|<sup>'''c'''</sup> The factors of safety for earthquake loading are 75% of that used in Load Case III. Battered piles are not recommended for use in seismic performance categories B, C, and D. Seismic design of retaining walls is not required in SPC A and B. Retaining walls in SPC B located under a bridge abutment shall be designed to AASHTO Specifications for SPC B.
|}


Rankine Formula: <math>P_p = \frac{1}{2}C_p\gamma_s[H^2-H_1^2]</math> where thefollowing variables are defined in the figure below
::'''Pile Properties and Capacities'''
:
:''C<sub>p</sub>'' = <math>\tan \big( 45^\circ + \frac{\phi}{2}\big)</math>


:''y<sub>1</sub> = <math>\frac{H_1y_2^2 + \frac{2}{3}y_2^3}{H^2 - H_1^2}</math>
::For Load Cases I-IV in the table above, the allowable compressive pile force may be taken from the pile capacity table in the Piling Section of the Bridge Manual which is based in part on AASHTO 4.5.7.3. Alternatively, the allowable compressive pile capacity of a friction pile may be determined from the ultimate frictional and bearing capacity between the soil and pile divided by a safety factor of 3.5 (AASHTO Table 4.5.6.2.A). The maximum amount of tension allowed on a heel pile is 3 tons.


:''P<sub>p</sub>'' = passive force at shear key in pounds per foot of wall length
::For Load Case V in the table above, the allowable compressive pile force may be taken from the pile capacity table in the Piling Section of the Bridge Manual multiplied by the appropriate factor (2.0 for steel bearing piles, 1.5 for friction piles). Alternatively, the allowable compressive pile capacity of a friction pile may be determined from the ultimate frictional and bearing capacity between the soil and pile divided by a safety factor of 2.0. The allowable tension force on a bearing or friction pile will be equal to the ultimate friction capacity between the soil and pile divided by a safety factor of 2.0.


:''C<sub>p</sub>'' = coefficient of passive earth pressure
::To calculate the ultimate compressive or tensile capacity between the soil and pile requires the boring data which includes the SPT blow counts, the friction angle, the water level, and the soil layer descriptions.


:<math>\boldsymbol{\gamma_s}</math> = unit weight of soil
::Assume the vertical load carried by battered piles is the same as it would be if the pile were vertical. The properties of piles may be found in the Piling Section of the Bridge Manual.


:''H'' = height of the front face fill less than 1 ft. min. for erosion
:::'''Neutral Axis of Pile Group'''


:''H<sub>1</sub>'' = H minus depth of shear key
:::Locate the neutral axis of the pile group in the repetitive strip from the toe of the footing at the bottom of the footing.


:''y<sub>1</sub>'' = location of ''P<sub>p</sub>'' from bottom of footing
:::'''Moment of Inertia of Pile Group'''


:<math>\boldsymbol{\phi}</math> = angle of internal friction of soil
:::The moment of inertia of the pile group in the repetitive strip about the neutral axis of the section may be determined using the parallel axis theorem:


[[image:751.24.3.2.1 passive.jpg|center|500px]]
::::I = Σ(I<sub>A</sub>) + Σ(Ad<sup>2</sup>) where :  
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.2
|}
The resistance due to passive pressure in front of the shear key shall be neglected unless the key extends below the depth of frost penetration.
{|style="padding: 0.3em; margin-right:7px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="left"
|-
|'''Additional Information'''
|-
|[http://wwwi/intranet/cm/ MoDOT Materials Division]
|}


Frost line is set at 36 in. at the north border of Missouri and at 18 in. at the south border.
::::''I<sub>A</sub>'' = moment of inertia of a pile about its neutral axis


'''Passive Pressure During Seismic Loading'''
::::''A'' = area of a pile


During an earthquake, the passive resistance of soil to lateral loads is slightly decreased. The Mononobe-Okabe static method is used to determine the equivalent fluid pressure.
::::''d'' = distance from a pile's neutral axis to pile group's neutral axis


:''P<sub>PE</sub>'' = equivalent passive earth pressure during an earthquake
:::''I<sub>A</sub>'' may be neglected so the equation reduces to:  
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|1992 AASHTO Div. IA Eqns. C6-5 and C6-6
|}
:<math>P_{PE} = \frac{1}{2}\gamma_sH^2(1 - k_v)K_{PE}</math> where:


:''K<sub>PE</sub>'' = seismic passive pressure coefficient
::::''I'' =  Σ(Ad<sup>2</sup>)


:<math>K_{PE} = \frac{\cos^2(\phi - \theta - \beta)}{\cos\theta\cos^2\beta\cos(\delta + \beta + \theta)\Bigg[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \theta - i)}{\cos(\delta + \beta + \theta)\cos(i - \beta)}}\Bigg]^2}</math>
::'''Resistance To Sliding'''


::<math>\boldsymbol{\gamma}_s</math> = unit weight of soil
::Any frictional resistance to sliding shall be ignored, such as would occur between the bottom of the footing and the soil on a spread footing.


:''H'' = height of soil at the location where the earth pressure is to be found
::'''Friction or Bearing Piles With Batter (Case 1)'''
 
::Retaining walls using friction or bearing piles with batter should develop lateral strength (resistance to sliding) first from the batter component of the pile and second from the passive pressure against the shear key and the piles.
 
::'''Friction or Bearing Piles Without Batter (Case 2)'''


:''k<sub>V</sub>'' = vertical acceleration coefficient
::Retaining walls using friction or bearing piles without batter due to site constrictions should develop lateral strength first from the passive pressure against the shear key and second from the passive pressure against the pile below the bottom of footing. In this case, the shear key shall be placed at the front face of the footing.


:<math>\boldsymbol{\phi}</math> = angle of internal friction of soil
::'''Concrete Pedestal Piles or Drilled Shafts (Case 3)'''


:<math>\boldsymbol{\theta} =  arctan \big[\frac{k_h}{1 - k_V}\big]</math>
::Retaining walls using concrete pedestal piles should develop lateral strength first from passive pressure against the shear key and second from passive pressure against the pile below the bottom of the footing. In this case, the shear key shall be placed at the front of the footing. Do not batter concrete pedestal piles.


:''k<sub>H</sub>'' = horizontal acceleration coefficient
[[image:751.24.3.2.2 cases.jpg|center|450px]]


:<math>\boldsymbol{\beta}</math> = slope of soil face in degrees
::'''Resistance Due to Passive Pressure Against Pile'''


:''i'' = backfill slope angle in degrees
::The procedure below may be used to determine the passive pressure resistance developed in the soil against the piles. The procedure assumes that the piles develop a local failure plane.


:<math>\boldsymbol{\delta}</math> = angle of friction between soil and wall
:::''F'' = the lateral force due to passive pressure on pile


'''Special Soil Conditions'''
:::<math>F = \frac{1}{2}\gamma_s C_P H^2 B </math> , where: <math> C_P = tan^2\Big[45 + \frac{\phi}{2}\Big]</math>


Due to creep, some soft clay soils have no passive resistance under a continuing load. Removal of undesirable material and replacement with suitable material such as sand or crushed stone is necessary in such cases. Generally, this condition is indicated by a void ratio above 0.9, an angle of internal friction (<math>\boldsymbol{\phi}</math>) less than 22°, or a soil shear less than 0.8 ksf. Soil shear is determined from a standard penetration test.
:::<math>\boldsymbol{\gamma_s}</math> = unit weight of soil


:Soil Shear <math>\Big(\frac{k}{ft^2}\Big) = \frac{blows \ per\ 12\ in.}{10}</math>
:::''H'' = depth of pile considered for lateral resistance (H<sub>max</sub>= 6B)


'''Friction'''
:::''C<sub>P</sub>'' = coefficient of active earth pressure


In the absence of tests, the total shearing resistance to lateral loads between the footing and a soil that derives most of its strength from internal friction may be taken as the normal force times a coefficient of friction. If the plane at
:::''B'' = width of pile
which frictional resistance is evaluated is not below the frost line then this resistance must be neglected.


[[image:751.24.3.2.1 friction 2016.jpg|center|450px|thumb|<center>'''When A Shear Key Is Not Used'''</center>]]
:::<math>\boldsymbol{\phi}</math> = angle of internal friction of soil
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
 
|-
[[image:751.24.3.2.2 resistance passive.jpg|center|450px]]
|'''Additional Information'''
|-
|AASHTO 5.5.2B
|}


Sliding is resisted by the friction force developed at the interface between the soil and the concrete footing along the failure plane. The coefficient of friction for soil against concrete can be taken from the table below. If soil data
::'''Resistance Due to Pile Batter'''
is not readily available or is inconsistent, the friction factor (f) can be taken as


: ''f'' =<math>tan \Big(\frac{2\phi}{3}\Big)</math> where <math>\boldsymbol{\phi}</math> is the angle of internal friction of the soil (''Civil Engineering Reference Manual'' by Michael R. Lindeburg, 6th ed., 1992).
::Use the horizontal component (due to pile batter) of the allowable pile load as the lateral resistance of the battered pile. (This presupposes that sufficient lateral movement of the wall can take place before failure to develop the ultimate strength of both elements.)


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
[[image:751.24.3.2.2 12.jpg|center|125px]]
|+
!style="background:#BEBEBE" colspan="2"|Coefficient of Friction Values for Soil Against Concrete
|-
! style="background:#BEBEBE" |Soil Type<sup>a</sup> !! style="background:#BEBEBE"|Coefficient of Friction
|-
|  align="center" |coarse-grained soil without silt || align="center"|0.55
|-
|  align="center"|coarse-grained soil with silt  || align="center"|0.45
|-
|align="center"|silt (only)||  align="center"|0.35
|-
|align="center"|clay||  align="center"|0.30<sup>b</sup>
|-
|colspan="2"|<sup>'''a'''</sup> It is not necessary to check rock or shale for sliding due to embedment.
|-
|colspan="2"|<sup>'''b'''</sup> Caution should be used with soils with <math>\boldsymbol{\phi}</math> < 22° or soil shear < 0.8 k/sq.ft. (soft clay soils). Removal and replacement of such soil with suitable material should be considered.
|}


[[image:751.24.3.2.1 soil and soil.jpg|center|450px|thumb|<center>'''When A Shear Key Is Used'''</center>]]
:::''b'' = the amount of batter per 12 inches.


When a shear key is used, the failure plane is located at the bottom of the shear key in the front half of the footing. The friction force resisting sliding in front of the shear key is provided at the interface between the stationary layer of soil and the moving layer of soil, thus the friction angle is the internal angle of friction of the soil (soil against soil). The friction force resisting sliding on the rest of the footing is of that between the concrete and soil. Theoretically
:::<math> c = \sqrt{(12 in.)^2 + b^2}</math>
the bearing pressure distribution should be used to determine how much normal load exists on each surface, however it is reasonable to assume a constant distribution. Thus the normal load to each surface can be divided out between the two surfaces based on the fractional length of each and the total frictional force will be the sum of the normal load on each surface
 
multiplied by the corresponding friction factor.
:::<math>P_{HBatter} = P_T \Big(\frac{b}{c}\Big)</math> (# of battered piles) where:


'''Bearing Pressure'''
:::''P<sub>HBatter</sub>'' = the horizontal force due to the battered piles
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 4.4.7.1.2 & 4.4.8.1.3
|}


:'''Group Loads I - VI'''
:::''P<sub>T</sub>'' = the allowable pile load


:The bearing capacity failure factor of safety for Group Loads I - VI must be greater than or equal to 3.0. This factor of safety is figured into the allowable bearing pressure given on the "Design Layout Sheet".
::Maximum batter is 4" per 12".


:The bearing pressure on the supporting soil shall not be greater than the allowable bearing pressure given on the "Design Layout Sheet".
::'''Resistance Due to Shear Keys'''


:'''Seismic Loads'''
::A shear key may be needed if the passive pressure against the piles and the horizontal force due to batter is not sufficient to attain the factor of safety against sliding. The passive pressure against the shear key on a pile footing is found in the same manner as for spread footings.
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO Div. IA 6.3.1(B) and AASHTO 5.5.6.2
|}


:When seismic loads are considered, AASHTO allows the ultimate bearing capacity to be used. The ultimate capacity of the foundation soil can be conservatively estimated as 2.0 times the allowable bearing pressure given on the "Design Layout".
::'''Resistance to Overturning'''


:'''Stem Design'''
::The resisting and overturning moments shall be computed at the centerline of the toe pile at a distance of 6B (where B is the width of the pile) below the bottom of the footing. A maximum of 3 tons of tension on each heel pile may be assumed to resist overturning. Any effects of passive pressure, either on the shear key or on the piles, which resist overturning, shall be ignored.
:The vertical stem (the wall portion) of a cantilever retaining wall shall be designed as a cantilever supported at the base.


:'''Footing Design'''
[[image:751.24.3.2.2 resistance overturning.jpg|center|450px]]
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.1
|}


::'''Toe'''
::'''Pile Properties'''


::The toe of the base slab of a cantilever wall shall be designed as a cantilever supported by the wall. The critical section for bending moments shall be taken at the front face of the stem. The critical section for shear shall be taken at a distance d (d = effective depth) from the front face of the stem.
:::'''Location of Resultant'''


::'''Heel'''
:::The location of the resultant shall be evaluated at the bottom of the footing and can be determined by the equation below:


::The rear projection (heel) of the base slab shall be designed to support the entire weight of the superimposed materials, unless a more exact method is used. The heel shall be designed as a cantilever supported by the wall. The critical section for bending moments and shear shall be taken at the back face of the stem.
::::<math>e = \frac{\Sigma M}{\Sigma V}</math>  where:


:'''Shear Key Design'''
::::e = the distance between the resultant and the neutral axis of the pile group


:The shear key shall be designed as a cantilever supported at the bottom of the footing.
::::''ΣM'' = the sum of the moments taken about the neutral axis of the pile group at the bottom of the footing


====751.24.3.2.2 Pile Footings====
::::''ΣV'' = the sum of the vertical loads used in calculating the moment


Footings shall be cast on piles when specified on the "Design Layout Sheet". If the horizontal force against the retaining wall cannot otherwise be resisted, some of the piles shall be driven on a batter.
:::'''Pile Loads'''


:'''Pile Arrangement'''
:::The loads on the pile can be determined as follows:


:For retaining walls subject to moderate horizontal loads (walls 15 to 20 ft. tall), the following layout is suggested.
::::<math>P = \frac{\Sigma V}{A} \pm \frac{Mc}{I}</math> where:


[[image:751.24.3.2.2 batter piles.jpg|center|300px|thumb|<center>'''Section'''</center>]]
:::::''P'' = the force on the pile


[[image:751.24.3.2.2 plan 2016.jpg|center|450px|thumb|<center>'''Plan'''</center>]]
:::::''A'' = the areas of all the piles being considered


:For higher walls and more extreme conditions of loading, it may be necessary to:
:::::''M'' = the moment of the resultant about the neutral axis


:* use the same number of piles along all rows
:::::''c'' = distance from the neutral axis to the centerline of the pile being investigated


:* use three rows of piles
:::::''I'' = the moment of inertia of the pile group
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.2
|}


:* provide batter piles in more than one row
:::'''Stem Design'''


::'''Loading Combinations for Stability and Bearing'''
:::The vertical stem (the wall portion) of a cantilever retaining wall shall be designed as a cantilever supported at the base.


::The following table gives the loading combinations to be checked for stability and pile loads. These abbreviations are used in the table:
:::'''Footing Design'''


:::DL = dead load weight of the wall elements
::::'''Toe'''
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.1
|}


:::SUR = two feet of live load surcharge
::::The toe of the base slab of a cantilever wall shall be designed as a cantilever supported by the wall. The critical section for bending moments shall be taken at the front face of the stem. The critical section for shear shall be taken at a distance d (d = effective depth) from the front face of the stem.


:::E = earth weight
::::'''Heel'''
::::The top reinforcement in the rear projection (heel) of the base slab shall be designed to support the entire weight of the superimposed materials plus any tension load in the heel piles (neglect compression loads in the pile), unless a more exact method is used. The bottom reinforcement in the heel of the base slab shall be designed to support the maximum compression load in the pile neglecting the weight of the superimposed materials. The heel shall be designed as a cantilever supported by the wall. The critical sections for bending moments and shear shall be taken at the back face of the stem.


:::EP = equivalent fluid earth pressure
:::'''Shear Key Design'''
:::The shear key shall be designed as a cantilever supported at the bottom of the footing.


:::COL = collision force
====751.24.3.2.3 Counterfort Walls====


:::EQ = earthquake inertial force of failure wedge
'''Assumptions:'''


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
(1) Stability
|+
The external stability of a counterfort retaining wall shall be determined in the same manner as described for cantilever retaining walls. Therefore refer to previous pages for the criteria for location of resultant, factor of safety for sliding and bearing pressures.
!style="background:#BEBEBE" rowspan="2"|Loading Case !!style="background:#BEBEBE" rowspan="2"|Vertical Loads !!style="background:#BEBEBE" rowspan="2"|Horizontal Loads !!style="background:#BEBEBE" rowspan="2"|Overturning Factor of Safety !!style="background:#BEBEBE" colspan="2"|Sliding Factor of Safety
|-
(2) Stem
!style="background:#BEBEBE" |Battered Toe Piles !!style="background:#BEBEBE" |Vertical Toe Piles
|-
[[image:751.24.3.2.3 counterfort.jpg|center|800px]]
|align="center"|I<sup>a</sup>||align="center"| DL+SUR+E ||align="center"|EP+SUR||align="center"| 1.5||align="center"| 1.5||align="center"|2.0
|-
|align="center"|II||align="center"| DL+SUR+E ||align="center"|EP+SUR+COL||align="center"| 1.2|| align="center"|1.2||align="center"| 1.2
|-
|align="center"|III||align="center"| DL+E||align="center"| EP||align="center"| 1.5||align="center"| 1.5||align="center"| 2.0
|-
|align="center"|IV<sup>b</sup>||align="center"| DL+E ||align="center"|None||align="center"| -||align="center"| -||align="center"| -
|-
|align="center"|V<sup>c</sup>||align="center"| DL+E||align="center"| EP+EQ||align="center"| 1.125||align="center"| 1.125||align="center"| 1.5
|-
|colspan="6"|<sup>'''a'''</sup> Load Case I should be checked with and without the vertical surcharge.
|-
|colspan="6"|<sup>'''b'''</sup> A 25% overstress is allowed on the heel pile in Load Case IV.
|-
|colspan="6"|<sup>'''c'''</sup> The factors of safety for earthquake loading are 75% of that used in Load Case III. Battered piles are not recommended for use in seismic performance categories B, C, and D. Seismic design of retaining walls is not required in SPC A and B. Retaining walls in SPC B located under a bridge abutment shall be designed to AASHTO Specifications for SPC B.
|}


::'''Pile Properties and Capacities'''
:<math>P = C_a \boldsymbol \gamma_s</math>


::For Load Cases I-IV in the table above, the allowable compressive pile force may be taken from the pile capacity table in the Piling Section of the Bridge Manual which is based in part on AASHTO 4.5.7.3. Alternatively, the allowable compressive pile capacity of a friction pile may be determined from the ultimate frictional and bearing capacity between the soil and pile divided by a safety factor of 3.5 (AASHTO Table 4.5.6.2.A). The maximum amount of tension allowed on a heel pile is 3 tons.
:where:
::''C<sub>a</sub>'' = coefficient of active earth pressure


::For Load Case V in the table above, the allowable compressive pile force may be taken from the pile capacity table in the Piling Section of the Bridge Manual multiplied by the appropriate factor (2.0 for steel bearing piles, 1.5 for friction piles). Alternatively, the allowable compressive pile capacity of a friction pile may be determined from the ultimate frictional and bearing capacity between the soil and pile divided by a safety factor of 2.0. The allowable tension force on a bearing or friction pile will be equal to the ultimate friction capacity between the soil and pile divided by a safety factor of 2.0.
::<math>\boldsymbol \gamma_s</math> = unit weigt of soil


::To calculate the ultimate compressive or tensile capacity between the soil and pile requires the boring data which includes the SPT blow counts, the friction angle, the water level, and the soil layer descriptions.
Design the wall to support horizontal load from the earth pressure and the liveload surcharge (if applicable) as outlined on the previous pages and as designated in AASHTD Section 3.20, except that maximum horizontal loads shall be the calculated equivalent fluid pressure at 3/4  height of wall [(0.75 H)P] which shall be considered applied uniformly from the lower quarter point to the bottom of wall.


::Assume the vertical load carried by battered piles is the same as it would be if the pile were vertical. The properties of piles may be found in the Piling Section of the Bridge Manual.
In addition, vertical steel In the fill face of the bottom quarter of the wall shall be that required by the vertical cantilever wall with the equivalent fluid pressure of that (0.25 H) height.


:::'''Neutral Axis of Pile Group'''
Maximum concrete stress shall be assumed as the greater of the two thus obtained.
The application of these horizontal pressures shall be as follows:
[[image:751.24.3.2.3 counterfort wall.jpg|center|800px|thumb|<center>'''Counterfort Wall Section'''</center> <center>Moments are to be determined by analysis as a continuous beam.  The counterforts are to be spaced so as to produce approximately equal positive and negative moments.</center>]]


:::Locate the neutral axis of the pile group in the repetitive strip from the toe of the footing at the bottom of the footing.
(3)  Counterfort
Counterforts shall be designed as T-beams, of which the wall is the flange and the counterfort is the stem.  For this reason the concrete stresses ane normally low and will not control.


:::'''Moment of Inertia of Pile Group'''
For the design of reinforcing steel in the back of the counterfort, the effective d shall be the perpendicular distance from the front face of the wall (at point that moment is considered), to center of reinforcing steel.


:::The moment of inertia of the pile group in the repetitive strip about the neutral axis of the section may be determined using the parallel axis theorem:
[[image:751.24.3.2.3 moment.jpg|center|500px]]


::::I = Σ(I<sub>A</sub>) + Σ(Ad<sup>2</sup>) where :
(4) Footing


::::''I<sub>A</sub>'' = moment of inertia of a pile about its neutral axis
The footing of the counterfort walls shall be designed as a continuous beam of spans equal to the distance between the counterforts.


::::''A'' = area of a pile
The rear projection or heel shall be designed to support the entire weight of the superimposed materials, unless a more exact method is used. Refer to AASHTD Section 5.5.6.


::::''d'' = distance from a pile's neutral axis to pile group's neutral axis
Divide footing (transversely) into four (4) equal sections for design footing pressures.


:::''I<sub>A</sub>'' may be neglected so the equation reduces to:
Counterfort walls on pile are very rare and are to be treated as special cases.  See Structural Project Manager.


::::''I'' =  Σ(Ad<sup>2</sup>)  
(5) Sign-Board type walls


::'''Resistance To Sliding'''
The Sign-Board type of retaining walls are a special case of the counterfort retaining walls.  This type of wall is used where the soiI conditions are such that the footings must be placed a great distance below the finished ground line.  For this situation, the wall is discontinued approximately 12 in. below the finished ground line or below the frost line.


::Any frictional resistance to sliding shall be ignored, such as would occur between the bottom of the footing and the soil on a spread footing.
Due to the large depth of the counterforts, it may be more economical to use a smaller number of counterforts than would otherwise be used.
All design assumptions that apply to counterfort walls will apply to sign-board walls with the exception of the application of horizontal forces for the stem (or wall design), and the footing design which shall be as follows:


::'''Friction or Bearing Piles With Batter (Case 1)'''
:'''Wall'''


::Retaining walls using friction or bearing piles with batter should develop lateral strength (resistance to sliding) first from the batter component of the pile and second from the passive pressure against the shear key and the piles.
[[image:751.24.3.2.3 load.jpg|center|550px]]


::'''Friction or Bearing Piles Without Batter (Case 2)'''
:'''Footing'''


::Retaining walls using friction or bearing piles without batter due to site constrictions should develop lateral strength first from the passive pressure against the shear key and second from the passive pressure against the pile below the bottom of footing. In this case, the shear key shall be placed at the front face of the footing.
:The individual footings shall be designed transversely as cantilevers supported by the wall.  Refer to AASHTO Section 5.


::'''Concrete Pedestal Piles or Drilled Shafts (Case 3)'''
===751.24.3.3 Example 1: Spread Footing Cantilever Wall===


::Retaining walls using concrete pedestal piles should develop lateral strength first from passive pressure against the shear key and second from passive pressure against the pile below the bottom of the footing. In this case, the shear key shall be placed at the front of the footing. Do not batter concrete pedestal piles.
[[image:751.24.3.3.jpg|center|750px|thumb|<Center>'''Typical Section thru Wall</center><center>(Spread Footing)</center>''']]


[[image:751.24.3.2.2 cases.jpg|center|450px]]
:f'<sub>c</sub> = 3,000 psi
:f<sub>y</sub> = 60,000 psi
:''φ'' = 24 in.
:''γ<sub>s</sub>'' = 120 pcf (unit wgt of soil)
:Allowable soil pressure = 2 tsf
:''γ<sub>c</sub>'' = 150 pcf (unit wgt of concr.)
:Retaining wall is located in Seismic Performance Category (SPC) B.
:A = 0.1 (A = seismic acceleration coefficient)


::'''Resistance Due to Passive Pressure Against Pile'''
{| style="margin: 1em auto 1em auto"
|-
|<math>P_a = \frac{1}{2}\gamma_s C_a H^2</math>||width=50| ||<math>P_p = \frac{1}{2}\gamma_s C_p H_2^2 - H_1^2</math>
|}


::The procedure below may be used to determine the passive pressure resistance developed in the soil against the piles. The procedure assumes that the piles develop a local failure plane.
'''Assumptions'''


:::''F'' = the lateral force due to passive pressure on pile
* Retaining wall is under an abutment or in a location where failure of the wall may affect the structural integrity of a bridge. Therefore, it must be designed for SPC B.


:::<math>F = \frac{1}{2}\gamma_s C_P H^2 B </math> , where: <math> C_P = tan^2\Big[45 + \frac{\phi}{2}\Big]</math>
* Design is for a unit length (1 ft.) of wall.


:::<math>\boldsymbol{\gamma_s}</math> = unit weight of soil
* Sum moments about the toe at the bottom of the footing for overturning.


:::''H'' = depth of pile considered for lateral resistance (H<sub>max</sub>= 6B)
*For Group Loads I-VI loading:
:* F.S. for overturning ≥ 2.0 for footings on soil.
:* F.S. for sliding ≥ 1.5.
* Resultant to be within middle 1/3 of footing.


:::''C<sub>P</sub>'' = coefficient of active earth pressure
* For earthquake loading:
:* F.S. for overturning ≥ 0.75(2.0) = 1.5.
:* F.S. for sliding ≥ 0.75(1.5) = 1.125.
:* Resultant to be within middle 1/2 of footing.


:::''B'' = width of pile
* Base of footing is below the frost line.


:::<math>\boldsymbol{\phi}</math> = angle of internal friction of soil
* Neglect top one foot of fill over toe when determining passive pressure and soil weight.


[[image:751.24.3.2.2 resistance passive.jpg|center|450px]]
* Use of a shear key shifts the failure plane to "B" where resistance to sliding is provided by passive pressure against the shear key, friction of soil along failure plane "B" in front of the key, and friction between soil and concrete along the footing behind the key.


::'''Resistance Due to Pile Batter'''
* Soil cohesion along failure plane is neglected.


::Use the horizontal component (due to pile batter) of the allowable pile load as the lateral resistance of the battered pile. (This presupposes that sufficient lateral movement of the wall can take place before failure to develop the ultimate strength of both elements.)
* Footings are designed as cantilevers supported by the wall.
:* Critical sections for bending are at the front and back faces of the wall.
:* Critical sections for shear are at the back face of the wall for the heel and at a distance d (effective depth) from the front face for the toe.


[[image:751.24.3.2.2 12.jpg|center|125px]]
* Neglect soil weight above toe of footing in design of the toe.


:::''b'' = the amount of batter per 12 inches.
* The wall is designed as a cantilever supported by the footing.


:::<math> c = \sqrt{(12 in.)^2 + b^2}</math>
* Load factors for AASHTO Groups I - VI for design of concrete:
:* ''γ'' = 1.3.
:* ''β<sub>E</sub>'' = 1.3 for horizontal earth pressure on retaining walls.
:* ''β<sub>E</sub>'' = 1.0 for vertical earth pressure.


:::<math>P_{HBatter} = P_T \Big(\frac{b}{c}\Big)</math> (# of battered piles) where:
* Load factor for earthquake loads = 1.0.


:::''P<sub>HBatter</sub>'' = the horizontal force due to the battered piles
'''Lateral Pressures Without Earthquake'''


:::''P<sub>T</sub>'' = the allowable pile load
:''C<sub>a</sub>'' = <math>\cos\delta\Bigg[\frac{\cos\delta - \sqrt{\cos^2\delta - \cos^2\phi}}{\cos\delta + \sqrt{\cos^2\delta - \cos^2\phi}}\Bigg]</math>


::Maximum batter is 4" per 12".
:''C<sub>a</sub>'' = <math>\cos 18.435^\circ \Bigg[\frac{\cos\ 18.435^\circ - \sqrt{\cos^2\ 18.435^\circ - \cos^2\ 24^\circ }}{\cos\ 18.435^\circ  + \sqrt{\cos^2\ 18.435^\circ  - \cos^2\ 24^\circ }}\Bigg]</math> = 0.546


::'''Resistance Due to Shear Keys'''
:<math>C_p = tan^2 \big( 45^\circ + \frac{\phi}{2}\big)  = tan^2 \big( 45^\circ + \frac{24^\circ}{2}\big) = 2.371</math>


::A shear key may be needed if the passive pressure against the piles and the horizontal force due to batter is not sufficient to attain the factor of safety against sliding. The passive pressure against the shear key on a pile footing is found in the same manner as for spread footings.
:<math>P_A = \frac{1}{2}\big[0.120\frac{k}{ft^3}\big](1 ft)(0.546)(10.667 ft)^2 = 3.726k</math>


::'''Resistance to Overturning'''
:<math>P_P = \frac{1}{2}\big[0.120\frac{k}{ft^3}\big](1 ft)(2.371)\big[(5.0)^2 - (2.5)^2\big] = 2.668k</math>


::The resisting and overturning moments shall be computed at the centerline of the toe pile at a distance of 6B (where B is the width of the pile) below the bottom of the footing. A maximum of 3 tons of tension on each heel pile may be assumed to resist overturning. Any effects of passive pressure, either on the shear key or on the piles, which resist overturning, shall be ignored.
:<math>P_{AV} = P_A (sin \delta) = 3.726k (sin 18.435^\circ ) = 1.178k</math>


[[image:751.24.3.2.2 resistance overturning.jpg|center|450px]]
:<math>P_{AH} = P_A (cos \delta) = 3.726k (cos 18.435^\circ ) = 3.534k</math>


::'''Pile Properties'''
{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
 
|+
:::'''Location of Resultant'''
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Area (ft<sup>2</sup>) !!style="background:#BEBEBE" |Force (k) = (Unit Wgt.)(Area) !!style="background:#BEBEBE" |Arm (ft.) !!style="background:#BEBEBE"|Moment (ft-k)
 
|-
:::The location of the resultant shall be evaluated at the bottom of the footing and can be determined by the equation below:
|align="center"|(1)||align="center"| (0.5)(6.667ft)(2.222ft) = 7.407||align="center"| 0.889||align="center"| 7.278 ||align="center"|6.469
 
|-
::::<math>e = \frac{\Sigma M}{\Sigma V}</mathwhere:
|align="center"|(2)||align="center"| (6.667ft)(6.944ft) = 46.296||align="center"| 5.556||align="center"| 6.167||align="center"| 34.259
 
|-
::::e = the distance between the resultant and the neutral axis of the pile group
|align="center"|(3) ||align="center"|(0.833ft)(8.000ft) + (0.5)(0.083ft)(8.000ft) = 7.000||align="center"|1.050||align="center"| 2.396||align="center"| 2.515
|-
|align="center"|(4) ||align="center"|(1.500ft)(9.500ft) = 14.250||align="center"| 2.138 ||align="center"|4.750 ||align="center"|10.153
|-
|align="center"|(5) ||align="center"|(2.500ft)(1.000ft) = 2.500||align="center"| 0.375||align="center"| 2.500||align="center"| 0.938
|-
|align="center"|(6) ||align="center"|(1.000ft)(1.917ft)+(0.5)(0.010ft)(1.000ft) = 1.922||align="center"|<u>0.231</u>||align="center"| 0.961||align="center"|<u>0.222</u>
|-
|align="center"|Σ ||align="center"| -  ||align="center"|ΣV = 10.239 ||align="center"| - ||align="center"|ΣM<sub>R</sub> = 54.556
|-
|align="center"|P<sub>AV</sub>||align="center"| -  ||align="center"|<u>1.178</u>||align="center"| 9.500 ||align="center"|<u>11.192</u>
|-
|align="center"|Σ resisting ||align="center"| - ||align="center"|ΣV = 11.417||align="center"| - ||align="center"| ΣM<sub>R</sub> = 65.748
|-
|align="center"|P<sub>AH</sub> ||align="center"| - ||align="center"|3.534 ||align="center"|3.556 ||align="center"|12.567
|-
|align="center"|P<sub>P</sub>||align="center"| - ||align="center"|2.668 ||align="center"|1.389<sup>1</sup>||align="center"| -
|-
|colspan="5"|'''<sup>1</sup>''' The passive capacity at the shear key is ignored in overturning checks,since this capacity is considered in the factor of safety against sliding. It is assumed that a sliding and overturning failure will not occur simultaneously. The passive capacity at the shear key is developed only if the wall does slide.
|}


::::''ΣM'' = the sum of the moments taken about the neutral axis of the pile group at the bottom of the footing
[[image:751.24.3.3 passive.jpg|right|150px]]
<math>\bar{y} = \frac{H_1y^2 + \frac{2}{3}y^3}{H_2^2 - H_1^2} = \frac{(2.5 ft)(2.5 ft)^2 + \frac{2}{3}(2.5 ft)^3}{(5.0 ft)^2 - (2.5 ft)^2}</math> = 1.389 ft.


::::''ΣV'' = the sum of the vertical loads used in calculating the moment
:'''Overturning'''


:::'''Pile Loads'''
:F.S. = <math>\frac{M_R}{M_{OT}} = \frac{65.748(ft-k)}{12.567(ft-k)} = 5.232 \ge 2.0 </math> <u>o.k.</u>


:::The loads on the pile can be determined as follows:
:where: M<sub>OT</sub> = overturning moment; M<sub>R</sub> = resisting moment


::::<math>P = \frac{\Sigma V}{A} \pm \frac{Mc}{I}</math> where:
:'''Resultant Eccentricity'''


:::::''P'' = the force on the pile
:<math>\bar{x} = \frac{(65.748 - 12.567)(ft-k)}{11.417k}</math> = 4.658 ft.


:::::''A'' = the areas of all the piles being considered
:<math>e = \frac{9.500 ft}{2} - 4.658 ft. = 0.092 ft.</math>
:<math>\frac{L}{6} =\frac{9.500 ft}{6} = 1.583 ft > e</math> <u>o.k.</u>


:::::''M'' = the moment of the resultant about the neutral axis
:'''Sliding'''


:::::''c'' = distance from the neutral axis to the centerline of the pile being investigated
:Check if shear key is required for Group Loads I-VI:


:::::''I'' = the moment of inertia of the pile group
:F.S. = <math>\frac{\Sigma V(tan\phi_{s-c})}{P_{AH}} = \frac{11.042k(tan \frac{2}{3}(24^\circ)}{3.534k} </math>= 0.896 <u>no good - shear key req'd</u>
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.2
|}


:::'''Stem Design'''
:where: ''φ<sub>s-c</sub>'' = angle of friction between soil and concrete = (2/3)''φ<sub>s-s</sub>''  


:::The vertical stem (the wall portion) of a cantilever retaining wall shall be designed as a cantilever supported at the base.
:F.S. = <math>\frac{P_P + (\Sigma V) \Big(\frac{L_2}{L_1} tan \phi_{s-s}+\frac{L_3}{L_1} tan \phi_{s-c}\Big)}{P_{AH}}</math>


:::'''Footing Design'''
:where: ''φ<sub>s-s</sub>'' = angle of internal friction of soil


::::'''Toe'''
:F.S. = <math>\frac{2.668k + (11.417k) \Big[\Big(\frac{2 ft}{9.50 ft}\Big) tan 24^\circ + \Big(\frac{7.50 ft}{9.50 ft} tan \Big(\frac{2}{3}(24^\circ)\Big)\Big]}{3.534 k}</math> = 1.789 ≥ 1.5 <u>o.k.</u>
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 5.5.6.1
|}


::::The toe of the base slab of a cantilever wall shall be designed as a cantilever supported by the wall. The critical section for bending moments shall be taken at the front face of the stem. The critical section for shear shall be taken at a distance d (d = effective depth) from the front face of the stem.
:'''Footing Pressure'''


::::'''Heel'''
:<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>
::::The top reinforcement in the rear projection (heel) of the base slab shall be designed to support the entire weight of the superimposed materials plus any tension load in the heel piles (neglect compression loads in the pile), unless a more exact method is used. The bottom reinforcement in the heel of the base slab shall be designed to support the maximum compression load in the pile neglecting the weight of the superimposed materials. The heel shall be designed as a cantilever supported by the wall. The critical sections for bending moments and shear shall be taken at the back face of the stem.


:::'''Shear Key Design'''
:P<sub>H</sub> = pressure at heel <math>P_H = \frac{11.417 k}{(1 ft)9.50 ft} \Big[1 - \frac{6 (0.092 ft)}{9.50 ft}\Big]</math> = 1.132 k/ft<sup>2</sup>
:::The shear key shall be designed as a cantilever supported at the bottom of the footing.


====751.24.3.2.3 Counterfort Walls====
:P<sub>T</sub> = pressure at toe <math>P_T = \frac{11.417 k}{(1 ft)9.50 ft} \Big[1 + \frac{6 (0.092 ft)}{9.50 ft}\Big]</math> = 1.272 k/ft<sup>2</sup>


'''Assumptions:'''
:Allowable pressure = 2 tons/ft<sup>2</sup> = 4 k/ft<sup>2</sup> ≥ 1.272 k/ft<sup>2</sup> <u>o.k.</u>


(1) Stability
'''Lateral Pressures With Earthquake'''
The external stability of a counterfort retaining wall shall be determined in the same manner as described for cantilever retaining walls. Therefore refer to previous pages for the criteria for location of resultant, factor of safety for sliding and bearing pressures.
(2) Stem
[[image:751.24.3.2.3 counterfort.jpg|center|800px]]


:<math>P = C_a \boldsymbol \gamma_s</math>
k<sub>h</sub> = 0.5A = 0.5 (0.1) = 0.05


:where:
k<sub>v</sub> = 0
::''C<sub>a</sub>'' = coefficient of active earth pressure


::<math>\boldsymbol \gamma_s</math> = unit weigt of soil
<math>\theta = arctan \Big[\frac{k_h}{1 - k_v}\Big] = arctan \Big[\frac{0.05}{1 - 0}\Big] = 2.862^\circ</math>


Design the wall to support horizontal load from the earth pressure and the liveload surcharge (if applicable) as outlined on the previous pages and as designated in AASHTD Section 3.20, except that maximum horizontal loads shall be the calculated equivalent fluid pressure at 3/4  height of wall [(0.75 H)P] which shall be considered applied uniformly from the lower quarter point to the bottom of wall.
:'''Active Pressure on Psuedo-Wall'''


In addition, vertical steel In the fill face of the bottom quarter of the wall shall be that required by the vertical cantilever wall with the equivalent fluid pressure of that (0.25 H) height.
:''δ'' = ''φ'' = 24° (''δ'' is the angle of friction between the soil and the wall. In this case, ''δ'' = ''φ'' = because the soil wedge considered is next to the soil above the footing.)


Maximum concrete stress shall be assumed as the greater of the two thus obtained.
:''i'' = 18.435°
The application of these horizontal pressures shall be as follows:
[[image:751.24.3.2.3 counterfort wall.jpg|center|800px|thumb|<center>'''Counterfort Wall Section'''</center> <center>Moments are to be determined by analysis as a continuous beam.  The counterforts are to be spaced so as to produce approximately equal positive and negative moments.</center>]]


(3)  Counterfort
:''β'' = 0°
Counterforts shall be designed as T-beams, of which the wall is the flange and the counterfort is the stem.  For this reason the concrete stresses ane normally low and will not control.


For the design of reinforcing steel in the back of the counterfort, the effective d shall be the perpendicular distance from the front face of the wall (at point that moment is considered), to center of reinforcing steel.
:<math>K_{AE} = \frac{cos^2(\phi - \theta - \beta)}{cos \theta cos^2 \beta cos(\delta + \beta + \theta)\Big(1 + \sqrt\frac{sin(\phi + \delta) sin (\phi - \theta - i)}{cos (\delta + \beta + \theta) cos(I - \beta)}\Big)^2}</math>


[[image:751.24.3.2.3 moment.jpg|center|500px]]
:<math>K_{AE} = \frac{cos^2(24^\circ - 2.862^\circ - 0^\circ)}{cos (2.862^\circ) cos^2 (0^\circ) cos(24^\circ + 0^\circ + 2.862^\circ)\Big(1 + \sqrt\frac{sin(24^\circ + 24^\circ) sin (24^\circ - 2.862^\circ - 18.435^\circ)}{cos (24^\circ + 0^\circ + 2.862^\circ) cos(18.435^\circ - 0^\circ)}\Big)^2}</math>
 
:K<sub>AE</sub> = 0.674


(4) Footing
:P<sub>AE</sub> = ½''γ<sub>s</sub>H<sup>2</sup>''(1 − ''k<sub>v</sub>'')''K<sub>AE</sub>''


The footing of the counterfort walls shall be designed as a continuous beam of spans equal to the distance between the counterforts.
:P<sub>AE</sub> =  ½[0.120 k/ft<sup>3</sup>](10.667 ft)<sup>2</sup>(1 ft.)(1 - 0)(0.674) = 4.602k


The rear projection or heel shall be designed to support the entire weight of the superimposed materials, unless a more exact method is used. Refer to AASHTD Section 5.5.6.
:P<sub>AEV</sub> = P<sub>AE</sub>(sin''δ'') = 4.602k(sin24°) = 1.872k


Divide footing (transversely) into four (4) equal sections for design footing pressures.
:P<sub>AEH</sub> = P<sub>AE</sub>(cos''δ'') = 4.602k(cos 24°) = 4.204k


Counterfort walls on pile are very rare and are to be treated as special cases. See Structural Project Manager.
:P'<sub>AH</sub> = P<sub>AEH</sub> − P<sub>AH</sub> = 4.204k − 3.534k = 0.670k


(5)  Sign-Board type walls
:P'<sub>AV</sub> = P<sub>AEV</sub> − P<sub>AV</sub> = 1.872k − 1.178k = 0.694k


The Sign-Board type of retaining walls are a special case of the counterfort retaining walls.  This type of wall is used where the soiI conditions are such that the footings must be placed a great distance below the finished ground line.  For this situation, the wall is discontinued approximately 12 in. below the finished ground line or below the frost line.
:where: P'<sub>AH</sub> and P'<sub>AV</sub> are the seismic components of the active force.


Due to the large depth of the counterforts, it may be more economical to use a smaller number of counterforts than would otherwise be used.
:'''Passive Pressure on Shear Key'''
All design assumptions that apply to counterfort walls will apply to sign-board walls with the exception of the application of horizontal forces for the stem (or wall design), and the footing design which shall be as follows:


:'''Wall'''
:''δ'' = ''φ'' = 24° (''δ'' = ''φ'' because the soil wedge considered is assumed to form in front of the footing.)


[[image:751.24.3.2.3 load.jpg|center|550px]]
:''i'' = 0


:'''Footing'''
:''β'' = 0


:The individual footings shall be designed transversely as cantilevers supported by the wall.  Refer to AASHTO Section 5.
:<math>K_{PE} = \frac{cos^2(\phi - \theta + \beta)}{cos \theta cos^2 \beta cos(\delta - \beta + \theta)\Big(1 - \sqrt\frac{sin(\phi - \delta) sin (\phi - \theta + i)}{cos (\delta - \beta + \theta) cos(I - \beta)}\Big)^2}</math>


===751.24.3.3 Example 1:  Spread Footing Cantilever Wall===
:<math>K_{PE} = \frac{cos^2(24^\circ - 2.862^\circ + 0^\circ)}{cos (2.862^\circ) cos^2 (0^\circ) cos(24^\circ - 0^\circ + 2.862^\circ)\Big(1 - \sqrt\frac{sin(24^\circ - 24^\circ) sin (24^\circ - 2.862^\circ + 0^\circ)}{cos (24^\circ - 0^\circ + 2.862^\circ) cos(0^\circ - 0^\circ)}\Big)^2}</math>


[[image:751.24.3.3.jpg|center|750px|thumb|<Center>'''Typical Section thru Wall</center><center>(Spread Footing)</center>''']]
:K<sub>PE</sub> = 0.976


:f'<sub>c</sub> = 3,000 psi
:P<sub>PE</sub> = ½''γ<sub>s</sub>H<sup>2</sup>''(1 − ''k<sub>v</sub>'')''K<sub>PE</sub>''
:f<sub>y</sub> = 60,000 psi
 
:''φ'' = 24 in.
:P<sub>PE</sub> = ½[0.120 k/ft<sup>3</sup>][(5.0 ft)<sup>2</sup> - (2.5 ft<sup>2</sup>)](1 ft.)(1 - 0)(0.976) = 1.098k
:''γ<sub>s</sub>'' = 120 pcf (unit wgt of soil)
:Allowable soil pressure = 2 tsf
:''γ<sub>c</sub>'' = 150 pcf (unit wgt of concr.)
:Retaining wall is located in Seismic Performance Category (SPC) B.
:A = 0.1 (A = seismic acceleration coefficient)


{| style="margin: 1em auto 1em auto"
{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
|+
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Force (k) !!style="background:#BEBEBE" |Arm (ft) !!style="background:#BEBEBE" |Moment (ft-k)
|-
|align="center"|Σ (1) thru (6) ||align="center"| 10.239||align="center"| - ||align="center"| 54.556
|-
|align="center"|P<sub>AV</sub>||align="center"| 1.178 ||align="center"|9.500||align="center"| 11.192
|-
|align="center"|P'<sub>AV</sub> ||align="center"|0.694 ||align="center"|9.500||align="center"| 6.593
|-
|align="center"|Σ<sub>resisting</sub> ||align="center"|ΣV = 12.111 ||align="center"| - ||align="center"|ΣM<sub>R</sub> = 72.341
|-
|-
|<math>P_a = \frac{1}{2}\gamma_s C_a H^2</math>||width=50| ||<math>P_p = \frac{1}{2}\gamma_s C_p H_2^2 - H_1^2</math>
|align="center"|P<sub>AH</sub> ||align="center"|3.534 ||align="center"|3.556 ||align="center"|12.567
|}
|-
 
|align="center"|P'<sub>AH</sub> ||align="center"|0.670||align="center"| 6.400<sup>a</sup>||align="center"| 4.288
'''Assumptions'''
|-
 
|align="center"|P<sub>PEV</sub> ||align="center"|0.447<sup>b</sup>||align="center"| 0.000||align="center"| 0.000
* Retaining wall is under an abutment or in a location where failure of the wall may affect the structural integrity of a bridge. Therefore, it must be designed for SPC B.
|-
 
|align="center"|P<sub>PEH</sub> ||align="center"|1.003<sup>b</sup> ||align="center"|1.389<sup>c</sup>||align="center"| <u>0.000</u>
* Design is for a unit length (1 ft.) of wall.
|-
 
|align="center"| - ||align="center"| - ||align="center"| - ||align="center"|ΣM<sub>OT</sub> = 16.855
* Sum moments about the toe at the bottom of the footing for overturning.
|-
 
|colspan="4"|<sup>'''a'''</sup> P'<sub>AH</sub> acts at 0.6H of the wedge face (1992 AASHTO Div. IA Commentary).
*For Group Loads I-VI loading:
|-
:* F.S. for overturning ≥ 2.0 for footings on soil.
|colspan="4"|<sup>'''b'''</sup> P<sub>PEH</sub> and P<sub>PEH</sub> are the components of P<sub>PE</sub> with respect to ''δ'' (the friction angle). P<sub>PE</sub> does not contribute to overturning.
:* F.S. for sliding ≥ 1.5.
|-
* Resultant to be within middle 1/3 of footing.
|colspan="4"|<sup>'''c'''</sup> The line of action of P<sub>PEH</sub> can be located as was done for P<sub>P</sub>.
|}


* For earthquake loading:
:'''Overturning'''
:* F.S. for overturning ≥ 0.75(2.0) = 1.5.
:* F.S. for sliding ≥ 0.75(1.5) = 1.125.
:* Resultant to be within middle 1/2 of footing.


* Base of footing is below the frost line.
:<math>F.S._{OT} = \frac{72.341ft-k}{16.855ft-k} = 4.292 > 1.5</math> <u>o.k.</u>


* Neglect top one foot of fill over toe when determining passive pressure and soil weight.
:'''Resultant Eccentricity'''


* Use of a shear key shifts the failure plane to "B" where resistance to sliding is provided by passive pressure against the shear key, friction of soil along failure plane "B" in front of the key, and friction between soil and concrete along the footing behind the key.
:<math>\bar{x} = \frac{72.341ft-k - 16.855ft-k}{12.111k} = 4.581 ft.</math>


* Soil cohesion along failure plane is neglected.
:<math>e = \frac{9.5 ft.}{2}\ - 4.581 ft. = 0.169 ft.</math>


* Footings are designed as cantilevers supported by the wall.
:<math>\frac{L}{4} = \frac{9.5 ft.}{4} = 2.375 ft. > e</math> <u>o.k.</u>
:* Critical sections for bending are at the front and back faces of the wall.
:* Critical sections for shear are at the back face of the wall for the heel and at a distance d (effective depth) from the front face for the toe.


* Neglect soil weight above toe of footing in design of the toe.


* The wall is designed as a cantilever supported by the footing.
:'''Sliding'''


* Load factors for AASHTO Groups I - VI for design of concrete:
:<math>F.S. = \frac{1.003k + 12.111k \Big[(\frac{2}{9.5})tan 24^\circ + (\frac{7.5}{9.5}) tan \Big( \frac{2}{3}(24^\circ) \Big)\Big]}{4.204 k} = 1.161 > 1.125</math> <u>o.k.</u>
:* ''γ'' = 1.3.
 
:* ''β<sub>E</sub>'' = 1.3 for horizontal earth pressure on retaining walls.
 
:* ''β<sub>E</sub>'' = 1.0 for vertical earth pressure.
:'''Footing Pressure'''
 
:for e ≤ L/6:


* Load factor for earthquake loads = 1.0.
:<math>P = \frac{\Sigma V}{bL} \Big[ 1 \pm \frac{6e}{L}\Big] </math>


'''Lateral Pressures Without Earthquake'''
:<math>P_H = pressure\ at\ heel\ P_H = \frac{12.111 k}{(1 ft.)9.50 ft.} \Big[1 - \frac{6(0.169 ft.)}{9.50 ft}\Big]</math> = 1.139 k/ft<sup>2</sup>


:''C<sub>a</sub>'' = <math>\cos\delta\Bigg[\frac{\cos\delta - \sqrt{\cos^2\delta - \cos^2\phi}}{\cos\delta + \sqrt{\cos^2\delta - \cos^2\phi}}\Bigg]</math>
:<math>P_TH = pressure\ at\ toe\ P_T = \frac{12.111 k}{(1 ft.)9.50 ft.} \Big[1 + \frac{6(0.169 ft.)}{9.50 ft}\Big]</math> = 1.411 k/ft<sup>2</sup>


:''C<sub>a</sub>'' = <math>\cos 18.435^\circ \Bigg[\frac{\cos\ 18.435^\circ - \sqrt{\cos^2\ 18.435^\circ - \cos^2\ 24^\circ }}{\cos\ 18.435^\circ  + \sqrt{\cos^2\ 18.435^\circ  - \cos^2\ 24^\circ }}\Bigg]</math> = 0.546
:Allowable soil pressure for earthquake = 2 (allowable soil pressure)


:<math>C_p = tan^2 \big( 45^\circ + \frac{\phi}{2}\big)  = tan^2 \big( 45^\circ + \frac{24^\circ}{2}\big) = 2.371</math>
:(2)[4 k/ft<sup>2</sup>] = 8 k/ft<sup>2</sup> > 1.411 k/ft<sup>2</sup> <u>o.k.</u>


:<math>P_A = \frac{1}{2}\big[0.120\frac{k}{ft^3}\big](1 ft)(0.546)(10.667 ft)^2 = 3.726k</math>
'''Reinforcement-Stem'''


:<math>P_P = \frac{1}{2}\big[0.120\frac{k}{ft^3}\big](1 ft)(2.371)\big[(5.0)^2 - (2.5)^2\big] = 2.668k</math>
[[image:751.24.3.3 reinforcement stem.jpg|center|200px]]


:<math>P_{AV} = P_A (sin \delta) = 3.726k (sin 18.435^\circ ) = 1.178k</math>
d = 11" - 2" - (1/2)(0.5") = 8.75"


:<math>P_{AH} = P_A (cos \delta) = 3.726k (cos 18.435^\circ ) = 3.534k</math>
b = 12"


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
f'<sub>c</sub> = 3,000 psi
|+
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Area (ft<sup>2</sup>) !!style="background:#BEBEBE" |Force (k) = (Unit Wgt.)(Area) !!style="background:#BEBEBE" |Arm (ft.) !!style="background:#BEBEBE"|Moment (ft-k)
|-
|align="center"|(1)||align="center"| (0.5)(6.667ft)(2.222ft) = 7.407||align="center"| 0.889||align="center"| 7.278 ||align="center"|6.469
|-
|align="center"|(2)||align="center"| (6.667ft)(6.944ft) = 46.296||align="center"| 5.556||align="center"| 6.167||align="center"| 34.259
|-
|align="center"|(3) ||align="center"|(0.833ft)(8.000ft) + (0.5)(0.083ft)(8.000ft) = 7.000||align="center"|1.050||align="center"| 2.396||align="center"| 2.515
|-
|align="center"|(4) ||align="center"|(1.500ft)(9.500ft) = 14.250||align="center"| 2.138 ||align="center"|4.750 ||align="center"|10.153
|-
|align="center"|(5) ||align="center"|(2.500ft)(1.000ft) = 2.500||align="center"| 0.375||align="center"| 2.500||align="center"| 0.938
|-
|align="center"|(6) ||align="center"|(1.000ft)(1.917ft)+(0.5)(0.010ft)(1.000ft) = 1.922||align="center"|<u>0.231</u>||align="center"| 0.961||align="center"|<u>0.222</u>
|-
|align="center"|Σ ||align="center"| -  ||align="center"|ΣV = 10.239 ||align="center"| - ||align="center"|ΣM<sub>R</sub> = 54.556
|-
|align="center"|P<sub>AV</sub>||align="center"| -  ||align="center"|<u>1.178</u>||align="center"| 9.500 ||align="center"|<u>11.192</u>
|-
|align="center"|Σ resisting ||align="center"| - ||align="center"|ΣV = 11.417||align="center"| - ||align="center"| ΣM<sub>R</sub> = 65.748
|-
|align="center"|P<sub>AH</sub> ||align="center"| - ||align="center"|3.534 ||align="center"|3.556 ||align="center"|12.567
|-
|align="center"|P<sub>P</sub>||align="center"| -  ||align="center"|2.668 ||align="center"|1.389<sup>1</sup>||align="center"| -
|-
|colspan="5"|'''<sup>1</sup>''' The passive capacity at the shear key is ignored in overturning checks,since this capacity is considered in the factor of safety against sliding. It is assumed that a sliding and overturning failure will not occur simultaneously. The passive capacity at the shear key is developed only if the wall does slide.
|}


[[image:751.24.3.3 passive.jpg|right|150px]]
:'''Without Earthquake'''
<math>\bar{y} = \frac{H_1y^2 + \frac{2}{3}y^3}{H_2^2 − H_1^2} = \frac{(2.5 ft)(2.5 ft)^2 + \frac{2}{3}(2.5ft)^3}{(5.0 ft)^2 − (2.5 ft)^2} = 1.389 ft.</math>


:'''Overturning'''
:P<sub>AH</sub> = ½ [0.120 k/ft<sup>3</sup>](0.546)(6.944 ft.)<sup>2</sup>(1 ft.)(cos 18.435°) = 1.499k


:F.S. = <math>\frac{M_R}{M_{OT}} = \frac{65.748(ft-k)}{12.567(ft-k)} = 5.232 \ge 2.0 </math> <u>o.k.</u>
:''γ'' = 1.3


:where: M<sub>OT</sub> = overturning moment; M<sub>R</sub> = resisting moment
:''β<sub>E</sub>'' = 1.3 (active lateral earth pressure)


:'''Resultant Eccentricity'''
:M<sub>u</sub> = (1.3)(1.3)(1.499k)(2.315ft) = 5.865 (ft-k)


:<math>\bar{x} = \frac{(65.748 - 12.567)(ft-k)}{11.417k}</math> = 4.658 ft.
:'''With Earthquake'''


:<math>e = \frac{9.500 ft}{2} - 4.658 ft. = 0.092 ft.</math>
:k<sub>h</sub> = 0.05
:<math>\frac{L}{6} =\frac{9.500 ft}{6} = 1.583 ft > e</math> <u>o.k.</u>


:'''Sliding'''
:k<sub>v</sub> = 0
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|1992 AASHTO Div. IA Commentary
|}


:Check if shear key is required for Group Loads I-VI:
:''θ'' = 2.862°


:F.S. = <math>\frac{\Sigma V(tan\phi_{s-c})}{P_{AH}} = \frac{11.042k(tan \frac{2}{3}(24^\circ)}{3.534k} </math>= 0.896 <u>no good - shear key req'd</u>
:''δ'' = ''φ''/2 = 24°/2 = 12° for angle of friction between soil and wall. This criteria is used only for seismic loading if the angle of friction is not known.


:where: ''φ<sub>s-c</sub>'' = angle of friction between soil and concrete = (2/3)''φ<sub>s-s</sub>''
:''φ'' = 24°


:F.S. = <math>\frac{P_P + (\Sigma V) \Big(\frac{L_2}{L_1} tan \phi_{s-s}+\frac{L_3}{L_1} tan \phi_{s-c}\Big)}{P_{AH}}</math>
:''i'' = 18.435°


:where: ''φ<sub>s-s</sub>'' = angle of internal friction of soil
:''β'' =


:F.S. = <math>\frac{2.668k + (11.417k) \Big[\Big(\frac{2 ft}{9.50 ft}\Big) tan 24^\circ + \Big(\frac{7.50 ft}{9.50 ft} tan \Big(\frac{2}{3}(24^\circ)\Big)\Big]}{3.534 k}</math> = 1.789 ≥ 1.5  <u>o.k.</u>
:K<sub>AE</sub> = 0.654


:'''Footing Pressure'''
:P<sub>AEH</sub> = 1/2 ''γ<sub>s''</sub>K<sub>AE</sub>H<sup>2</sup>cos''δ''


:<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>
:P<sub>AEH</sub> = 1/2 [0.120k/ft](0.654)(6.944 ft.)<sup>2</sup>(1 ft.) cos(12°) = 1.851k


:P<sub>H</sub> = pressure at heel <math>P_H = \frac{11.417 k}{(1 ft)9.50 ft} \Big[1 - \frac{6 (0.092 ft)}{9.50 ft}\Big]</math> = 1.132 k/ft<sup>2</sup>
:M<sub>u</sub> = (1.499k)(2.315 ft.) + (1.851k − 1.499k)(0.6(6.944 ft.)) = 4.936(ft−k)


:P<sub>T</sub> = pressure at toe <math>P_T = \frac{11.417 k}{(1 ft)9.50 ft} \Big[1 + \frac{6 (0.092 ft)}{9.50 ft}\Big]</math> = 1.272 k/ft<sup>2</sup>
:The moment without earthquake controls:


:Allowable pressure = 2 tons/ft<sup>2</sup> = 4 k/ft<sup>2</sup> ≥ 1.272 k/ft<sup>2</sup> <u>o.k.</u>
:<math>R_n = \frac{M_u}{\phi bd^2} = \frac{5.865(ft-k)}{0.9(1 ft.)(8.75 in.)^2}\Big(1000 \frac{lb}{k}\Big)</math> = 85.116 psi


'''Lateral Pressures With Earthquake'''
:''ρ'' = <math>\frac{0.85f'_c}{f_y} \Big[1 - \sqrt{1 - \frac{2R_n}{0.85f'_c}}\Big]</math>


k<sub>h</sub> = 0.5A = 0.5 (0.1) = 0.05
:''ρ'' = <math>\frac{0.85 (3.000 psi}{60,000 psi} \Bigg[1 - \sqrt{1 - \frac{2 (85.116 psi}{0.85 (3000 psi)}}\Bigg]</math> = 0.00144


k<sub>v</sub> = 0
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|AASHTO 8.17.1.1 & 8.15.2.1.1
|}


<math>\theta = arctan \Big[\frac{k_h}{1 - k_v}\Big] = arctan \Big[\frac{0.05}{1 - 0}\Big] = 2.862^\circ</math>
:''ρ<sub>min</sub>'' = <math> 1.7 \Bigg[\frac{h}{d}\Bigg]^2 \frac{\sqrt{f'_c}}{f_y} = 1.7 \Bigg[\frac{11 in.}{8.75 in.}^2 \frac{\sqrt{3000 psi}}{60,000 psi}\Bigg]</math> = 0.00245


:'''Active Pressure on Psuedo-Wall'''
:Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.00144) = 0.00192


:''δ'' = ''φ'' = 24° (''δ'' is the angle of friction between the soil and the wall. In this case, ''δ'' = ''φ'' = because the soil wedge considered is next to the soil above the footing.)
:''A<sub>S<sub>Req</sub></sub>'' = ''ρbd'' = 0.00192 (12 in.)(8.75 in.) = 0.202 in.<sup>2</sup>/ft


:''i'' = 18.435°
:One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>


:''β'' =
:<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.202 in.^2}</math>


:<math>K_{AE} = \frac{cos^2(\phi -−\theta−-\beta)}{cos \theta cos^2 \beta cos(\delta + \beta + \theta)\Big(1 + \sqrt\frac{sin(\phi + \delta) sin (\phi -\theta - i)}{cos (\delta + \beta + \theta) cos(i−-\beta)}\Big)^2}</math>
:''s'' = 11.64 in.


:<math>K_{AE} = \frac{cos^2(24^\circ -−2.862^\circ−-0^\circ)}{cos (2.862^\circ) cos^2 (0^\circ) cos(24^\circ + 0^\circ + 2.862^\circ)\Big(1 + \sqrt\frac{sin(24^\circ + 24^\circ) sin (24^\circ -2.862^\circ - 18.435^\circ)}{cos (24^\circ + 0^\circ + 2.862^\circ) cos(18.435^\circ−-0^\circ)}\Big)^2}</math>
:<u>Use #4's @ 10" cts.</u>


:K<sub>AE</sub> = 0.674
:'''Check Shear'''


:P<sub>AE</sub> = ½''γ<sub>s</sub>H<sup>2</sup>''(1 − ''k<sub>v</sub>'')''K<sub>AE</sub>''
:V<sub>u</sub> ''φ'' V<sub>n</sub>


:P<sub>AE</sub> =  ½[0.120 k/ft<sup>3</sup>](10.667 ft)<sup>2</sup>(1 ft.)(1 - 0)(0.674) = 4.602k
::'''Without Earthquake'''


:P<sub>AEV</sub> = P<sub>AE</sub>(sin''δ'') = 4.602k(sin24°) = 1.872k
::V<sub>u,</sub> = (1.3)(1.3)(1.499k) = 2.533k


:P<sub>AEH</sub> = P<sub>AE</sub>(cos''δ'') = 4.602k(cos 24°) = 4.204k
::'''With Earthquake'''


:P'<sub>AH</sub> = P<sub>AEH</sub> − P<sub>AH</sub> = 4.204k − 3.534k = 0.670k
::V<sub>u</sub> = 1.851k


:P'<sub>AV</sub> = P<sub>AEV</sub> − P<sub>AV</sub> = 1.872k − 1.178k = 0.694k
:The shear force without earthquake controls.


:where: P'<sub>AH</sub> and P'<sub>AV</sub> are the seismic components of the active force.
:<math>\frac{\nu_u}{\phi} = \frac{2.533k}{0.85(12 in.)(8.75 in.)} (1000 lb/k)</math> = 28.4 psi
 
:<math>\nu_c = 2 \sqrt{3,000 psi}</math> = 109.5 psi > 28.4 psi <u>o.k.</u>
 
'''Reinforcement-Footing-Heel'''
 
[[image:751.24.3.3 heel.jpg|center|250px]]
 
Note: Earthquake will not control and will not be checked.
 
'<sub>E</sub>'' = 1.0 (vertical earth pressure)
 
d = 18" - 3" - (1/2)(0.750") = 14.625"


:'''Passive Pressure on Shear Key'''
b = 12"


:''δ'' = ''φ'' = 24° (''δ'' = ''φ'' because the soil wedge considered is assumed to form in front of the footing.)
''f'<sub>c</sub>'' = 3,000 psi


:''i'' = 0
''M<sub>u</sub>'' = 1.3 [(5.556k + 1.500k)(3.333ft) + 0.889k(4.444ft) + 1.178k(6.667ft)]


:''β'' = 0
''M<sub>u</sub>'' = 45.919(ft−k)


:<math>K_{PE} = \frac{cos^2(\phi -−\theta + \beta)}{cos \theta cos^2 \beta cos(\delta - \beta + \theta)\Big(1 - \sqrt\frac{sin(\phi - \delta) sin (\phi -\theta + i)}{cos (\delta - \beta + \theta) cos(i−-\beta)}\Big)^2}</math>
<math>R_n = \frac{45.919(ft-k)}{0.9(1 ft.)(14.625 in.)^2}(1000\frac{lb}{k})</math> = 238.5 psi


:<math>K_{PE} = \frac{cos^2(24^\circ -−2.862^\circ + 0^\circ)}{cos (2.862^\circ) cos^2 (0^\circ) cos(24^\circ - 0^\circ + 2.862^\circ)\Big(1 - \sqrt\frac{sin(24^\circ - 24^\circ) sin (24^\circ -2.862^\circ + 0^\circ)}{cos (24^\circ - 0^\circ + 2.862^\circ) cos(0^\circ−-0^\circ)}\Big)^2}</math>
''ρ'' = <math>\frac{0.85(3000)psi}{60,000 psi} \Bigg[ 1 - \sqrt{1 - \frac{2(238.5 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.00418


:K<sub>PE</sub> = 0.976
''ρ<sub>min</sub>'' = <math> 1.7 \Big[\frac{18 in.}{14.625 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00235


:P<sub>PE</sub> = ½''γ<sub>s</sub>H<sup>2</sup>''(1 − ''k<sub>v</sub>'')''K<sub>PE</sub>''
''A<sub>S<sub>Req</sub></sub>'' = 0.00418 (12 in.) (14.625 in.) = 0.734 in<sup>2</sup>/ft.


:P<sub>PE</sub> =  ½[0.120 k/ft<sup>3</sup>][(5.0 ft)<sup>2</sup> - (2.5 ft<sup>2</sup>)](1 ft.)(1 - 0)(0.976) = 1.098k


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
<u>Use #6's @ 7" cts.</u>
|+
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Force (k) !!style="background:#BEBEBE" |Arm (ft) !!style="background:#BEBEBE" |Moment (ft-k)
|-
|align="center"|Σ (1) thru (6) ||align="center"| 10.239||align="center"| - ||align="center"| 54.556
|-
|align="center"|P<sub>AV</sub>||align="center"| 1.178 ||align="center"|9.500||align="center"| 11.192
|-
|align="center"|P'<sub>AV</sub> ||align="center"|0.694 ||align="center"|9.500||align="center"| 6.593
|-
|align="center"|Σ<sub>resisting</sub> ||align="center"|ΣV = 12.111 ||align="center"| - ||align="center"|ΣM<sub>R</sub> = 72.341
|-
|align="center"|P<sub>AH</sub> ||align="center"|3.534 ||align="center"|3.556 ||align="center"|12.567
|-
|align="center"|P'<sub>AH</sub> ||align="center"|0.670||align="center"| 6.400<sup>a</sup>||align="center"| 4.288
|-
|align="center"|P<sub>PEV</sub> ||align="center"|0.447<sup>b</sup>||align="center"| 0.000||align="center"| 0.000
|-
|align="center"|P<sub>PEH</sub> ||align="center"|1.003<sup>b</sup> ||align="center"|1.389<sup>c</sup>||align="center"| <u>0.000</u>
|-
|align="center"| - ||align="center"| - ||align="center"| - ||align="center"|ΣM<sub>OT</sub> = 16.855
|-
|colspan="4"|<sup>'''a'''</sup> P'<sub>AH</sub> acts at 0.6H of the wedge face (1992 AASHTO Div. IA Commentary).
|-
|colspan="4"|<sup>'''b'''</sup> P<sub>PEH</sub> and P<sub>PEH</sub> are the components of P<sub>PE</sub> with respect to ''δ'' (the friction angle). P<sub>PE</sub> does not contribute to overturning.
|-
|colspan="4"|<sup>'''c'''</sup> The line of action of P<sub>PEH</sub> can be located as was done for P<sub>P</sub>.
|}


:'''Overturning'''
:'''Check Shear'''


:<math>F.S._{OT} = \frac{72.341(ft−-k)}{16.855(ft-−k)} = 4.292 > 1.5</math> <u>o.k.</u>
:Shear shall be checked at back face of stem.


:'''Resultant Eccentricity'''
:''V<sub>u</sub>'' = 1.3 (5.556k + 1.500k + 0.889k + 1.178k) = 11.860k


:<math>\bar{x} = \frac{72.341(ft−-k) - 16.855(ft-−k)}{12.111k} = 4.581 ft.</math>
:<math>\frac{\nu_u}{\phi} = \frac{11.860k}{0.85(12 in.)(14.625 in.)}(1000 \frac{lb}{k} ) = 79.5 psi < 2 \sqrt{3,000 psi}</math> = 109.5 psi  o.k.


:<math>e = \frac{9.5 ft.}{2} − 4.581 ft. = 0.169 ft.</math>
'''Reinforcement-Footing-Toe'''


:<math>\frac{L}{4} = \frac{9.5 ft.}{4} = 2.375 ft. > e</math> <u>o.k.</u>
[[image:751.24.3.3. toe.jpg|center|350px]]


d = 18" - 4" = 14"


:'''Sliding'''
b = 12"


:<math>F.S. = \frac{1.003k + 12.111k \Big[(\frac{2}{9.5})tan 24^\circ + (\frac{7.5}{9.5}) tan \Big( \frac{2}{3}(24^\circ) \Big)\Big]}{4.204 k} = 1.161 > 1.125</math> <u>o.k.</u>
:'''Without Earthquake'''


::'''Apply Load Factors'''


:'''Footing Pressure'''
::load 4 (weight) = 0.431k(1.3)(1.0) = 0.560k


:for e ≤ L/6:
::''β<sub>E</sub>'' = 1.3 for lateral earth pressure for retaining walls.


:<math>P = \frac{\Sigma V}{bL} \Big[ 1 \pm \frac{6e}{L}\Big] </math>
::''β<sub>E</sub>'' = 1.0 for vertical earth pressure.


:<math>P_H = pressure\ at\ heel\ P_H = \frac{12.111 k}{(1 ft.)9.50 ft.} \Big[1 −- \frac{6(0.169 ft.)}{9.50 ft}\Big]</math> = 1.139 k/ft<sup>2</sup>
::''ΣM<sub>OT</sub>'' = 12.567(ft−k)(1.3)(1.3) = 21.238(ft−k)


:<math>P_TH = pressure\ at\ toe\ P_T = \frac{12.111 k}{(1 ft.)9.50 ft.} \Big[1 + \frac{6(0.169 ft.)}{9.50 ft}\Big]</math> = 1.411 k/ft<sup>2</sup>
::''ΣM<sub>R</sub>'' = [54.556(ft−k) + 11.192(ft−k)](1.3)(1.0) = 85.472(ft−k)


:Allowable soil pressure for earthquake = 2 (allowable soil pressure)
::''ΣV'' = 11.417k(1.3)(1.0) = 14.842k


:(2)[4 k/ft<sup>2</sup>] = 8 k/ft<sup>2</sup> > 1.411 k/ft<sup>2</sup> <u>o.k.</u>
:<math>\bar{x} = \frac{85.472(ft-k) - 21.238(ft-k)}{14.842k}</math> = 4.328 ft.


'''Reinforcement-Stem'''
:''e'' = (9.5 ft./2) − 4.328 ft. = 0.422 ft.


[[image:751.24.3.3 reinforcement stem.jpg|center|200px]]
:<math>P_H = \frac{14.842k}{(1 ft.)(9.5 ft.)} \Big[1 - \frac{6(0.422 ft.)}{9.5 ft.}\Big]</math> = 1.146k/ft<sup>2</sup>


d = 11" - 2" - (1/2)(0.5") = 8.75"
:<math>P_T = \frac{14.842k}{(1 ft.)(9.5 ft.)} \Big[1 + \frac{6(0.422 ft.)}{9.5 ft.}\Big]</math> = 1.979k/ft<sup>2</sup>


b = 12"
:<math>P =\Bigg[\frac{1.979 \frac{k}{ft.} - 1.146 \frac{k}{ft.}}{9.5 ft.}\Bigg](7.583 ft.) + 1.146\frac{k}{ft.}</math> = 1.811k/ft.


f'<sub>c</sub> = 3,000 psi
:<math>M_u = 1.811\frac{k}{ft.}\frac{(1.917 ft.)^2}{2} + \frac{1}{2}(1.917 ft.)^2\Big[1.979\frac{k}{ft.} - 1.811\frac{k}{ft.}\Big]\frac{2}{3} - 0.560k(0.958 ft.)</math>


:'''Without Earthquake'''
:''M<sub>u</sub>'' = 2.997(ft−k)


:P<sub>AH</sub> = ½ [0.120 k/ft<sup>3</sup>](0.546)(6.944 ft.)<sup>2</sup>(1 ft.)(cos 18.435°) = 1.499k
:'''With Earthquake'''


:''γ'' = 1.3
:''P<sub>H</sub>'' = 1.139 k/ft


:''β<sub>E</sub>'' = 1.3 (active lateral earth pressure)
:''P<sub>T</sub>'' = 1.411 k/ft


:M<sub>u</sub> = (1.3)(1.3)(1.499k)(2.315ft) = 5.865 (ft-k)
:<math>P = \Bigg[\frac{1.411\frac{k}{ft.} - 1.139\frac{k}{ft.}}{9.5 ft.}\Bigg](7.583 ft.) + 1.139\frac{k}{ft.}</math> = 1.356 k/ft


:'''With Earthquake'''
:<math>M_u = 1.356\frac{k}{ft.}\frac{(1.917 ft.)^2}{2} + \frac{1}{2}(1.917 ft.)^2 \Bigg[1.411\frac{k}{ft.} - 1.356\frac{k}{ft.}\Bigg]\frac{2}{3} - 0.431k (0.958 ft.)</math>


:k<sub>h</sub> = 0.05
:''M<sub>u</sub>'' = 2.146(ft−k)


:k<sub>v</sub> = 0
:The moment without earthquake controls.
{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
|-
|'''Additional Information'''
|-
|1992 AASHTO Div. IA Commentary
|}


:''θ'' = 2.862°
:<math>R_n = \frac{2.997(ft-k)}{0.9(1 ft.)(14.0 in.)^2}(1000\frac{lb}{k})</math> = 16.990 psi


:''δ'' = ''φ''/2 = 24°/2 = 12° for angle of friction between soil and wall. This criteria is used only for seismic loading if the angle of friction is not known.
:''ρ'' = <math>\frac{0.85(3000 psi)}{60,000 psi}\Bigg[1 - \sqrt{1 - \frac{2(16.990 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.000284


:''φ'' = 24°
:''ρ<sub>min</sub>'' = <math>1.7\Big[\frac{18 in.}{14.0 in.}\Big]^2 \frac{\sqrt{3,000 psi}}{60,000 psi}</math> = 0.00257


:''i'' = 18.435°
:Use ''ρ'' = 4/3 ''ρ'' = <math>\frac{4}{3}(0.000284)</math> = 0.000379


:''β'' =
:''A<sub>S<sub>Req</sub></sub>'' = 0.000379 (12 in.)(14.0 in.) = 0.064 in.<sup>2</sup>/ft.


:K<sub>AE</sub> = 0.654


:P<sub>AEH</sub> = 1/2 ''γ<sub>s''</sub>K<sub>AE</sub>H<sup>2</sup>cos''δ''
:<math>\frac{12 in.}{0.064 in.^2} = \frac{s}{0.196 in.^2}</math>


:P<sub>AEH</sub> = 1/2 [0.120k/ft](0.654)(6.944 ft.)<sup>2</sup>(1 ft.) cos(12°) = 1.851k
:''s'' = 36.8 in.


:M<sub>u</sub> = (1.499k)(2.315 ft.) + (1.851k − 1.499k)(0.6(6.944 ft.)) = 4.936(ft−k)
:Minimum is # 4 bars at 12 inches. These will be the same bars that are in the back of the stem. Use the smaller of the two spacings.


:The moment without earthquake controls:
:<u>Use # 4's @ 10" cts.</u>


:<math>R_n = \frac{M_u}{\phi bd^2} = \frac{5.865(ft−-k)}{0.9(1 ft.)(8.75 in.)^2}\Big(1000 \frac{lb}{k}\Big)</math> = 85.116 psi
:'''Check Shear'''


:''ρ'' = <math>\frac{0.85f'_c}{f_y} \Big[1 −- \sqrt{1 - \frac{2R_n}{0.85f'_c}}\Big]</math>
:Shear shall be checked at a distance "d" from the face of the stem.


:''ρ'' = <math>\frac{0.85 (3.000 psi}{60,000 psi} \Bigg[1 −- \sqrt{1 - \frac{2 (85.116 psi}{0.85 (3000 psi)}}\Bigg]</math> = 0.00144
::'''Without Earthquake'''


{|style="padding: 0.3em; margin-left:5px; border:2px solid #a9a9a9; text-align:center; font-size: 95%; background:#f5f5f5" width="160px" align="right"
::<math>P_d =\Bigg[\frac{1.979\frac{k}{ft.} - 1.146\frac{k}{ft.}}{9.5 ft.}\Bigg](8.750 ft.) + 1.146\frac{k}{ft.}</math> = 1.913k/ft.
|-
|'''Additional Information'''
|-
|AASHTO 8.17.1.1 & 8.15.2.1.1
|}


:''ρ<sub>min</sub>'' = <math> 1.7 \Bigg[\frac{h}{d}\Bigg]^2 \frac{\sqrt{f'_c}}{f_y} = 1.7 \Bigg[\frac{11 in.}{8.75 in.}^2 \frac{\sqrt{3000 psi}}{60,000 psi}\Bigg]</math> = 0.00245
::<math>V_u =\frac{1.979\frac{k}{ft.} + 1.913\frac{k}{ft.}}{2}(0.750 ft.) - 1.3\Big[0.225\frac{k}{ft.}\Big](0.750 ft.)</math> = 1.240k


:Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.00144) = 0.00192
::'''With Earthquake'''


:''A<sub>S<sub>Req</sub></sub>'' = ''ρbd'' = 0.00192 (12 in.)(8.75 in.) = 0.202 in.<sup>2</sup>/ft
::<math>P_d =\Bigg[\frac{1.411\frac{k}{ft.} - 1.139\frac{k}{ft.}}{9.5 ft.}\Bigg](8.750 ft.) + 1.139\frac{k}{ft.}</math> = 1390k/ft.


:One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>
::<math>V_u =\frac{1.411\frac{k}{ft.} + 1.139\frac{k}{ft.}}{2}(0.750 ft.) - \Big[0.225\frac{k}{ft.}\Big](0.750 ft.)</math> = 0.788k


:<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.202 in.^2}</math>
:Shear without earthquake controls.


:''s'' = 11.64 in.
:<math>\frac{\nu_u}{\phi} = \frac{1.240k}{0.85(12 in.)(14.0 in.)}(1000\frac{lb}{k} ) = 8.7 psi < 2\sqrt{3000 psi}</math> = 109.5 psi <u>o.k.</u>


:<u>Use #4's @ 10" cts.</u>
'''Reinforcement-Shear Key'''


:'''Check Shear'''
[[image:751.24.3.3 shear key.jpg|center|250px]]


:V<sub>u</sub> ≥ ''φ'' V<sub>n</sub>
The passive pressure is higher without earthquake loads.


::'''Without Earthquake'''
''γ'' = 1.3


::V<sub>u,</sub> = (1.3)(1.3)(1.499k) = 2.533k
''β<sub>E</sub>'' = 1.3 (lateral earth pressure)


::'''With Earthquake'''
d = 12"-3"-(1/2)(0.5") = 8.75"


::V<sub>u</sub> = 1.851k
b = 12"


:The shear force without earthquake controls.
''M<sub>u</sub> = (3.379k)(1.360 ft.)(1.3)(1.3) = 7.764(ft−k)


:<math>\frac{\nu_u}{\phi} = \frac{2.533k}{0.85(12 in.)(8.75 in.)} (1000 lb/k)</math> = 28.4 psi
<math>R_n = \frac{7.764(ft-k)}{0.9(1 ft.)(8.75 in.)^2} (1000\frac{lb}{k})</math> = 112.677 psi


:<math>\nu_c = 2 \sqrt{3,000 psi}</math> = 109.5 psi > 28.4 psi <u>o.k.</u>
''ρ'' = <math>\frac{0.85(3000 psi)}{60,000 psi}\Bigg[1 - \sqrt{1 - \frac{2(112.677 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.00192


'''Reinforcement-Footing-Heel'''
''ρ<sub>min</sub> = <math>1.7\Big[\frac{12 in.}{8.75 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00292


[[image:751.24.3.3 heel.jpg|center|250px]]
Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.00192) = 0.00256


Note: Earthquake will not control and will not be checked.
A<sub>S<sub>Req</sub></sub> = 0.00256(12 in.)(8.75 in.) = 0.269 in.<sup>2</sup>/ft.


''β<sub>E</sub>'' = 1.0 (vertical earth pressure)


d = 18" - 3" - (1/2)(0.750") = 14.625"
<u>Use # 4 @ 8.5 in cts.</u>


b = 12"
Check Shear


''f'<sub>c</sub>'' = 3,000 psi
:<math>\frac{\nu_u}{\phi} = \frac{1.3(3.379k)(1.3)}{0.85(12 in.)(8.75.)}(1000\frac{lb}{k} ) = 64.0 psi < 2\sqrt{3000 psi}</math> = 109.5 psi <u>o.k.</u>


''M<sub>u</sub>'' = 1.3 [(5.556k + 1.500k)(3.333ft) + 0.889k(4.444ft) + 1.178k(6.667ft)]
'''Reinforcement Summary'''


''M<sub>u</sub>'' = 45.919(ft−k)
[[Image:751.24.3.3 summary.jpg|500px|center]]


<math>R_n = \frac{45.919(ft−-k)}{0.9 (1 ft.)(14.625 in.)^2}(1000 \frac{lb}{k})</math> = 238.5 psi
===751.24.3.4 Example 2: L-Shaped Cantilever Wall===


''ρ'' = <math>\frac{0.85 (3000)psi}{60,000 psi} \Bigg[ 1 −- \sqrt{1 − \frac{2(238.5 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.00418
[[image:751.24.3.4.jpg|center|650px|thumb|<center>'''Typical Section thru Wall</center><center>(Spread Footing)</center>''']]


''ρ<sub>min</sub>'' = <math> 1.7 \Big[\frac{18 in.}{14.625 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00235
''f'<sub>c</sub>'' = 4000 psi  


''A<sub>S<sub>Req</sub></sub>'' = 0.00418 (12 in.) (14.625 in.) = 0.734 in<sup>2</sup>/ft.
''f<sub>y</sub>'' = 60,000 psi


''φ'' = 29°


<u>Use #6's @ 7" cts.</u>
''γ<sub>s</sub> = 120 pcf


:'''Check Shear'''
Allowable soil pressure = 1.5 tsf = 3.0 ksf


:Shear shall be checked at back face of stem.
Retaining wall is located in Seismic Performance Category (SPC) A.


:''V<sub>u</sub>'' = 1.3 (5.556k + 1.500k + 0.889k + 1.178k) = 11.860k
<math>\delta = tan^{-1}\frac{1}{2.5}</math> = 21.801°


:<math>\frac{\nu_u}{\phi} = \frac{11.860k}{0.85(12 in.)(14.625 in.)}(1000 \frac{lb}{k} ) = 79.5 psi < 2 \sqrt{3,000 psi}</math> = 109.5 psi  o.k.
<math>C_a = cos \delta\Bigg[\frac{cos \delta - \sqrt{cos^2\delta - cos^2\phi}}{cos \delta + \sqrt{cos^2\delta - cos^2\phi}}\Bigg]</math> = 0.462


'''Reinforcement-Footing-Toe'''
<math>C_p = tan^2\Big[45 + \frac{\phi}{2}\Big]</math> = 2.882


[[image:751.24.3.3. toe.jpg|center|350px]]
''P<sub>A</sub>'' = 1/2 ''γ<sub>s</sub>'' C<sub>a</sub>H<sup>2</sup> = 1/2 (0.120 k/ft<sup>3</sup>)(0.462)(4.958 ft.)<sup>2</sup> = 0.681k


d = 18" - 4" = 14"
For sliding, P<sub>P</sub> is assumed to act only on the portion of key below the frost line that is set at an 18 in. depth on the southern border.


b = 12"
''P<sub>P</sub>'' = 1/2 (0.120 k/ft<sup>3</sup>)(2.882)[(2.458 ft.)<sup>2</sup> − (1.500 ft.)<sup>2</sup>] = 0.656k


:'''Without Earthquake'''
'''Assumptions'''


::'''Apply Load Factors'''
* Design is for a unit length (1 ft.) of wall.


::load 4 (weight) = 0.431k(1.3)(1.0) = 0.560k
* Sum moments about the toe at the bottom of the footing for overturning.


::''β<sub>E</sub>'' = 1.3 for lateral earth pressure for retaining walls.
* F.S. for overturning ≥ 2.0 for footings on soil.


::''β<sub>E</sub>'' = 1.0 for vertical earth pressure.
* F.S. for sliding ≥ 1.5 for footings on soil.


::''ΣM<sub>OT</sub>'' = 12.567(ft−k)(1.3)(1.3) = 21.238(ft−k)
* Resultant of dead load and earth pressure to be in back half of the middle third of the footing if subjected to frost heave.


::''ΣM<sub>R</sub>'' = [54.556(ft−k) + 11.192(ft−k)](1.3)(1.0) = 85.472(ft−k)
* For all loading combinations the resultant must be in the middle third of the footing except for collision loads.


::''ΣV'' = 11.417k(1.3)(1.0) = 14.842k
* The top 12 in. of the soil is not neglected in determining the passive pressure because the soil there will be maintained.


:<math>\bar{x} = \frac{85.472(ft-−k) −- 21.238(ft-−k)}{14.842k}</math> = 4.328 ft.
* Frost line is set at 18 in. at the south border for Missouri.


:''e'' = (9.5 ft./2) − 4.328 ft. = 0.422 ft.
* Portions of shear key which are above the frost line are assumed not to resist sliding by passive pressure.


:<math>P_H = \frac{14.842k}{(1 ft.)(9.5 ft.)} \Big[1 -− \frac{6(0.422 ft.)}{9.5 ft.}\Big]</math> = 1.146k/ft<sup>2</sup>
* Use of a shear key shifts the failure plane to "B" where resistance to sliding is also provided by friction of soil along the failure plane in front of the shear key. Friction between the soil and concrete behind the shear key will be neglected.


:<math>P_T = \frac{14.842k}{(1 ft.)(9.5 ft.)} \Big[1 + \frac{6(0.422 ft.)}{9.5 ft.}\Big]</math> = 1.979k/ft<sup>2</sup>
* Soil cohesion along the failure plane is neglected.


:<math>P =\Bigg[\frac{1.979 \frac{k}{ft.} -− 1.146 \frac{k}{ft.}}{9.5 ft.}\Bigg](7.583 ft.) + 1.146\frac{k}{ft.}</math> = 1.811k/ft.
* Live loads can move to within 1 ft. of the stem face and 1 ft. from the toe.


:<math>M_u = 1.811\frac{k}{ft.}\frac{(1.917 ft.)^2}{2} + \frac{1}{2}(1.917 ft.)^2\Big[1.979\frac{k}{ft.} -− 1.811\frac{k}{ft.}\Big]\frac{2}{3} -− 0.560k(0.958 ft.)</math>
* The wall is designed as a cantilever supported by the footing.


:''M<sub>u</sub>'' = 2.997(ft−k)
* Footing is designed as a cantilever supported by the wall. Critical sections for bending and shear will be taken at the face of the wall.


:'''With Earthquake'''
* Load factors for AASHTO Groups I-VI for design of concrete are:


:''P<sub>H</sub>'' = 1.139 k/ft
::*''γ'' = 1.3.


:''P<sub>T</sub>'' = 1.411 k/ft
::*''β<sub>E</sub>'' = 1.3 for horizontal earth pressure on retaining walls.


:<math>P = \Bigg[\frac{1.411\frac{k}{ft.} -− 1.139\frac{k}{ft.}}{9.5 ft.}\Bigg](7.583 ft.) + 1.139\frac{k}{ft.}</math> = 1.356 k/ft
::*''β<sub>E</sub>'' = 1.0 for vertical earth pressure.


:<math>M_u = 1.356\frac{k}{ft.}\frac{(1.917 ft.)^2}{2} + \frac{1}{2}(1.917 ft.)^2 \Bigg[1.411\frac{k}{ft.} -− 1.356\frac{k}{ft.}\Bigg]\frac{2}{3} - 0.431k (0.958 ft.)</math>
::*''β<sub>LL</sub>'' = 1.67 for live loads and collision loads.


:''M<sub>u</sub>'' = 2.146(ft−k)
'''Dead Load and Earth Pressure - Stabilty and Pressure Checks'''


:The moment without earthquake controls.
{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
|+
|-
!colspan="4" style="background:#BEBEBE" |Dead Load and Earth Pressure - Stabilty and Pressure Checks
|-
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Force (k) !!style="background:#BEBEBE" |Arm (in.) !!style="background:#BEBEBE"|Moment (ft-k)
|-
|align="center"|(1)||align="center"| (0.833 ft.)(5.167 ft.)(0.150k/ft<sup>3</sup>) = 0.646||align="center"| 5.333||align="center"| 3.444
|-
|align="center"|(2)||align="center"| (0.958ft)(5.750ft)(0.150k/ft3) = 0.827||align="center"| 2.875||align="center"| 2.376
|-
|align="center"| (3)||align="center"|  (1.000ft)(1.500ft)(0.150k/ft3) = 0.22534.259||align="center"| 4.250 ||align="center"| 0.956
|-
|align="center" colspan="3"|ΣV = 1.698 ||align="center"| ΣM<sub>R</sub> = 6.776
|-
|align="center"| P<sub>AV</sub>||align="center"|  0.253 ||align="center"| 5.750 ||align="center"| 1.455
|-
|align="center" colspan="3"| ΣV = 1.951||align="center"|  ΣM<sub>R</sub> = 8.231
|-
|align="center"| P<sub>AH</sub> ||align="center"| 0.633 ||align="center"| 1.653 ||align="center"| 1.045
|-
|align="center"| P<sub>P</sub>||align="center"|  0.656 ||align="center"| 1.06<sup>1</sup>||align="center"| -
|-
|colspan="4" align="right"|ΣM<sub>OT</sub> = 1.045
|-
|colspan="4"|<sup>'''1'''</sup> The passive pressure at the shear key is ignored in overturning checks.
|}


:<math>R_n = \frac{2.997(ft-−k)}{0.9(1 ft.)(14.0 in.)^2}(1000\frac{lb}{k})</math> = 16.990 psi
:'''Overturning'''


:''ρ'' = <math>\frac{0.85(3000 psi)}{60,000 psi}\Bigg[1 -− \sqrt{1 − \frac{2(16.990 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.000284
:<math>F.S. = \frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{8.231(ft-k)}{1.045(ft-k)}</math> = 7.877 ≥ 2.0 <u>o.k.</u>


:''ρ<sub>min</sub>'' = <math>1.7\Big[\frac{18 in.}{14.0 in.}\Big]^2 \frac{\sqrt{3,000 psi}}{60,000 psi}</math> = 0.00257
:'''Location of Resultant'''


:Use ''ρ'' = 4/3 ''ρ'' = <math>\frac{4}{3}(0.000284)</math> = 0.000379
:MoDOT policy is that the resultant must be in the back half of the middle third of the footing when considering dead and earth loads:


:''A<sub>S<sub>Req</sub></sub>'' = 0.000379 (12 in.)(14.0 in.) = 0.064 in.<sup>2</sup>/ft.
:<math>\Bigg[\frac{5.750 ft.}{2} = 2.875 ft.\Bigg] \le \bar{x} \le \Bigg[\Bigg(\frac{5.750 ft.}{2} + \frac{5.750 ft.}{6}\Bigg) = 3.833 ft.\Bigg] </math>


:<math>\bar{x} = \frac{M_{NET}}{\Sigma V} = \frac{8.231(ft-k) - 1.045(ft-k)}{1.951k}</math> = 3.683 ft. <u>o.k.</u>


:<math>\frac{12 in.}{0.064 in.^2} = \frac{s}{0.196 in.^2}</math>
:'''Sliding'''


:''s'' = 36.8 in.
:<math>F.S. = \frac{P_P + \Sigma V \Bigg[\Big(\frac{L_2}{L_1}\Big)tan\phi_{s-s} + \Big(\frac{L_3}{L_1}\Big)tan\phi_{s-c}\Bigg]}{P_{AH}}</math>


:Minimum is # 4 bars at 12 inches. These will be the same bars that are in the back of the stem. Use the smaller of the two spacings.
:where:
::''φ<sub>s-s</sub>'' = angle of internal friction of soil


:<u>Use # 4's @ 10" cts.</u>
::''φ<sub>s-c</sub>'' = angle of friction between soil and concrete = (2/3)''φ<sub>s-s</sub>''


:'''Check Shear'''
:<math>F.S. = \frac{0.656k +(1.951k)\Big[\Big(\frac{3.75 ft.}{5.75 ft.}\Big)tan 29^\circ + \Big(\frac{1 ft.}{5.75 ft.}\Big) tan\Big(\frac{2}{3}(29^\circ)\Big)\Big]}{0.633 k}</math> = 2.339 ≥ 1.5 <u>o.k.</u>


:Shear shall be checked at a distance "d" from the face of the stem.
:'''Footing Pressure'''


::'''Without Earthquake'''
:<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>


::<math>P_d =\Bigg[\frac{1.979\frac{k}{ft.} -− 1.146\frac{k}{ft.}}{9.5 ft.}\Bigg](8.750 ft.) + 1.146\frac{k}{ft.}</math> = 1.913k/ft.
:<math>e = \bar{x} - \frac{L}{2} = 3.683 ft. - \frac{5.75 ft.}{2}</math> = 0.808 ft.


::<math>V_u =\frac{1.979\frac{k}{ft.} + 1.913\frac{k}{ft.}}{2}(0.750 ft.) - 1.3\Big[0.225\frac{k}{ft.}\Big](0.750 ft.)</math> = 1.240k
:Heel: <math>P_H = \frac{1.951k}{(1 ft.)(5.75 ft.)}\Big[1 + \frac{6(0.808 ft.)}{5.75 ft.}\Big]</math> = 0.625 ksf < 3.0 ksf <u>o.k.</u>


::'''With Earthquake'''
:Toe: <math>P_T = \frac{1.951k}{(1 ft.)(5.75 ft.)}\Big[1 - \frac{6(0.808 ft.)}{5.75 ft.}\Big]</math> = 0.053 ksf < 3.0 ksf <u>o.k.</u>


::<math>P_d =\Bigg[\frac{1.411\frac{k}{ft.} -− 1.139\frac{k}{ft.}}{9.5 ft.}\Bigg](8.750 ft.) + 1.139\frac{k}{ft.}</math> = 1390k/ft.
'''Dead Load, Earth Pressure, and Live Load - Stability and Pressure Checks'''


::<math>V_u =\frac{1.411\frac{k}{ft.} + 1.139\frac{k}{ft.}}{2}(0.750 ft.) - \Big[0.225\frac{k}{ft.}\Big](0.750 ft.)</math> = 0.788k
Stability is not an issue because the live load resists overturning and increases the sliding friction force.


:Shear without earthquake controls.
[[image:751.24.3.4 checks.jpg|center|250px]]


:<math>\frac{\nu_u}{\phi} = \frac{1.240k}{0.85(12 in.)(14.0 in.)}(1000\frac{lb}{k} ) = 8.7 psi < 2\sqrt{3000 psi}</math> = 109.5 psi <u>o.k.</u>
The live load will be distributed as:


'''Reinforcement-Shear Key'''
<math> F_{LL} = \frac{LL_{WL}}{E}</math>


[[image:751.24.3.3 shear key.jpg|center|250px]]
:where E = 0.8X + 3.75


The passive pressure is higher without earthquake loads.
::X = distance in feet from the load to the front face of wall


''γ'' = 1.3
The live load will be positioned as shown by the dashed lines above. The bearing pressure and resultant location will be determined for these two positions.


''β<sub>E</sub>'' = 1.3 (lateral earth pressure)
:'''Live Load 1 ft From Stem Face'''


d = 12"-3"-(1/2)(0.5") = 8.75"
::'''Resultant Eccentricity'''


b = 12"
::X = 1 ft.


''M<sub>u</sub> = (3.379k)(1.360 ft.)(1.3)(1.3) = 7.764(ft−k)
::E = 0.8(1 ft.) + 3.75 = 4.55 ft.


<math>R_n = \frac{7.764(ft−-k)}{0.9(1 ft.)(8.75 in.)^2} (1000\frac{lb}{k})</math> = 112.677 psi
::<math>F_{LL} = \frac{16k}{4.55 ft.} (1 ft.)</math> = 3.516k


''ρ'' = <math>\frac{0.85(3000 psi)}{60,000 psi}\Bigg[1 -− \sqrt{1 − \frac{2(112.677 psi)}{0.85(3000psi)}}\Bigg]</math> = 0.00192
::<math>\bar{x} = \frac{M_{NET}}{\Sigma V} = \frac{8.231(ft-k) + (3.516k)(3.917 ft.) - 1.045(ft-k)}{1.951k + 3.516k}</math> = 3.834 ft.


''ρ<sub>min</sub> = <math>1.7\Big[\frac{12 in.}{8.75 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00292
::<math>e = \bar{x} - \frac{L}{2} = 3.834 ft. - \frac{5.75 ft.}{2} = 0.959 ft. \le \frac{L}{6}</math> = 5.75 ft. <u>o.k.</u>


Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.00192) = 0.00256
::'''Footing Pressure'''


A<sub>S<sub>Req</sub></sub> = 0.00256(12 in.)(8.75 in.) = 0.269 in.<sup>2</sup>/ft.
::<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>


::Allowable Pressure = 3.0 ksf


<u>Use # 4 @ 8.5 in cts.</u>
::Heel: <math>P_H = \frac{5.467k}{(1 ft.)(5.75 ft.)}\Big[1 + \frac{6(0.959 ft.)}{5.75 ft.}\Big]</math> = 1.902 ksf


Check Shear
::Toe: <math>P_T = \frac{5.467k}{(1 ft.)(5.75 ft.)}\Big[1 - \frac{6(0.959 ft.)}{5.75 ft.}\Big]</math> = 0.000ksf


:<math>\frac{\nu_u}{\phi} = \frac{1.3(3.379k)(1.3)}{0.85(12 in.)(8.75.)}(1000\frac{lb}{k} ) = 64.0 psi < 2\sqrt{3000 psi}</math> = 109.5 psi <u>o.k.</u>
:'''Live Load 1 ft From Toe'''


'''Reinforcement Summary'''
::'''Resultant Eccentricity'''


[[Image:751.24.3.3 summary.jpg|500px|center]]
::X = 3.917 ft.


===751.24.3.4 Example 2: L-Shaped Cantilever Wall===
::E = 0.8(3.917 ft.) + 3.75 = 6.883 ft.


[[image:751.24.3.4.jpg|center|650px|thumb|<center>'''Typical Section thru Wall</center><center>(Spread Footing)</center>''']]
::<math>F_{LL} = \frac{16k}{6.883 ft} (1 ft.)</math> = 2.324k


''f'<sub>c</sub>'' = 4000 psi
::<math>x = \frac{8.231(ft-k) + (2.324k)(1ft.) - 1.045(ft-k)}{1.951k + 2.324k}</math> = 2.225 ft.


''f<sub>y</sub>'' = 60,000 psi
::<math>e = \frac{L}{2} - \bar{x} = \frac{5.75 ft.}{2} - 2.225 ft. = 0.650 ft. \le \frac{L}{6} = \frac{5.75 ft.}{6}</math> = 0.958 ft. <u>o.k.</u>


''φ'' = 29°
::'''Footing Pressure'''


''γ<sub>s</sub> = 120 pcf
::Allowable Pressure = 3.0ksf


Allowable soil pressure = 1.5 tsf = 3.0 ksf
::Heel: <math>P_H = \frac{4.275k}{(1 ft.)(5.75 ft.}\Big[1 - \frac{6 (0.650 ft.)}{5.75 ft.}\Big]</math> = 0.239ksf <u>o.k.</u>


Retaining wall is located in Seismic Performance Category (SPC) A.
::Toe: <math>P_T = \frac{4.275k}{(1 ft.)(5.75 ft.}\Big[1 + \frac{6 (0.650 ft.)}{5.75 ft.}\Big]</math> = 1.248ksf <u>o.k.</u>


<math>\delta = tan^{−-1}\frac{1}{2.5}</math> = 21.801°
'''Dead Load, Earth Pressure, Collision Load, and Live Load - Stability and Pressure Checks'''


<math>C_a = cos \delta\Bigg[\frac{cos \delta -− \sqrt{cos^2\delta -− cos^2\phi}}{cos \delta + \sqrt{cos^2\delta -− cos^2\phi}}\Bigg]</math> = 0.462
During a collision, the live load will be close to the wall so check this combination when the live load is one foot from the face of the stem. Sliding (in either direction) will not be an issue. Stability about the heel should be checked although it is unlikely to be a problem. There are no criteria for the location of the resultant, so long as the footing pressure does not exceed 125% of the allowable. It is assumed that the distributed collision force will develop an equal and opposite force on the fillface of the back wall unless it exceeds the passive pressure that can be developed by soil behind the wall.


<math>C_p = tan^2\Big[45 + \frac{\phi}{2}\Big]</math> = 2.882
''F<sub>LL</sub>'' = 3.516k


''P<sub>A</sub>'' = 1/2 ''γ<sub>s</sub>'' C<sub>a</sub>H<sup>2</sup> = 1/2 (0.120 k/ft<sup>3</sup>)(0.462)(4.958 ft.)<sup>2</sup> = 0.681k
[[image:751.24.3.4 collision.jpg|center|250px]]


For sliding, P<sub>P</sub> is assumed to act only on the portion of key below the frost line that is set at an 18 in. depth on the southern border.
''F<sub>COLL</sub>'' = <math>\frac{10k}{2(3 ft.)}(1 ft.)</math> = 1.667k


''P<sub>P</sub>'' = 1/2 (0.120 k/ft<sup>3</sup>)(2.882)[(2.458 ft.)<sup>2</sup> − (1.500 ft.)<sup>2</sup>] = 0.656k
<math>C_P = cos \delta \Bigg[\frac{cos \delta + \sqrt{cos^2 \delta - cos^2 \phi}}{cos \delta - \sqrt{cos^2 \delta - cos^2 \phi}}\Bigg]</math> = 1.867


'''Assumptions'''
<math>P_{PH} = \frac{1}{2}\gamma_s C_P H^2 cos\delta = \frac{1}{2}(0.120kcf)(1.867)(4.958ft)^2 cos(21.801^\circ)</math>


* Design is for a unit length (1 ft.) of wall.
''P<sub>PH</sub>'' = 2.556k > ''F<sub>COLL</sub>''  Thus the soil will develop an equal but opp. force.


* Sum moments about the toe at the bottom of the footing for overturning.
:'''Overturning About the Heel'''


* F.S. for overturning ≥ 2.0 for footings on soil.
:F.S. = <math>\frac{(0.646k)(0.417 ft.) + (0.827k)(2.875 ft.) + (0.225k)(1.500 ft.) + (3.516k)(1.833 ft.) + (1.667k)\big(\frac{4.958 ft.}{3}\big)}{(1.667k)(3.958 ft.)}</math>


* F.S. for sliding ≥ 1.5 for footings on soil.
:F.S. = <math>\frac{12.184(ft-k)}{6.598(ft-k)}</math> = 1.847 ≥ 1.2 <u>o.k.</u>


* Resultant of dead load and earth pressure to be in back half of the middle third of the footing if subjected to frost heave.
:'''Footing Pressure'''


* For all loading combinations the resultant must be in the middle third of the footing except for collision loads.
:<math>\bar{x} = \frac{12.184(ft-k) - 6.598(ft-k)}{1.951k + 3.516k}</math> = 1.022 ft. from heel


* The top 12 in. of the soil is not neglected in determining the passive pressure because the soil there will be maintained.
:''e'' = <math>\frac{5.75 ft.}{2} - 1.022 ft.</math> = 1.853 ft.


* Frost line is set at 18 in. at the south border for Missouri.
:Allowable Pressure = (1.25)(3.0ksf) = 3.75ksf


* Portions of shear key which are above the frost line are assumed not to resist sliding by passive pressure.
:Heel: <math> P_H =\frac {2(\Sigma V)}{3b[\frac{L}{2} - e]} = \frac {2(5.467k)}{3(1 ft.)\big[\frac{5.75 ft.}{2} - 1.853 ft.\big]}</math> = 3.566ksf <u>o.k.</u>


* Use of a shear key shifts the failure plane to "B" where resistance to sliding is also provided by friction of soil along the failure plane in front of the shear key. Friction between the soil and concrete behind the shear key will be neglected.
'''Stem Design-Steel in Rear Face'''


* Soil cohesion along the failure plane is neglected.
[[image:751.24.3.4 steel in rear face.jpg|center|250px]]
 
''γ'' = 1.3
 
''β<sub>E</sub>'' = 1.3 (active lateral earth pressure)
 
d = 10 in. − 2 in. − (0.5 in./2) = 7.75 in.
 
<math>P_{AH} = \frac{1}{2}\gamma_s C_a H^2 cos\delta = \frac{1}{2}\Bigg[0.120 \frac{k}{ft^3}\Bigg](0.462)(4 ft.)^2(1 ft.) cos 21.801^\circ</math>


* Live loads can move to within 1 ft. of the stem face and 1 ft. from the toe.
''P<sub>AH</sub>'' = 0.412k


* The wall is designed as a cantilever supported by the footing.
''M<sub>u</sub>'' = (1.333 ft.)(0.412k)(1.3)(1.3) = 0.928(ft−k)


* Footing is designed as a cantilever supported by the wall. Critical sections for bending and shear will be taken at the face of the wall.
<math>R_n = \frac{M_u}{\phi b d^2} = \frac{0.928(ft-k)}{(0.9)(1 ft.)(7.75 in.)^2}\Big(1000\frac{lb}{k}\Big)</math> = 17.160psi


* Load factors for AASHTO Groups I-VI for design of concrete are:
<math>\rho = \frac{0.85f_c}{f_y}\Bigg[1 - \sqrt{1 - \frac{2R_n}{0.85 f_c}}\Bigg]</math>


::*''γ'' = 1.3.
<math>\rho = \frac{4000 psi}{60,000 psi}\Bigg[1 - \sqrt{1 - \frac{2(17.160 psi)}{0.85 (4000psi)}}\Bigg]</math> = 0.000287


::*''β<sub>E</sub>'' = 1.3 for horizontal earth pressure on retaining walls.
<math>\rho_{min} = 1.7 \Bigg[\frac{h}{d}\Bigg]^2 \frac{\sqrt{f_c}}{f_y}</math>


::*''β<sub>E</sub>'' = 1.0 for vertical earth pressure.
<math>\rho_{min} = 1.7 \Bigg[\frac{10 in.}{7.75 in.}\Bigg]^2 \frac{\sqrt{4000 psi}}{60000 psi}</math> = 0.00298


::*''β<sub>LL</sub>'' = 1.67 for live loads and collision loads.
Use ''ρ'' = (4/3)ρ = (4/3)(0.000287) = 0.000382


'''Dead Load and Earth Pressure - Stabilty and Pressure Checks'''
<math>A_{S_{Req}} = \rho bd = 0.000382(12 in.)(7.75 in.) = 0.036 \frac{in^2}{ft.}</math>


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>, so the required minimum of one #4 bar every 12 in. controls.
|+
|-
!colspan="4" style="background:#BEBEBE" |Dead Load and Earth Pressure - Stabilty and Pressure Checks
|-
!style="background:#BEBEBE" |Load !!style="background:#BEBEBE" |Force (k) !!style="background:#BEBEBE" |Arm (in.) !!style="background:#BEBEBE"|Moment (ft-k)
|-
|align="center"|(1)||align="center"| (0.833 ft.)(5.167 ft.)(0.150k/ft<sup>3</sup>) = 0.646||align="center"| 5.333||align="center"| 3.444
|-
|align="center"|(2)||align="center"| (0.958ft)(5.750ft)(0.150k/ft3) = 0.827||align="center"| 2.875||align="center"| 2.376
|-
|align="center"| (3)||align="center"|  (1.000ft)(1.500ft)(0.150k/ft3) = 0.22534.259||align="center"| 4.250 ||align="center"| 0.956
|-
|align="center" colspan="3"|ΣV = 1.698 ||align="center"| ΣM<sub>R</sub> = 6.776
|-
|align="center"| P<sub>AV</sub>||align="center"|  0.253 ||align="center"| 5.750 ||align="center"| 1.455
|-
|align="center" colspan="3"| ΣV = 1.951||align="center"|  ΣM<sub>R</sub> = 8.231
|-
|align="center"| P<sub>AH</sub> ||align="center"| 0.633 ||align="center"| 1.653 ||align="center"| 1.045
|-
|align="center"| P<sub>P</sub>||align="center"|  0.656 ||align="center"| 1.06<sup>1</sup>||align="center"| -
|-
|colspan="4" align="right"|ΣM<sub>OT</sub> = 1.045
|-
|colspan="4"|<sup>'''1'''</sup> The passive pressure at the shear key is ignored in overturning checks.
|}


:'''Overturning'''
<u>Use #4's @ 12 in. (min)</u>


:<math>F.S. = \frac{ΣM_R}{ΣM_{OT}} = \frac{8.231(ft−-k)}{1.045(ft−-k)}</math> = 7.877 ≥ 2.0 <u>o.k.</u>
(These bars are also the bars in the bottom of the footing so the smaller of the two required spacings will be used.)


:'''Location of Resultant'''
:'''Check Shear'''


:MoDOT policy is that the resultant must be in the back half of the middle third of the footing when considering dead and earth loads:
:<math>\frac{\nu_u}{\phi} \le V_n</math>


:<math>\Bigg[\frac{5.750 ft.}{2} = 2.875 ft.\Bigg] \le \bar{x} \le \Bigg[\Bigg(\frac{5.750 ft.}{2} + \frac{5.750 ft.}{6}\Bigg) = 3.833 ft.\Bigg] </math>
:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.3)(0.412k)}{0.85(12 in.)(7.75 in.)}(1000\frac{lb}{k})</math> = 8.8 psi


:<math>\bar{x} = \frac{M_{NET}}{\Sigma V} = \frac{8.231(ft-−k) −- 1.045(ft-−k)}{1.951k}</math> = 3.683 ft. <u>o.k.</u>
:<math>\nu_c = 2 \sqrt{f'_c}</math>


:'''Sliding'''
:<math>\nu_c = 2 \sqrt{4, 000 psi}</math> = 126.5 psi > 8.8 psi <u>o.k.</u>


:<math>F.S. = \frac{P_P + \Sigma V \Bigg[\Big(\frac{L_2}{L_1}\Big)tan\phi_{s−-s} + \Big(\frac{L_3}{L_1}\Big)tan\phi_{s-−c}\Bigg]}{P_{AH}}</math>
'''Stem Design-Steel in Front Face (Collision Loads)'''


:where:
[[image:751.24.3.4 steel in front face.jpg|center|300px]]
::''φ<sub>s-s</sub>'' = angle of internal friction of soil


::''φ<sub>s-c</sub>'' = angle of friction between soil and concrete = (2/3)''φ<sub>s-s</sub>''


:<math>F.S. = \frac{0.656k +(1.951k)\Big[\Big(\frac{3.75 ft.}{5.75 ft.}\Big)tan 29^\circ + \Big(\frac{1 ft.}{5.75 ft.}\Big) tan\Big(\frac{2}{3}(29^\circ)\Big)\Big]}{0.633 k}</math> = 2.339 ≥ 1.5 <u>o.k.</u>
The soil pressure on the back of the stem becomes passive soil pressure during a collision, however this pressure is ignored for reinforcement design.


:'''Footing Pressure'''
''γ'' = 1.3


:<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>
''β<sub>LL</sub>'' = 1.67


:<math>e = \bar{x} − \frac{L}{2} = 3.683 ft. \frac{5.75 ft.}{2}</math> = 0.808 ft.
<math>d = 10 in. - 1.5 in. - 0.5 in. - \frac{0.5 in.}{2}</math> = 7.75 in.


:Heel: <math>P_H = \frac{1.951k}{(1 ft.)(5.75 ft.)}\Big[1 + \frac{6(0.808 ft.)}{5.75 ft.}\Big]</math> = 0.625 ksf < 3.0 ksf <u>o.k.</u>
<math>F_{COLL} = \frac{10k}{2L} = \frac{10k}{(2)(3 ft.)}</math> = 1.667 k/ft.


:Toe: <math>P_T = \frac{1.951k}{(1 ft.)(5.75 ft.)}\Big[1 - \frac{6(0.808 ft.)}{5.75 ft.}\Big]</math> = 0.053 ksf < 3.0 ksf <u>o.k.</u>
''M<sub>u</sub>'' = 1.667k/ft. (1 ft.)(3 ft.)(1.3)(1.67) = 10.855(ft−k)


'''Dead Load, Earth Pressure, and Live Load - Stability and Pressure Checks'''
<math>R_n = \frac{10.855(ft-k)}{0.9(1 ft.)(7.75 in.)^2} (1000\frac{lb}{k})</math> = 200.809 psi


Stability is not an issue because the live load resists overturning and increases the sliding friction force.
<math>\rho = \frac{0.85(4000 psi)}{60,000 psi}\Bigg[1 - \sqrt{1 - \frac{2(200.809 psi)}{0.85(4000psi)}}\Bigg]</math> = 0.00345


[[image:751.24.3.4 checks.jpg|center|250px]]
<math>\rho_{min} = 1.7\Bigg[\frac{10 in.}{7.75 in.}\Bigg]^2 \frac{\sqrt{4000 psi}}{60,000 psi}</math> = 0.00298


The live load will be distributed as:
<math>A_{S_{Req}} = 0.00345 (12 in.)(7.75 in.) = 0.321 \frac{in.^2}{ft.}</math>


<math> F_{LL} = \frac{LL_{WL}}{E}</math>  
One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>.


:where E = 0.8X + 3.75
<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.321 in.^2}</math>


::X = distance in feet from the load to the front face of wall
''s'' = 7.3 in.


The live load will be positioned as shown by the dashed lines above. The bearing pressure and resultant location will be determined for these two positions.
<u>Use #4's @ 7 in.</u>


:'''Live Load 1 ft From Stem Face'''
:'''Check Shear'''


::'''Resultant Eccentricity'''
:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.67)(1.667k)}{(0.85)(12 in.)(7.75 in.)} (1000\frac{lb}{k})</math> = 45.8 psi < 126.5 psi <u>o.k.</u>


::X = 1 ft.
'''Footing Design - Bottom Steel'''


::E = 0.8(1 ft.) + 3.75 = 4.55 ft.
It is not considered necessary to design footing reinforcement based upon a load case which includes collision loads.


::<math>F_{LL} = \frac{16k}{4.55 ft.} (1 ft.)</math> = 3.516k
:'''Dead Load and Earth Pressure Only'''


::<math>\bar{x} = \frac{M_{NET}}{\Sigma V} = \frac{8.231(ft−-k) + (3.516k)(3.917 ft.) -− 1.045(ft-−k)}{1.951k + 3.516k}</math> = 3.834 ft.
[[image:751.24.3.4 dead load.jpg|center|250px]]


::<math>e = \bar{x} −- \frac{L}{2} = 3.834 ft. −- \frac{5.75 ft.}{2} = 0.959 ft. \le \frac{L}{6}</math> = 5.75 ft. <u>o.k.</u>
:''Footing wt.'' = <math>\Big[\frac{11.5}{12}ft.\Big](4.917 ft.)\Big[0.150 \frac{k}{ft.^3}\Big](1 ft.)</math> = 0.707k


::'''Footing Pressure'''
:''β<sub>E</sub>'' = 1.3 (lateral earth pressure)


::<math>P = \frac{\Sigma V}{bL} \Big[1 \pm \frac{6e}{L}\Big]</math>
:''γ'' = 1.3


::Allowable Pressure = 3.0 ksf
:Apply Load Factors:


::Heel: <math>P_H = \frac{5.467k}{(1 ft.)(5.75 ft.)}\Big[1 + \frac{6(0.959 ft.)}{5.75 ft.}\Big]</math> = 1.902 ksf
:''ΣV'' = 1.951k (1.3) = 2.536k


::Toe: <math>P_T = \frac{5.467k}{(1 ft.)(5.75 ft.)}\Big[1 - \frac{6(0.959 ft.)}{5.75 ft.}\Big]</math> = 0.000ksf
:''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) = 10.700(ft−k)


:'''Live Load 1 ft From Toe'''
:''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)


::'''Resultant Eccentricity'''
:''Footing wt.'' = 0.707k (1.3) = 0.919k


::X = 3.917 ft.
:<math>\bar{x} = \frac{10.700(ft-k) - 1.766(ft-k)}{2.536k}</math> = 3.523 ft.


::E = 0.8(3.917 ft.) + 3.75 = 6.883 ft.
:<math>e = 3.523 ft. - \frac{5.75ft}{2}</math> = 0.648 ft.


::<math>F_{LL} = \frac{16k}{6.883 ft} (1 ft.)</math> = 2.324k
:<math>P_H = \frac{2.536k}{(1 ft.)(5.75 ft.)}\Bigg[1 + \frac{6(0.648 ft.)}{5.75 ft.}\Bigg]</math> = 0.739 ksf


::<math>x = \frac{8.231(ft-−k) + (2.324k)(1ft.) - 1.045(ft-−k)}{1.951k + 2.324k}</math> = 2.225 ft.  
:<math>P_T = \frac{2.536k}{(1 ft.)(5.75 ft.)}\Bigg[1 - \frac{6(0.648 ft.)}{5.75 ft.}\Bigg]</math> = 0.143ksf


::<math>e = \frac{L}{2} - \bar{x} = \frac{5.75 ft.}{2} -− 2.225 ft. = 0.650 ft. \le \frac{L}{6} = \frac{5.75 ft.}{6}</math> = 0.958 ft. <u>o.k.</u>
:<math>P_W = 0.143 ksf + [0.739 ksf - 0.143 ksf]\Bigg[\frac{4.917 ft.}{5.75 ft.}\Bigg]</math> = 0.653 ksf


::'''Footing Pressure'''
:Moment at Wall Face:


::Allowable Pressure = 3.0ksf
:<math>M_W = \Big[0.143\frac{k}{ft.}\Big]\Bigg[\frac{(4.917 ft.)^2}{2}\Bigg] + \frac{1}{3}(4.917 ft.)^2 \Bigg[0.653\frac{k}{ft.} - 0.143\frac{k}{ft.}\Bigg]\frac{1}{2} -  0.919k \Bigg[\frac{4.917 ft.}{2}\Bigg]</math> = 1.524(ft−k)


::Heel: <math>P_H = \frac{4.275k}{(1 ft.)(5.75 ft.}\Big[1 -− \frac{6 (0.650 ft.)}{5.75 ft.}\Big]</math> = 0.239ksf <u>o.k.</u>
:'''Dead Load, Earth Pressure, and Live Load'''


::Toe: <math>P_T = \frac{4.275k}{(1 ft.)(5.75 ft.}\Big[1 + \frac{6 (0.650 ft.)}{5.75 ft.}\Big]</math> = 1.248ksf <u>o.k.</u>
::'''Live Load 1 ft. From Stem Face'''


'''Dead Load, Earth Pressure, Collision Load, and Live Load - Stability and Pressure Checks'''
[[image:751.24.3.4 live load.jpg|center|300px]]


During a collision, the live load will be close to the wall so check this combination when the live load is one foot from the face of the stem. Sliding (in either direction) will not be an issue. Stability about the heel should be checked although it is unlikely to be a problem. There are no criteria for the location of the resultant, so long as the footing pressure does not exceed 125% of the allowable. It is assumed that the distributed collision force will develop an equal and opposite force on the fillface of the back wall unless it exceeds the passive pressure that can be developed by soil behind the wall.
::''β<sub>E</sub>'' = 1.3 (lateral earth pressure)


''F<sub>LL</sub>'' = 3.516k
::''β<sub>LL</sub>'' = 1.67


[[image:751.24.3.4 collision.jpg|center|250px]]
::''γ'' = 1.3


''F<sub>COLL</sub>'' = <math>\frac{10k}{2(3 ft.)}(1 ft.)</math> = 1.667k
::Apply Load Factors:


<math>C_P = cos \delta \Bigg[\frac{cos \delta + \sqrt{cos^2 \delta −- cos^2 \phi}}{cos \delta -− \sqrt{cos^2 \delta -− cos^2 \phi}}\Bigg]</math> = 1.867
::''F<sub>LL</sub>'' = 3.516k(1.3)(1.67) = 7.633k


<math>P_{PH} = \frac{1}{2}\gamma_s C_P H^2 cos\delta = \frac{1}{2}(0.120kcf)(1.867)(4.958ft)^2 cos(21.801^\circ)</math>
::''ΣV'' = 7.633k + 1.951k(1.3) = 10.169k


''P<sub>PH</sub>'' = 2.556k > ''F<sub>COLL</sub>''  Thus the soil will develop an equal but opp. force.
::''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)


:'''Overturning About the Heel'''
::''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) + 3.917 ft.(7.633k) = 40.599(ft−k)


:F.S. = <math>\frac{(0.646k)(0.417 ft.) + (0.827k)(2.875 ft.) + (0.225k)(1.500 ft.) + (3.516k)(1.833 ft.) + (1.667k)\big(\frac{4.958 ft.}{3}\big)}{(1.667k)(3.958 ft.)}</math>
::<math>\bar{x} = \frac{40.599(ft-k) - 1.766(ft-k)}{10.169k}</math> = 3.819 ft.


:F.S. = <math>\frac{12.184(ft-−k)}{6.598(ft−-k)}</math> = 1.847 ≥ 1.2 <u>o.k.</u>
::''e'' = 3.819 ft. (5.75 ft./2) = 0.944 ft.


:'''Footing Pressure'''
::<math>P_T = \Bigg[\frac{10.169k}{(1 ft.)(5.75 ft.)}\Bigg]\Bigg[{1 - \frac{ 6(0.944 ft.)}{5.75 ft.}}\Bigg]</math> = 0.026 ksf


:<math>\bar{x} = \frac{12.184(ft-−k) −- 6.598(ft-−k)}{1.951k + 3.516k}</math> = 1.022 ft. from heel
::<math>P_H = \Bigg[\frac{10.169k}{(1 ft.)(5.75 ft.)}\Bigg]\Bigg[{1 + \frac{ 6(0.944 ft.)}{5.75 ft.}}\Bigg]</math> = 3.511 ksf


:''e'' = <math>\frac{5.75 ft.}{2} − 1.022 ft.</math> = 1.853 ft.
::<math>P_W = 0.026 ksf + [3.511 ksf - 0.026 ksf]\Big[\frac{4.917 ft.}{5.75 ft.}\Big]</math> = 3.006 ksf


:Allowable Pressure = (1.25)(3.0ksf) = 3.75ksf
::<math>P_{LL} = 0.026 ksf + [3.511 ksf - 0.026 ksf]\Bigg[\frac{3.917 ft.}{5.75 ft.}\Bigg] </math> = 2.400 ksf


:Heel: <math> P_H =\frac {2(\Sigma V)}{3b[\frac{L}{2} -− e]} = \frac {2(5.467k)}{3(1 ft.)\big[\frac{5.75 ft.}{2} -− 1.853 ft.\big]}</math> = 3.566ksf <u>o.k.</u>
::Footing wt. from face of wall to toe:


'''Stem Design-Steel in Rear Face'''
::''Footing wt.'' = <math>1.3\Bigg[\frac{11.5}{12} ft.\Bigg](4.917 ft.)\Bigg[0.150 \frac{k}{ft^3}\Bigg](1 ft.)</math> = 0.919k


[[image:751.24.3.4 steel in rear face.jpg|center|250px]]
::Footing wt. from LL<sub>WL</sub> to toe:


''γ'' = 1.3
::''Footing wt.'' = <math>1.3\Bigg[\frac{11.5}{12} ft.\Bigg](3.917 ft.)\Bigg[0.150 \frac{k}{ft^3}\Bigg](1 ft.)</math> = 0.732k


''β<sub>E</sub>'' = 1.3 (active lateral earth pressure)
::Moment at Wall Face:


d = 10 in. 2 in. (0.5 in./2) = 7.75 in.
::''M<sub>W</sub> = <math>0.026\frac{k}{ft} \frac{(4.917 ft.)^2}{2} - 7.633k (1 ft.) + \frac{1}{2}\Bigg[3.006\frac{k}{ft} - 0.026\frac{k}{ft}\Bigg](4.917 ft.)^2\Big[\frac{1}{3}\Big] - 0.919k\frac{(4.917 ft.)}{2}</math>


<math>P_{AH} = \frac{1}{2}\gamma_s C_a H^2 cos\delta = \frac{1}{2}\Bigg[0.120 \frac{k}{ft^3}\Bigg](0.462)(4 ft.)^2(1 ft.) cos 21.801^\circ</math>
::M<sub>W</sub> = 2.430(ft−k)


''P<sub>AH</sub>'' = 0.412k
::Moment at LL<sub>WL</sub>:


''M<sub>u</sub>'' = (1.333 ft.)(0.412k)(1.3)(1.3) = 0.928(ft−k)
::''M<sub>LL</sub>'' = <math>0.026\frac{k}{ft} \frac{(3.917 ft.)^2}{2} - 0.732k \frac{(3.917 ft.)}{2} + \frac{1}{2}\Bigg[2.400\frac{k}{ft} - 0.026\frac{k}{ft}\Bigg](3.917 ft.)^2\Big[\frac{1}{3}\Big] </math> = 4.837(ft−k)


<math>R_n = \frac{M_u}{\phi b d^2} = \frac{0.928(ft−-k)}{(0.9)(1 ft.)(7.75 in.)^2}\Big(1000\frac{lb}{k}\Big)</math> = 17.160psi
::'''Live Load 1 ft. From Toe'''


<math>\rho = \frac{0.85f_c}{f_y}\Bigg[1 -− \sqrt{1 -− \frac{2R_n}{0.85 f_c}}\Bigg]</math>
[[image:751.24.3.4 toe.jpg|center|250px]]


<math>\rho = \frac{4000 psi}{60,000 psi}\Bigg[1 -− \sqrt{1 -− \frac{2(17.160 psi)}{0.85 (4000psi)}}\Bigg]</math> = 0.000287
::Apply Load Factors:


<math>\rho_{min} = 1.7 \Bigg[\frac{h}{d}\Bigg]^2 \frac{\sqrt{f_c}}{f_y}</math>
::''F<sub>LL</sub>'' = 2.324k(1.3)(1.67) = 5.045k


<math>\rho_{min} = 1.7 \Bigg[\frac{10 in.}{7.75 in.}\Bigg]^2 \frac{\sqrt{4000 psi}}{60000 psi}</math> = 0.00298
::''ΣV'' = 5.045k + 1.951k(1.3) = 7.581k


Use ''ρ'' = (4/3)ρ = (4/3)(0.000287) = 0.000382
::''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)


<math>A_{S_{Req}} = \rho bd = 0.000382(12 in.)(7.75 in.) = 0.036 \frac{in^2}{ft.}</math>
::''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) + 5.045k(1ft.) = 15.745(ft−k)


One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>, so the required minimum of one #4 bar every 12 in. controls.
::<math>\bar{x} = \frac{15.745(ft-k)- 1.766(ft-k)}{7.581k}</math> = 1.844 ft.


<u>Use #4's @ 12 in. (min)</u>
::<math>e = \frac{5.75 ft.}{2} - 1.844 ft.</math> = 1.031 ft.


(These bars are also the bars in the bottom of the footing so the smaller of the two required spacings will be used.)
::''P<sub>H</sub>'' = 0 ksf


:'''Check Shear'''
::<math>P_T = \frac{2(7.581k)}{3(1 ft.)\big[\frac{5.75 ft.}{2} - 1.031 ft.\big]}</math> = 2.741 ksf


:<math>\frac{\nu_u}{\phi} \le V_n</math>
::''L<sub>1</sub>'' = 3[(L/2)− e]


:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.3)(0.412k)}{0.85(12 in.)(7.75 in.)}(1000\frac{lb}{k})</math>  = 8.8 psi
::''L<sub>1</sub>'' = 3[(5.75 ft./2)− 1.031 ft.] = 5.532 ft.


:<math>\nu_c = 2 \sqrt{f'_c}</math>
::<math>P_W = 2.741 ksf \Big[\frac{0.615 ft.}{5.532 ft.}\Big]</math> = 0.305 ksf


:<math>\nu_c = 2 \sqrt{4, 000 psi}</math> = 126.5 psi > 8.8 psi <u>o.k.</u>
::<math>P_{LL} = 2.741 ksf \Big[\frac{4.432 ft.}{5.532 ft.}\Big]</math> = 2.196 ksf


'''Stem Design-Steel in Front Face (Collision Loads)'''
::Moment at Wall Face:


[[image:751.24.3.4 steel in front face.jpg|center|300px]]
::''M<sub>W</sub>'' = <math> -5.045k (3.917 ft.) - 0.919k\Bigg[\frac{4.917 ft.}{2}\Bigg] + \frac{1}{2}(0.305\frac{k}{ft.})(4.917 ft.)^2 + \frac{1}{2}(4.917 ft.)^2 \Bigg[2.741\frac{k}{ft.} - 0.305\frac{k}{ft.}\Bigg]\Bigg[\frac{2}{3}\Bigg]</math> = 1.298(ft−k)


::Moment at LL<sub>WL</sub>:


The soil pressure on the back of the stem becomes passive soil pressure during a collision, however this pressure is ignored for reinforcement design.
::''M<sub>LL</sub>'' = <math>-0.187k(0.5 ft.) + 2.196\frac{k}{ft.}\frac{(1 ft.)^2}{2} +\frac{1}{2}(1 ft.)\Bigg[2.741\frac{k}{ft.}  - 2.196\frac{k}{ft.}\Bigg]\Bigg[\frac{2}{3}\Bigg](1 ft.)</math> = 1.186(ft−k)


''γ'' = 1.3
:'''Design Flexural Steel in Bottom of Footing'''


''β<sub>LL</sub>'' = 1.67
:''d'' = 11.5 in. − 4 in. = 7.500 in.


<math>d = 10 in. −- 1.5 in. -− 0.5 in. −- \frac{0.5 in.}{2}</math> = 7.75 in.
:''M<sub>u</sub>'' = 4.837(ft−k) (controlling moment)


<math>F_{COLL} = \frac{10k}{2L} = \frac{10k}{(2)(3 ft.)}</math> = 1.667 k/ft.
:<math>R_n = \frac{4.837(ft-k)}{0.9(1 ft.)(7.5 in.)^2}</math> = 0.096 ksi


''M<sub>u</sub>'' = 1.667k/ft. (1 ft.)(3 ft.)(1.3)(1.67) = 10.855(ft−k)
:<math>\rho = \frac{0.85(4000 psi)}{60,000 psi}\Bigg[1 - \sqrt{1 - \frac{2(0.096 ksi)}{0.85(4 ksi)}}\Bigg] </math> = 0.00162


<math>R_n = \frac{10.855(ft-−k)}{0.9(1 ft.)(7.75 in.)^2} (1000\frac{lb}{k})</math> = 200.809 psi
:<math>\rho_{min} = 1.7\Bigg[\frac{11.5 in.}{7.5 in.}\Bigg]^2\frac{\sqrt{4000 psi}}{60,000 psi}</math> = 0.00421


<math>\rho = \frac{0.85(4000 psi)}{60,000 psi}\Bigg[1 -− \sqrt{1 − \frac{2(200.809 psi)}{0.85(4000psi)}}\Bigg]</math> = 0.00345
:Use ''ρ'' = (4/3)''ρ'' = (4/3)(0.00162) = 0.00216


<math>\rho_{min} = 1.7\Bigg[\frac{10 in.}{7.75 in.}\Bigg]^2 \frac{\sqrt{4000 psi}}{60,000 psi}</math> = 0.00298
:''A<sub>S<sub>Req</sub></sub>'' = 0.00216(12 in.)(7.5 in.) = 0.194 in<sup>2</sup>/ft.


<math>A_{S_{Req}} = 0.00345 (12 in.)(7.75 in.) = 0.321 \frac{in.^2}{ft.}</math>


One #4 bar has A<sub>S</sub> = 0.196 in<sup>2</sup>.
:<math>\frac{s}{0.196 in^2} = \frac{12 in.}{0.194 in^2}</math>


<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.321 in.^2}</math>
:''s'' = 12.1 in.


''s'' = 7.3 in.
:<u>Use #4's @ 12 in. cts.</u> (Also use this spacing in the back of the stem.)
 
<u>Use #4's @ 7 in.</u>


:'''Check Shear'''
:'''Check Shear'''


:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.67)(1.667k)}{(0.85)(12 in.)(7.75 in.)} (1000\frac{lb}{k})</math> = 45.8 psi < 126.5 psi <u>o.k.</u>
::'''Dead Load and Earth Pressure Only'''


'''Footing Design - Bottom Steel'''
::<math>V_W = 0.143\frac{k}{ft.}(4.917 ft.) + \frac{1}{2}(4.917 ft.)\Big[0.653\frac{k}{ft.} - 0.143\frac{k}{ft.}\Big] - 0.919k</math>


It is not considered necessary to design footing reinforcement based upon a load case which includes collision loads.
::''V<sub>W</sub>'' = 1.038k


:'''Dead Load and Earth Pressure Only'''
::'''Live Load 1 ft. From Stem Face'''


[[image:751.24.3.4 dead load.jpg|center|250px]]
::Shear at the wall can be neglected for this loading case.


:''Footing wt.'' = <math>\Big[\frac{11.5}{12}ft.\Big](4.917 ft.)\Big[0.150 \frac{k}{ft.^3}\Big](1 ft.)</math> = 0.707k
::<math>V_{LL} = 0.026\frac{k}{ft.}(3.917 ft.) + \frac{1}{2}(3.917 ft.)\Big[2.400\frac{k}{ft.} - 0.026\frac{k}{ft.}\Big] - 0.732k</math>


:''β<sub>E</sub>'' = 1.3 (lateral earth pressure)
::''V<sub>LL</sub>'' = 4.019k


:''γ'' = 1.3
::'''Live Load 1 ft. From Toe'''


:Apply Load Factors:
::<math>V_W = 0.305\frac{k}{ft.}(4.917 ft.) + \frac{1}{2}(4.917 ft.)\Big[2.741\frac{k}{ft.} - 0.305\frac{k}{ft.}\Big] - 0.919k - 5.045k</math>


:''ΣV'' = 1.951k (1.3) = 2.536k
::''V<sub>W</sub>'' = 1.525k


:''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) = 10.700(ft−k)
::<math>V_{LL} = 2.196\frac{k}{ft.}(1ft) + \frac{1}{2}(1ft)\Big[2.741\frac{k}{ft.} - 2.196\frac{k}{ft.}\Big] - 0.187k</math>


:''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)
::''V<sub>LL</sub>'' = 2.282k


:''Footing wt.'' = 0.707k (1.3) = 0.919k
:Use ''V<sub>U</sub>'' = 4.019k


:<math>\bar{x} = \frac{10.700(ft-−k) -− 1.766(ft-−k)}{2.536k}</math> = 3.523 ft.
:<math>\frac{\nu_u}{\phi} = \frac{4019(lbs)}{0.85(12 in.)(7.5 in.)} = 52.5 psi < 2\sqrt{4000 psi}</math> = 126.5 psi


:<math>e = 3.523 ft. −- \frac{5.75ft}{2}</math> = 0.648 ft.
'''Shear Key Design'''


:<math>P_H = \frac{2.536k}{(1 ft.)(5.75 ft.)}\Bigg[1 + \frac{6(0.648 ft.)}{5.75 ft.}\Bigg]</math> = 0.739 ksf
[[image:751.24.3.4 shear key.jpg|center|300px]]


:<math>P_T = \frac{2.536k}{(1 ft.)(5.75 ft.)}\Bigg[1 - \frac{6(0.648 ft.)}{5.75 ft.}\Bigg]</math> = 0.143ksf
For concrete cast against and permanently exposed to earth, minimum cover for reinforcement is 3 inches.


:<math>P_W = 0.143 ksf + [0.739 ksf − 0.143 ksf]\Bigg[\frac{4.917 ft.}{5.75 ft.}\Bigg]</math> = 0.653 ksf
<math>d = 12 in. - 3 in. - \frac{1}{2}\Big[\frac{1}{2}in.\Big]</math> = 8.75 in.


:Moment at Wall Face:
<math>P_1 = 0.120\frac{k}{ft^3}(1 ft.)(2.882)\Big[\frac{11.5}{12}ft.\Big]</math> = 0.331 k/ft.


:<math>M_W = \Big[0.143\frac{k}{ft.}\Big]\Bigg[\frac{(4.917 ft.)^2}{2}\Bigg] + \frac{1}{3}(4.917 ft.)^2 \Bigg[0.653\frac{k}{ft.} −- 0.143\frac{k}{ft.}\Bigg]\frac{1}{2} -  0.919k \Bigg[\frac{4.917 ft.}{2}\Bigg]</math> = 1.524(ft−k)
<math>P_2 = 0.120\frac{k}{ft^3}(1 ft.)(2.882)\Big[\frac{29.5}{12}ft.\Big]</math> = 0.850 k/ft.


:'''Dead Load, Earth Pressure, and Live Load'''
<math>M_u = (1.3)(1.3)\Bigg\{0.331\frac{k}{ft.}\frac{(1.5 ft.)^2}{2} + \frac{1}{2}(1.5 ft.)\Big[0.850\frac{k}{ft.} - 0.331\frac{k}{ft}\Big]\Big[\frac{2}{3}\Big](1.5 ft.)\Bigg\}</math>


::'''Live Load 1 ft. From Stem Face'''
''M<sub>u</sub>'' = 1.287(ft−k)


[[image:751.24.3.4 live load.jpg|center|300px]]
<math>R_n = \frac{1.287(ft-k)}{0.9(1ft.)(8.75in.)^2}</math> = 0.0187 ksi


::''β<sub>E</sub>'' = 1.3 (lateral earth pressure)
<math>\rho = \frac{0.85(4000psi)}{60,000psi}\Bigg[1 - \sqrt{1 - \frac{2(0.0187ksi)}{0.85(4ksi)}}\Bigg]</math> = 0.000312


::''β<sub>LL</sub>'' = 1.67
<math>\rho_{min} = 1.7\Big[\frac{12in.}{8.75in.}\Big]^2\frac{\sqrt{4000psi}}{60,000psi}</math> = 0.00337


::''γ'' = 1.3
Use ''ρ'' = (4/3)''ρ'' = (4/3)(0.000312) = 0.000416


::Apply Load Factors:
''A<sub>S<sub>Req</sub></sub>'' = 0.000416 (12 in.)(8.75 in.) = 0.0437 in<sup>2</sup>/ft.


::''F<sub>LL</sub>'' = 3.516k(1.3)(1.67) = 7.633k


::''ΣV'' = 7.633k + 1.951k(1.3) = 10.169k
<math>\frac{s}{0.196 in.^2} = \frac{12in.}{0.0437in.^2}</math>


::''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)
''s'' = 53.8 in.


::''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) + 3.917 ft.(7.633k) = 40.599(ft−k)
<u>Use #4's @ 18 in. cts. (min)</u>


::<math>\bar{x} = \frac{40.599(ft-−k) -− 1.766(ft−-k)}{10.169k}</math> = 3.819 ft.
:'''Check Shear'''


::''e'' = 3.819 ft. − (5.75 ft./2) = 0.944 ft.
:''V'' = 0.886k


::<math>P_T = \Bigg[\frac{10.169k}{(1 ft.)(5.75 ft.)}\Bigg]\Bigg[{1 −- \frac{ 6(0.944 ft.)}{5.75 ft.}}\Bigg]</math> = 0.026 ksf
:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.3)(886 lbs)}{0.85(12 in.)(8.75 in.)}</math> = 16.8 psi < 126.5 psi <u>o.k.</u>


::<math>P_H = \Bigg[\frac{10.169k}{(1 ft.)(5.75 ft.)}\Bigg]\Bigg[{1 + \frac{ 6(0.944 ft.)}{5.75 ft.}}\Bigg]</math> = 3.511 ksf
'''Reinforcement Summary'''


::<math>P_W = 0.026 ksf + [3.511 ksf -− 0.026 ksf]\Big[\frac{4.917 ft.}{5.75 ft.}\Big]</math> = 3.006 ksf
[[image:751.24.3.4 summary.jpg|center|400px]]


::<math>P_{LL} = 0.026 ksf + [3.511 ksf − 0.026 ksf]\Bigg[\frac{3.917 ft.}{5.75 ft.}\Bigg] </math> = 2.400 ksf
===751.24.3.5 Example 3: Pile Footing Cantilever Wall===


::Footing wt. from face of wall to toe:
[[image:751.24.3.5.jpg|center|850px]]


::''Footing wt.'' = <math>1.3\Bigg[\frac{11.5}{12} ft.\Bigg](4.917 ft.)\Bigg[0.150 \frac{k}{ft^3}\Bigg](1 ft.)</math> = 0.919k
''f’<sub>c</sub>'' = 3,000 psi


::Footing wt. from LL<sub>WL</sub> to toe:
''f<sub>y</sub>'' = 60,000 psi


::''Footing wt.'' = <math>1.3\Bigg[\frac{11.5}{12} ft.\Bigg](3.917 ft.)\Bigg[0.150 \frac{k}{ft^3}\Bigg](1 ft.)</math> = 0.732k
''φ'' = 27°


::Moment at Wall Face:
''γ<sub>s</sub>'' = 120 pcf


::''M<sub>W</sub> = <math>0.026\frac{k}{ft} \frac{(4.917 ft.)^2}{2} −- 7.633k (1 ft.) + \frac{1}{2}\Bigg[3.006\frac{k}{ft} -− 0.026\frac{k}{ft}\Bigg](4.917 ft.)^2\Big[\frac{1}{3}\Big] -− 0.919k\frac{(4.917 ft.)}{2}</math>
Pile type: HP 10 x 42


::M<sub>W</sub> = 2.430(ft−k)
Allowable pile bearing = 56 tons


::Moment at LL<sub>WL</sub>:
Pile width = 10 inches


::''M<sub>LL</sub>'' = <math>0.026\frac{k}{ft} \frac{(3.917 ft.)^2}{2} −- 0.732k \frac{(3.917 ft.)}{2} + \frac{1}{2}\Bigg[2.400\frac{k}{ft} -− 0.026\frac{k}{ft}\Bigg](3.917 ft.)^2\Big[\frac{1}{3}\Big] </math> = 4.837(ft−k)
Toe pile batter = 1:3


::'''Live Load 1 ft. From Toe'''
See [[751.12 Barriers, Railings, Curbs and Fences|EPG 751.12 Barriers, Railings, Curbs and Fences]] for weight and centroid of barrier.  


[[image:751.24.3.4 toe.jpg|center|250px]]
'''Assumptions'''


::Apply Load Factors:
:* Retaining wall is located such that traffic can come within half of the wall height to the plane where earth pressure is applied.


::''F<sub>LL</sub>'' = 2.324k(1.3)(1.67) = 5.045k
:* Reinforcement design is for one foot of wall length.


::''ΣV'' = 5.045k + 1.951k(1.3) = 7.581k
:* Sum moments about the centerline of the toe pile at a distance of 6B (where B is the pile width) below the bottom of the footing for overturning.


::''ΣM<sub>OT</sub>'' = 1.045(ft−k)(1.3)(1.3) = 1.766(ft−k)
:* Neglect top one foot of fill over toe in determining soil weight and passive pressure on shear key.


::''ΣM<sub>R</sub>'' = 8.231(ft−k)(1.3) + 5.045k(1ft.) = 15.745(ft−k)
:* Neglect all fill over toe in designing stem reinforcement.


::<math>\bar{x} = \frac{15.745(ft−-k)-− 1.766(ft-−k)}{7.581k}</math> = 1.844 ft.
:* The wall is designed as a cantilever supported by the footing.


::<math>e = \frac{5.75 ft.}{2} -− 1.844 ft.</math> = 1.031 ft.
:* Footing is designed as a cantilever supported by the wall.


::''P<sub>H</sub>'' = 0 ksf
:* Critical sections for bending are at the front and back faces of the wall.


::<math>P_T = \frac{2(7.581k)}{3(1 ft.)\big[\frac{5.75 ft.}{2} -− 1.031 ft.\big]}</math> = 2.741 ksf
:* Critical sections for shear are at the back face of the wall for the heel and at a distance d (effective depth) from the front face for the toe.


::''L<sub>1</sub>'' = 3[(L/2)− e]
:* For load factors for design of concrete, see [[#Group Loads|EPG 751.24.1.2 Group Loads]].


::''L<sub>1</sub>'' = 3[(5.75 ft./2)− 1.031 ft.] = 5.532 ft.
<math>C_A = cos\delta\Bigg[\frac{cos\delta - \sqrt{cos^2\delta - cos^2\phi}}{cos\delta + \sqrt{cos^2\delta - cos^2\phi}}\Bigg]</math>


::<math>P_W = 2.741 ksf \Big[\frac{0.615 ft.}{5.532 ft.}\Big]</math> = 0.305 ksf
''δ'' = 0, ''ϕ'' = 27° so ''C<sub>A</sub>'' reduces to:


::<math>P_{LL} = 2.741 ksf \Big[\frac{4.432 ft.}{5.532 ft.}\Big]</math> = 2.196 ksf
<math>C_A = \frac{1 - sin\phi}{1 + sin\phi} = \frac{1 - sin 27^\circ}{1 + sin 27^\circ}</math> = 0.376
<math>C_P = tan^2\Bigg[45^\circ + \frac{\phi}{2}\Bigg] = tan^2\Bigg[ 45^\circ + \frac{27^\circ}{2}\Bigg]</math> = 2.663


::Moment at Wall Face:
Table 751.24.3.5.1 is for stability check (moments taken about C.L. of toe pile at a depth of 6B below the bottom of the footing).


::''M<sub>W</sub>'' = <math> -−5.045k (3.917 ft.) -− 0.919k\Bigg[\frac{4.917 ft.}{2}\Bigg] + \frac{1}{2}(0.305\frac{k}{ft.})(4.917 ft.)^2 + \frac{1}{2}(4.917 ft.)^2 \Bigg[2.741\frac{k}{ft.} -− 0.305\frac{k}{ft.}\Bigg]\Bigg[\frac{2}{3}\Bigg]</math> = 1.298(ft−k)
{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
 
|+ '''''Table 751.24.3.5.1'''''
::Moment at LL<sub>WL</sub>:
! style="background:#BEBEBE" colspan="2"|Load !! style="background:#BEBEBE"|Force (kips/ft) !! style="background:#BEBEBE"|Arm about C.L. of toe pile at 6B below footing (ft.) !! style="background:#BEBEBE"|Moment (ft-kips) per foot of wall length
 
|-
::''M<sub>LL</sub>'' = <math>-−0.187k(0.5 ft.) + 2.196\frac{k}{ft.}\frac{(1 ft.)^2}{2} +\frac{1}{2}(1 ft.)\Bigg[2.741\frac{k}{ft.} -2.196\frac{k}{ft.}\Bigg]\Bigg[\frac{2}{3}\Bigg](1 ft.)</math> = 1.186(ft−k)
|rowspan="5"|'''Dead Load'''||(1)|| 0.340|| 2.542|| 0.864
 
|-
:'''Design Flexural Steel in Bottom of Footing'''
|(2)|| (1.333 ft.)(7.000 ft.)(0.150k/ft<sup>3</sup>) = 1.400 ||2.833|| 3.966
 
|-
:''d'' = 11.5 in. 4 in. = 7.500 in.
|(3)|| (3.000 ft.)(8.500 ft.)(0.150k/ft<sup>3</sup>) = 3.825|| 4.417|| 16.895
 
|-
:''M<sub>u</sub>'' = 4.837(ft−k) (controlling moment)
|(4)|| (1.000 ft.)(1.750 ft.)(0.150k/ft<sup>3</sup>) = <u>0.263</u>|| 4.417|| <u>1.162</u>
 
|-
:<math>R_n = \frac{4.837(ft-−k)}{0.9(1 ft.)(7.5 in.)^2}</math> = 0.096 ksi
|Σ||ΣV = 5.828 || - ||ΣM<sub>R</sub> = 22.887
|-
|rowspan="3"|'''Earth Load'''||(5)|| (7.000 ft.)(5.167 ft.)(0.120k/ft<sup>3</sup>) = 4.340|| 6.083|| 26.400
|-
|(6)|| (2.000 ft.)(2.000 ft.)(0.120k/ft<sup>3</sup>) = <u>0.480</u>|| 1.167|| <u>0.560</u>
|-
|Σ ||ΣV = 4.820|| - ||ΣM<sub>R</sub> = 26.960
|-
|rowspan="2"|'''Live Load Surcharge'''||P<sub>SV</sub>|| (2.000 ft.)(5.167 ft.)(0.120k/ft<sub>3</sub>) = 1.240|| 6.083|| M<sub>R</sub> = 7.543
|-
|P<sub>SH</sub>||(2.000 ft.)(0.376)(10.000 ft.)(0.120k/ft<sup>3</sup>) = 0.902||10.000|| M<sub>OT</sub> = 9.020
|-
|rowspan="2"|'''Earth Pressure'''||P<sub>A</sub>||2.256<sup>'''1'''</sup>|| 8.333|| M<sub>OT</sub> = 18.799
|-
|P<sub>P</sub>|| 3.285<sup>'''2'''</sup> || - || -
|-
|colspan="2"|'''Collision Force''' (F<sub>COL</sub>)||(10.000k)/[2(7.000 ft.)] = 0.714|| 18.000 ||M<sub>OT</sub> = 12.852
|-
|colspan="2"|'''Heel Pile Tension''' (P<sub>HV</sub>)||(3.000 tons)(2 k/ton)(1 pile)/(12.000 ft.) = 0.500|| 7.167|| M<sub>R</sub> = 3.584
|-
|colspan="2"|'''Toe Pile Batter''' (P<sub>BH</sub>)|| 5.903<sup>'''3'''</sup>|| - || -
|-
|colspan="2"|'''Passive Pile Pressure''' (P<sub>pp</sub>)|| 0.832<sup>'''4'''</sup>|| - || -
|-
|colspan="5" align="left"|<sup>'''1'''</sup> <math>P_A = \frac{1}{2}\gamma_S C_A H^2 = \frac{1}{2}\Bigg[0.120\frac{k}{ft^3}\Bigg](0.376)(10 ft.)^3 = 2.256\frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''2'''</sup> <math>P_P = \frac{1}{2}\gamma_S C_A\Big[H_2^2 - H_1^2\Big] = \frac{1}{2}\Bigg[0.120\frac{k}{ft^3}\Bigg](2.663)[(6.75 ft.)^2 - (5 ft.)^2] = 3.285\frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''3'''</sup> <math>P_{BH} = \Big(56 \frac{tons}{pile}\Big)\Big( 2 \frac{k}{ton}\Big)(2 piles)\Bigg(\frac{4 in.}{\sqrt{(12 in.)^2 + (4 in.)^2}}\Bigg)\Big(\frac{1}{12 ft.}\Big) = 5.903 \frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''4'''</sup> <math>P_{PP} = \frac{1}{2}(2.663)(5 ft.)^2\Big(0.120 \frac{k}{ft^3}\Big)(0.833 ft.)(3 piles)\Big(\frac{1}{12 ft.}\Big) = 0.832\frac{k}{ft}</math>
|}


:<math>\rho = \frac{0.85(4000 psi)}{60,000 psi}\Bigg[1 -− \sqrt{1 −- \frac{2(0.096 ksi)}{0.85(4 ksi)}}\Bigg] </math> = 0.00162


:<math>\rho_{min} = 1.7\Bigg[\frac{11.5 in.}{7.5 in.}\Bigg]^2\frac{\sqrt{4000 psi}}{60,000 psi}</math> = 0.00421
Table 751.24.3.5.2 is for bearing pressure checks (moments taken about C.L of toe pile at the bottom of the footing).


:Use ''ρ'' = (4/3)''ρ'' = (4/3)(0.00162) = 0.00216
{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
 
|+ '''''Table 751.24.3.5.2'''''
:''A<sub>S<sub>Req</sub></sub>'' = 0.00216(12 in.)(7.5 in.) = 0.194 in<sup>2</sup>/ft.
! style="background:#BEBEBE" colspan="2"|Load !! style="background:#BEBEBE"|Force (kips/ft) !! style="background:#BEBEBE"|Arm about C.L. of toe pile at 6B below footing (ft.) !! style="background:#BEBEBE"|Moment (ft-kips) per foot of wall length
 
|-
 
|rowspan="5"|'''Dead Load'''||(1)|| 0.340|| 0.875|| 0.298
:<math>\frac{s}{0.196 in^2} = \frac{12 in.}{0.194 in^2}</math>
|-
 
|(2)|| (1.333 ft.)(7.000 ft.)(0.150k/ft<sup>3</sup>) = 1.400 ||1.167|| 1.634
:''s'' = 12.1 in.
|-
 
|(3)|| (3.000 ft.)(8.500 ft.)(0.150k/ft<sup>3</sup>) = 3.825|| 2.750|| 10.519
:<u>Use #4's @ 12 in. cts.</u> (Also use this spacing in the back of the stem.)
|-
 
|(4)|| (1.000 ft.)(1.750 ft.)(0.150k/ft<sup>3</sup>) = <u>0.263</u>|| 2.750|| <u>0.723</u>
:'''Check Shear'''
|-
 
|Σ||ΣV = 5.828 || - ||ΣM<sub>R</sub> = 13.174
::'''Dead Load and Earth Pressure Only'''
|-
 
|rowspan="3"|'''Earth Load'''||(5)|| (7.000 ft.)(5.167 ft.)(0.120k/ft<sup>3</sup>) = 4.340|| 4.417|| 19.170
::<math>V_W = 0.143\frac{k}{ft.}(4.917 ft.) + \frac{1}{2}(4.917 ft.)\Big[0.653\frac{k}{ft.} -− 0.143\frac{k}{ft.}\Big] -− 0.919k</math>
|-
 
|(6)|| (2.000 ft.)(2.000 ft.)(0.120k/ft<sup>3</sup>) = <u>0.480</u>|| -0.500|| <u>-0.240</u>
::''V<sub>W</sub>'' = 1.038k
|-
|Σ ||ΣV = 4.820|| - ||ΣM<sub>R</sub> = 18.930
|-
|rowspan="2"|'''Live Load Surcharge'''||P<sub>SV</sub>|| (2.000 ft.)(5.167 ft.)(0.120k/ft<sub>3</sub>) = 1.240|| 4.417|| M<sub>R</sub> = 5.477
|-
|P<sub>SH</sub>||(2.000 ft.)(0.376)(10.000 ft.)(0.120k/ft<sup>3</sup>) = 0.902||5.000|| M<sub>OT</sub> = 4.510
|-
|rowspan="2"|'''Earth Pressure'''||P<sub>A</sub>||2.256|| 3.333|| M<sub>OT</sub> = 7.519
|-
|P<sub>P</sub>|| 3.285 || - || -
|-
|colspan="2"|'''Collision Force''' (F<sub>COL</sub>)||(10.000k)/[2(7.000 ft.)] = 0.714|| 13.000 ||M<sub>OT</sub> = 9.282
|-
|colspan="2"|'''Heel Pile Tension''' (P<sub>HV</sub>)||(3.000 tons)(2 k/ton)(1 pile)/(12.000 ft.) = 0.500|| 5.500|| M<sub>R</sub> = 2.750
|-
|colspan="2"|'''Toe Pile Batter''' (P<sub>BH</sub>)|| 5.903|| - || -
|-
|colspan="2"|'''Passive Pile Pressure''' (P<sub>pp</sub>)|| 0.832|| - || -
|}


::'''Live Load 1 ft. From Stem Face'''
Investigate a representative 12 ft. strip. This will include one heel pile and two toe piles. The assumption is made that the stiffness of a batter pile in the vertical direction is the same as that of a vertical pile.


::Shear at the wall can be neglected for this loading case.
Neutral Axis Location = [2piles(1.5 ft.) + 1pile(7 ft.)] / (3 piles) = 3.333 ft. from the toe.


::<math>V_{LL} = 0.026\frac{k}{ft.}(3.917 ft.) + \frac{1}{2}(3.917 ft.)\Big[2.400\frac{k}{ft.} -− 0.026\frac{k}{ft.}\Big] -− 0.732k</math>
[[image:751.24.3.5 neutral axis.jpg|center|350px]]


::''V<sub>LL</sub>'' = 4.019k
''I ''= Ad<sup>2</sup>


::'''Live Load 1 ft. From Toe'''
For repetitive 12 ft. strip:


::<math>V_W = 0.305\frac{k}{ft.}(4.917 ft.) + \frac{1}{2}(4.917 ft.)\Big[2.741\frac{k}{ft.} -− 0.305\frac{k}{ft.}\Big] -− 0.919k −- 5.045k</math>
:Total pile area = 3A


::''V<sub>W</sub>'' = 1.525k
:''I ''= 2A(1.833 ft.)<sup>2</sup> + A(3.667 ft.)<sup>2</sup> = 20.167(A)ft.<sup>2</sup>


::<math>V_{LL} = 2.196\frac{k}{ft.}(1ft) + \frac{1}{2}(1ft)\Big[2.741\frac{k}{ft.} −- 2.196\frac{k}{ft.}\Big] -− 0.187k</math>
For a 1 ft. unit strip:


::''V<sub>LL</sub>'' = 2.282k
:<math>I = \frac{20.167(A)ft.^2}{12 ft.} = 1.681(A)ft.^2</math>


:Use ''V<sub>U</sub>'' = 4.019k
:Total pile area = (3A/12 ft.) = 0.250A


:<math>\frac{\nu_u}{\phi} = \frac{4019(lbs)}{0.85(12 in.)(7.5 in.)} = 52.5 psi < 2\sqrt{4000 psi}</math> = 126.5 psi
:'''Case I'''


'''Shear Key Design'''
:F.S. for overturning ≥ 1.5


[[image:751.24.3.4 shear key.jpg|center|300px]]
:F.S. for sliding ≥ 1.5


For concrete cast against and permanently exposed to earth, minimum cover for reinforcement is 3 inches.
::'''Check Overturning'''


<math>d = 12 in. -− 3 in. − \frac{1}{2}\Big[\frac{1}{2}in.\Big]</math> = 8.75 in.
::Neglect resisting moment due to P<sub>SV</sub> for this check.


<math>P_1 = 0.120\frac{k}{ft^3}(1 ft.)(2.882)\Big[\frac{11.5}{12}ft.\Big]</math> = 0.331 k/ft.
::''ΣM<sub>R</sub>'' = 22.887(ft−k) + 26.960(ft−k) + 3.584(ft−k)


<math>P_2 = 0.120\frac{k}{ft^3}(1 ft.)(2.882)\Big[\frac{29.5}{12}ft.\Big]</math> = 0.850 k/ft.
::''ΣM<sub>R</sub>'' = 53.431(ft−k)


<math>M_u = (1.3)(1.3)\Bigg\{0.331\frac{k}{ft.}\frac{(1.5 ft.)^2}{2} + \frac{1}{2}(1.5 ft.)\Big[0.850\frac{k}{ft.} -− 0.331\frac{k}{ft}\Big]\Big[\frac{2}{3}\Big](1.5 ft.)\Bigg\}</math>
::''ΣM<sub>OT</sub>'' = 9.020(ft−k) + 18.799(ft−k) = 27.819(ft−k)


''M<sub>u</sub>'' = 1.287(ft−k)
::''F.S.<sub>OT</sub>'' = <math>\frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{53.431(ft-k)}{27.819(ft-k)}</math> = 1.921 > 1.5 <u>o.k.</u>


<math>R_n = \frac{1.287(ft−-k)}{0.9(1ft.)(8.75in.)^2}</math> = 0.0187 ksi
::'''Check Pile Bearing'''


<math>\rho = \frac{0.85(4000psi)}{60,000psi}\Bigg[1 -− \sqrt{1 − \frac{2(0.0187ksi)}{0.85(4ksi)}}\Bigg]</math> = 0.000312
::Without P<sub>SV</sub> :


<math>\rho_{min} = 1.7\Big[\frac{12in.}{8.75in.}\Big]^2\frac{\sqrt{4000psi}}{60,000psi}</math> = 0.00337
::''ΣV'' = 5.828k + 4.820k = 10.648k


Use ''ρ'' = (4/3)''ρ'' = (4/3)(0.000312) = 0.000416
::''e'' = <math>\frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft-k) - (4.510 + 7.519)(ft-k)}{10.648k}</math> = 1.885 ft.


''A<sub>S<sub>Req</sub></sub>'' = 0.000416 (12 in.)(8.75 in.) = 0.0437 in<sup>2</sup>/ft.
::Moment arm = 1.885 ft. - 1.833 ft. = 0.052 ft.


::<math>P_T = \frac{\Sigma V}{A} - \frac{M_c}{I} = \frac{10.648k}{0.250A} - \frac{10.648k(0.052 ft.)(1.833 ft.)}{1.681(A)ft^2}</math>


<math>\frac{s}{0.196 in.^2} = \frac{12in.}{0.0437in.^2}</math>
::<math>P_T = \frac{41.988}{A} k</math>


''s'' = 53.8 in.
::<math>P_H = \frac{10.648k}{0.250A} + \frac{10.648k(0.052 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>


<u>Use #4's @ 18 in. cts. (min)</u>
::<math>P_H = \frac{43.800}{A} k</math>


:'''Check Shear'''
::Allowable pile load = 56 tons/pile. Each pile has area A, so:


:''V'' = 0.886k
::<math>P_T = 41.988\frac{k}{pile} = 20.944\frac{tons}{pile} </math> <u> o.k.</u>


:<math>\frac{\nu_u}{\phi} = \frac{(1.3)(1.3)(886 lbs)}{0.85(12 in.)(8.75 in.)}</math> = 16.8 psi < 126.5 psi <u>o.k.</u>
::<math>P_H = 43.800\frac{k}{pile} = 21.900\frac{tons}{pile} </math> <u> o.k.</u>


'''Reinforcement Summary'''
::With P<sub>SV</sub>:


[[image:751.24.3.4 summary.jpg|center|400px]]
::''ΣV'' = 5.828k + 4.820k + 1.240k = 11.888k


===751.24.3.5 Example 3: Pile Footing Cantilever Wall===
::<math>e = \frac{(13.174 + 18.930 + 5.477)(ft-k) - (4.510 + 7.519)(ft-k)}{11.888k}</math> = 2.149 ft.


[[image:751.24.3.5.jpg|center|850px]]
::Moment arm = 2.149 ft. - 1.833 ft. = 0.316 ft.


''f’<sub>c</sub>'' = 3,000 psi
::<math>P_T = \frac{11.888k}{0.250A} - \frac{11.888k(0.316 ft.)(1.833 ft.)}{1.681(A)ft^2} = 43.456k = 21.728\frac{tons}{pile}</math> <u> o.k.</u>


''f<sub>y</sub>'' = 60,000 psi
::<math>P_H = \frac{11.888k}{0.250A} + \frac{11.888k(0.316 ft.)(3.667 ft.)}{1.681(A)ft^2} = 55.747k = 27.874\frac{tons}{pile}</math> <u> o.k.</u>


''φ'' = 27°
::'''Check Sliding'''


''γ<sub>s</sub>'' = 120 pcf
::<math>F.S._{Sliding} = \frac{3.285k + 5.903k + 0.832k}{0.902 k + 2.256k}</math> = 3.173 ≥ 1.5 <u> o.k.</u>


Pile type: HP 10 x 42
:'''Case II'''


Allowable pile bearing = 56 tons
:F.S. for overturning ≥ 1.2


Pile width = 10 inches
:F.S. for sliding ≥ 1.2


Toe pile batter = 1:3
::'''Check Overturning'''


Barrier curb weight = 340 lbs/ft. of length
::''ΣM<sub>R</sub> ''= (22.887 + 26.960 + 7.543 + 3.584)(ft−k) = 60.974(ft−k)


Barrier curb resultant = 0.375 ft. from its flat back
::''ΣM<sub>OT</sub>'' = (9.020 + 18.799 + 12.852)(ft−k) = 40.671(ft−k)


'''Assumptions'''
::<math>F.S._{OT} = \frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{60.974(ft-k)}{40.671(ft-k)}</math> = 1.499 ≥ 1.2  <u> o.k.</u>


:* Retaining wall is located such that traffic can come within half of the wall height to the plane where earth pressure is applied.
::'''Check Pile Bearing'''


:* Reinforcement design is for one foot of wall length.
::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930 + 5.477)(ft-k) - (4.510 + 7.519 + 9.282)(ft-k)}{(5.828 + 4.820 + 1.240)k}</math> = 1.369 ft.


:* Sum moments about the centerline of the toe pile at a distance of 6B (where B is the pile width) below the bottom of the footing for overturning.
::Moment arm = 1.833 ft. - 1.369 ft. = 0.464 ft.


:* Neglect top one foot of fill over toe in determining soil weight and passive pressure on shear key.
::<math>P_T = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{11.888k}{0.250A} + \frac{11.888k(0.464 ft.)(1.833 ft.)}{1.681(A)ft^2}</math>
 
::<math>P_T = 53.567\frac{k}{pile} = 26.783\frac{tons}{pile} \le 56\frac{tons}{pile}</math> <u> o.k.</u>
 
::<math>P_H = \frac{11.888k}{0.250A} - \frac{11.888k(0.464 ft.)(3.667 ft.)}{1.681(A)ft^2}</math> = 35.519k
 
::<math>P_H = 17.760\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>
 
::'''Check Sliding'''


:* Neglect all fill over toe in designing stem reinforcement.
::<math>F.S._{Sliding} = \frac{3.285k + 5.903k + 0.832k}{0.902k + 2.256k + 0.714k}</math> = 2.588 ≥ 1.2 <u> o.k.</u>


:* The wall is designed as a cantilever supported by the footing.
:'''Case III'''


:* Footing is designed as a cantilever supported by the wall.
:F.S. for overturning ≥ 1.5


:* Critical sections for bending are at the front and back faces of the wall.
:F.S. for sliding ≥ 1.5


:* Critical sections for shear are at the back face of the wall for the heel and at a distance d (effective depth) from the front face for the toe.
::'''Check Overturning'''


:* For load factors for design of concrete, see [[#Group Loads|EPG 751.24.1.2 Group Loads]].
::''ΣM<sub>R</sub>'' = (22.887 + 26.960 + 3.584)(ft−k) = 53.431(ft−k)


<math>C_A = cos\delta\Bigg[\frac{cos\delta - \sqrt{cos^2\delta - cos^2\phi}}{cos\delta + \sqrt{cos^2\delta - cos^2\phi}}\Bigg]</math>
::''ΣM<sub>OT</sub>'' = 18.799(ft−k)


''δ'' = 0, ''ϕ'' = 27° so ''C<sub>A</sub>'' reduces to:
::<math>F.S._{OT} = \frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{53.431(ft-k)}{18.799(ft-k)}</math> = 2.842 ≥ 1.5 <u> o.k.</u>


<math>C_A = \frac{1 - sin\phi}{1 + sin\phi} = \frac{1 - sin 27^\circ}{1 + sin 27^\circ}</math> = 0.376
::'''Check Pile Bearing'''
<math>C_P = tan^2\Bigg[45^\circ + \frac{\phi}{2}\Bigg] = tan^2\Bigg[ 45^\circ + \frac{27^\circ}{2}\Bigg]</math> = 2.663


Table 751.24.3.5.1 is for stability check (moments taken about C.L. of toe pile at a depth of 6B below the bottom of the footing).
::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft-k) - 7.519(ft-k)}{(5.828 + 4.820)k}</math> = 2.309 ft.


{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
::Moment arm = 2.309 ft. - 1.833 ft. = 0.476 ft.
|+ '''''Table 751.24.3.5.1'''''
 
! style="background:#BEBEBE" colspan="2"|Load !! style="background:#BEBEBE"|Force (kips/ft) !! style="background:#BEBEBE"|Arm about C.L. of toe pile at 6B below footing (ft.) !! style="background:#BEBEBE"|Moment (ft-kips) per foot of wall length
::<math>P_T = \frac{10.648k}{0.250A} - \frac{10.648k(0.476 ft.)(1.833 ft.)}{1.681(A)ft^2}</math> = 37.065k
|-
 
|rowspan="5"|'''Dead Load'''||(1)|| 0.340|| 2.542|| 0.864
::<math>P_T = 18.532\frac{tons}{pile} \le 56\frac{tons}{pile}</math> <u> o.k.</u>
|-
 
|(2)|| (1.333 ft.)(7.000 ft.)(0.150k/ft<sup>3</sup>) = 1.400 ||2.833|| 3.966
::<math>P_H = \frac{10.648k}{0.250A} + \frac{10.648k(0.476 ft.)(3.667 ft.)}{1.681(A)ft^2}</math> = 53.649k
|-
 
|(3)|| (3.000 ft.)(8.500 ft.)(0.150k/ft<sup>3</sup>) = 3.825|| 4.417|| 16.895
::<math>P_H = 26.825\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>
|-
 
|(4)|| (1.000 ft.)(1.750 ft.)(0.150k/ft<sup>3</sup>) = <u>0.263</u>|| 4.417|| <u>1.162</u>
::'''Check Sliding'''
|-
 
|Σ||ΣV = 5.828 || - ||ΣM<sub>R</sub> = 22.887
::<math>F.S._{Sliding} = \frac{3.285k+5.903k+0.832k}{2.256k}</math> = 4.441 ≥ 1.5 <u> o.k.</u>
|-
 
|rowspan="3"|'''Earth Load'''||(5)|| (7.000 ft.)(5.167 ft.)(0.120k/ft<sup>3</sup>) = 4.340|| 6.083|| 26.400
:'''Case IV'''
|-
 
|(6)|| (2.000 ft.)(2.000 ft.)(0.120k/ft<sup>3</sup>) = <u>0.480</u>|| 1.167|| <u>0.560</u>
::'''Check Pile Bearing'''
|-
 
|Σ ||ΣV = 4.820|| - ||ΣM<sub>R</sub> = 26.960
::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft-k)}{5.828k + 4.820k}</math> = 3.015 ft.
|-
 
|rowspan="2"|'''Live Load Surcharge'''||P<sub>SV</sub>|| (2.000 ft.)(5.167 ft.)(0.120k/ft<sub>3</sub>) = 1.240|| 6.083|| M<sub>R</sub> = 7.543
::Moment arm = 3.015 ft. - 1.833 ft. = 1.182 ft.
|-
 
|P<sub>SH</sub>||(2.000 ft.)(0.376)(10.000 ft.)(0.120k/ft<sup>3</sup>) = 0.902||10.000|| M<sub>OT</sub> = 9.020
::<math>P_H = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{10.648k}{0.250A} + \frac{10.648k(1.182 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>
|-
|rowspan="2"|'''Earth Pressure'''||P<sub>A</sub>||2.256<sup>'''1'''</sup>|| 8.333|| M<sub>OT</sub> = 18.799
|-
|P<sub>P</sub>|| 3.285<sup>'''2'''</sup> || - || -
|-
|colspan="2"|'''Collision Force''' (F<sub>COL</sub>)||(10.000k)/[2(7.000 ft.)] = 0.714|| 18.000 ||M<sub>OT</sub> = 12.852
|-
|colspan="2"|'''Heel Pile Tension''' (P<sub>HV</sub>)||(3.000 tons)(2 k/ton)(1 pile)/(12.000 ft.) = 0.500|| 7.167|| M<sub>R</sub> = 3.584
|-
|colspan="2"|'''Toe Pile Batter''' (P<sub>BH</sub>)|| 5.903<sup>'''3'''</sup>|| - || -
|-
|colspan="2"|'''Passive Pile Pressure''' (P<sub>pp</sub>)|| 0.832<sup>'''4'''</sup>|| - || -
|-
|colspan="5" align="left"|<sup>'''1'''</sup> <math>P_A = \frac{1}{2}\gamma_S C_A H^2 = \frac{1}{2}\Bigg[0.120\frac{k}{ft^3}\Bigg](0.376)(10 ft.)^3 = 2.256\frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''2'''</sup> <math>P_P = \frac{1}{2}\gamma_S C_A\Big[H_2^2 - H_1^2\Big] = \frac{1}{2}\Bigg[0.120\frac{k}{ft^3}\Bigg](2.663)[(6.75 ft.)^2 - (5 ft.)^2] = 3.285\frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''3'''</sup> <math>P_{BH} = \Big(56 \frac{tons}{pile}\Big)\Big( 2 \frac{k}{ton}\Big)(2 piles)\Bigg(\frac{4 in.}{\sqrt{(12 in.)^2 + (4 in.)^2}}\Bigg)\Big(\frac{1}{12 ft.}\Big) = 5.903 \frac{k}{ft}</math>
|-
|colspan="5" align="left"|<sup>'''4'''</sup> <math>P_{PP} = \frac{1}{2}(2.663)(5 ft.)^2\Big(0.120 \frac{k}{ft^3}\Big)(0.833 ft.)(3 piles)\Big(\frac{1}{12 ft.}\Big) = 0.832\frac{k}{ft}</math>
|}


::<math>P_H = 70.047k = 35.024 \frac{tons}{pile}</math>


Table 751.24.3.5.2 is for bearing pressure checks (moments taken about C.L of toe pile at the bottom of the footing).
::25% overstress is allowed on the heel pile:


{| border="1" class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
::<math>P_H = 35.024\frac{tons}{pile} \le 1.25 (56\frac{tons}{pile}) = 70 \frac{tons}{pile}</math> <u> o.k.</u>
|+ '''''Table 751.24.3.5.2'''''
! style="background:#BEBEBE" colspan="2"|Load !! style="background:#BEBEBE"|Force (kips/ft) !! style="background:#BEBEBE"|Arm about C.L. of toe pile at 6B below footing (ft.) !! style="background:#BEBEBE"|Moment (ft-kips) per foot of wall length
|-
|rowspan="5"|'''Dead Load'''||(1)|| 0.340|| 0.875|| 0.298
|-
|(2)|| (1.333 ft.)(7.000 ft.)(0.150k/ft<sup>3</sup>) = 1.400 ||1.167|| 1.634
|-
|(3)|| (3.000 ft.)(8.500 ft.)(0.150k/ft<sup>3</sup>) = 3.825|| 2.750|| 10.519
|-
|(4)|| (1.000 ft.)(1.750 ft.)(0.150k/ft<sup>3</sup>) = <u>0.263</u>|| 2.750|| <u>0.723</u>
|-
|Σ||ΣV = 5.828 || - ||ΣM<sub>R</sub> = 13.174
|-
|rowspan="3"|'''Earth Load'''||(5)|| (7.000 ft.)(5.167 ft.)(0.120k/ft<sup>3</sup>) = 4.340|| 4.417|| 19.170
|-
|(6)|| (2.000 ft.)(2.000 ft.)(0.120k/ft<sup>3</sup>) = <u>0.480</u>|| -0.500|| <u>-0.240</u>
|-
|Σ ||ΣV = 4.820|| - ||ΣM<sub>R</sub> = 18.930
|-
|rowspan="2"|'''Live Load Surcharge'''||P<sub>SV</sub>|| (2.000 ft.)(5.167 ft.)(0.120k/ft<sub>3</sub>) = 1.240|| 4.417|| M<sub>R</sub> = 5.477
|-
|P<sub>SH</sub>||(2.000 ft.)(0.376)(10.000 ft.)(0.120k/ft<sup>3</sup>) = 0.902||5.000|| M<sub>OT</sub> = 4.510
|-
|rowspan="2"|'''Earth Pressure'''||P<sub>A</sub>||2.256|| 3.333|| M<sub>OT</sub> = 7.519
|-
|P<sub>P</sub>|| 3.285 || - || -
|-
|colspan="2"|'''Collision Force''' (F<sub>COL</sub>)||(10.000k)/[2(7.000 ft.)] = 0.714|| 13.000 ||M<sub>OT</sub> = 9.282
|-
|colspan="2"|'''Heel Pile Tension''' (P<sub>HV</sub>)||(3.000 tons)(2 k/ton)(1 pile)/(12.000 ft.) = 0.500|| 5.500|| M<sub>R</sub> = 2.750
|-
|colspan="2"|'''Toe Pile Batter''' (P<sub>BH</sub>)|| 5.903|| - || -
|-
|colspan="2"|'''Passive Pile Pressure''' (P<sub>pp</sub>)|| 0.832|| - || -
|}


Investigate a representative 12 ft. strip. This will include one heel pile and two toe piles. The assumption is made that the stiffness of a batter pile in the vertical direction is the same as that of a vertical pile.
::<math>P_T = \frac{10.648k}{0.250A} - \frac{10.648k(1.182 ft.)(1.833 ft.)}{1.681(A)ft^2}</math> = 28.868k


Neutral Axis Location = [2piles(1.5 ft.) + 1pile(7 ft.)] / (3 piles) = 3.333 ft. from the toe.
::<math>P_T = 14.434\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>


[[image:751.24.3.5 neutral axis.jpg|center|350px]]
:'''Reinforcement - Stem'''


''I ''= Ad<sup>2</sup>
[[image:751.24.3.5 reinforcement stem.jpg|300px|center]]


For repetitive 12 ft. strip:
:b = 12 in.


:Total pile area = 3A
:cover = 2 in.


:''I ''= 2A(1.833 ft.)<sup>2</sup> + A(3.667 ft.)<sup>2</sup> = 20.167(A)ft.<sup>2</sup>
:h = 16 in.


For a 1 ft. unit strip:
:d = 16 in. - 2 in. - 0.5(0.625 in.) = 13.688 in.


:<math>I = \frac{20.167(A)ft.^2}{12 ft.} = 1.681(A)ft.^2</math>
:''F<sub>Collision</sub>'' = 0.714k/ft


:Total pile area = (3A/12 ft.) = 0.250A
::<math>P_{LL} = \gamma_s C_A H(2.000 ft.) = (2.000 ft.)(0.376)(7.000 ft.)(0.120 \frac{k}{ft^3}) = 0.632\frac{k}{ft}</math>


:'''Case I'''
::<math>P_{A_{Stem}} = \frac{1}{2} \gamma_s C_A H^2 = \frac{1}{2}\Big[0.120 \frac{k}{ft^3}\Big](0.376)(7.000 ft.)^2 = 1.105\frac{k}{ft} </math>


:F.S. for overturning ≥ 1.5
:::'''Apply Load Factors'''


:F.S. for sliding ≥ 1.5
:::''F<sub>Col.</sub>'' = ''γβ<sub>LL</sub>''(0.714k) = (1.3)(1.67)(0.714k) = 1.550k


::'''Check Overturning'''
:::''P<sub>LL</sub>'' = ''γβ<sub>E</sub>'' (0.632k) = (1.3)(1.67)(0.632k) = 1.372k


::Neglect resisting moment due to P<sub>SV</sub> for this check.
:::''P<sub>A<sub>Stem</sub></sub>'' = ''γβ<sub>E</sub>'' (1.105k) = (1.3)(1.3)(1.105k) = 1.867k


::''ΣM<sub>R</sub>'' = 22.887(ft−k) + 26.960(ft−k) + 3.584(ft−k)
::''M<sub>u</sub>'' = (10.00 ft.)(1.550k) + (3.500 ft.)(1.372k) + (2.333 ft.)(1.867k)


::''ΣM<sub>R</sub>'' = 53.431(ft−k)
::''M<sub>u</sub>'' = 24.658(ft−k)


::''ΣM<sub>OT</sub>'' = 9.020(ft−k) + 18.799(ft−k) = 27.819(ft−k)
::<math>R_n = \frac{M_u}{\phi b d^2} = \frac{24.658(ft-k)}{(0.9)(1 ft.)(13.688 in.)^2}</math> = 0.146ksi


::''F.S.<sub>OT</sub>'' = <math>\frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{53.431(ft−-k)}{27.819(ft-−k)}</math> = 1.921 > 1.5 <u>o.k.</u>
::<math>\rho = \frac{0.85f'_c}{f_y}\Bigg[1 - \sqrt{1 - \frac{2R_n}{0.85f'_c}}\Bigg] =  
\frac{0.85(3 ksi)}{60 ksi}\Bigg[1 - \sqrt{1 - \frac{2(0.146 ksi)}{0.85(3 ksi)}}\Bigg]</math> = 0.00251


::'''Check Pile Bearing'''
::<math>\rho_{min} = 1.7\Big[\frac{h}{d}\Big]^2 \frac{\sqrt{f'_c}}{f_y} = 1.7\Big[\frac{16 in.}{13.688 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00212


::Without P<sub>SV</sub> :
::''ρ'' = 0.00251


::''ΣV'' = 5.828k + 4.820k = 10.648k
::<math>A_{S_{Req.}} = \rho bd = (0.00251)(12 in.)(13.688 in.) = 0.412 \frac{in^2}{ft.}</math>


::''e'' = <math>\frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft−-k) − (4.510 + 7.519)(ft-−k)}{10.648k}</math> = 1.885 ft.
::One #5 bar has A<sub>S</sub> = 0.307 in<sup>2</sup>


::Moment arm = 1.885 ft. - 1.833 ft. = 0.052 ft.
::<math>\frac{s}{0.307 in^2} = \frac{12 in.}{0.412 in^2}</math>


::<math>P_T = \frac{\Sigma V}{A} -− \frac{M_c}{I} = \frac{10.648k}{0.250A} -− \frac{10.648k(0.052 ft.)(1.833 ft.)}{1.681(A)ft^2}</math>
::''s'' = 8.9 in.


::<math>P_T = \frac{41.988}{A} k</math>
::<u>Use # 5 bars @ 8.5 in. cts.</u>


::<math>P_H = \frac{10.648k}{0.250A} + \frac{10.648k(0.052 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>
:::'''Check Shear'''


::<math>P_H = \frac{43.800}{A} k</math>
:::''V<sub>u</sub>'' ≤ ''φV<sub>n</sub>''


::Allowable pile load = 56 tons/pile. Each pile has area A, so:
:::''V<sub>u</sub>'' = ''F<sub>Collision</sub>'' + ''P<sub>LL</sub>'' + ''P<sub>A<sub>Stem</sub></sub>'' = 1.550k + 1.372k + 1.867k = 4.789k


::<math>P_T = 41.988\frac{k}{pile} = 20.944\frac{tons}{pile} </math> <u> o.k.</u>


::<math>P_H = 43.800\frac{k}{pile} = 21.900\frac{tons}{pile} </math> <u> o.k.</u>
:::<math>\frac{\nu_u}{\phi} = \frac{v_u}{\phi bd} = \frac{4789 lbs}{0.85(12 in.)(13.688 in.)}</math> = 34.301 psi


::With P<sub>SV</sub>:
:::<math> \nu_n = \nu_c = 2\sqrt{f'_c} = 2\sqrt{3000psi}</math> = 109.5 psi > 34.3 psi <u>o.k.</u>


::''ΣV'' = 5.828k + 4.820k + 1.240k = 11.888k
::'''Reinforcement - Footing - Top Steel'''


::<math>e = \frac{(13.174 + 18.930 + 5.477)(ft-−k) -− (4.510 + 7.519)(ft-−k)}{11.888k}</math> = 2.149 ft.
[[image:751.24.3.5 footing.jpg|300px|center]]


::Moment arm = 2.149 ft. - 1.833 ft. = 0.316 ft.
::b = 12 in.


::<math>P_T = \frac{11.888k}{0.250A} -− \frac{11.888k(0.316 ft.)(1.833 ft.)}{1.681(A)ft^2} = 43.456k = 21.728\frac{tons}{pile}</math> <u> o.k.</u>
::cover = 3 in.


::<math>P_H = \frac{11.888k}{0.250A} + \frac{11.888k(0.316 ft.)(3.667 ft.)}{1.681(A)ft^2} = 55.747k = 27.874\frac{tons}{pile}</math> <u> o.k.</u>
::h = 36 in.


::'''Check Sliding'''
::d = 36 in. - 3 in. - 0.5(0.5 in.) = 32.750 in.


::<math>F.S._{Sliding} = \frac{3.285k + 5.903k + 0.832k}{0.902 k + 2.256k}</math> = 3.173 ≥ 1.5 <u> o.k.</u>
::Design the heel to support the entire weight of the superimposed materials.


:'''Case II'''
::Soil(1) = 4.340k/ft.


:F.S. for overturning ≥ 1.2
::LL<sub>s</sub> = 1.240k/ft.


:F.S. for sliding ≥ 1.2
::<math>Slab \ wt. = (3.000 ft.)\Big[0.150 \frac{k}{ft^3}\Big](5.167 ft.)</math> = 2.325k/ft.


::'''Check Overturning'''
:::'''Apply Load Factors'''


::''ΣM<sub>R</sub> ''= (22.887 + 26.960 + 7.543 + 3.584)(ft−k) = 60.974(ft−k)
:::Soil(1) = ''γβ<sub>E</sub>''(4.340k) = (1.3)(1.0)(4.340k) = 5.642k


::''ΣM<sub>OT</sub>'' = (9.020 + 18.799 + 12.852)(ft−k) = 40.671(ft−k)
:::''LL<sub>s</sub>'' = ''γβ<sub>E</sub>''(1.240k) = (1.3)(1.67)(1.240k) = 2.692k


::<math>F.S._{OT} = \frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{60.974(ft-−k)}{40.671(ft-−k)}</math> = 1.499 ≥ 1.2 <u> o.k.</u>
:::Slab wt. = ''γβ<sub>D</sub>''(2.325k) = (1.3)(1.0)(2.325k) = 3.023k


::'''Check Pile Bearing'''
::''M<sub>u</sub>'' = (2.583 ft.)(5.642k + 2.692k + 3.023k) = 29.335(ft−k)


::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930 + 5.477)(ft-−k) -− (4.510 + 7.519 + 9.282)(ft-−k)}{(5.828 + 4.820 + 1.240)k}</math> = 1.369 ft.
::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{29.335(ft-k)}{(0.9)(1 ft.)(32.750 in.)^2}</math> = 0.0304 ksi
::<math>\rho = \frac{0.85(3ksi)}{60ksi}\Bigg[1 - \sqrt{1 - \frac{2(0.0304ksi)}{0.85(3ksi)}}\Bigg]</math> = 0.000510


::Moment arm = 1.833 ft. - 1.369 ft. = 0.464 ft.
::<math>\rho_{min} = 1.7\Big[\frac{36 in.}{32.750 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000psi}</math> = 0.00188


::<math>P_T = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{11.888k}{0.250A} + \frac{11.888k(0.464 ft.)(1.833 ft.)}{1.681(A)ft^2}</math>
::Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.000510) = 0.000680


::<math>P_T = 53.567\frac{k}{pile} = 26.783\frac{tons}{pile} \le 56\frac{tons}{pile}</math> <u> o.k.</u>
::<math>A_{S_{Req}} = \rho bd = (0.000680)(12 in.)(32.750 in.) = 0.267\frac{in^2}{ft.}</math>


::<math>P_H = \frac{11.888k}{0.250A} -− \frac{11.888k(0.464 ft.)(3.667 ft.)}{1.681(A)ft^2}</math> = 35.519k
::One #4 bar has A<sub>s</sub> = 0.196 in.<sup>2</sup>


::<math>P_H = 17.760\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>
::<math>\frac{s}{0.196 in^2} = \frac{12 in}{0.267 in.^2}</math>


::'''Check Sliding'''
::''s'' = 8.8 in.


::<math>F.S._{Sliding} = \frac{3.285k + 5.903k + 0.832k}{0.902k + 2.256k + 0.714k}</math> = 2.588 ≥ 1.2 <u> o.k.</u>
::<u>Use #4 bars @ 8.5 in. cts.</u>


:'''Case III'''
:::'''Check Shear'''


:F.S. for overturning ≥ 1.5
:::<math>V_u = Soil(1) + LL_s + Slab \ wt. = 5.642k + 2.692k + 3.023k = 11.357k</math>


:F.S. for sliding ≥ 1.5
:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = \frac{11357 lbs}{(0.85)(12 in.)(32.750 in.)}</math> = 33.998 psi ≤ 109.5 psi = ''ν<sub>c</sub>''  <u>o.k.</u>


::'''Check Overturning'''
::'''Reinforcement - Footing - Bottom Steel'''


::''ΣM<sub>R</sub>'' = (22.887 + 26.960 + 3.584)(ft−k) = 53.431(ft−k)
::Design the flexural steel in the bottom of the footing to resist the largest moment that the heel pile could exert on the footing. The largest heel pile bearing force was in Case IV. The heel pile will cause a larger moment about the stem face than the toe pile (even though there are two toe piles for every one heel pile) because it has a much longer moment arm about the stem face.


::''ΣM<sub>OT</sub>'' = 18.799(ft−k)
[[image: 751.24.3.5 heel pile.jpg|center|300px]]


::<math>F.S._{OT} = \frac{\Sigma M_R}{\Sigma M_{OT}} = \frac{53.431(ft−-k)}{18.799(ft−-k)}</math> = 2.842 ≥ 1.5 <u> o.k.</u>
::Pile is embedded into footing 12 inches.


::'''Check Pile Bearing'''
::''b'' = 12 in.


::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft-−k) -− 7.519(ft-−k)}{(5.828 + 4.820)k}</math> = 2.309 ft.
::''h'' = 36 in.


::Moment arm = 2.309 ft. - 1.833 ft. = 0.476 ft.
::''d'' = 36 in. - 4 in. = 32 in.


::<math>P_T = \frac{10.648k}{0.250A} -− \frac{10.648k(0.476 ft.)(1.833 ft.)}{1.681(A)ft^2}</math> = 37.065k
:::'''Apply Load Factors to Case IV Loads'''


::<math>P_T = 18.532\frac{tons}{pile} \le 56\frac{tons}{pile}</math> <u> o.k.</u>
:::<math>\Sigma V = \gamma \beta_D\Big[5.828 \frac{k}{ft.}\Big] + \gamma \beta_E \Big[4.820 \frac{k}{ft.}\Big]</math>


::<math>P_H = \frac{10.648k}{0.250A} + \frac{10.648k(0.476 ft.)(3.667 ft.)}{1.681(A)ft^2}</math> = 53.649k
:::<math>\Sigma V = 1.3(1.0)\Big[5.828\frac{k}{ft.}\Big] + 1.3(1.0)\Big[4.820\frac{k}{ft.}\Big]</math>


::<math>P_H = 26.825\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>
:::''ΣV'' = 13.842 k/ft.


::'''Check Sliding'''
:::<math>\Sigma M = \gamma \beta_D\Big[13.174\frac{(ft-k)}{ft.}\Big] + \gamma \beta_E\Big[18.930\frac{(ft-k)}{ft.}\Big]</math>


::<math>F.S._{Sliding} = \frac{3.285k+5.903k+0.832k}{2.256k}</math> = 4.441 ≥ 1.5 <u> o.k.</u>
:::<math>\Sigma M = (1.3)(1.0)\Big[13.174\frac{(ft-k)}{ft.}\Big] + (1.3)(1.0)\Big[18.930\frac{(ft-k)}{ft.}\Big]</math>


:'''Case IV'''
:::''ΣM'' = 41.735 (ft−k)/ft.


::'''Check Pile Bearing'''
::e = <math>\frac{\Sigma M}{\Sigma V} = \frac{41.735 (ft-k)}{13.842k}</math> = 3.015 ft.
 
::<math>e = \frac{\Sigma M}{\Sigma V} = \frac{(13.174 + 18.930)(ft-−k)}{5.828k + 4.820k}</math> = 3.015 ft.


::Moment arm = 3.015 ft. - 1.833 ft. = 1.182 ft.
::Moment arm = 3.015 ft. - 1.833 ft. = 1.182 ft.


::<math>P_H = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{10.648k}{0.250A} + \frac{10.648k(1.182 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>
::<math>P_H = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{13.842k}{0.250A} + \frac{13.842k (1.182 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>


::<math>P_H = 70.047k = 35.024 \frac{tons}{pile}</math>
::<math>P_H = 91.059 \frac{k}{pile}\Big(\frac{1}{12 ft.}\Big)</math> = 7.588 k/ft.


::25% overstress is allowed on the heel pile:
::<math>M_u = \Big(7.588\frac{k}{ft.}\Big)(3.667 ft.)</math> = 27.825(ft−k)/ft.


::<math>P_H = 35.024\frac{tons}{pile} \le 1.25 (56\frac{tons}{pile}) = 70 \frac{tons}{pile}</math> <u> o.k.</u>
::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{27.825(ft-k)}{(0.9)(1 ft.)(32 in.)^2}</math> = 0.0301 ksi


::<math>P_T = \frac{10.648k}{0.250A} -\frac{10.648k(1.182 ft.)(1.833 ft.)}{1.681(A)ft^2}</math> = 28.868k
::<math>\rho = \frac{0.85(3 ksi)}{60ksi}\Bigg[1 - \sqrt{1 - \frac{2(0.0301 ksi)}{0.85(3 ksi)}}\Bigg]</math> = 0.000505


::<math>P_T = 14.434\frac{tons}{pile} \le 56\frac{tons}{pile} </math> <u> o.k.</u>
::<math>\rho_{min} = 1.7\Big[\frac{36 in.}{32 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00196


:'''Reinforcement - Stem'''
::Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.000505) = 0.000673


[[image:751.24.3.5 reinforcement stem.jpg|300px|center]]
::''A<sub>S<sub>Req</sub></sub> = ρbd'' = (0.000673)(12 in.)(32 in.) = 0.258 in<sup>2</sup>/ft.''


:b = 12 in.


:cover = 2 in.
::One #4 bar has A<sub>s</sub> = 0.196 in<sup>2</sup>.


:h = 16 in.
::<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.258 in.^2}</math>


:d = 16 in. - 2 in. - 0.5(0.625 in.) = 13.688 in.
::''s'' = 9.1 in.


:''F<sub>Collision</sub>'' = 0.714k/ft
::<u>Use #4 bars @ 9 in. cts.</u>


::<math>P_{LL} = \gamma_s C_A H(2.000 ft.) = (2.000 ft.)(0.376)(7.000 ft.)(0.120 \frac{k}{ft^3}) = 0.632\frac{k}{ft}</math>
:::'''Check Shear'''


::<math>P_{A_{Stem}} = \frac{1}{2} \gamma_s C_A H^2 = \frac{1}{2}\Big[0.120 \frac{k}{ft^3}\Big](0.376)(7.000 ft.)^2 = 1.105\frac{k}{ft} </math>
:::The critical section for shear for the toe is at a distance d = 21.75 inches from the face of the stem. The toe pile is 6 inches from the stem face so the toe pile shear does not affect the shear at the critical section. The critical section for shear is at the stem face for the heel so all of the force of the heel pile affects the shear at the critical section. The worst case for shear is Case IV.


:::'''Apply Load Factors'''
:::''V<sub>u</sub>'' = 7.588k


:::''F<sub>Col.</sub>'' = ''γβ<sub>LL</sub>''(0.714k) = (1.3)(1.67)(0.714k) = 1.550k
:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = {7588 lbs}{0.85(12 in.)(32 in.)}</math> = 23.248 psi ≤ 109.5 psi = ''ν<sub>c</sub>'' <u>o.k.</u>


:::''P<sub>LL</sub>'' = ''γβ<sub>E</sub>'' (0.632k) = (1.3)(1.67)(0.632k) = 1.372k
::'''Reinforcement - Shear Key'''


:::''P<sub>A<sub>Stem</sub></sub>'' = ''γβ<sub>E</sub>'' (1.105k) = (1.3)(1.3)(1.105k) = 1.867k
::''b'' = 12 in.


::''M<sub>u</sub>'' = (10.00 ft.)(1.550k) + (3.500 ft.)(1.372k) + (2.333 ft.)(1.867k)
::''h'' = 12 in.


::''M<sub>u</sub>''  = 24.658(ft−k)
::cover = 3 in.


::<math>R_n = \frac{M_u}{\phi b d^2} = \frac{24.658(ft−-k)}{(0.9)(1 ft.)(13.688 in.)^2}</math> = 0.146ksi
::''d'' = 12 in. - 3 in. - 0.5(0.5 in.) = 8.75 in.


::<math>\rho = \frac{0.85f'_c}{f_y}\Bigg[1 -− \sqrt{1 -− \frac{2R_n}{0.85f'_c}}\Bigg] =
:::'''Apply Load Factors'''
\frac{0.85(3 ksi)}{60 ksi}\Bigg[1 -− \sqrt{1 −- \frac{2(0.146 ksi)}{0.85(3 ksi)}}\Bigg]</math> = 0.00251


::<math>\rho_{min} = 1.7\Big[\frac{h}{d}\Big]^2 \frac{\sqrt{f'_c}}{f_y} = 1.7\Big[\frac{16 in.}{13.688 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00212
:::''P<sub>P</sub> = γβ<sub>E</sub>'' (3.845k) = (1.3)(1.3)(3.845k) = 6.498k


::''ρ'' = 0.00251
::''M<sub>u</sub>'' = (0.912 ft.)(6.498k) = 5.926(ft−k)


::<math>A_{S_{Req.}} = \rho bd = (0.00251)(12 in.)(13.688 in.) = 0.412 \frac{in^2}{ft.}</math>
::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{5.926(ft-k)}{(0.9)(1 ft.)(8.75 in.)^2}</math> = 0.0860 ksi


::One #5 bar has A<sub>S</sub> = 0.307 in<sup>2</sup>
::<math>\rho = \frac{0.85(3ksi)}{60ksi}\Bigg[1 - \sqrt{1 - \frac{2(0.0860ksi)}{0.85(3ksi)}}\Bigg]</math> = 0.00146


::<math>\frac{s}{0.307 in^2} = \frac{12 in.}{0.412 in^2}</math>
::<math>\rho_{min} = 1.7\Big[\frac{12 in.}{8.75 in}\Big]^2\frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00292


::''s'' = 8.9 in.
::Use ''ρ'' = 4/3 ''ρ'' = 4/3(0.00146) = 0.00195


::<u>Use # 5 bars @ 8.5 in. cts.</u>
::''A<sub>S<sub>Req</sub></sub> = ρbd'' = (0.00195)(12 in.)(8.75 in.) = 0.205 in.<sup>2</sup>/ft.


:::'''Check Shear'''


:::''V<sub>u</sub>'' ≤ ''φV<sub>n</sub>''
::One #4 bar has A<sub>s</sub> = 0.196 in<sup>2</sup>


:::''V<sub>u</sub>'' = ''F<sub>Collision</sub>'' + ''P<sub>LL</sub>'' + ''P<sub>A<sub>Stem</sub></sub>'' = 1.550k + 1.372k + 1.867k = 4.789k


::<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.205 in.^2}</math>


:::<math>\frac{\nu_u}{\phi} = \frac{v_u}{\phi bd} = \frac{4789 lbs}{0.85(12 in.)(13.688 in.)}</math> = 34.301 psi
::''s'' = 11.5 in.


:::<math> \nu_n = \nu_c = 2\sqrt{f'_c} = 2\sqrt{3000psi}</math> = 109.5 psi > 34.3 psi <u>o.k.</u>
::<u>Use #4 bars @ 11 in. cts.</u>


::'''Reinforcement - Footing - Top Steel'''
:::'''Check Shear'''


[[image:751.24.3.5 footing.jpg|300px|center]]
:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = \frac{6498 lbs}{0.85(12 in.)(8.75 in.)}</math> = 72.807 psi < 109.5 psi = ''ν<Sub>c</sub>''


::b = 12 in.
::'''Reinforcement Summary'''


::cover = 3 in.
[[image:751.24.3.5 summary.jpg|center|350px]]


::h = 36 in.
===751.24.3.6 Dimensions===
'''Cantilever Walls'''
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 friction or bearing piles.jpg|center|800px]]


::d = 36 in. - 3 in. - 0.5(0.5 in.) = 32.750 in.


::Design the heel to support the entire weight of the superimposed materials.
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 pile footing.jpg|center|800px]]


::Soil(1) = 4.340k/ft.


::LL<sub>s</sub> = 1.240k/ft.
'''Cantilever Walls - L-Shaped'''


::<math>Slab \ wt. = (3.000 ft.)\Big[0.150 \frac{k}{ft^3}\Big](5.167 ft.)</math> = 2.325k/ft.
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 L shaped.jpg|center|800px]]


:::'''Apply Load Factors'''


:::Soil(1) = ''γβ<sub>E</sub>''(4.340k) = (1.3)(1.0)(4.340k) = 5.642k
'''Counterfort Walls'''
[[image:751.24.3.6 counterfort part elev.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Dimension "A"'''
|-
|• Maximum length = 28'-0".
|-
|• Each section to be in 4'-0" increments.
|-
|• (See [[#Rustication Recess|rustication recess details]].)
|-
|'''Dimensions "B" & "C"'''
|-
|• As required by the design to balance the negative and positive moments. (See the design assumptions).
|}]]


:::''LL<sub>s</sub>'' = ''γβ<sub>E</sub>''(1.240k) = (1.3)(1.67)(1.240k) = 2.692k
[[image:751.24.3.6 counterfort typ section.jpg|center|800px|thumb|
 
{| style="margin: 1em auto 1em auto style="text-align:left""
:::Slab wt. = ''γβ<sub>D</sub>''(2.325k) = (1.3)(1.0)(2.325k) = 3.023k
|-
 
|'''Notes:'''
::''M<sub>u</sub>'' = (2.583 ft.)(5.642k + 2.692k + 3.023k) = 29.335(ft−k)
|-
|'''Batter  "D":'''
|-
|* As required to maintain 9" minimum at the top of the counterfort and 12" minimum edge distance at the top of the footing, between counterfort and footing edge.
|-
|* Batter to be given an eighth of an inch per foot of counterfort height.
|-
|'''Dimension "L":'''
|-
|* As required for stability.
|-
|* As an estimate, use "L" equal to 1/2 the height of "H".
|}]]


::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{29.335(ft-−k)}{(0.9)(1 ft.)(32.750 in.)^2}</math> = 0.0304 ksi
'''Sign-Board Type Counterfort Walls'''
[[image:751.24.3.6 sign board part elev.jpg|center|800px|thumb|
::<math>\rho = \frac{0.85(3ksi)}{60ksi}\Bigg[1 -− \sqrt{1 -− \frac{2(0.0304ksi)}{0.85(3ksi)}}\Bigg]</math> = 0.000510
{| style="margin: 1em auto 1em auto style="text-align:left""
 
|-
::<math>\rho_{min} = 1.7\Big[\frac{36 in.}{32.750 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000psi}</math> = 0.00188
|'''Notes:'''
 
|-
::Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.000510) = 0.000680
|'''Dimension "A"'''
 
|-
::<math>A_{S_{Req}} = \rho bd = (0.000680)(12 in.)(32.750 in.) = 0.267\frac{in^2}{ft.}</math>
|* Maximum length = 28'-0".
 
|-
::One #4 bar has A<sub>s</sub> = 0.196 in.<sup>2</sup>
|* Each section to be in 4'-0" increments.
 
|-
::<math>\frac{s}{0.196 in^2} = \frac{12 in}{0.267 in.^2}</math>
|* (See [[#Rustication Recess|rustication recess details]].)
|-
|'''Dimensions "B" & "C"'''
|-
|* As required by the design to balance the negative and positive moments. (See the design assumptions).
|-
|'''Dimension "E"'''
|-
|* (Sign-board type only)
|-
|* As required to maintain footing pressure within the allowable for existing foundation material. 12" minimum.


::''s'' = 8.8 in.
|}]]


::<u>Use #4 bars @ 8.5 in. cts.</u>
[[image:751.24.3.6 sign board typ section.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Batter  "D":'''
|-
|* As required to maintain 9" minimum at the top of the counterfort and 12" minimum edge distance at the top of the footing, between counterfort and footing edge.
|-
|* Batter to be given an eighth of an inch per foot of counterfort height.
|-
|'''Dimension "L":'''
|-
|* As required for stability.
|-
|* As an estimate, use "L" equal to 1/2 the height of "H".
|}]]


:::'''Check Shear'''
===751.24.3.7 Reinforcement===
'''Cantilever Walls'''
[[image:751.24.3.7 friction.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Alternate long and short bars at equal spaces.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.8.1 Development and Lap Splice General|EPG 751.5.9.2.8.1 Development and Lap Splice General]].)
|-
|'''(***)''' Theo. cut-off for bending + development length.  (Wall height over 10' only.)
|}
]]


:::<math>V_u = Soil(1) + LL_s + Slab \ wt. = 5.642k + 2.692k + 3.023k = 11.357k</math>
[[image:751.24.3.7 pile footing.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Alternate long and short bars at equal spaces.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.8.1 Development and Lap Splice General|EPG 751.5.9.2.8.1 Development and Lap Splice General]].)
|-
|'''(***)''' Theo. cut-off for bending + development length.  (Wall height over 10' only.)
|-
|'''(****)''' Due to site constriction.
|}
]]


:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = \frac{11357 lbs}{(0.85)(12 in.)(32.750 in.)}</math> = 33.998 psi ≤ 109.5 psi = ''ν<sub>c</sub>'' <u>o.k.</u>
'''Cantilever Walls - L-Shaped'''  


::'''Reinforcement - Footing - Bottom Steel'''
[[image:751.24.3.7 L shaped.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Do not splice stress bars in the fill face at top of footing.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.8.1 Development and Lap Splice General|EPG 751.5.9.2.8.1 Development and Lap Splice General]].)
|}
]]


::Design the flexural steel in the bottom of the footing to resist the largest moment that the heel pile could exert on the footing. The largest heel pile bearing force was in Case IV. The heel pile will cause a larger moment about the stem face than the toe pile (even though there are two toe piles for every one heel pile) because it has a much longer moment arm about the stem face.
'''Counterfort Walls'''
:'''Wall and Stem'''
[[image:751.24.3.7 counterfort.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|<center>(For footing reinforcement, see the "Footing" diagram, below)</center>
|-
|'''(*)''' Use development length or standard hook in accordance with [[751.5 Structural Detailing Guidelines#751.5.9.2.8.1 Development and Lap Splice General|EPG 751.5.9.2.8.1 Development and Lap Splice General]].
|-
|'''(**)''' See lap splices Class B.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.8.1 Development and Lap Splice General|EPG 751.5.9.2.8.1 Development and Lap Splice General]].)
|}
]]


[[image: 751.24.3.5 heel pile.jpg|center|300px]]
:'''Footing'''
[[image:751.24.3.7 footing.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' By design for loads and footing pressures on section under consideration.  (#5 @ 12" cts. is the minimum.)
|}
]]


::Pile is embedded into footing 12 inches.
'''Counterfort Walls - Sign-Board Type'''
:'''Wall and Stem'''
:Refer to "Counterfort Walls, Wall and Stem", above.


::''b'' = 12 in.
:'''Spread Footing'''
[[image:751.24.3.7 sign board.jpg|center|800px]]


::''h'' = 36 in.
:If the shear line is within the counterfort projected (longitudinally or transversely), the footing may be considered satisfactory for all conditions.  If outside of the counterfort projected, the footing must be analyzed and reinforced for bending and checked for bond stress and for diagonal tension stress.


::''d'' = 36 in. - 4 in. = 32 in.
[[image:751.24.3.7 sign board footing.jpg|center|800px]]


:::'''Apply Load Factors to Case IV Loads'''
===751.24.3.8 Details===
'''Non-Keyed Joints'''


:::<math>\Sigma V = \gamma \beta_D\Big[5.828 \frac{k}{ft.}\Big] + \gamma \beta_E \Big[4.820 \frac{k}{ft.}\Big]</math>
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.8 nonkeyed.jpg|center|800px]]
<center>See [[751.50 Standard Detailing Notes|EPG 751.50 Standard Detailing Notes]] for appropriate notes.</center>


:::<math>\Sigma V = 1.3(1.0)\Big[5.828\frac{k}{ft.}\Big] + 1.3(1.0)\Big[4.820\frac{k}{ft.}\Big]</math>
'''Keyed Joints'''
[[image:751.24.3.8 keyed.jpg|center|800px]]
<center>See [[751.50 Standard Detailing Notes|EPG 751.50 Standard Detailing Notes]] for appropriate notes.</center>


:::''ΣV'' = 13.842 k/ft.


:::<math>\Sigma M = \gamma \beta_D\Big[13.174\frac{(ft-−k)}{ft.}\Big] + \gamma \beta_E\Big[18.930\frac{(ft-−k)}{ft.}\Big]</math>
<div id="Rustication Recess"></div>
'''Rustication Recess'''
[[image:751.24.3.8 rustication.jpg|center|800px]]


:::<math>\Sigma M = (1.3)(1.0)\Big[13.174\frac{(ft-−k)}{ft.}\Big] + (1.3)(1.0)\Big[18.930\frac{(ft-−k)}{ft.}\Big]</math>


:::''ΣM'' = 41.735 (ft−k)/ft.
'''Drains'''
[[image:751.24.3.8 drains.jpg|center|800px]]
<center>Note: French drains shall be used on all retaining walls, unless otherwise specified on the Design Layout.</center>


::e = <math>\frac{\Sigma M}{\Sigma V} = \frac{41.735 (ft-−k)}{13.842k}</math> = 3.015 ft.
[[image:751.24.3.8 drop inlet.jpg|center|800px]]


::Moment arm = 3.015 ft. - 1.833 ft. = 1.182 ft.


::<math>P_H = \frac{\Sigma V}{A} + \frac{M_c}{I} = \frac{13.842k}{0.250A} + \frac{13.842k (1.182 ft.)(3.667 ft.)}{1.681(A)ft^2}</math>
'''Construction Joint Keys:
:'''Cantilever Walls'''
[[image:751.24.3.8 cantilever.jpg|center|800px]]


::<math>P_H = 91.059 \frac{k}{pile}\Big(\frac{1}{12 ft.}\Big)</math> = 7.588 k/ft.


::<math>M_u = \Big(7.588\frac{k}{ft.}\Big)(3.667 ft.)</math> = 27.825(ft−k)/ft.
:'''Counterfort Walls'''
[[image:751.24.3.8 counterfort.jpg|center|800px]]


::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{27.825(ft−-k)}{(0.9)(1 ft.)(32 in.)^2}</math> = 0.0301 ksi


::<math>\rho = \frac{0.85(3 ksi)}{60ksi}\Bigg[1 -− \sqrt{1 -− \frac{2(0.0301 ksi)}{0.85(3 ksi)}}\Bigg]</math> = 0.000505
::Key length:  Divide the length "A" into an odd number of spaces of equal lengths. Each space shall not exceed a length of 24 inches. Use as few spaces as possible with the minimum number of spaces equal to three (or one key).


::<math>\rho_{min} = 1.7\Big[\frac{36 in.}{32 in.}\Big]^2 \frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00196
::Key width = Counterfort width/3 (to the nearest inch)


::Use ''ρ'' = 4/3 ''ρ'' = 4/3 (0.000505) = 0.000673
::Key depth = 2" (nominal)


::''A<sub>S<sub>Req</sub></sub> = ρbd'' = (0.000673)(12 in.)(32 in.) = 0.258 in<sup>2</sup>/ft.''
:'''Sign-Board Walls'''
[[image:751.24.3.8 sign board.jpg|center|800px]]


::Key length = divide length "A" or "B" into an odd number of spaces of equal lengths.  Each space length shall not exceed 24 inches.  Use as few spaces as possible with the minimum number of spaces equal to three (or one key).


::One #4 bar has A<sub>s</sub> = 0.196 in<sup>2</sup>.
::<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.258 in.^2}</math>
::''s'' = 9.1 in.
::<u>Use #4 bars @ 9 in. cts.</u>
:::'''Check Shear'''
:::The critical section for shear for the toe is at a distance d = 21.75 inches from the face of the stem. The toe pile is 6 inches from the stem face so the toe pile shear does not affect the shear at the critical section. The critical section for shear is at the stem face for the heel so all of the force of the heel pile affects the shear at the critical section. The worst case for shear is Case IV.
:::''V<sub>u</sub>'' = 7.588k
:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = {7588 lbs}{0.85(12 in.)(32 in.)}</math> = 23.248 psi ≤ 109.5 psi = ''ν<sub>c</sub>'' <u>o.k.</u>
::'''Reinforcement - Shear Key'''
::''b'' = 12 in.
::''h'' = 12 in.
::cover = 3 in.
::''d'' = 12 in. - 3 in. - 0.5(0.5 in.) = 8.75 in.
:::'''Apply Load Factors'''
:::''P<sub>P</sub> = γβ<sub>E</sub>'' (3.845k) = (1.3)(1.3)(3.845k) = 6.498k
::''M<sub>u</sub>'' = (0.912 ft.)(6.498k) = 5.926(ft−k)
::<math>R_n = \frac{M_u}{\phi bd^2} = \frac{5.926(ft-−k)}{(0.9)(1 ft.)(8.75 in.)^2}</math> = 0.0860 ksi
::<math>\rho = \frac{0.85(3ksi)}{60ksi}\Bigg[1 -− \sqrt{1 -− \frac{2(0.0860ksi)}{0.85(3ksi)}}\Bigg]</math> = 0.00146
::<math>\rho_{min} = 1.7\Big[\frac{12 in.}{8.75 in}\Big]^2\frac{\sqrt{3000 psi}}{60,000 psi}</math> = 0.00292
::Use ''ρ'' = 4/3 ''ρ'' = 4/3(0.00146) = 0.00195
::''A<sub>S<sub>Req</sub></sub> = ρbd'' = (0.00195)(12 in.)(8.75 in.) = 0.205 in.<sup>2</sup>/ft.
::One #4 bar has A<sub>s</sub> = 0.196 in<sup>2</sup>
::<math>\frac{s}{0.196 in.^2} = \frac{12 in.}{0.205 in.^2}</math>
::''s'' = 11.5 in.
::<u>Use #4 bars @ 11 in. cts.</u>
:::'''Check Shear'''
:::<math>\frac{\nu_u}{\phi} = \frac{V_u}{\phi bd} = \frac{6498 lbs}{0.85(12 in.)(8.75 in.)}</math> = 72.807 psi < 109.5 psi = ''ν<Sub>c</sub>''
::'''Reinforcement Summary'''
[[image:751.24.3.5 summary.jpg|center|350px]]
===751.24.3.6 Dimensions===
'''Cantilever Walls'''
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 friction or bearing piles.jpg|center|800px]]
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 pile footing.jpg|center|800px]]
'''Cantilever Walls - L-Shaped'''
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.6 L shaped.jpg|center|800px]]
'''Counterfort Walls'''
[[image:751.24.3.6 counterfort part elev.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Dimension "A"'''
|-
|* Maximum length = 28'-0".
|-
|* Each section to be in 4'-0" increments.
|-
|* (See [[#Rustication Recess|rustication recess details]].)
|-
|'''Dimensions "B" & "C"'''
|-
|* As required by the design to balance the negative and positive moments. (See the design assumptions).
|}]]
[[image:751.24.3.6 counterfort typ section.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Batter  "D":'''
|-
|* As required to maintain 9" minimum at the top of the counterfort and 12" minimum edge distance at the top of the footing, between counterfort and footing edge.
|-
|* Batter to be given an eighth of an inch per foot of counterfort height.
|-
|'''Dimension "L":'''
|-
|* As required for stability.
|-
|* As an estimate, use "L" equal to 1/2 the height of "H".
|}]]
'''Sign-Board Type Counterfort Walls'''
[[image:751.24.3.6 sign board part elev.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Dimension "A"'''
|-
|* Maximum length = 28'-0".
|-
|* Each section to be in 4'-0" increments.
|-
|* (See [[#Rustication Recess|rustication recess details]].)
|-
|'''Dimensions "B" & "C"'''
|-
|* As required by the design to balance the negative and positive moments. (See the design assumptions).
|-
|'''Dimension "E"'''
|-
|* (Sign-board type only)
|-
|* As required to maintain footing pressure within the allowable for existing foundation material.  12" minimum.
|}]]
[[image:751.24.3.6 sign board typ section.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto style="text-align:left""
|-
|'''Notes:'''
|-
|'''Batter  "D":'''
|-
|* As required to maintain 9" minimum at the top of the counterfort and 12" minimum edge distance at the top of the footing, between counterfort and footing edge.
|-
|* Batter to be given an eighth of an inch per foot of counterfort height.
|-
|'''Dimension "L":'''
|-
|* As required for stability.
|-
|* As an estimate, use "L" equal to 1/2 the height of "H".
|}]]
===751.24.3.7 Reinforcement===
'''Cantilever Walls'''
[[image:751.24.3.7 friction.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Alternate long and short bars at equal spaces.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.7.1 Development and Lap Splice General|EPG 751.5.9.2.7.1 Development and Lap Splice General]].)
|-
|'''(***)''' Theo. cut-off for bending + development length.  (Wall height over 10' only.)
|}
]]
[[image:751.24.3.7 pile footing.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Alternate long and short bars at equal spaces.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.7.1 Development and Lap Splice General|EPG 751.5.9.2.7.1 Development and Lap Splice General]].)
|-
|'''(***)''' Theo. cut-off for bending + development length.  (Wall height over 10' only.)
|-
|'''(****)''' Due to site constriction.
|}
]]
'''Cantilever Walls - L-Shaped'''
[[image:751.24.3.7 L shaped.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' Do not splice stress bars in the fill face at top of footing.
|-
|'''(**)''' If collision forces are assumed, use #4 @ 12" cts. min. and extend at least development length into footing.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.7.1 Development and Lap Splice General|EPG 751.5.9.2.7.1 Development and Lap Splice General]].)
|}
]]
'''Counterfort Walls'''
:'''Wall and Stem'''
[[image:751.24.3.7 counterfort.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|<center>(For footing reinforcement, see the "Footing" diagram, below)</center>
|-
|'''(*)''' Use development length or standard hook in accordance with [[751.5 Structural Detailing Guidelines#751.5.9.2.7.1 Development and Lap Splice General|EPG 751.5.9.2.7.1 Development and Lap Splice General]].
|-
|'''(**)''' See lap splices Class B.  (See [[751.5 Structural Detailing Guidelines#751.5.9.2.7.1 Development and Lap Splice General|EPG 751.5.9.2.7.1 Development and Lap Splice General]].)
|}
]]
:'''Footing'''
[[image:751.24.3.7 footing.jpg|center|800px|thumb|
{| style="margin: 1em auto 1em auto"
|-
|'''(*)''' By design for loads and footing pressures on section under consideration.  (#5 @ 12" cts. is the minimum.)
|}
]]
'''Counterfort Walls - Sign-Board Type'''
:'''Wall and Stem'''
:Refer to "Counterfort Walls, Wall and Stem", above.
:'''Spread Footing'''
[[image:751.24.3.7 sign board.jpg|center|800px]]
:If the shear line is within the counterfort projected (longitudinally or transversely), the footing may be considered satisfactory for all conditions.  If outside of the counterfort projected, the footing must be analyzed and reinforced for bending and checked for bond stress and for diagonal tension stress.
[[image:751.24.3.7 sign board footing.jpg|center|800px]]
===751.24.3.8 Details===
'''Non-Keyed Joints'''
Each section of wall shall be in increments of 4 ft. with a maximum length of 28'-0".
[[image:751.24.3.8 nonkeyed.jpg|center|800px]]
<center>See [[751.50 Standard Detailing Notes|EPG 751.50 Standard Detailing Notes]] for appropriate notes.</center>
'''Keyed Joints'''
[[image:751.24.3.8 keyed.jpg|center|800px]]
<center>See [[751.50 Standard Detailing Notes|EPG 751.50 Standard Detailing Notes]] for appropriate notes.</center>
<div id="level of service (LOS)"></div>
'''Rustication Recess'''
[[image:751.24.3.8 rustication.jpg|center|800px]]
'''Drains'''
[[image:751.24.3.8 drains.jpg|center|800px]]
<center>Note: French drains shall be used on all retaining walls, unless otherwise specified on the Design Layout.</center>
[[image:751.24.3.8 drop inlet.jpg|center|800px]]
'''Construction Joint Keys:
:'''Cantilever Walls'''
[[image:751.24.3.8 cantilever.jpg|center|800px]]
:'''Counterfort Walls'''
[[image:751.24.3.8 counterfort.jpg|center|800px]]
::Key length:  Divide the length "A" into an odd number of spaces of equal lengths.  Each space shall not exceed a length of 24 inches.  Use as few spaces as possible with the minimum number of spaces equal to three (or one key).
::Key width = Counterfort width/3 (to the nearest inch)
::Key depth = 2" (nominal)
:'''Sign-Board Walls'''
[[image:751.24.3.8 sign board.jpg|center|800px]]
::Key length = divide length "A" or "B" into an odd number of spaces of equal lengths.  Each space length shall not exceed 24 inches.  Use as few spaces as possible with the minimum number of spaces equal to three (or one key).




-->
[[Category:751 LRFD Bridge Design Guidelines]]
[[Category:751 LRFD Bridge Design Guidelines]]

Latest revision as of 15:13, 27 November 2024

751.24.1 General

Additional Information
LRFD 11

For understanding the equivalency of seismic design category (SDC) and seismic zone for LRFD, see EPG 751.9.1.1 and Bridge Seismic Design Flowchart.

Retaining wall shall be designed to withstand lateral earth and water pressures, including any live and dead load surcharge, the self weight of the wall, temperature and shrinkage effect, live load and collision forces, and earthquake loads in accordance with the general principles of LRFD Section 11 and the general principles specified in this article.

Seismic analysis provisions shall not be ignored for walls that support another structure (i.e. support abutment fill or building) in SDC B or C (seismic zone 2 or 3). No-seismic-analysis provisions may be considered for walls that do not support another structure (i.e. most of District walls) in SDC B or C (seismic zone 2 or 3) in accordance with LRFD 11.5.4.2 and Geotech report. Seismic analysis provisions shall not be ignored for walls in SDC D (seismic zone 4).

751.24.1.1 Wall Type Selection

Additional Information
LRFD 11

Selection of wall type shall be based on an assessment of the magnitude and direction of loading, depth to suitable foundation support, potential for earthquake loading, presence of deleterious environmental factors, wall site cross-sectional geometry, proximity of physical constraints, tolerable and differential settlement, facing appearance and ease and cost of construction.

The following wall types are the most commonly used in MoDOT projects

  • Mechanically Stabilized Earth Retaining Walls
  • Cast-In-Place Concrete Cantilever Retaining Walls
▪ Cantilever Walls on Spread Footings
▪ Cantilever Wall on Pile Footings
▪ L-Shaped Walls on Spread Footings

Mechanically Stabilized Earth (MSE) Retaining Walls

Additional Information
LRFD 11.10,
FHWA-NHI-10-024 and 025

MSE retaining walls use precast block or panel like facing elements combined with either metallic or geosynthetic tensile reinforcements in the soil mass. MSE walls are preferred over cast-in-place walls because they are usually more economical. Other advantages include a wide variety of design styles, ease and speed of installation, and their ability to accommodate total and differential settlements. Wall design heights upwards of 80 ft. are technically feasible (FHFW-SA-96-071). MSE walls may be used to retain fill for end bents of bridge structures.

Situations exist where the use of MSE walls is either limited or not recommended. Some obstacles such as drop inlets, sign truss pedestals or footings, and fence posts may be placed within the soil reinforcement area, however, these obstacles increase the difficulty and expense of providing sufficient soil reinforcement for stability. Box culverts and highway drainage pipes may run through MSE walls, but it is preferable not to run the pipes close to or parallel to the walls. Utilities other than highway drainage should not be constructed within the soil reinforcement area. Be cautious when using MSE walls in a floodplain. A flood could cause scouring around the reinforcement and seepage of the backfill material. Soil reinforcements should not be used where exposure to ground water contaminated by acid mine drainage or other industrial pollutants as indicated by a low pH and high chlorides and sulfates exist. Galvanized metallic reinforcements shall not be used where stray electrical ground currents could occur as would be present near an electrical substation.

Sufficient right of way is required to install the soil reinforcement which extends into the backfill area at least 8 feet, 70 percent of the wall height or as per design requirements set forth in EPG 751.6.2.17 Excavation, whichever is greater. For more information regarding soil reinforcement length, excavation limits and Minimum Embedment Depth of MSEW, see EPG 751.6.2.17 Excavation.

Finally, barrier curbs constructed over or in line with the front face of the wall shall have adequate room provided laterally between the back of the wall facing and the curb or slab so that load is not directly transmitted to the top of MSE wall or facing units.



Concrete Cantilever Wall on Spread Footing

Concrete cantilever walls derive their capacity through combinations of dead weight and structural resistance. These walls are constructed of reinforced concrete.

Concrete cantilever walls are used when MSE walls are not a viable option. Cantilever walls can reduce the rock cut required and can also provide solutions when there are right of way restrictions. Concrete walls also provide better structural capacity when barrier or railing on top of the walls are required.

Counterforts are used on rare occasions. Sign-board type retaining walls are a special case of counterfort retaining walls. They are used where the soil conditions are such that the footings must be placed well below the finished ground line. For these situations the wall is discontinued 12 in. below the ground line or below the frost line. Counterforts may also be a cost-savings option when the wall height approaches 20 ft. (Foundation Analysis and Design by Joseph E. Bowles, 4th ed., 1988). However, other factors such as poor soil conditions, slope of the retained soil, wall length and uniformity in wall height should also be considered before using counterforts.

Concrete Cantilever Wall on Pile Footing

Concrete cantilever walls on pile footings are used when the soil conditions do not permit the use of spread footings. These walls are also used when an end bent requires wings longer than 22 feet for seismic category A or 17 ft. for seismic category B, C or D. In these cases a stub wing is left attached to the end bent and the rest of the wing is detached to become a retaining wall as shown in 751.35.3.5 Wing and Detached Wing Walls.

Concrete L-Shaped Retaining Wall on Spread Footings

Concrete L-Shaped walls are cantilever walls without heels. These walls are used when there are space limitations for cantilever walls. Since there is no heel the height of these walls is limited to about 7 ft. depending on the soil conditions and the slope of the retained soil.

L-Shaped Walls are often used next to roadways where the footings are frequently used as shoulders and where the wall will require structural capacity for collision forces.

751.24.1.2 Loads

Conventional retaining walls: Loads shall be determined in accordance with LRFD 3 and 11.6.

MSE retaining walls: Loads shall be determined in accordance with LRFD 3 and 11.10.

Note: For guidance, follow the 751.40.8.15 Cast -In-Place Concrete Retaining Walls and modify guidance of ASD as necessary to meet LRFD requirements until this section is modified for LRFD.

Dead Loads

Dead loads shall be determined from the unit weights in EPG 751.2.1.1 Dead Load.

751.24.2 Mechanically Stabilized Earth (MSE) Walls

751.24.2.1 Design

Designs of Mechanically Stabilized Earth (MSE) walls shall be completed by consultants or contractors in accordance with Section 11.10 of LRFD specifications, FHWA-NHI-10-024 and FHWA-NHI-10-025 for LRFD. Bridge Pre-qualified Products List (BPPL) provided on MoDOT's web page and in Sharepoint contains a listing of facing unit manufacturers, soil reinforcement suppliers, and wall system suppliers which have been approved for use. See Sec 720 and Sec 1010 for additional information. The Geotechnical Section is responsible for checking global stability of permanent MSE wall systems, which should be reported in the Foundation Investigation Geotechnical Report. For MSE wall preliminary information, see EPG 751.1.4.3 MSE Walls. For design requirements of MSE wall systems and temporary shoring (including temporary MSE walls), see EPG 720 Mechanically Stabilized Earth Wall Systems. For staged bridge construction, see EPG 751.1.2.11 Staged Construction.

For seismic design requirements, see Bridge Seismic Design Flowchart. References for consultants and contractors include Section 11.10 of LRFD, FHWA-NHI-10-024 and FHWA-NHI-10-025.

Design Life

  • 75 year minimum for permanent walls (if retained foundation require 100 year than consider 100 year minimum design life for wall).

Global stability:

Global stability will be performed by Geotechnical Section or their agent.

MSE wall contractor/designer responsibility:

MSE wall contractor/designer shall perform following analysis in their design for all applicable limit states.

  • External Stability
  • Limiting Eccentricity
  • Sliding
  • Factored Bearing Pressure/Stress ≤ Factored Bearing Resistance
  • Internal Stability
  • Tensile Resistance of Reinforcement
  • Pullout Resistance of Reinforcement
  • Structural Resistance of Face Elements
  • Structural Resistance of Face Element Connections
  • Compound Stability
Capacity/Demand ratio (CDR) for bearing capacity shall be ≥ 1.0
Strength Limit States:
Factored bearing resistance = Nominal bearing resistance from Geotech report X Minimum Resistance factor (0.65, Geotech report) LRFD Table 11.5.7-1
Extreme Event I Limit State:
Factored bearing resistance = Nominal bearing resistance from Geotech report X Resistance factor
Resistance factor = 0.9 LRFD 11.8.6.1
Factored bearing stress shall be computed using a uniform base pressure distribution over an effective width of footing determined in accordance with the provisions of LRFD 10.6.3.1 and 10.6.3.2, 11.10.5.4 and Figure 11.6.3.2-1 for foundation supported on soil or rock.
B’ = L – 2e
Where,
L = Soil reinforcement length (For modular block use B in lieu of L as per LRFD 11.10.2-1)
B’ = effective width of footing
e = eccentricity
Note: When the value of eccentricity e is negative then B´ = L.
Capacity/Demand ratio (CDR) for overturning shall be ≥ 1.0
Capacity/Demand ratio (CDR) for eccentricity shall be ≥ 1.0
Capacity/Demand ratio (CDR) for sliding shall be ≥ 1.0      LRFD 11.10.5.3 & 10.6.3.4
Capacity/Demand ratio (CDR) for internal stability shall be ≥ 1.0
Eccentricity, (e) Limit for Strength Limit State:      LRFD 11.6.3.3 & C11.10.5.4
For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, L or (e ≤ 0.33L).
Eccentricity, (e) Limit for Extreme Event I (Seismic):      LRFD 11.6.5.1
For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, L or (e ≤ 0.33L) for γEQ = 0.0 and middle eight-tenths of the base width, L or (e ≤ 0.40L) for γEQ = 1.0. For γEQ between 0.0 and 1.0, interpolate e value linearly between 0.33L and 0.40L. For γEQ refer to LRFD 3.4.
Note: Seismic design shall be performed for γEQ = 0.5
Eccentricity, (e) Limit for Extreme Event II:
For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle eight-tenths of the base width, L or (e ≤ 0.40L).

General Guidelines

  • Drycast modular block wall (DMBW-MSE) systems are limited to a 10 ft. height in one lift.
  • Wetcast modular block wall (WMBW-MSE) systems are limited to a 15 ft. height in one lift.
  • For Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems, top cap units shall be used and shall be permanently attached by means of a resin anchor system.
  • For precast modular panel wall (PMPW-MSE) systems, capstone may be substituted for coping and either shall be permanently attached to wall by panel dowels.
  • For precast modular panel wall (PMPW-MSE) systems, form liners are required to produce all panels. Using form liner to produce panel facing is more cost effective than producing flat panels. Standard form liners are specified on the MSE Wall Standard Drawings. Be specific regarding names, types and colors of staining, and names and types of form liner.
  • MSE walls shall not be used where exposure to acid water may occur such as in areas of coal mining.
  • MSE walls shall not be used where scour is a problem.
  • MSE walls with metallic soil reinforcement shall not be used where stray electrical ground currents may occur as would be present near electrical substations.
  • No utilities shall be allowed in the reinforced earth if future access to the utilities would require that the reinforcement layers be cut, or if there is a potential for material, which can cause degradation of the soil reinforcement, to leak out of the utilities into the wall backfill, with the exception of storm water drainage.
  • All vertical objects shall have at least 4’-6” clear space between back of the wall facing and object for select granular backfill compaction and soil reinforcement skew limit requirements. For piles, see pipe pile spacers guidance.
  • The interior angle between two MSE walls should be greater than 70°. However, if unavoidable, then place EPG 751.50 J1.41 note on the design plans.
  • Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems may be battered up to 1.5 in. per foot. Modular blocks are also known as “segmental blocks”.
  • The friction angle used for the computation of horizontal forces within the reinforced soil shall be greater than or equal to 34°.
  • All concrete except facing panels or units shall be CLASS B or B-1.
  • The friction angle of the soil to be retained by the reinforced earth shall be listed on the plans as well as the friction angle for the foundation material the wall is to rest on.
  • The following requirement shall be considered (from 2009_FHWA-NHI-10-024 MSE wall 132042.pdf, page 200-201) when seismic design is required:
  • For seismic design category, SDC C or D (Zones 3 or 4), facing connections in modular block faced walls (MBW) shall use shear resisting devices (shear keys, pin, etc.) between the MBW units and soil reinforcement, and shall not be fully dependent on frictional resistance between the soil reinforcement and facing blocks. For connections partially dependent on friction between the facing blocks and the soil reinforcement, the nominal long-term connection strength Tac, should be reduced to 80 percent of its static value.
  • Seismic design category and acceleration coefficients shall be listed on the plans for categories B, C and D. If a seismic analysis is required that shall also be noted on the plans. See EPG 751.50 A1.1 note.
  • Plans note (EPG 751.50 J1.1) is required to clearly identify the responsibilities of the wall designer.
  • Do not use Drycast modular block wall (DMBW-MSE) systems in the following locations:
  • Within the splash zone from snow removal operations (assumed to be 15 feet from the edge of the shoulder).
  • Where the blocks will be continuously wetted, such as around sources of water.
  • Where blocks will be located behind barrier or other obstacles that will trap salt-laden snow from removal operations.
  • Do not use Drycast modular block wall (DMBW-MSE) systems or Wetcast modular block wall (WMBW-MSE) systems in the following locations:
  • For structurally critical applications, such as containing necessary fill around structures.
  • In tiered wall systems.
  • For locations where Drycast modular block wall (DMBW-MSE) systems and Wetcast modular block wall (WMBW-MSE) systems are not desirable, consider coloring agents and/or architectural forms using precast modular panel wall (PMPW-MSE) systems for aesthetic installations.
  • Roadway runoff should be directed away from running along face of MSE walls used as wing walls on bridge structures.
  • Drainage:
  • Gutter type should be selected at the core team meeting.
  • When gutter is required without fencing, use Type A or Type B gutter (for detail, see Std. Plan 609.00).
  • When gutter is required with fencing, use Modified Type A or Modified Type B gutter (for detail, see Std. Plan 607.11).
  • When fencing is required without gutter, place in tube and grout behind the MSE wall (for detail, see MSE Wall Standard Drawings - MSEW, Fence Post Connection Behind MSE Wall (without gutter).
  • Lower backfill longitudinal drainage pipes behind all MSE walls shall be two-6” (Min.) diameter perforated PVC or PE pipe (See Sec 1013) unless larger sizes are required by design which shall be the responsibility of the District Design Division. Show drainage pipe size on plans. Outlet screens and cleanouts should be detailed for any drain pipe (shown on MoDOT MSE wall plans or roadway plans). Lateral non-perforated drain pipes (below leveling pad) are permitted by Standard Specifications and shall be sized by the District Design Division if necessary. Lateral outlet drain pipe sloped at 2% minimum.
  • Identify on MSE wall plans or roadway plans drainage pipe point of entry, point of outlet (daylighting), 2% min. drainage slopes in between points to ensure positive flow and additional longitudinal drainage pipes if required to accommodate ground slope changes and lateral drainage pipes if required by design.
  • Adjustment in the vertical alignment of the longitudinal drainage pipes from that depicted on the MSE wall standard drawings may be necessary to ensure positive flow out of the drainage system.
  • Identify on MSE wall plans or roadway plans the outlet ends of pipes which shall be located to prevent clogging or backflow into the drainage system. Outlet screens and cleanouts should be detailed for any drain pipe.

MSE Wall Construction: Pipe Pile Spacers Guidance

For bridges not longer than 200 feet, pipe pile spacers or pile jackets shall be used at pile locations behind mechanically stabilized earth walls at end bents. Corrugated pipe pile spacers are required when the wall is built prior to driving the piles to protect the wall reinforcement when driving pile for the bridge substructure at end bents(s). Pile spacers or pile jackets may be used when the piles are driven before the wall is built. Pipe pile spacers shall have an inside diameter greater than that of the pile and large enough to avoid damage to the pipe when driving the pile. Use EPG 751.50 Standard Detailing Note E1.2a on bridge plans.

For bridges longer than 200 feet, pipe pile spacers are required and the pile spacer shall be oversized to mitigate the effects of bridge thermal movements on the MSE wall. For HP12, HP14, CIP 14” and CIP 16” piles provide 24-inch inside diameter of pile spacer for bridge movement. Minimum pile spacing shall be 5 feet to allow room for compaction of the soil layers. Use EPG 751.50 Standard Detailing Note E1.2b on bridge plans.

The bottom of the pipe pile spacers shall be placed 5 ft. min. below the bottom of the MSE wall leveling pad. The pipe shall be filled with sand or other approved material after the pile is placed and before driving. Pipe pile spacers shall be accurately located and capped for future pile construction.

Alternatively, for bridges shorter than or equal to 200 feet, the contractor shall be given the option of driving the piles before construction of the mechanically stabilized earth wall and placing the soil reinforcement and backfill material around the piling. In lieu of pipe pile spacers contractor may place pile jackets on the portion of the piles that will be in the MSE soil reinforced zone prior to placing the select granular backfill material and soil reinforcement. The contractor shall adequately support the piling to ensure that proper pile alignment is maintained during the wall construction. The contractor’s plan for bracing the pile shall be submitted to the engineer for review.

Piling shall be designed for downdrag (DD) loads due to either method. Oversized pipe pile spacers with sand placed after driving or pile jacket may be considered to mitigate some of the effects of downdrag (DD) loads. Sizing of pipe pile spacers shall account for pile size, thermal movements of the bridge, pile placement plan, and vertical and horizontal placement tolerances.

When rock is anticipated within the 5 feet zone below the MSE wall leveling pad, prebore into rock and prebore holes shall be sufficiently wide to allow for a minimum 10 feet embedment of pile and pipe pile spacer. When top of rock is anticipated within the 5 to 10 feet zone below the MSE wall leveling pad, prebore into rock to achieve a minimum embedment (pile only) of 10 feet below the bottom of leveling pad. Otherwise, the pipe pile spacer requires a minimum 5 feet embedment below the levelling pad. Consideration shall also be given to oversizing the prebore holes in rock to allow for temperature movements at integral end bents.

For bridges not longer than 200 feet, the minimum clearance from the back face of MSE walls to the front face of the end bent beam, also referred to as setback, shall be 4 ft. 6 in. (typ.) unless larger than 18-inch pipe pile spacer required. The 4 ft. 6 in. dimension is based on the use of 18-inch inside diameter pipe pile spacers & FHWA-NHI-10-24, Figure 5-17C, which will help ensure that soil reinforcement is not skewed more than 15° for nut and bolt reinforcement connections. Similarly, the minimum setback shall be determined when larger diameter pile spacers are required. For bridges longer than 200 feet, the minimum setback shall be 5 ft. 6 in. based on the use of 24-inch inside diameter of pipe pile spacers. Other types of connections may require different methods for splaying. In the event that the minimum setback cannot be used, the following guidance for pipe pile spacers clearance shall be used: pipe pile spacers shall be placed 18 in. clear min. from the back face of MSE wall panels; 12 in. minimum clearance is required between pipe pile spacers and leveling pad and 18 in. minimum clearance is required between leveling pad and pile.

MSE Wall Plan and Geometrics

  • A plan view shall be drawn showing a baseline or centerline, roadway stations and wall offsets. The plan shall contain enough information to properly locate the wall. The ultimate right of way shall also be shown, unless it is of a significant distance from the wall and will have no effect on the wall design or construction.
  • Stations and offsets shall be established between one construction baseline or roadway centerline and a wall control line (baseline). Some wall designs may contain a slight batter, while others are vertical. A wall control line shall be set at the front face of the wall, either along the top or at the base of the wall, whichever is critical to the proposed improvements. For battered walls, in order to allow for batter adjustments of the stepped level pad or variation of the top of the wall, the wall control line (baseline) is to be shown at a fixed elevation. For battered walls, the offset location and elevation of control line shall be indicated. All horizontal breaks in the wall shall be given station-offset points, and walls with curvature shall indicate the station-offsets to the PC and PT of the wall, and the radius, on the plans.
  • Any obstacles which could possibly interfere with the soil reinforcement shall be shown. Drainage structures, lighting, or truss pedestals and footings, etc. are to be shown, with station offset to centerline of the obstacle, with obstacle size. Skew angles are shown to indicate the angle between a wall and a pipe or box which runs through the wall.
  • Elevations at the top and bottom of the wall shall be shown at 25 ft. intervals and at any break points in the wall.
  • Curve data and/or offsets shall be shown at all changes in horizontal alignment. If battered wall systems are used on curved structures, show offsets at 10 ft. (max.) intervals from the baseline.
  • Details of any architectural finishes (formliners, concrete coloring, etc.).
  • Details of threaded rod connecting the top cap block.
  • Estimated quantities, total sq. ft. of mechanically stabilized earth systems.
  • Proposed grade and theoretical top of leveling pad elevation shall be shown in constant slope. Slope line shall be adjusted per project. Top of wall or coping elevation and stationing shall be shown in the developed elevation per project. If leveling pad is anticipated to encounter rock, then contact the Geotechnical Section for leveling pad minimum embedment requirements.

MSE Wall Cross Sections

  • A typical wall section for general information is shown.
  • Additional sections are drawn for any special criteria. The front face of the wall is drawn vertical, regardless of the wall type.
  • Any fencing and barrier or railing are shown.
  • Barrier if needed are shown on the cross section. Barriers are attached to the roadway or shoulder pavement, not to the MSE wall. Standard barriers are placed along wall faces when traffic has access to the front face of the wall over shoulders of paved areas.

Drainage at MSE Walls

  • Drainage Before MSE Wall
Drainage is not allowed to be discharged within 10 ft. from front of MSE wall in order to protect wall embedment, prevent erosion and foundation undermining, and maintain soil strength and stability.
  • Drainage Behind MSE Wall
Internal (Subsurface) Drainage
Groundwater and infiltrating surface waters are drained from behind the MSE wall through joints between the face panels or blocks (i.e. wall joints) and two-6 in. (min.) diameter pipes located at the base of the wall and at the basal interface between the reinforced backfill and the retained backfill.
Excessive subsurface draining can lead to increased risk of backfill erosion/washout through the wall joints and erosion at the bottom of walls and at wall terminal ends. Excessive water build-up caused by inadequate drainage at the bottom of the wall can lead to decreased soil strength and wall instability. Bridge underdrainage (vertical drains at end bents and at approach slabs) can exacerbate the problem.
Subsurface drainage pipes should be designed and sized appropriately to carry anticipated groundwater, incidental surface run-off that is not collected otherwise including possible effects of drainage created by an unexpected rupture of any roadway drainage conveyance or storage as an example.
External (Surface) Drainage
External drainage considerations deal with collecting water that could flow externally over and/or around the wall surface taxing the internal drainage and/or creating external erosion issues. It can also infiltrate the reinforced and retained backfill areas behind the MSE wall.
Diverting water flow away from the reinforced soil structure is important. Roadway drainage should be collected in accordance with roadway drainage guidelines and bridge deck drainage should be collected similarly.
  • Guidance
ALL MSE WALLS
1. Appropriate measures to prevent surface water infiltration into MSE wall backfill should be included in the design and detail layout for all MSE walls and shown on the roadway plans.
2. Gutters behind MSE walls are required for flat or positive sloping backfills to prevent concentrated infiltration behind the wall facing regardless of when top of backfill is paved or unpaved. This avoids pocket erosion behind facing and protection of nearest-surface wall connections which are vulnerable to corrosion and deterioration. Drainage swales lined with concrete, paved or precast gutter can be used to collect and discharge surface water to an eventual point away from the wall. If rock is used, use impermeable geotextile under rock and align top of gutter to bottom of rock to drain. (For negative sloping backfills away from top of wall, use of gutters is not required.)
District Design Division shall verify the size of the two-6 in. (min.) diameter lower perforated MSE wall drain pipes and where piping will daylight at ends of MSE wall or increase the diameters accordingly. This should be part of the preliminary design of the MSE wall. (This shall include when lateral pipes are required and where lateral drain pipes will daylight/discharge).
BRIDGE ABUTMENTS WITH MSE WALLS
Areas of concern: bridge deck drainage, approach slab drainage, approach roadway drainage, bridge underdrainage: vertical drains at end bents and approach slab underdrainage, showing drainage details on the roadway and MSE wall plans
3. Bridge slab drain design shall be in accordance with EPG 751.10.3 Bridge Deck Drainage – Slab Drains unless as modified below.
4. Coordination is required between the Bridge Division and District Design Division on drainage design and details to be shown on the MSE wall and roadway plans.
5. Bridge deck, approach slab and roadway drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.
  • (Recommended) Use of a major bridge approach slab and approach pavement is ideal because bridge deck, approach slab and roadway drainage are directed using curbs and collected in drain basins for discharge that protect MSE wall backfill. For bridges not on a major roadway, consideration should be given to requiring a concrete bridge approach slab and pavement incorporating these same design elements (asphalt is permeable).
  • (Less Recommended) Use of conduit and gutters:
  • Conduit: Drain away from bridge and bury conduit daylighting to natural ground or roadway drainage ditch at an eventual point beyond the limits of the wall. Use expansion fittings to allow for bridge movement and consider placing conduit to front of MSE wall and discharging more than 10 feet from front of wall or using lower drain pipes to intercept slab drainage conduit running through backfill.
  • Conduit and Gutters: Drain away from bridge using conduit and 90° elbow (or 45° bend) for smoothly directing drainage flow into gutters and that may be attached to inside of gutters to continue along downward sloping gutters along back of MSE wall to discharge to sewer or to natural drainage system, or to eventual point beyond the limits of the wall. Allow for independent bridge and wall movements by using expansion fittings where needed. See EPG 751.10.3.1 Type, Alignment and Spacing and EPG 751.10.3.3 General Requirements for Location of Slab Drains.
6. Vertical drains at end bents and approach slab underdrainage should be intercepted to drain away from bridge end and MSE wall.
7. Discharging deck drainage using many slab drains would seem to reduce the volume of bridge end drainage over MSE walls.
8. Drain flumes at bridge abutments with MSE walls do not reduce infiltration at MSE wall backfill areas and are not recommended.
DISTRICT DESIGN DIVISION MSE WALLS
Areas of concern: roadway or pavement drainage, MSE wall drainage, showing drainage details on the roadway and MSE wall plans.
9. For long MSE walls, where lower perforated drain pipe slope become excessive, non-perforated lateral drain pipes, permitted by Standard Specifications, shall be designed to intercept them and go underneath the concrete leveling pad with a 2% minimum slope. Lateral drain pipes shall daylight/discharge at least 10 ft. from front of MSE wall. Screens should be installed and maintained on drain pipe outlets.
10. Roadway and pavement drainage shall not be allowed to be discharged to MSE wall backfill area or within 10 feet from front of MSE wall.
11. For district design MSE walls, use roadway or pavement drainage collection pipes to transport and discharge to an eventual point outside the limits of the wall.
Example: Showing drain pipe details on the MSE wall plans.

Notes:
(1) To be designed by District Design Division.
(2) To be designed by District Design Division if needed. Provide non-perforated lateral drain pipe under leveling pad at 2% minimum slope. (Show on plans).
(3) Discharge to drainage system or daylight screened outlet at least 10 feet away from end of wall (typ.). (Skew in the direction of flow as appropriate).
(4) Discharge to drainage system or daylight screened outlet at least 10 feet away from front face of wall (typ.). (Skew in the direction of flow as appropriate).
(5) Minimum backfill cover = Max(15”, 1.5 x diameter of drain pipe).

751.24.2.2 Excavation

For estimating excavation and minimum soil reinforcement length, see EPG 751.6.2.17 Excavation.

For division responsibilities for preparing MSE wall plans, computing excavation class, quantities and locations, see EPG 747.2.6.2 Mechanically Stabilized Earth (MSE) Wall Systems.

751.24.2.3 Details

Bridge Standard Drawings
MSE Wall - MSEW

(1) Minimum embedment = maximum (2 feet; or embedment based on Geotechnical Report and global stability requirements;
or FHWA-NH1-10-0124, Table 2-2); or as per Geotechnical Report if rock is known to exist from Geotechnical Report.

Drycast Modular Block Wall Systems and Wetcast Modular Block Wall Systems

Battered mechanically stabilized earth wall systems may be used unless the design layout specifically calls for a vertical wall (precast modular panel wall systems shall not be battered and drycast modular block wall systems or wetcast modular block wall systems may be built vertical). If a battered MSE wall system is allowed, then EPG 751.50 J1.19 note shall be placed on the design plans:

For battered walls, note on the plans whether the horizontal offset from the baseline is fixed at the top or bottom of the wall. Horizontal offset and corresponding vertical elevation shall be noted on plans.

* The maximum vertical spacing of reinforcement should be limited to two times the block depth or 32 in., whichever is less.
For large modular block (block height > 16 in.), maximum vertical spacing of reinforcement equal to the block height.

Fencing (See Bridge Standard Drawing for details)

Fencing may be installed on the Modified Type A or Modified Type B Gutter or behind the MSE Wall.

For Modified Type A and Modified Type B Gutter and Fence Post Connection details, see Standard Plan 607.11.

751.24.3 Cast-In-Place Concrete Retaining Walls

751.24.3.1 Unit Stresses

Concrete Concrete for retaining walls shall be Class B Concrete (f'c = 3000 psi) unless the footing is used as a riding surface in which case Class B-1 Concrete (f'c = 4000 psi) shall be used.

Reinforcing Steel

Reinforcing Steel shall be Grade 60 (fy = 60,000 psi).

Pile Footing

For steel piling material requirements, see the unit stresses in EPG 751.50 A1.3 note.

Spread Footing

For foundation material capacity, see Foundation Investigation Geotechnical Report.

751.24.3.2 Design

Note: For design concepts and guidance, follow the design process (EPG 751.40.8.15) and modify design/details of ASD as necessary to meet LRFD requirements until EPG 751.24 is updated for LRFD.

Capacity/Demand ratio (CDR) for bearing capacity shall be ≥ 1.0

Strength Limit States:
Factored bearing resistance = Nominal bearing resistance from Geotech report X
Minimum Resistance factor (0.55, Geotech report)      LRFD Table 11.5.7
Extreme Event I and II Limit State:
Factored bearing resistance = Nominal bearing resistance from Geotech report X Resistance factor
Resistance factor = 0.8      LRFD 11.5.8
When wall is supported by soil:
Factored bearing stress per LRFD eq. 11.6.3.2-1
When wall is supported by a rock foundation:
Factored bearing stress per LRFD eq. 11.6.3.2-2 and 11.6.3.2-3
Note: When the value of eccentricity e is negative then use e = 0.

Capacity/Demand ratio (CDR) for overturning shall be ≥ 1.0

Capacity/Demand ratio (CDR) for eccentricity shall be ≥ 1.0

Capacity/Demand ratio (CDR) for sliding shall be ≥ 1.0

Sliding shall be checked in accordance with LRFD 11.6.3.6 and 10.6.3.4

Eccentricity, (e) Limit for Strength Limit State:      LRFD 11.6.3.3

  • For foundations supported on soil, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, B or (e ≤ 0.33B).
  • For foundations supported on rock, the location of the resultant of the reaction forces shall be within the middle nine-tenths of the base width, B or (e ≤ 0.45B).

Eccentricity, (e) Limit for Extreme Event I (Seismic):      LRFD 11.6.5.1

  • For foundations supported on soil or rock, the location of the resultant of the reaction forces shall be within the middle two-thirds of the base width, B or (e ≤ 0.33B) for γEQ = 0.0 and middle eight-tenths of the base width, B or (e ≤ 0.40B) for γEQ = 1.0. For γEQ between 0.0 and 1.0, interpolate e value linearly between 0.33B and 0.40B. For γEQ refer to LRFD 3.4.
Note: Seismic design shall be performed for γEQ = 0.5

Eccentricity, (e) Limit for Extreme Event II:

  • For foundations supported on soil or/and rock, the location of the resultant of the reaction forces shall be within the middle eight-tenths of the base width, B or (e ≤ 0.40B).

For epoxy coated reinforcement requirements, see EPG 751.5.9.2.2 Epoxy Coated Reinforcement Requirements.

If the height of the wall or fill is a variable dimension, then base the structural design of the wall, toe, and heel on the high quarter point between expansion joints.

Fig. 751.24.3.2