## Factor of safety against bearing capacity failure

EM 1110-2-2502 requires the following minimum factors of safety against bearing capacity failure FSbc for cantilever retaining walls EM 1110-2-2502 provides the following expression for the normal component to the base of the structure of the ultimate bearing capacity for strip footings Q B ( ci- cg-c-Nc) + ( qd- qt-Wqo-Nq) +-2- However, for the wall being analyzed, only the last term is nonzero, and thus, this expression reduces to The FSbc for usual loadings is computed as follows _ 5 yd yi...

## Factor of safety against sliding

EM 1110-2-2502 requires the following minimum factors of safety against sliding (FSsliding) for cantilever retaining walls b. 1.33 short-duration (unusual) loads, such as those that might occur during high winds, construction activities, or the Operational Basis Earthquake (OBE). c. 1.1 extreme loads, such as those that might occur during the Maximum Design Earthquake (MDE). The FSsliding for usual loadings is computed as follows Tult N' tan(Sbase) FSsliding T 26,625 lbs tan(35 ) 10,137.5 lbs...

## Moment and shear capacity of the heel

The heel is analyzed as being singly reinforced, with the steel along the top face and 4 in. coverage. The critical section for both moment and shear capacity in the heel is at the interface of the heel and the stem. 4' (18,000 lbs + 2400 lbs) - 3.38' (12,694 lbs) - Mheel 0 Mheel 38,694 ft-lbs 1.7 1.3 (38,694 ft-lbs) 85,514 ft-lbs Because the dimensions (i.e., d) for both the stem and heel and Mn are essentially the same, the reinforcement used for the stem is also used for the heel (i.e., 11 9...

## Moment capacity of the stem

The stem is analyzed as being singly reinforced with the critical section for moment capacity being at the base of the stem, as illustrated below. Mstem 4.667' 8302.5 lbs 38,748 ft-lbs 1.7 1.3 (38,748 ft-lbs) 85,633 ft-lbs 0.425 4 ksi-12 (19.5) 0.85-4 ksi-0.0765-12-19.5 Use 11 9 c-c (conservative)

## Percentage of the base area in compression ie overturning stability

The global stability of cantilever retaining walls to overturning is quantified by the percentage of the base area in compression. EM 1110-2-2502 requires the following minimum percentages cantilever walls on soil foundations c. Resultant within base extreme loadings. The percentage of the base area that is in compression for usual loadings is computed as follows cmin > 0 .'. 100 base area in compression okay Note Per Corps design criteria, a linear effective base pressure is assumed.

## Shear capacity of the stem

The critical section for shear in the stem is taken as 19.5 above the interface of the base and stem, where 19.5 is d at the base of the stem. However, the d at the critical section is only 19, due to the taper of the wall. Vu 1.7 (D + L) 1.7 (6871 lbs) 11,681 lbs 0.85 2 - 4000 psi 12 19 24,514 lbs Appendix A Static Design of the Cantilever Retaining Wall 1.3-(Vu - Vc) 1.3 (11,681 lbs - 24,514 lbs) < 0 okay

## Shear capacity of the toe

The critical section for shear capacity in the toe is at a distance d from the interface of the toe and the stem, where d 19.5. Because the length of the toe is short, moment capacity is not checked. Vtoe 4293 lbs - 413 lbs 3880 lbs Kear -2- fe b-d Vs > 1.3-(Vu - Vc) 0.85 2 4000 psi 12-19.5 L3-(Vu - -Vc) 13-(6596 lbs - 25,159 lbs) Use 11 9 c-c As 2.08 in2 per ft of wall Figure A-5a shows the steel reinforcing detailing, determined previously (note development lengths need to be checked),...

## Stage Sizing of the Cantilever Retaining Wall

As stated previously, the first design stage consists of sizing the cantilever wall such that global stability requirements are satisfied (i.e., sliding, overturning, and bearing capacity), in general accordance with EM 1110-2-2502 (HQUSACE 1989). The structural wedge of the proposed wall and backfill is shown in Figure A-1, as well as the backfill and foundation material properties. To assess the global stability of the wall, the external forces and corresponding points of action acting on the...

## Displacement Controlled Design Procedure

The following is a summary of the steps listed in Ebeling and Morrison (1992), Section 6.3.2, for the displacement-controlled design procedure for a wall retaining dry backfill, for which kv is assumed equal to zero. The procedure pertains to the global stability of the wall and is repeated here for reference purposes only. It is applicable to walls whose geometry has already been established (e.g., an existing wall or established during a static design step). These steps are geared toward hand...

## Characteristics of Ground Motion Selected

As stated previously, at least five time-histories (for each component of motion) meeting the selection criteria should be used in nonlinear dynamic analyses (EC 1110-2-6051 (HQUSACE 2000)). However, for the first phase of this study, only SG3351 was used, which was recorded during the 1989 Loma Prieta earthquake in California. The basis for selecting SG3351 was that it was estimated, using CWROTATE (Ebeling and White, in preparation), to induce the greatest permanent relative displacement of...

## Determination of forces assuming linearly varying stress distribution

In the second approach used to determine the forces acting on the stem and heel sections, linearly varying stress distributions across the elements were assumed, as illustrated in Figure 4-2. This approach is analogous to using a firstorder shape function in the FEM. To apply this approach, the only actual nodes considered were the nodes at the top and bottom of the wall, shown as red dots in Figure 4-2. In addition to these two actual nodes, the centers of the elements were treated as nodes,...

## FLAC Data Reduction Discussion of Results

In the previous chapter, an overview was given of the numerical model used to analyze the cantilever retaining wall. In this chapter an overview is given on how the FLAC data were reduced, followed by a presentation and discussion of the reduced data. Two FLAC analyses were performed as part of the first phase of this research effort, one using the uncracked properties of the concrete wall, and the other using the fully cracked properties (refer to Table 3-1 for the listing of the respective...

## Incremental dynamic forces

In addition to computing the total resultant forces acting on the stem and heel sections, the incremental dynamic forces APj at time increment j were computed. APj is the difference between the total resultant force Pj at time increment j minus the total resultant force prior to shaking (i.e., Pj at j 0, designated as Pstatic) Because Pj values computed assuming constant and linearly varying stress distributions were essentially identical, APj was computed only using Pj for the constant stress...

## Incremental resultant forces and points of action

As an alternate to presenting the total resultant force of the lateral earth pressures, Seed and Whitman (1970) expressed the resolved lateral earth pressures in terms of a static active resultant (Pstatic) and a dynamic incremental resultant (AP), as discussed previously in Section 4.1.3. Seed and Whitman's (1970) procedure is outlined in Ebeling and Morrison (1992), Section 4.2.2. Using Equations 4-5 and 4-7, time-histories for AP acting on the stem and heel sections and their corresponding...

## Interface elements

Interface elements were used to model the interaction between the concrete retaining wall and the soil. However, FLAC does not allow interface elements to be used at the intersection of branching structures (e.g., the intersection of the stem and base of the cantilever wall). Of the several attempts by the authors to circumvent this limitation in FLAC, the simplest and best approach found is illustrated in Figure 3-6. As shown in this figure, three very short beam elements, oriented in the...

## List of Candidate Motions

Based on the selection criteria, the motions listed in Table 2-2 were considered as candidates for use in the numerical analyses. Closest to surface projection of rupture 32.6 km Closest to surface projection of rupture 0.9 km Closest to surface projection of rupture 15.6 km Closest to surface projection of rupture 19.9 km Closest to surface projection of rupture 34.1 km Note Ms surface wave magnitude of earthquake M moment magnitude of earthquake. These records were obtained by searching the...

## Notation Sign Convention and Earth Pressure Expressions

The notation shown in Figure B-1 is used throughout this report. All the variables shown in this figure are presented in their positive orientation. Additionally, expressions for the classical Mononobe-Okabe active and passive dynamic earth pressures are presented (e.g., Ebeling and Morrison 1992, Chapter 4),1 as well as expressions for the slope of the corresponding failure planes. 1 References cited in this appendix are included in the References section at the end of the main text. Figure...

## Numerical Model Parameters

The previous section gave an overview of the physical system being analyzed and its numerical model counterpart. This section focuses on the specific constitutive models used for the soil, retaining wall, and their interface, with particular attention given to how to determine the various model parameters. An elastoplastic constitutive model, in conjunction with Mohr-Coulomb failure criteria, was used to model the soil. Elastic beam elements were used to model the concrete retaining wall, and...

## Organization of Report

The organization of the report follows the sequence in which the work was performed. Chapter 2 outlines the process of selecting the ground motions (e.g., acceleration time-histories) used in the FLAC analyses. Chapter 3 gives a brief overview of the numerical algorithms in FLAC and outlines how the various numerical model parameters were determined. Chapter 4 describes the data reduction and interpretation of the FLAC results, followed by the References. Appendix A provides detailed...

## Overview of FLAC

As stated in Chapter 1, the detailed numerical analyses of the cantilever retaining walls were performed using FLAC, a commercially available, two-dimensional, explicit finite difference program, which was written primarily for geotechnical engineering applications. The basic formulation of FLAC is planestrain, which is the condition associated with long structures perpendicular to the analysis plane (e.g., retaining wall systems). The following is a brief overview of FLAC and is largely based...

## Permanent relative displacement of the wall

Using an acceleration time-history computed by FLAC at approximately middepth of the backfill and located near the free-field boundary in the FLAC model, a Newmark sliding block-type analysis was performed on the structural wedge. This analysis was similar to those performed using CWROTATE (Ebeling and White, in preparation) with the SHAKE computed time-histories, which are presented in Appendix C. The results from the sliding block-type analysis were compared to wall movements computed by...

## Preface

The study documented herein was undertaken as part of Work Unit 387-9456h, Seismic Design of Cantilever Retaining Walls, funded by the Headquarters, U.S. Army Corps of Engineers (HQUSACE) Civil Works Earthquake Engineering Research Program (EQEN) under the purview of the Geotechnical and Structures Laboratory (GSL), Vicksburg, MS, U.S. Army Engineer Research and Development Center (ERDC). Technical Director for this research area was Dr. Mary Ellen Hynes, GSL. The HQUSACE Program Monitor for...

## Presentation and Discussion of Reduced Data

Using the procedures described in the preceding section, the total and dynamic incremental resultant forces acting on the stem and heel sections were determined from the FLAC computed stresses, as well as the corresponding vertical distance above the base at which the resultant forces act. Additionally, the permanent relative displacements of the wall computed in the FLAC analysis are compared with those predicted using a Newmark sliding block-type analysis procedure, e.g., CWROTATE (Ebeling...

## Processing of the Selected Ground Motion

Although motion SG3351 met the selection criteria, several stages of processing were required before it could be used as an input motion in the FLAC analyses. The first stage was simply scaling the record. As a general rule, ground motions can be scaled upward by a factor of two without distorting the realistic characteristics of the motion (EC 1110-2-6051 (HQUSACE 2000)). The upward scaling was desired because although the motion induced the largest permanent relative displacement dr of the...

## Ratio of total resultant forces and points of action

Given the difference in the phasing of the stem and heel section time-histories, it is only of limited value to examine the ratios of the respective time-histories for the stem and heel sections, which are presented in Figure 4-7. For example, from examination of Figure 4-7b, PstemIPheei, one might conclude that the maximum value for Pstem is larger than the maximum value for Pheel, which is clearly not the case, as may be seen from Figure 4-4c and d. Similar misinterpretations can be made in...

## Reaction height of forces

The points of application of the total and incremental dynamic resultant forces were computed for the stem and heel sections in terms of their vertical distances above the base of the wall. For the total resultant forces, the vertical distances Y were computed using the following relation Yj vertical distance from the base of the retaining wall to the point of application of the total resultant force acting on the stem or heel section at time increment j yi vertical distance from the base of...

## Representative magnitude and sitetosource distance

As stated in Chapter 1, the objective of this study is to determine the seismic structural design loads for the stem portion of a cantilever retaining wall. Accordingly, the magnitude Mand site-to-source distance R of the ground motion used in the numerical analyses should be representative of an actual design earthquake, which will depend on several factors including geographic location and consequences of failure. In an effort to select a representative M and R for a design event, the...

## Selection Criteria

The selection of an earthquake acceleration time-history for use in the numerical analyses was guided by the following criteria a. A real earthquake motion was desired, not a synthetic motion. b. The earthquake magnitude and site-to-source distance corresponding to the motion should be representative of design ground motions. c. The motion should have been recorded on rock or stiff soil. These criteria were used to assemble a list of candidate acceleration time-histories, while additional...

## Specifying Ground Motions in FLAC

As briefly outlined in Chapter 3, dynamic analyses can be performed with FLAC, wherein user-specified acceleration, velocity, stress, or force time-histories can be input as an exterior boundary condition or as an interior excitation. A parametric study was performed to determine the best way to specify the ground motions in FLAC for earthquake analyses. The parametric study involved performing a series of one-dimensional site response analyses using consistently generated acceleration,...

## Structural elements

The concrete retaining wall was modeled using elastic beam elements approximately 1 ft (0.3 m) long. In FLAC, four parameters are required to define the mechanical properties of the beam elements cross-sectional area Ag mass density p elastic modulus Ec and second moment of area I, commonly referred to as moment of inertia. The wall was divided into five segments having constant parameters, as illustrated in Figure 3-5, with each segment consisting of several 1-ft (0.3-m) beam elements. The...

## Total resultant forces and points of action

The horizontal acceleration ah and the corresponding dimensionless horizontal inertial coefficient kh at approximately the middle of the backfill portion of the structural wedge were computed during the FLAC analyses, as shown in Figure 4-3. Appendix B gives the appropriate sign convention related to ah and kh. In this figure, the potential active and passive failure planes are shown for illustration only. The kh time-history shown in this figure is that to which reference is made during the...

## Retaining Wall Model

The retaining wall-soil system analyzed in the first phase of this investigation is depicted in Figure 3-2. As shown in this figure, the FLAC model is only the top 30 ft (9 m) of a 225-ft (69-m) profile. Although the entire profile, to include the retaining wall, can be modeled in FLAC, the required computational time would be exorbitant, with little to no benefit added. To account for the influence of the soil profile below 30 ft (9 m), the entire profile without the retaining wall was modeled...

## Stage Structural Design of Concrete Cantilever Retaining Wall

As stated in the introduction to this appendix, the second stage of the wall design entails the structural design of the concrete wall, to include the dimensioning of the concrete base slab the toe and heel elements and stem, and the detailing of the reinforcing steel. All reinforced-concrete hydraulic structures must satisfy both strength and serviceability requirements. In the strength design method, this is accomplished by multiplying the service loads by appropriate load factors and by a...

## Research into the Seismic Response of a Cantilever Retaining Wall

The seismic loads acting on the structural wedge of a cantilever retaining wall are illustrated in Figure 1-3. The structural wedge consists of the concrete wall and the backfill above the base of the wall i.e., the backfill to the left of a vertical section through the heel of the cantilever wall . The resultant force of the static and dynamic stresses acting on the vertical section through the heel i.e., heel section is designated as PaE, heel, and the normal and shear base reactions are N'...

## References

Building code requirements for reinforced concrete and commentary, ACI 318-02, Detroit, MI. Aitken, G. H., Elms, D. G., and Berrill, J. B. 1982 . Seismic response of retaining walls, Research Report 82-5, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, 87 pp. Clough, G. W., and Duncan, J. M. 1991 . Earth pressures. Foundation engineering handbook. 2nd ed., H.Y. Fang, ed., Van Nostrand Reinhold, New York, Chapter 6,...

## Background

Formal consideration of the permanent seismic wall displacement in the seismic design process for Corps-type retaining structures is given in Ebeling and Morrison 1992 . The key aspect of this engineering approach is that simplified procedures for computing the seismically induced earth loads on retaining structures are dependent upon the amount of permanent wall displacement that is expected to occur for each specified design earthquake. The Corps uses two design earthquakes as stipulated in...