AD-A265 707,• ,"• I; •,' ',• ,R -9 3 I I January 1993 By: Robert M. Ebeling, PhD N C EL and Ernest E. Morrison Sponsored by: Technical Report Office of Navy Technology and Department of the Army THE SEISMIC DESIGN OF WATERFRONT RETAINING "STRUCTURES JUN 11193 U' 4< P IWI93-13063 ABSTRACT This technical report deals with the soil mechanics aspects of the design of waterfront retaining structures built to withstand the effects of earthquake loadings. It addresses the stability and movement of gravity retaining walls and anchored sheet pile walls and the dynamic forces against the walls of drydocks. The effects of wall displacements. submergence, liquefaction potential, and excess pore water pres- sures, as well as inertial and hydrodynamic forces, are incorporated in the design procedures. Several new computational procedures are described. The procedures used to calculate the dynamic earth pressures acting on retaining structures consider the magnitude of wall displacements. For example, dynamic active earth pressures are computed for walls that retain yielding backfills, i.e., backfills that undergo sufficient displacements during seismic events to mobilize fully the shear resistance of the soil. For smaller wall movements, the shear resistance of the soil is not fully mobilized and the dynamic earth pressures acting on those walls are greater because the soil comprising the backfill does not yield, i.e., a non-yielding backfill. Procedures for incorporating the effects of submergence within the earth pressure computations, including consideration of excess pore water pressure, are described. NAVAL CIVIL ENGINEEING LABORATORY PORT HUENEME CALIFORNIA 93043-4328 Approved for public -elease, d,;;t.Aion is unlimited
Transcript
AD-A265 707,•,"• I; •,' ',• ,R -9 3
I I January 1993
By: Robert M. Ebeling, PhD
N C EL and Ernest E. Morrison
Sponsored by:Technical Report Office of Navy Technology
and Department of the Army
THE SEISMIC DESIGNOF WATERFRONT RETAINING
"STRUCTURES
JUN 11193 U'4<
P IWI93-13063
ABSTRACT This technical report deals with the soil mechanics aspects of the design of waterfrontretaining structures built to withstand the effects of earthquake loadings. It addresses the stability andmovement of gravity retaining walls and anchored sheet pile walls and the dynamic forces against thewalls of drydocks.
The effects of wall displacements. submergence, liquefaction potential, and excess pore water pres-sures, as well as inertial and hydrodynamic forces, are incorporated in the design procedures. Several newcomputational procedures are described.
The procedures used to calculate the dynamic earth pressures acting on retaining structures considerthe magnitude of wall displacements. For example, dynamic active earth pressures are computed forwalls that retain yielding backfills, i.e., backfills that undergo sufficient displacements during seismicevents to mobilize fully the shear resistance of the soil. For smaller wall movements, the shear resistanceof the soil is not fully mobilized and the dynamic earth pressures acting on those walls are greaterbecause the soil comprising the backfill does not yield, i.e., a non-yielding backfill. Procedures forincorporating the effects of submergence within the earth pressure computations, including considerationof excess pore water pressure, are described.
NAVAL CIVIL ENGINEEING LABORATORY PORT HUENEME CALIFORNIA 93043-4328
Approved for public -elease, d,;;t.Aion is unlimited
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P.4b•bl e0orting bu roor tThis coIOllton of Inforrmatiort i estr .ead to average • hour Pe responae. ku the uns. tae viewng kalsudiona, iiritl" axaIat•lg dae sioJrces.gatr'ehnn and mainta roriq the data heae and corr*WV and feywiONg the collaicon of Irrnhataon. Send convnh; regarding VhM bu et ifat. or " ty ht aspeit of this
olteC¶Oi ;' man non '.; nclnq sugestmoa for reducng ts tnburdn. to Washington Headuartr Service. Dictoral k Informall on end Rev". 1215 Jeffron Dvis Highway.SuItO 1204, ArlingTon. VA 22202-4302 anld to the Ofte ol Managenent and Budget. Perww•r Redibd ProncP (0701-$a). Washington, DC 20603.
1. AGENCY USE ONLY ILeavganbaf) Z REPORT DATE S. EPOT ITYPE A0 DATES COVERED
January 1993 Final Report
4. TITLE AND SiJBltl..7E &. N•NO NUM Kobe"
THE SEISMIC DESIGN OF WATERFRONTRETAINING STRUCTURES RM - 33F60-001-0106
WU- DN387338& AUTHOR1S)
Robert M. Ebeling and Ernest E. Morrison, Jr.
7. PERFORMING ORGANIZAInON NAME(S 4NO AGORESSEIS) L PEAFOWINO ORGANLZATION
U.S. Arm) Waterways Experiment Station VPoRTWBEAInformation Technology Laboratory Technical Report ITL-92-113909 Halls Ferry Road NCEL TR-939Vickshurg, MS 39180-6199
9. SPONSO•HNOMOtNTORING AGENCY NAMES AND ADORE8SE1I 10L SPONSOWN"00ONTOFII•
AGENCY REPORT NUMBERChief of Naval Research / Department of the ArmyOffice of Naval Technology U.S. Army Corps of Engineers800 No. Quincy Street Washington, DC 20314-1000Arlington, VA 22217-5000
11t. SUPPLEMENTARY NOTES
Available from National Technical Information Service. 5285 Port Royal Road, Springfield, VA 22161
Approved for public release; distribution is unlimited.
13. ABSTRACT :ar~ia,, 2 ,wrtm)
This technical report deals with the soil mechanics aspects of the design of waterfront retaining structures built towithstand the effects of earthquake loadings. It addresses the stability and movement of gravity retaining walls andanchored sheet pile walls and the dynamic forces against the wails of drydocks.
The effects of wall displacements, submergence, liquefaction potential, and excess pore water pressures, as well asinertial and hydrodynamic forces, are incorporated in the design procedures. Several new computational procedures aredescribed.
The procedures used to calculate the dynamic earth pressures acting on retaining structures consider the magnitudeof wall displacements. For example, dynamic active earth pressures are computed for walls that retain yieldingbackfills, i.e.. backfills that undergo sufficient displacements during seismic events to mobilize fully the shear resis-tance of the soil. For smaller wall movements, the shear resistance of the soil is not fully mobilized and the dynamicearth pressures acting on those walls are greater because the soil comprising the backfill does not yield, i.e.. a non-yielding backfill. Procedures for incorporating the effects of submergence within the earth pressure computations, in-cluding consideration of excess pore water pressure, are described.
This report describes procedures used in the seismic design ofwaterfront retaining structures. Funding for the preparation of this re.porlwas provided by the US Naval Civil Engineering Laboratory through the tollow-ing instruments: NAVCOMPT Form N6830591WR00011, dated 24 October 1990; Amend-ment #1 to that form, dated 30 November 1990; NAVCOMPT Form N6830592WRO001(,dated 10 October 1991; Amendment #1 to the latter, dated 3 February 19)2: aindthe Computer-Aided Structural Engineering Program sponsored by the Director-ate, Headquarters, US Army Corps of Engineers (HQUSACE), under the Strrlct,,ralEngineering Research Program. Supplemental support was also provided by thtUS Army Civil Works Guidance Update Program toward cooperative product ion ofgeotechnical seismic design guidance for the Corps of Engineers, Generalproject management was provided by Dr. Mary Ellen iiyn~s and Dr. Joseph P_Koester, both of the Earthquake Engineering and Seismology Branch (EESKI,Earthquake Engineering and Geosciences Division (EEGD), Geotechnical L;ahora-tory (GL), under the gcnrral .p .Jibion ot Dr. Wiltiam F. Marcuson Ill.Director, GL. Mr. John Ferritto of the Naval Civil Engineering Labor;ulorv.Port Hueneme, CA, was the Project Monitor.
The work was performed at the US Army Engineer Waterways Experim-litStation (WES) by Dr. Robert M. Ebeling and Mr. Ernest E. Morrison,Interdisciplinary Research Group, Computer-Aided Engineering Division (CAED).Information Technology Laboratory (ITL). This report was prepared byDr. Ebeling and Mr. Morrison with contributions provided by Professor RobertV. Whitman of Massachusetts Institute of Technology and Professor W. D. LiamnFinn of University of British Columbia. Review commentary was also providedby Dr. Paul F. Hadala, Assistant Director, GL, Professor William P. Dawkins ofOklahoma State University, Dr. John Christian of Stone & Webster EngineeringCorporation, and Professor Raymond B. Seed of University of California.Berkeley. The work was accomplished under the general direction of Dr. ReedL. Mosher, Acting Chief, CAED and the general supervision of Dr. N.Radhakrishnan, Director, ITL.
At the time of publication of this report, Director of WES wasDr. Robert W. Whalin. Commander was COL Leonard G. Hassell, EN.
Acceioi For
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SA0 A b
-I.
PROCEDURAL SUMMARY
This section summarizes the computational procedures described in this reportto compute dynamic earth pressures. The procedures for computing dynamicearth pressures are grouped according to the expected displacement of thebackfill and wall during seismic events. A yielding backfill displacessufficiently (refer to the values given in Table 1, Chapter 2) to mobilizefully the shear resistance of the soil, with either dynamic active earth pres-sures or dynamic passive earth pressures acting on the wall, depending uponthe direction of wall movement. When the displacement of the backfill (aridwall) is less than one-fourth to one-half of the Table 1 values, the term non-yielding backfill is used because the shear strength of the soil is not fullymobilized.
The procedures for computing dynamic active and passive earth pressuresfor a wall retaining a dry yielding backfill or a submerged yielding backfillare discussed in detail in Chapter 4 and summarized in Table i and Table ii,respectively. The procedures for computing dynamic earth pressures for a wallretaining a non-yielding backfill are discussed in Chapter 5 and summarized inTable i.
The assignment of the seismic coefficient in the design procedures forwalls retaining yielding backfills are discussed in detail in Chapter 6 andsummarized in Table iii. The assignment of the seismic coefficient in thedesign procedures for walls retaining non-yielding backfills are discussed indetail in Chapter 8 and summarized in Table iii.
ii
TABLE I
DETERMINATION OF DYNAMIC EARTH PRESSURES FOR DRY BACKFILLS
YIELDING BACKFILL
DYNAMIC ACTIVE EARTH PRESSURES
MONONOBE - OKABE
Equivalent Static Formulation (Arango)
Simplified Procedure (Seed and Whitman)- restricted to: vertical wall and level backfills.- approximate if: 4 35°, kv o 0.
DYNAMIC PASSIVE EARTH PRESSURES
MONONOBE - OKABE- approximate for 6 > 0.- inaccurate for some wall
geometries and loading conditions.
Equivalent Static Formulation- approximate if: Kp(#*,O*) is computed
using Coulomb's equation, see above comments.approximate if: Kp(o*,O*) is computed usingLog-Spiral solutions.
Simplified Procedure (Towhata and Islam)- restricted to: vertical walls and level backfills
and S = 0'.- approximate if: 0 o 350. k, o 0.
iii
TABLE i - Continued
DETERMINATION OF DYNAMIC EARTH PRESSURES FOR DRY BACKFILLS
NON-YIELDING BACKFILL
LATERAL SEISMIC FORCE
Wood's Simplified Procedure- restricted to: kh constant with depth and k, - 0.
Soil-Structure Interaction Analysis Using the Finite Element Method
TABLE ii
DETERMINATION OF DYNAMIC EARTH PRESSURESFOR SUBMERGED OR PARTIALLY SUBMERGED BACKFILLS
Select the appropriate technique for either yielding backfill or non-yielding backfill with additional computations as specified by one of thefollowing procedures:
Restrained water case
Free water case- restricted to soils of high permeability
(e.g. k > I cm/sec)
iv
TABLE iii
DESIGN PROCEDURES - ASSIGNMENT OF SEISMIC COEFFICIENT
YIELDING BACKFILL
Preselected Seismic Coefficient Method
- The approximate value of horizontal displacementis related to the value of the horizontal seismiccoefficient.
Displacement Controlled Approach
The seismic coefficient is computed based uponan explicit choice of an allowable level ofpermanent horizontal wall displacement.
NON-YIELDING BACKFILL
Displacement Of The Wall Is Not Allowed
The seismic coefficient is set equal to the peakhorizontal acceleration coefficient, assumingacceleration within the backfill to be constantwith depth. Otherwise, consider dynamic finiteelement method of analysis.
v
TABLE OF CONTENTS
PAGE
PREFACE .......................... .... ................................ i
PROCEDURAL SUMMARY ....................... .......................... ii
TABLE i ........................ ................................ . ii
TABLE ii ..................... ....................................... iv
TABLE iii .............................. ............................... v
CONVERSION FACTORS, NON-SI TO SI (METRIC) UNITS OF MEASUREMENT . . .. xvi
CHAPTER 1 GENERAL DESIGN CONSIDERATIONS FOR WATERFRONT SITES ..... I
1.1 Scope and Applicability .......... .................. . i1
1.2 Limit States ....................... ........................ 3
1.3 Key Role of Liquefaction Hazard Assessment ........ ......... 3
1.4 Choice of Design Ground Motions ............. .............. 5
6.2 Procedure Based upon Preselected Seismic Coefficient . . . . 140
6.2.1 Stability of Rigid Walls Retaining Dry Backfills whichUndergo Movements during Earthquakes .... ........... .. 142
6.2.2 Stability of Rigid Walls Retaining Submerged Backfillswhich Undergo Movements During Earthquakes No ExcessPore Water Pressures ............ ................... 148
6.2.3 Stability of Rigid Walls Retaining Submerged Backfillswhich Undergo Movements During Earthquakes - Excess PoreWater Pressures ................... ..................... l
6.2.4 Stability of Rigid Walls Retaining Submerged Backfills
which Undergo Movements During Earthquakes - LiquifiedBackfill. ............... ......................... .. 155
6.3.4 Analysis of Earthquake Induced Displacements for a WallRetaining Submerged Backfill - No Excess Pore WaterPressures ................. ........................ .. 165
viii
PACE
6.3.5 Displacement Controlled Design Procedure for a Wall
D.1 Some Key References ............ .................... D2
D.2 Principal Issues .............. .. .................... D2
D.2.1 Total Versus Effective Stress Analysis ... .......... .. D3
D.2.2 Modeling Versus Nonlinear Behavior .... ............ .. D3
D.2.3 Time Versus Frequency Domain Analysis ... .......... D3
D.2.4 1-D Versus 2-D Versus 3-D ........ ................ .. D4
D.2.5 Nature of Input Ground Motion ...... .............. .. D4
D.2.6 Effect of Free Water .......... ................... .. D4
DA3 A Final Perspective .................. .................... D4
APPENDIX E: NOTATION ..................... ......................... El
xi
LIST OF TABLES
No.Page
1 Approximate Magnitudes of Movements Required to Reach MinimumActive and Maximum Passive Earth Pressure Conditions ..... 16
2 Ultimate Friction Factors for Dissimilar Materials ....... .. 313 Valves of KA and KP for Log-Spiral Failure Surface .. ....... . 444 Section Numbers That Outline Each of the Two Design Procedures
for Yielding Walls for the Four Categories of RetainingWalls Identified in Figure 6.1 ......... ................ .. 142
5 Minimum Factors of Safety When Using the Preselected SeismicCoefficient Method of Analysis ......... ................ .. 143
6 Qualitative and Quantitative Description of the ReportedDegrees of Damage ............... ....................... .. 207
7 Ten Stages of the Analyses in the Design of Anchored Wallsfor Seismic Loadings ............ ..................... .. 218
C.1 Horizontal Force Components ............... ................... C3C.2 Moments About Tie Rod Due to Active Earth Pressures ... ....... C4C.3 Moments About Tie Rod Due to Passive Earth Pressures ..... ...... C4C.4 Calculation of the Depth of Penetration ......... ............. C5C.5 Horizontal Force Components for D = 10 Feet ........ ........... C5C.6 Moment Internal to the Sheet Pile at y - 12.79 Feet Below
the Water Table and About the Elevation of the Tie Rod . . .. C6C.7 Design Moment for Sheet Pile Wall in Dense Sand ..... ......... C7C.8 Allowable Bending Moment for Four ASTM A328 Grade Sheet Pile
Sections (Oallowable = 0.65 "yield) ...... . . . . . .. . . ............ C8C.9 Five Horizontal Static Active Earth Pressure Force Components
of PA with D - 20.24 Feet ........... .................. C14C.10 Summary of Depth of Penetration Calculations ... .......... .. C18C.11 Tie Rod Force TFES ................... ....................... C19C.12 Moment of Forces Acting Above y = 15.32 Feet Below the Water
Table and About the Tie Rod ......... ................... C21C.13 Design Moment for Sheet Pile Wall in Dense Sand ... .......... C22C.14 Allowable Bending Moment for Four ASTM A328 Grade Sheet
Pile Sections (Ljalnowable = 0.9 . Oyield) .... ... ............ C22C.15 Required Geometry of Tie Rod .......... ................... C23D.1 Partial Listing of Computer-Based Codes for Dynamic Analysis
of Soil Systems ........................ ......................... Dl
LIST OF FIGURES
No. Page
1.1 Overall limit states at waterfronts ............. ............... 42.1 Potential soil and structural failure modes due to earthquake
shaking of an anchored sheet pile wall .... ............ .. 122.2 Rigid walls retaining backfills which undergo movements during
earthquakes .................. .......................... .. 132.3 Horizontal pressure components and anchor force acting on
sheet pile wall ................ ......................... 142.4 Effect of wall movement on static horizontal earth pressures 15
xii
No. Pare
2.5 Effect of wall movement on static and dynamic horizontal earthpressures ................. ........................... .. 17
2.6 Effect of wall movement on static and dynamic horizontalearth pressures ............... ........................ .. 18
2.7 Failure surface below wall .......... ................... .. 193.1 Three earth pressure theories for active and passive earth
pressures ................. ........................... ..... 223.2 Computation of Rankine active and passive earth pressures for
level backfills ............... ........................ .. 243.3 Rankine active and passive earth pressures for inclined
backfills ................. ........................... .. 263.4 Coulomb active and passive earth pressures for inclined
backfills and inclined walls ........ ................. .. 293.5 Coulomb active earth pressures for a partially submerged
backfill and a uniform surcharge ...... ............... .. 323.6 Coulomb active earth pressures for a backfill subjected to
steady state flow ............. ....................... .. 343.7 Coulomb and log-spiral passive earth pressure coefficients
with 6-0/2 - vertical wall and level backfill ........... .. 373.8 Coulomb and log-spiral passive earth pressure coefficients
with 6-0 - vertical wall and level backfill ... .......... .. 373.9 Example of trial wedge procedure ........ ................ .. 383.10 Example of trial wedge procedure, hydrostatic water table . . .. 403.11 Active and passive earth pressure coefficients with wall
friction-sloping wall ........... ..................... .. 423.12 Active and passive earth pressure coefficients with wall
friction-sloping backfill ......... ................... ... 433.13 Theory of elasticity equations for pressures on wall due to
surcharge loads ............... ........................ .. 463.14 Use of an imaginary load to enforce a zero-displacement
condition at the soil-structure interface ... .......... 474.1 Driving and resisting seismic wedges, no saturation ....... .. 564.2 Variation in KA and KA *cos 6 with kh ..... ............. .. 584.3 Variation in KA1 .cos 6 with kh, 0, and .. .... ............ .. 584.4 Variation in apE with * for 6 equal to 4 vertical wall and
level backfill ................ ........................ .. 604.5 Variation in a with t for 6 equal to zero degrees, vertical
wall and level backfill ......... .................... .. 604.6 Variation in dynamic active horizontal earth pressure
coefficient with peak horizontal acceleration ........... .. 614.7 Equivalent static formulation of the Mononobe-Okobe active
dynamic earth pressure problem ........ ................ .. 624.8 Values of factor FAE for determination of KA. ... .......... .. 634.9 Point of action of P.. ........... ..................... .. 654.10 Static active earth pressure force and incremental dynamic
active earth pressure force for dry backfill .. ......... .. 664.11 Limiting values for horizontal acceleration equal to k g . . 674.12 Modified effective friction angle ....... ................ .. 714.13 Effective unit weight for partially submerged backfills ...... 734.14 Variation aPE with * for 6 equal to 0/2, vertical wall and
level backfill ................ ........................ 754.15 Variation in aPE with * for 6 equal to zero degrees, vertical
wall and level backfill ........... .................... .. 75
xiii
No. Pare
4.16 Equivalent static formulation of the Mononobe-Okabe passivedynamic earth pressure problem ........ ................ .. 76
loading ................... ............................ .. 804.19 Static active earth pressure force including surcharge ..... 814.20 Static active earth pressure force and incremental dynamic
active earth pressure force including surcharge .......... .. 835.1 Model of elastic backfill behind a rigid wall .... .......... .. 1345.2 Pressure distributions on smooth rigid wall for l-g static
horizontal body force .............. ..................... .. 1355.3 Resultant force and resultant moment on smooth rigid wall
for l-g static horizontal body force ....... ............. .. 1366.1 Rigid walls retaining backfills which undergo movements during
during earthquakes ............... ...................... 1446.3 Linear and uniform base pressure distributions ... ......... .. 1476.4 Rigid wall retaining submerged backfill which undergo movements
during no excess pore water pressures ...... ............. .. 1506.5 Rigid wall retaining submerged backfill which undergo movements
during earthquakes, including excess pore water pressures . . 1536.6 Rigid wall retaining submerged backfill which undergo
movements during earthquakes-liquified backfill ... ........ .. 1566.7 Gravity retaining wall and failure wedge treated as a sliding
block ..................... ............................. .. 1596.8 Incremental displacement ............ .................... 1596.9 Forces acting on a gravity wall for a limiting acceleration
equal to N*,g ............... ......................... .. 1627.1 Decrease in failure surface slope of the active and passive
sliding wedges with increasing lateral accelerations ..... .. 2077.2 Reduction in bending moments in anchored bulkhead from wall
flexibility ................. .......................... .. 2097.3 Free earth support analysis distribution of earth pressures,
moments and displacements, and design moment distributions 2107.4 Two distributions for unbalanced water pressures .. ........ .. 2117.5 Measured distributions of bending moment in three model tests
on anchored bulkhead ............ ..................... .. 2137.6 Anchored sheet pile walls retaining backfills which undergo
movements during earthquakes ......... ................. .. 2157.7 Anchored sheet pile wall with no excess pore water pressure due
to earthquake shaking ............. ..................... .. 2177.8 Static and inertial horizontal force components of the Mononobe-
Okabe earth pressure forces .......... .................. .. 2197.9 Distributions of horizontal stresses corresponding to AP . . 222/.10 Horizontal pressure components and anchor force acting on
sheet pile wall ................ ........................ .. 2227.11 Dynamic forces acting on an anchor block .... ............ 2247.12 Anchored sheet pile wall with excess pore water pressures
generated during earthquake shaking ..... .............. 2268.1 Simplified procedure for dynamic analysis of a wall
retaining nonyielding backfill ......... ................ 2328.2 Linear and uniform base pressure distributions ... ......... .. 234
xiv
No. Page
A.1 Dynamic active wedge analysis with excess pore water pressures AlA.2 Equilibrium of horizontal hydrostatic water pressure forces
acting on backfill wedge ........... ................... A3A.3 Dynamic active wedge analysis including a surcharge loading . A7A.4 Dynamic active wedge analysis including a surcharge loading . ASA.5 Dynamic passive wedge analysis with excess pore water
pressure ..................... ........................... A9A.6 Dynamic passive wedge analysis including a surcharge load . ... A12
A.] Dynamic passive wedge analysis including a surcharge load . ... A13B.1 Hydrostatic and westergaard hydrodynamic water pressures
acting along vertical wall during earthquakes ........... .. B2C.A Anchored sheet pile wall design problem .......... ............. ClC.2 Horizontal earth pressure components in free earth support
design .......................... ............................ C3C.3 Horizontal active and passive earth pressure components acting
on a continuous slender anchor ......... ................ C10C.4 Design criteria for deadman anchorage ........ ............... CI1C.5 Distribution of horizontal stresses corresponding to APAE .... C20c.6 Seismic design problem for a continuous anchor blast .. ...... C24C.7 Simplified procedure for siting a continuous anchor wall . ... C28D.1 Earth retaining structure, soil-structure interaction ...... .. D5
xv
CONVERSION FACTORS, NON-SI TO SI (METRIC)UNITS OF MEASUREMENT
CHAPTER I GENERAL DESIGN CONSIDERATIONS FOR WATERFRONT SITES
1.1 Scope and Applicability
This manual deals with the soil mechanics aspects of the seismic designof waterfront earth retaining structures. Specifically, this reportaddresses:
"* The stability and movement of gravity retaining walls and
anchored bulkheads.
"* Dynamic forces against subsurface structures such aswalls of dry docks and U-frame locks.
The report does not address the seismic design of structural frameworks ofbuildings or structures such as docks and cranes. It also does not considerthe behavior or design of piles or pile groups.
The design of waterfront retaining structures against earthquakes isstill an evolving art. The soils behind and beneath such structures often arecohesionless and saturated with a relatively high water table, and hence thereis a strong possibility of pore pressure buildup and associated liquefactionphenomena during strong ground shaking. There have been numerous instances offailure or unsatisfactory performance. However, there has been a lack ofdetailed measurements and observations concerning such failures. There alsoare very few detailed measurements at waterfront structures that have per-formed well during major earthquakes. A small number of model testing pro-grams have filled in some of the blanks in the understanding of dynamicresponse of such structures. Theoretical studies have been made, but withvery limited opportunities to check the results of these calculations againstactual, observed behavior. As a result, there are still major gaps in know-ledge concerning proper methods for analysis and design.
The methods set forth in this report are hence based largely uponjudgement. It is the responsibility of the reader to judge the validity ofthese methods, and no responsibility is assumed for the design of any struc-ture based on the methods described herein.
The methods make use primarily of simplified procedures for evaluatingforces and deformations. There is discussion of the use of finite elementmodels, and use of the simpler finite element methods is recommended in somecircumstances. The most sophisticated analyses using finite element codes andcomplex stress-strain relations are useful mainly for understanding patternsof behavior, but quantitative results from such analyses should be used withconsiderable caution.
This report is divided into eight chapters and five appendixes. Thesubsequent sections in Chapter 1 describe the limit states associated with theseismic stability of waterfront structures during earthquake loadings, the keyrole of liquefaction hazard assessment, and the choice of the design groundmotion(s).
Chapter 2 describes the general design considerations for retainingstructures, identifying the interdependence between wall defornt~tiuns andforces acting on the wall. Additional considerations such as failure surfaces
I
pa-' ing below the wall, failure of anchoring systems for sheet pile walls, andanalysis of the post-seismic condition are also discussed.
The procedures for calculating static earth pressures acting on wallsreLalning yielding backfills are described in Chapter 3. A wall retaining ayielding backfill is defined as a wall with movements greater than or equal tothe values given in Table 1 (Chapter 2). These movements allow the fullmobilization of the shearing resistance within the backfill. For a wall thatmoves away from the backfill, active earth pressures act along the soil-wallinterface. In the case of a wall that moves towards the backfill, displacingthe soil, passive earth pressures act along the interface.
Chapter 4 describes the procedures for calculating seismic earth pres-sures acting on walls retaining yielding backfills. The Mononobe-Okabe theoryfor calculating the dynamic active earth pressure force and dynamic passiveearth pressure force is described. Two limiting cases used to incorporate theeffect of submergence of the backfill in the Mononobe-Okabe method of analysisare discussed: (1) the restrained water case and (2) the free water case.These procedures include an approach for incorporating excess pore water pres-sures generated during earthquake shaking within each of the analyses.
The procedures for calculating dynamic earth pressures acting on wallsretaining nonyielding backfills are described in Chapter 5. A wall retaininga nonyielding backfill is one that does not develop the limiting dynamicactive or passive earth pressures because sufficient wall movements do notoc .. th. -L, r strength of the backfill is not fully mobilized - wallmovements that are less than one-fourth to one-half of Table 1 (Chapter 2)wall movement values. The simplified analytical procedure due to Wood (1973)and a complete soil-structure interaction analysis using the finite elementmethod are discussed.
The analysis and design of gravity walls retaining yielding backfill aredescribed in Chapter 6. Both the preselected seismic coefficient method ofanalysis and the Richards and Elms (1979) procedure based on displacementcontrol are discussed.
Chapter 7 discusses the analysis and design of anchored sheet pilewalls.
The analysis and design of gravity walls retaining nonyielding backfillusing the Wood (1973) simplified procedure is described in Chapter 8.
Appendix A describes the computation of the dynamic active and passiveearth pressure forces for partially submerged backfills using the wedgemethod.
Appendix B describes the Westergaard procedure for computing hydro-dynamic water pressures along vertical walls during earthquakes.
Appendix C contains a design example of an anchored sheet pile wall.
Appeiidiix D is a brief guide to the several types of finite elementmethods that might be used when considered appropriate,
Appendix E summarizes the notation used in this report.
2
1.2 Limit States
A broad look at the problem of seismic safety of waterfront structuresinvolves the three general limit states shown in Figure 1.1 which should beconsidered in design.
1) Gross site instability: This limit state involves lateral earthmovements exceeding several feet. Such instability would be the result ofliquefaction of a site, together with failure of an edge retaining structureto hold the liquefied soil mass in place. Liquefaction of backfill is a prob-lem associated with the site, mostly independent of the type of retainingstructure. Failure of the retaining structure might result from overturning,sliding, or a failure surface passing beneath the structure. Any of thesemodes might be triggered by liquefaction of soil beneath or behind the retain-ing structure. There might also be a structural failure, such as failure ofan anchorage which is a common problem if there is liquefaction of thebackfill.
2) Unacceptable movement of retaining structure: Even if a retainingstructure along the waterfront edge of a site remains essentially in place,too much permanent movement of the structure may be the cause of damage tofacilities immediately adjacent to the quay. Facilities of potential concerninclude cranes and crane rails, piping systems, warehouses, or otherbuildings. An earthquake-induced permanent movement of an inch will seldom beof concern. There have been several cases where movements as large as4 inches have not seriously interrupted operations or caused material damage,and hence have not been considered failures. The level of tolerable displace-ment is usually specific to the planned installation.
Permanent outward movement of retaining structures may be caused bytilting and/or sliding of massive walls or excessive deformations of anchoredbulkheads. Partial liquefaction of backfill will make such movements morelikely, but this limit state is of concern even if there are no problems withliquefaction.
3) Local instabilities and settlements: If a sit- experiences liquefac-tion and yet is contained against major lateral flow, buildings and otherstructures founded at the site may still experience unacceptable damage.Possible modes of failure include bearing capacity failure, excessive seLtle-ments, and tearing apart via local lateral spreading. Just the occurrence ofsand boils in buildings can seriously interrupt operations and lead to costlyclean-up operations.
This document addresses the first two of these limit states. The thirdlimit state is discussed in the National Research Council (1985), Seed (1987),and Tokimatsu and Seed (1987).
1.3 Key Role of Liquefaction Hazard Assessment
The foregoing discussion of general limit states has emphasized problemsdue to soil liquefaction. Backfills behind waterfront retaining structuresoften iro cohesionless soils, and by their location have relatively high watertables. Cohesionless soils may also exist beneath the base or on the water-side of such structures. Waterfront sites are often developed by hydraulicfilling using cohesionless soils, resulting in low density fills that are
3
(a.) WATERFRONT STRUCTURES
(b.) GROSS SITE INSTABILITY
III~ ~ ~ ~ El S."1 l
(c.) UNACCEPTABLE LATERAL MOVEMENT
(d.) LOCAL INSTABILITY; SETTLEMENT
Figure 1.1 Overall limiit states at waterfronts
4
susceptible to liquefaction. Thus, liquefaction may be a problem for build-ings or other structures located well away from the actual waterfront.Hence, evaluation of potential liquefaction should be the first step in analy-sis of any existing or new site, and the first step in establishing criteriafor control of newly-placed fill. Methods for such evaluation are set forthin numerous articles, including the National Research Council (1985) and Seed,Tokimatsu, Harder and Chung (1985).
The word "liquefaction" has been applied to different but relatedphenomena (National Research Council 1985). To some, it implies a flow fail-ure of an earthen mass in the form of slope failure or lateral spreading,bearing capacity failure, etc. Others use the word to connote a number ofphenomena related to the buildup of pore pressures within soil, including theappearance of -and boils and excessive movements of buildings, structures, orslopes. Situations in which there is a loss of shearing resistance, resultingin flow slides or bearing capacity failures clearly are unacceptable. How-ever, some shaking-induced increase in pore pressure may be acceptable, pro-vided it does not lead to excessive movements or settlements.
Application of the procedures set forth in this manual may require evil-uation of: (a) residual strength for use in analyzing for flow or bearingcapacity failure; or (b) buildup of excess pore pressure during shaking. As ageneral design principle, the predicted buildup of excess pore pressurk shouldInot exceed 30 to 40 percent of the initial vertical effective stress, exceptin cases where massive walls have been designed to resist larger pore pres-sures and where there are no nearby buildings or other structures that wouldbe damaged by excessive settlements or bearing capacity failures. With -ervloose and contractive cohesionless soils, flow failures occur when the resid-ual excess pore pressure ratio reaches about 40 percent (Vasquez and Dobry1988, or Marcuson, Hynes, and Franklin 1990).* Even with soils lesssusceptible to flow failures, the actual level of pore pressure buildupbecomes uncertain and difficult to predict with confidence when the excesspore pressure ratio reaches this level.
Remedial measures for improving seismic stability to resistliquefaction, the buildup of excess pore water pressures, or unacceptablemovements, are beyond the scope of this report. Remedial measures are dis-cussed in numerous publications, including Chapter 5 of the National ResearchCouncil (1985).
1.4 Choice of Design Ground Motions
A key requirement for any analysis for purposes of seismic design is aquantitative specification of the design ground motion. In this connection,
* The word "contractive" reflects the tendency of a soil specimen to decreasein volume during a drained shear test. During undrained shearing of a con-tractive soil specimen, the pore water pressure increases, in excess of thepre-sheared pore water pressure value. "Dilative" soil specimens exhibitthe opposite behavior; an increase in volume during drained shear testingand negative excess pore water pressures during undrained shear testing.Loose sands and dense sands are commonly used as examples ofsoils exhibiting contractive and dilative behavior, respectively, duringshear.
it is important to distinguish between the level of ground shaking that astructure or facility is to resist safely and a parameter, generally called aseismic coefficient that is used as input to a simplified, pseudo-staticanalysis.
1.4.1 Design Seismic Event
Most often a design seismic event is specified by a peak acceleration.However, more information concerning the ground motion often is necessary.Duration of shaking is an important parameter for analysis of liquefaction.Magnitude is used as an indirect measure of duration. For estimatingpermanent displacements, specification of either peak ground velocity orpredominant period of the ground motion is essential. Both duration andpredominant periods are influenced strongly by the magnitude of the causativeearthquake, and hence magnitude sometimes is used as a parameter in analyses.
Unless the design event is prescribed for the site in question, peakaccelerations and peak velocities may be selected using one of the followingapproaches:
(I) By using available maps for the contiguous 48 states. Such maps maybe found in National Earthquake Hazards Reduction Program (1988). Such maps
are available for several different levels of risk, expressed as probability
of non-exceedance in a stated time interval or mean recurrence interval. A
probability of non-exceedance of 90 percent in 50 years (mean recurrenceintt-rval of 475 years) is considered normal for ordinary buildings.
(2) By using attenuation relations giving ground motion as a function of
magnitude and distance (e.g. attenuation relationships for various tectonic
environments and site conditions are summarized in Joyner and Boore (1988).
This approach requires a specific choice of a magnitude of the causative
earthquake, requiring expertise in engineering seismology. Once this choice
is made, the procedure is essentially deterministic. (;enerally it is neces-
sary to consider various combinations of magnitude and distance.
(3) By a site-specific probabilistic seismic hazard assessment (e.g.National Research Council 1988). Seismic source zones must be identified and
characterized, and attenuation relations must be chosen. Satisfactory accom-
plishment of such an analysis requires considerable expertise and experience,with input from both experienced engineers and seismologists. This approach
requires selection of a level of risk.
It is of greatest importance to recognize that, for a given site, the
ground motion description suitable for design of a building may not be appro-
priate for analysis of liquefaction.
Locai soil conditions: The soil conditions at a site should be con-
sidered when selecting the design ground motion. Attenuation relations are
available for several different types of ground conditions, and hence the
analyses in items (2) and (3) might be made for any of these particular sitecondition,;. However, attenuation relations applicable to the soft groundconditions often found at waterfront sites are the least reliable. The maps
referred to unLder item (1) apply for a specific type of ground condition:
soft rock. More recent maps will apply for devp, firm alluvium, after
revision of the document referenced in item (1). Hence, it generally i.: nec-
6
essary to make a special analysis to establish the effects of local soil con-ditions.
A site-specific site response study is made using one-dimensional analy-ses that model the vertical propagation of shear waves through a column ofsoil. Available models include the computer codes SHAKE (Schnabel, Lysmer,and Seed 1972), DESRA (Lee and Finn 1975, 1978) and CHARSOL (Streeter, Wylie,and Richart 1974). These programs differ in that SHAKE and CHARSOL are for-mulated using the total stress procedures, while DESRA is formulated usingboth total and effective stress procedures. All three computer codesincorporate the nonlinear stress-strain response of the soil during shaking intheir analytical formulation, which has been shown to be an essentialrequirement in the dynamic analysis of soil sites.
For any site-specific response study, it first will be necessary todefine the ground motion at the base of the soil column. This will require atlestablishment of a peak acceleration for firm ground using one of the threemethods enumerated above, and the selection of several representatives timehistories of motion scaled to the selected peak acceleration. These timehistories must be selected with considerable care, taking into account themagnitude of the causative earthquake and the distance from the epicenter.Procedures for choosing suitable time histories are set forth in Seed andIdriss (1982). Green (1992), and procedures are also under dewvlopment by thel.US Army Corps of Engineers.
If a site response analysis is made, the peak ground motions will ingeneral vary vertically along the soil column. Depending upon the type ofanalysis being made, it may be desirable to average the motions over depth toprovide a single input value. At each depth, the largest motion computed inany of the several analyses using different time histories should be used.
If finite element analyses are made, it will again be necessary toselect several time histories to use as input at the base of the grid, or atime history corresponding to a target spectra (retfer to page 54 ot Seed andIdriss 1982 or Green 1992).
1.4.2 Seismic Coefficients
A seismic coefficient (typical symbols are kh and k) is a dimensionl.essnumber that, when muultiplied times the weight of some body, gives a pse,.udo-
static inertia force for use in analysis and desigl. The coeteficieont s k, midk, are, in effect, decimal fractions of the acceleration of gravity (g). Fulrsome analyses, it is appropriate to use values of khg or k~g smaller thian thepeak accelerations anticipated during the design earthquake event.
For analys;is of liquefaction, it is conlVenti onal to utSe 0.,65 ltes the
peak accel erationi. The reason is.; that liquefaction is controlled by theamplitude ot as succession of cycles of motion, rather than just by thbe i iltlyl'
largest peak. The most common, empirical ,-thbds of. analys;iq described in theNational Research Council (1985) and Seed, Tokimatsu, Harder, and Chung (I485)presume use of this reduction factor.
In design of buildings, it is; common practice to bas;e dveign upon a
sei.tsmic coefficient corresponding to a ground motion smaller th an the designground motion. It is recognized that a building designed on this; basis may
7
likely yield and even experience some notltif-threateuing dimae,- if the dt-ri gti
ground motion actually occurs. The permitted reduction depends upon the duoc-
tility of the structural system; that is, the ability of tO,- structure to
undergo yielding and yet remain intact so as to continue to support saI(-ly the
normal dead and live loads, This approach represents a compromise bhtweviidesirable performance and cost of earthquake resistance.
The same principle applies to earth structures, once it has been estab-lished that site instability caused by liquefaction is riot a prob)lem. It aretaining wall system yields, some permanent outward displacement will occur,which often is an acceptable alternative to significantly increased cost ofconstruction. However, there is no generally accepted set of ruleS forselecting an appropriate seismic coefficient. The displacement controlledapproach to design (Section 6.3) is in effect a systematic and rationai method
for evaluating a seismic coefficient based upon allowable permanent displace -
ment. The AASHTO seismic design for highway bridges (1483) is an example of'
design guidance using the seismic coefficient method for earth retaining
structures.* AASHTO reconunends that a value of kt, = 0,SA be used for most
cases if the wall is designed to move up to IOA (in.) where A is peak groundacceleration coefficient for a site (acceleration = Ag). However, use of kht
O.5A is not necessarily conservative for areas of high seis•micity (se't. Thitman
and Liao 1985).
Various relationships have been proposed for estimat ing permaneut dis-placements, as a function of the ratio kh/A and parameters describing thek
ground motion. Richards and Elms (1979) and Whitman and Liao (1985) use peak
ground acceleration and velocity, while Makdisi and Seed (1979) use pe-ak
ground acceleration and magnitude. Values for the ratio V/dm,', may he used,both for computations and to relate the several methods. Typical values forthe ratio V/am,,ý are provided in numerous publications discussing ground shak-ing, including the 1982 Seed and Idriss, and the 1983 Newmark and Hall FIERImonographs, and Sadigh (1983). Seed and Idriss (1982), Newmark and Hall(1983), and Sadigh (1973) report that values for the ratio V/a..• varies withgeologic conditions at the site. Additionally, Sadigh (1973) reports that thevalues for the ratio V/area, varies with earthquake magnitude, the ratio in-
creasing in value with increasing magnitude earthquake.
Based upon simplified assumptions and using the Whitman ind Liao rela-tionship for earthquakes to magnitude 7, kh values were computed:
A = 0.2 A ().A
Displacement < 1 in. k, = 0.13 kh O.30
Displacement < 4 in. kh 0.10 kh = 025
These numbers are based upon V/Ag = 50 in/sec/g (Sadigh 1981), which ap'lliesto deep stiff soil sites (geologic condition); smalle-r kh would bh appropriatefor hard (e.g. rock) sites. The Whitman and Liao study did not directlyaddress the special case of sites located within opicentral regions.
* The map in AASHTO (1983) is not accepted widely as being representat ive of
the ground shaking hazard.
8
The value assigned to kh is to be established by the seismic design teamfor the project considering the seismotectonic structures within the region,or as specified by the design agency.
1.4.3 Vertical Ground Accelerations
The effect of vertical ground accelerations upon response of waterfrontstructures is quite complex. Peak vertical accelerations can equal or exceedpeak horizontal accelerations, especially in epicentral regions. However, thepredominant frequencies generally differ in the vertical and horizontal com-ponents, and phasing relationships are very complicated. Where retainingstructures support dry backfills, studies have shown that vertical motionshave little overall influences (Whitman and Liao 1985). :Towever, the Whitmanand Liao study did not directly address the special case of sites locatedwithin epicentral regions. For cases where water is present within soils oragainst walls, the possible influence of vertical motions have received littlestudy. It is very difficult to represent adequately the effect of verticalmotions in pseudo-static analyses, such as those set forth in this manual.
The value absigned to kv is to be established by the seismic design teamfor the project considering the seismotectonic structures within the region,or as specified by the design agency. However, pending the results of furtherstudies and in the absence of specific guidance for the choice of k, forwaterfront structures the following guidance has been expressed in literature:A vertical seismic coefficient be used in situations where the horizontalseismic coefficient is 0.1 or greater for gravity walls and 0.05 or greaterfor anchored sheet pile walls. This rough guidance excludes the special caseof structures located within epicentral regions for the reasons discussedpreviously. It is recommended that three solutions should be made: one assum-ing the acceleration upward, one assuming it downward, and the other assumingzero vertical acceleration. If the vertical seismic coefficient is found tohave a major effect and the use of the most conservative assumption has amajor cost implication, more sophisticated dynamic analyses should probably beconsidered.
I9
CHAPTER 2 GENERAL DESIGN CONSIDERATIONS FOR RETAINING WALLS
2,1 Approaches to Design for Various Classes of Structure
The basic elements of seismic design of waterfront retaining structuresare a set of design criteria, specification of the static and seismic forcesacting on the structure in terms of magnitude, direction and point of applica-tion, and a procedure for estimating whether the structure satisfies thedesign criteria.
The criteria are related to the type of structure and its function.Limits of tolerable deformations may be specified, or it may be sufficient toassure the gross stability of the structure by specifying factors of safetyagainst rotational and sliding failure and overstressing the foundation. Inaddition, the structural capacity of the wall to resist internal moments andshears with adequate safety margins must be assured. Structural capacity is acontrolling factor in design for tied-back or anchored walls of relativelythin section such as sheet pile walls. Crib walls, or gravity walls composedof blocks of rock are examples of structures requiring a check for safetyagainst sliding and tipping at each level of interface between structuralcomponents.
Development of design criteria begins with a clear concept of the fail-ure modes of the retaining structure. Anchored sheet pile walls display themost varied modes of failure as shown in Figure 2.1, which illustrates bothgross stability problems and potential structural failure modes. The morerestricted failure modes of a gravity wall are shown in Figure 2.2. A faiiuresurface passing below a wall can occur whenever there is weak soil in thefoundation, and not just when there is a stratum of liquified soil.
Retaining structures must be designed for the static soil and waterpressures existing before the earthquake and for superimposed dynamic andinertia forces generated by seismic excitation, and for post seismic condi-tions, since strengths of soils may be altered as a result of an earthquake.Figure 2.3 shows the various force components using an anchored sheet pilewall example from Chapter 7. With massive walls, it is especially importantto include the inertia force acting on the wall itself. There are super-imposed inertia forces from water as well as from soil. Chapters 3, 4, and 5consider the evaluation of static and dynamic earth and water pressures.
2.2 Interdependence between Wall Deformations and Forces Acting on the Wall
The interdependence between wall deformations and the static and dynamicearth pressure forces acting on the wall has been demonstrated in a number oftests on model retaining walls at various scales. An understanding of thisinterdependence is fundamental to the proper selection of earth pressures foranalysis and design of walls, The results from these testing programs aresummarized in the following two sections.
2.2.1 Wall Deformations and Static Earth Pressure Forces
The relationships between the movement of the sand backfills and themeasured static earth pressure forces acting on the wall are shown in Fig-ure 2.4. The figure is based on data from the model retaining wall tests con-ducted by Terzaghi (19"34, 1936, and 1954) at MIT and the tests by Johnson
II
CU
44
00
-4
%C D
II Q t-
.0&400
i~j CU1
Li ý
'Li 0
-4
0 c
CO
'-4
-j 0
I&J 0
0i. -4
12
Movement
"N /./
Sliding Rotation
"N, AM
Overturning Rotaton
v \,,, " 10"")
IN
N //
N.
Liquified Substrata
Slip Within Substrata
Figure 2.2 Rigid walls retaining backfills whichundergo movements during earthquakes
ACTER ACTIVE PASIVE WATEREARTH PRESSURE EARTH EARTH PRESSURE
PRESSURE PRESSURE PRESSURE
Figure 2.3 Horizontal pressure components and anchor force acting onsheet pile wall
(1953) at Princeton University, conducted under the direction ofTschebotarioff. The backfill movements are presented as the movement at thetop of the wall, Y, divided by the height of the wall, H, and the earth pres-sure forces -ire expressed in terms of an equivalent horizontal earth pressurecoefficient, Kh. Kh is equal to the horizontal effective stress, Oh', dividedby the vertical effective stress, a,'.
The test results in Figure 2.4 show that as the wall is rotated fromvertical (Y = 0) and away from the backfill, the horizontal earth pressurecoefficient acting on the wall decreases from the value recorded prior tomovement of the wall. The zero wall movement horizontal earth pressure coef-ficient is equal to the at-rest value, K,. When the backfill movements at thetop of the wall, Y, attain a value equal to 0.004 times the height of thewall, H, the earth pressure force acting on the wall decreases to the limitingvalue of the active earth pressure force, PA, and the earth pressure coeffi-cient reduces to the active coefficient, KA.
In a second series of tests, the wall was rotated from vertical in theopposite direction, displacing the backfill. The horizontal earth pressurecoefficient acting on the wall increased from the K, value. When the backfillmovements at the top of the wall, Y, attain a value equal to 0.04 times theheight of the wall, H, the earth pressure force acting on the wall increasesto the other limiting value of the passive earth pressure force, Pp, with acorresponding passive earth pressure coefficient, Kp. The movements requiredto develop passive earth pressures are on the order of ten times the movementsrequired to develop active earth pressures.
With the soil in either the active or passive state, the magnitude ofthe backfill displacements are sufficient to fully miobil]ize the shear strength
14
r MITPRINCETON TEST RESULTS TEST
RESULTS
10K p , -' - K A
S8 o. 8-"• --DENSESN
tt SAN D -Y,~ 6i Pp • 5
"" I y '2 LUL 4
V) 3
t-..-LOOSE SAND -""LJi
S1.0z 0.80
"2Y 0.6I 0 .~i SAND y,q • .
u- 0 ., V.' Kl' •u0.3 •
oL .ULOOSE SANVD - -L( 0.2
0 MEDIUM SAND
4p -YH0.1
LEGEND 0.06 0.04 0.02 0 0.002 0.004
Y- HORIZONTAL DISPLACEMENT WALL ROTATION, Y/HH - HEIGHT OF THE WALL
After NAVFAC DM-7.2Figure 2.4 Effect of wall movement on static horizontal earth pressures
of soil within a wedge of backfill located directly behind the heel of thewall. With the soil wedge in a state of plastic equilibrium, PA or Pp may becomputed using either Rankine's or Coulomb's theory for earth pressures or thelogarithmic spiral procedures, as described in Chapter 3. The values for KA
and Kp measured in above tests using backfills placed at a range of densitiesagree with the values computed using the appropriate earth pressure theories.
The test results show that the relationship between backfill displace-ments and earth pressures varies with the relative density of the backfill.Table I lists the minimum wall movements required to reach active and passiveearth pressure conditions for various types of backfills. Clough and Duncan,(1991) and Duncan, Clough, and Ebeling (1990) give the following easy-to-remember guidelines for the amounts of movements required to reach the pres-sure extremes;; for a cohesionless backfill the movement required to reach theminimum active condition is no more than about I inch in 20 feet (A/H = 0.004)
15
and the movement required to reach the minimum passive condition is no more
than about I inch in 2 feet (A/H - 0.04).
Table 1
Approximate Magnitudes of Movements Required to Reach MinimumActive and Maximum Passive Earth Pressure Conditions
aY - movement of top of wall required to reach minimum active or maximumpassive pressure, by tilting or lateral translation.H - height of wall.
2.2.2 Wall Deformations and Dynamic Earth Pressure Forces
The interdependence between wall deformations and the forces acting onthe wall has been extended to problems involving dynamic earth pressures intests on model retaining walls conducted at the University of Washington andat research laboratories in Japan. The University of Washington studiesinvolved a series of static and dynamic tests using an instrumented modelretaining wall mounted on a shaking table, as described by Sherif, Ishibashiand Lee (1982), Sherif and Fang (1984a), :nerif and Fang (1984b), andIshibashi and Fang (1987). The shaking table used in this testing program iscapable of applying a harmonic motion of constant amplitude to the base of thewall and the backfill. In each of the tests, the wall was constrained eitherto translate without rotation, to rotate about either the base or the top ofthe wall, or some combination of translation and rotation. During the courseof the dynamic earth pressure tests, the wall was moved away from the backfillin a prescribed manner while the base was vibrated. Movement of the wall con-tinued until active dynamic earth pressures acted along the back of the wall.Static tests were also carried out for comparison.
The active state during the dynamic tests occurred at almost the samewall displacement as in the static tests, at a value of wall rotation equal to0.001 for the static and dynamic test results that are shown in Figure 2.5 ondense Ottawa sand. This was also the finding in a similar program of testingusing a model wall retaining dense sand, as reported by Ichihara and Matsuzawa(1973) and shown in Figure 2.6. The magnitude of these wall movements are ingeneral agreement with those measured in the MIT testing program shown inFigure 2.4 and those values reported in Table 1.
There has been relatively little experimental investigation of thedynamic pas:ive case, however, the available results indicate that consider-able wall moaements are required to reach the full passive condition.
16
0.7
0.6
0.5k TEST NO.1125DENSE OTTAWA SAND (0-1.6 CR/CC)
\0.3 ....
0.2
0.1
00 0.0005 0.0010 0.0015
WALL ROTATION, Y/H
(a). Static Horizontal Earth Pressure
0.5o_ ,DENSE OTTAWA SAND0. (Y . _MX--"1.66 GR/CC)
o 0.3
v 0.2\ K -M l.. . . . . . . . . .
0,1
0T0 0.0005 0.0010 0.0015
WALL ROTATION, Y/H
(b). Dynamic HorizontalEarth Pressure
-0.6
-0.4
-0.2
0
0.2
1.4
0.60 0.0005 0.0010 0.0015
WALL ROTATION, Y/H(c). Base Acceleration
From Sherif And Fanq (1983)
Figure 2.5 Effect of wall movement on static and dynamic
horizontal earth pressures
17
WALL ROTATION, Y/H
0 0.0005 0,00101.0 1
- - - --- RANGE OF STATIC EARTH
- PRESSURE
0.90.9_ MEAN EARTH PRESSURE
DURING VIBRATION
, 0 .8
Lj
V)0.7
M.
0.6
W
0 \\I\"_____ ______ __N 0.5
VALUE OF ACCELERATION
0.4 488 GALS48
.A 0.2 STATI HORZOTA ---------- W ý -84
EARTH PRESSURE._ ,0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
MEAN WALL DISPLACEMENT, Y (mm)
From Ichihora and Motsuzowa (1973).
Figure 2.6 Effect of wall movement on static and
dynamic horizontal earth pressures
The Table 1 values are used as rough guidance throughout this report,pending the results from additional research into the relationships betweendynamic earth pressures and wall displacements.
2.3 Comments on Analyses for Various Cases
The greatest part of this report is devoted to the evaluation of static
and dynamic earth and water pressures against walls, and the use of thesepressures in the analysis of the equilibrium of such walls. Such analyses arepresented and discussed in Chapters 6, 7, and 8. The examples and discussiongenerally presume uniform and cohesionless backfills.
The soil strength parameters used in the analysis must be consistentwith the displacements. Large displacements, or an accumulation of smaller
18
INERTIA FORCES
IN E R T IA O C S 'P O E T A/ FAILURE
• •.• 'E•.• /SURF ACE
Figure 2.7 Failure surface below wall
displacements tend to support the use of residual strength parameters, ascompared to peak values. Wall displacements must also be considered whenassigning the foundation to structure interface strength parameters.
There are two potentially important situations that are not discussed orillustrated in detail in this manual. A brief treatment of these casesappears in the following subsections.
2.3.1 Analysis of Failure Surf-es Passing below Wall
This situation may be a problem if soils of low strength exist below awall, either because the before-earthquake strength of this material is smallor because the strength of the soil decreases as a result of earthquakeshaking.
Such cases may be studied using principles from the analysis of slopestability (e.g. Edris and Wright 1987). Figure 2.7 shows again the diagramfrom Figure 2.1, and indicates the inertia forces that must be considered inaddition to the static forces. Evaluation of suitable strengths may requirecareful consideration. Appropriate excess pore pressures should be appliedwhere the failure surface passes through cohesionless soils; see Seed andHarder (1990), Marcuson, Hynes, and Franklin (1990). With cohesive soils, thepossibility of degradation of strength by cyclic straining should beconsidered. A safety factor ranging from 1.1 to 1.2 is considered satis-factory: provided that reasonable conservative strengths and seismiccoefficients have been assigned. With a smaller safety factor, permanentdisplacements may be estimated using the Makdisi-Seed procedure (Makdisi andSeed 1979) or the Sarma-Ambraseys procedure (Hynes-Griffin and Franklin 1984).
2.3.2 Analysis of Post-Seismic Condition
There are four circumstances that may cause the safety of a retainingstructure to be less following an earthquake than prior to the earthquake.
1. Persistent excess pore pressures on the landside of the wall. Anysuch buildup may be evaluated using procedures described in Seed and Harder(1990) and Marcuson, Hynes, and Franklin (1990). The period of time duringwhich such excess pressures will persist can be estimated using appropriateconsolidation theory.
19
2. Residual earth pressures as a result of seismic straining. There isevidence that such residual pressures may reach those associated with theat-rest condition (see Whitman 1990).
3. Reduction in strength of backfill (or soils beneath or outside oftoe of wall) as a result of earthquake shaking. In the cxtrzime case, only theresidual strength (see the National Research Council 1985; Seed 1987; Seed andHarder 1990; Marcuson, Hynes, and Franklin 1990; Poulos, Castro, and France1985; and Stark and Mesri 1992) may be available in some soils. Residualstrengths may be treated as cohesive shear strengths for evaluation of corre-sponding earth pressures.
4. Lowering of water level on waterside of wzll during the fallingwater phase of a tsunami. Estimates of possible water level decrease duringtsunamis require expert input.
The possibility that each of these situations may occur must be considered,and where appropriate the adjusted earth and fluid pressures must beintroduced into an analysis of static equilibrium of the wall. Safety factorssomewhat less than those for the usual static case are normally consideredappropriate.
Methods for evaluating static earth pressures are essential for design.They also form the basis for simplified methods for determining dynamic earthpressures associated with earthquakes. This chapter describes analyticalprocedures for computing earth pressures for earth retaining structures withstatic loadings. Three methods are described: the classical earth pressuretheories of Rankine and Coulomb and the results of logarithmic spiral failuresurface analyses. The three failure mechanisms are illustrated in Figure 3.1.
The Rankine theory of active and passive earth pressures (Rankine 1857)determines the state of stress within a semi-infinite (soil) mass that,because of expansion or compression of the (soil) mass, is transformed from anelastic state to a state of plastic equilibrium. The orientation of thelinear slip lines within the (soil mass) are also determined in the analysis.The shear stress at failure within the soil is defined by a Mohr-Coulomb shearstrength relationship. The resulting failure surfaces within the soil massand the corresponding Rankine active and passive earth pressures are shown inFigure 3.1 for a cohesionless soil.
The wedge theory, as developed by Coulomb (1776), looks at the equili-brium of forces acting upon a soil wedge without regard to the state of stresswithin the soil mass. This wedge theory assumes a linear slip plane withinthe backfill and the full mobilization of the shear strength of the soil alongthis plane. Interface friction between the wall and the backfill may be con-sidered in the analysis.
Numerous authors have developed relationships for active and passiveearth pressure coefficients based upon an assumption of a logarithmic failuresurface, as illustrated in Figure 3.1. One of the most commonly used sets ofcoefficients was tabulated by Caquot and Kerisel (1948). Representative KA
and Kp values from that effort are illustrated in Table 3 and discussed inSection 3.5. NAVFAC developed nomographs from the Caquot and Kerisel efforts,and are also included in this chapter (Figures 3.11 and 3.12).
Rankine's theory, Coulomb's wedge theory, and the logarithmic spiralprocedure result in similar values for active and passive thrust when theinterface friction between the wall and the backfill is equal to zero. Forinterface friction angles greater than zero, the wedge method and the loga-rithmic spiral procedure result in nearly the same values for active thrust.The logarithmic spiral procedure results in accurate values for passive thrustfor all values of interface friction between the wall and the backfill. Theaccuracy of the passive thrust values computed using the wedge methoddiminishes with increasing values of interface friction because the boundaryof the failure block becomes increasingly curved.
This procedure is illustrated in example 1 at the end of this chapter.
21
0)
0 CL0
Z W4
CC C--. 0zzjtD
-2 24
0 0 u
0.)
U <,< CL
(0
CXC
22
3.2 Rankine Theory
The Rankine theory of active and passive earth pressures is the simplestof the earth pressure theories. It is assumed that the vertical stress at anydepth is equal to the depth times the unit weight of the overlying soil plusany surcharge on the surface of the ground. Horizontal stresses are thenfound assuming that shear resistance is fully mobilized within the soil. Theforces and stresses corresponding to these two limiting states are shown inFigure 3.2 for a vertical retaining wall of height H. The effects of sur-charge and groundwater pressures may be incorporated into the theory.
The backfill in Figure 3.2 is categorized as one of three types, accord-ing to the strength parameters assigned for the soil: frictional (c - 0, 0 >0), cohesive (c = S 4, 0 - 0) or a combination of the two (c > 0, 4 > 0). Botheffective and total stress methods are used in stability analyses of earthretaining structures. In an effective stress analysis the Mohr-Coulomb shearstrength relationship defines the ultimate shearing resistance, Tf, of thebackfill as
rf = c + a I (1)
where c is the effective cohesion, a,' is the effective normal stress on thefailure plane, and 4 is the effective angle of internal friction. The effec-tive stress, a', is equal to the difference between the total stress, a, andthe pore water pressure, u.
a = a - u (2)
The effective stress is the portion of total stress that is carried by thesoil skeleton. The internal pore water pressures, as governed by seepageconditions, are considered explicitly in the effective stress analysis. Forthe total stress methods of analysis, the strength of the soil is equal to theundrained strength of the soil, S,.
•f = Sf (3)
The internal pore water pressures are not considered explicitly in the totalstress analysis, but the effects of the pore water are reflected in the valueof S".
3.2.1 Rankine Theory - Active Earth Pressures - Cohesionless Soils
Active earth pressures result when the wall movements away from thebackfill are sufficient to mobilize fully the shearing resistance within thesoil mass behind the wall.
If the soil is frictional and dry, the horizontal effective stress atany depth is obtained from the vertical effective stress, -yZ, using the activecoefficient KA:
A = KA-Yz (4)
23
(a) FRICTIONAL RESISTANCE. I COHESIVE SOLNO COMBINED COHESION AND FRICTION
NO COHESION FRICTIONAL RESISTANCE
ACTIVE PRESSURES
to C -- VEaE ,WZl1TA.--7 (b) IcI
._i..-, 'cgTESO ZCV .o jjirA// )" SCR
i' -. FAILURE G / % .C
SURFRLTANT -, SUNEI
CA YC
KA -TAN2145- @/2) *' 2C/Y' *0 o TAN (4,5. */21
C-0 KAZA IlkZ-C 2ALR f UFC
KoA - T' 2 P"A 012) 2C _,
2 0A * YZ TAN2
(45-0/2)-2C TM4C45-*/2)
PA KAYH2/2 PA YH2/2-2CH°- 5 - PA "I-) TAN2 (45-'0/2)-2CH TA/N(45-"/2)y 2C 2
/ T
PASSIVE PRESSURESI• f /t VE MENT I 0 2C M,. 2C) rT• 4 - #12J
Kp TAN2 (45. 0/2) 0'p YZ.2C 0'I, - YZ TAN2(45.0/21.2C TAN'd45.*/2)011 Kp YZ Pp. '/2 YH2.2CH Pp - ( I TAN 2 (4541/212CH0 7AN(45*0/2)
Pp Kp-fH 2/2 I1-
After NAVFAC DM-7.2
Figure 3.2 Computation of Rankine active and passive earth pressures forlevel backfills
24
If there are zero shear stresses on vertical and horizontal planes, theRankine active earth pressure coefficient, KA, is equal to
KA = tan2 (45 - 0/2). (5)
The variation in the active earth pressure is linear with z, as shown inFigure 3.2 (a). A planar slip surface extends upwards from the heel of thewall through the backfill, inclined at an angle OA from horizontal. For fric-tional backfills, aA is equal to
a^ = 45 + 0/2. (6)
PA is the resultant force of the a, distribution and is equal to
PA = KA yH2 (7)
acting normal to the back of the wall at one-third H above the heel of wall.In these expressions, I is the dry unit weight.
If the soil is saturated with water table at the surface, the foregoingequations still apply but - is replaced by 1b, the buoyant unit weight.Equations 4 and 7 give the effective stresses and the active thrust from themineral skeleton, and water pressures must be added.
The Rankine active earth pressure coefficient for a dry frictional back-fill inclined at an angle 6 from horizontal is determined by computing theresultant forces acting on vertical planes within an infinite slope verging oninstability, as described by Terzaghi (1943) and Taylor (1948). KA is equalto
KA = coso cos# - Vcos 2p - cos 2 (8)cosp + cos 2# - cos 2
with the limitation that 6 is less than or equal to 0. Equation 4 stillapplies but is inclined at the backfill slope angle P, as shown in Figure 3.3.The distribution of oa is linear with depth along the back of the wall. Thus,there are shear stresses on vertical (and hence horizontal) planes. PA iscomputed using Equation 7. It is inclined at an angle P from the normal tothe back of the wall, and acts at one-third H above the heel of the wall.
3.2.2 Rankine Theory - Active Earth Pressures - Cohesive Soils -
General Case
For the cases shown in Figure 3.2 (b) and (c), the active earth pres-sure, a., normal to the back of the wall at depth z is equal to
25
Z~ V
A HH •
3 0 Foilure 3 alr" Surface 0.....oPA oA' " P ;sr,
Active Pressures Passive Pressures
H - Height of WallS- Slope AngleFor Granular BackFill 4 > 0, C - 0
Figure 3.3 Rankine active and passive earth pressures for inclinedbackfills
Oa = 1tzKA - 2 cv/X (9)
The PA and cxA relationships for backfills whose strengths are defined using Suor an effective cohesion and effective angle of internal friction are given inthe figure.
According to Equation 9, tensile stresses develop to a depth Z0 at thetop of the backfill to wall interface in a backfill whose shear strength iseither fully or partially attributed to the cohesion or undrained strength. Agap may form within this region over time. During rainstorms, these gaps willfill with water, resulting in hydrostatic water pressures along the back ofthe wall to depth Zo. Tensile stresses are set equal to zero over the depthZ, when applying this theory to long term wall designs because c' goes to zerowith time for clayey soils due to changes in water content. For clayey back-fills, retaining walls are designed using Terzaghi and Peck's (1967) equiva-lent fluid pressure values rather than active earth pressures because earthpressure theories do not account for the effects of creep in clayey backfills(Clough and Duncan 1991).
3.2.3 Rankine Theory - Passive Earth Pressures
The derivation of the Rankine theory of passive earth pressures followsthe same steps as were used in the derivation of the active earth pressurerelationships. The forces and stresses corresponding to this limiting stateare shown in Figure 3.2 (d), (e), and (f) for a vertical wall retaining thethree types of soil backfill. The effects of surcharge and groundwaterpressures are not included in this figure. To develop passive earthpressures, the wall moves towards the backfill, with the resultingdisplacements sufficient to fully mobilize the shear resistance within the
26
soil mass (Section 2.2.1). The passive earth pressure, op, normal to theback of the wall at depth z is equal to
or y= tzKp + 2cy- (10)
and the Rankine passive earth pressure coefficient, Kp, for level backfill isequal to
Kp = tan2 (45 + 0/2). (11)
A planar slip surface extends upwards from the heel of the wall through thebackfill and is inclined at an angle ap from horizontal, where ap is equal to
ap = 45 - 0/2. (12)
Pp is the resultant force of the ap distribution and is equal to
Pp = K H2 (13)
for dry frictional backfills and is normal to the back of the wall at one-third H above the heel of the wall. The Pp and ap relationships for back-fills whose strengths are defined using S,, or an effective cohesion andeffective angle of internal friction are given in Figure 3.2.
This procedure is illustrated in example 2 at the end of this chapter.
Kp for a frictional backfill inclined at an angle 0 from horizontal isequal to
Kp = coso cos_ + _cossfl - cos 2 0 (14)
cosp - ýcos 2f - cos 2 0
with the limitation that P is less than or equal to 0. Pp is computed usingEquation 13. It is inclined at an angle 6 from the normal to the back of thevertical wall, and acts at one-third H above the back of the wall as shown inFigure 3.3. With c - 0, ap from Equation 10 becomes
u = 7tzKp. (15)
The distribution of ap is linear with depth along the back of the wall and isinclined at the backfill slope angle /3, as shown in Figure 3.3.
27
3.3 Coulomb Theory
The Coulomb theory of active and passive earth pressures looks at theequilibrium of the forces acting on a soil wedge, assuming that the wall move-ments are sufficient to fully mobilize the shear resistance along a planarsurface that extends from the heel of the wall into the backfill as shown inFigure 3.4. Coulomb's wedge theory allows for shear stresses along the wallto backfill interface. The forces corresponding to the active and passivestates of stress are shown in Figure 3.4 for a wall with a face inclined atangle +0 from vertical, retaining a frictional backfill inclined at angle +f.The effects of surcharge and groundwater pressures are not included in thisfigure.
3.3.1 Coulomb Theory - Active Earth Pressures
In the active case the wall movements away from the backfill are suffi-cient to fully mobilize the shear resistance within a soil wedge. Coulomb'stheory assumes that the presence of the wall introduces shearing stress alongthe interface, due to the downward movement of the backfill along the back ofthe wall as the wall moves away from the backfill. The active earth pressureforce PA is computed using Equation 7 and is oriented at an angle 6 to thenormal along the back of the wall at a height equal to H/3 above the heel, asshown in Figure 3.4. The shear component of PA acts upward on the soil wedgedue to the downward movement of the soil wedge along the face of the wall. KA
for frictional backfills. The active earth pressure, aa, along the back ofthe wall at depth z is computed using Equation 4 and oriented at an angle 6 tothe normal along the back of the wall. The variation in a, is assumed linearwith depth for a dry backfill, as shown in Figure 3.4.
The planar slip surface extends upwards from the heel of the wallthrough the backfill and is inclined at an angle OA from horizontal. aA is
One widely quoted reference for effective angles of friction alonginterfaces between various types of materials, 6, is Table 2. Potyondy (1961)and Peterson et. al. (1976) also provide recommendations for 8 values fromstatic direct shear test results.
This procedure is illustrated in example 3 at the end of this chapter.
29
3.3.2 Coulomb Active Pressures - Hydrostatic Water Table Within Backfill andSurcharge
The distribution of Coulomb active earth pressures for a partially sub-merged wall retaining a fricti.onal backfill and supporting a uniform sur-charge, q, is shown in Figure 3.5. With a hydrostatic water table at heightHw above the base of the wall, the resulting pressures acting along the backof the wall are equal to the sum of (1) the thrust of the soil skeleton as aresult of its unit weight, (2) the thrust of the soil skeleton as a result ofthe surcharge, q, and (3) the thrust of the pore water. The effective weightof the backfill, a'wt, above the water table is equal to
/ = - z (18)
and below the water table, a'wt is equal to
a/t = t -(H - Hiw) + -y'[z - (H - Hw). (19)
where y' is the effective unit weight at depth z. For hydrostatic pore waterpressures, 7' is equal to the buoyant unit weight, 1b.The buoyant unit weight, 1b, is equal to
'Yb = ft -(20)
ca is equal to the sum of the thrust of the soil skeleton as a result of itsunit weight and the thrust of the soil skeleton as a result of the surcharge,
/ + q).KA (21)
and is inclined at an angle 6 from the normal to the back of the wall. KA iscomputed using Equation 16 for a level backfill (0 = 0) and a vertical wall,ace (0 = 0). The hydrostatic water pressures are equal to
u = 7w.[ - (H - Hw)] (22)
and is normal to the back of the wall. The total thrust on the wall, P, isequal to the sum of the equivalent forces for the three pressure distribu-tions. Due to the shape of the three pressure distributions, its point ofaction is higher up the back of the wall than one-third H above the heel. Theorientation of the failure surface is not affected by the hydrostatic waterpressures and is calculated using Equation 17.
30
Table 2. Ultimate Friction Factors for Dissimilar MaterialsFrom NAVFAC DM-7.2
Friction Friction
Interface Materials Factor, angle, 6
tan 5 deRrees
Mass concrete on the following foundation 0a.er0als:Clean sound rock ..................................... 0.70 35Clean grave., gravei-sana mixtures, coarse sanao.. U.5i to 0.u6 Zý to 31Clean fine to medium sand, silty medium to coarse
sand, silty or clayey gravel .................... 0.45 to 0.55 24 to 29Clean fine sand, silty or clayey fine to medium
sand ................ .......................... 0.35 to 0.45 19 to 24Fine sandy silt, nonplastic silt ................. 0.30 to 0.35 17 to 19Very stiff and hard residual or preconsolidated
clay ........... ............................... 0.40 to 0. 50 22 to 26Medium stiff and stiff clay and silty clay ........ 0.30 to 0.35 17 to 19(Hasonry on foundation materials has same friction
-actors.)Steel sheet piles against t.'e following soils:
Clean gravel, gravei-sand mixtures, well-gradedrock fill with spalls ........................... 0.40 22
Clean sand, silty sana-gravel mixture, single sizehard rock fill ................................ 0.30 17
Silty sand, gravel or sand mixed with silt or clay 0.25 14Fine sandy silt, noupiastic silt .................. U.ZO 11
formed concrete or concrete sheet piling against thefollowing soils:
Clean gravel, gravel-sand mixture, well-gradedrock fill with spalls ........................... 0.40 to 0.50 22 to 26
Clean sand, silty sand-gravel mixture, single sizehard rock fill .................... i.. ........... 0.30 to 0.40 17 to 22
Silty sand, gravel or sand mixed with silt or clay 0.30 17Fine sandy silt, rionplastic silt ......... 02....... 0.25 14
Various structural materials:Masonry on masonry, igneous and metamorphic rocks:
Dressed soft rock on dressed soft rock .......... 0.70 35Dressed hard rock on dressed soft rock .......... 0.65 33Dressed hard rock on dressed hard rock .......... 0.55 29
Masonry on wood (cross grain) ..................... 0.50 26Steel on steel at sheet pile interlocks ........... 0.30 17
o Vertcol effective stress due to weight of boc.kdill
k= A kA+ q 0 kA
Figure 3.5 Coulomb active earth pressures for a partially submergedbackfill and a uniform surcharge
The equation for a. of a soil whose shear strength is defined in termsof the effective strength parameters c and 4 would be equal to
a = (aWt + q)-K^ - 2cF-A (23)
and inclined at an angle 6 from the normal to the back of the wall.
32
3.3.3 Coulomb Active Pressures - Steady State Seepage Within Backfill
This section summarizes the equations for determining the Coulomb activeearth pressure forces and pore water pressures acting on the back of a wallretaining a drained backfill that is subjected to steady state flow. Fig-ure 3.6 shows a wall with a vertical face retaining a level backfill, support-ing a uniform surcharge load, q, and subjected to a constant water infiltra-tion. The wall has a drainage system consisting of a gravel drain below thesand backfill, with weep holes through the wall. Steady state flow maydevelop during a rainstorm of sufficient intensity and duration. The result-ing flownet is shown in Figure 3.6, consisting of vertical flow lines andhorizontal equipotential lines, assuming the drain has sufficient permeabilityand thickness to be free draining (i.e. with zero pressure head within thedrain). Adjacent to the back of the wall, the flow net has five head drops.With the datum at the base of the wall, the total head at the top of the back-fill is equal to the height of the wall, H, and a total head is equal to zeroat the weep holes. The drop in total head between each of the five equipoten-tial lines is equal to H/5. Neglecting the velocity head, the total head, h,is equal to
h = he + h (24)
where he is the elevation head, and hp is the pressure head equal to
Su(25)
Tyw
With the total head equal to the elevation head for each of the equipotentiallines, hp and the pore water pressure, u, are equal to zero. The seepagegradient, i, at any point in the backfill is equal to
i = Ah (26)
where Ah is the change in total head and Al the length of the flowpath over which the incremental head drop occurs. With horizontalequipotential lines, the flow is vertical and directed downward (iy = +i).For steady state seepage conditions, the effective unit weight is equal to
I1 = 7b ± Iw. " (27)
The seepage force is added to the buoyant unit weight when flow is downwardand subtracted with upward flow. For the example shown in Figure 3.6 with iequal to positive unity and directed downward, I' is equal to the total unitweight, It. The effective weight of the backfill, c'wt, is equal to
t = 7•(28)zwt Z = (B + 'yw) *z = . z ( Z
An alternative procedure for calculating a'wt is using the total overburdenpressure, awt, and pore water pressures, u. By Equation 7, we see that with
33
Surcharge, qMovements
SandS lBockFill
H
Gravel Drain
h -415 H -- FLOW NEThh Tiotl Head
h -315 H __ _
h HFlow Lne
h - //5 H EqulIpotentll Lne
Datum
H+Ph
P t
a-*,' k kA
a. = Vertical effective stress due to weight of backfill
ho "/5 9 k kA+ q k
After Lambe and Whitman (1969).
Figure 3.6 Coulomb active earth pressures for a backfillsubjected to steady state flow
34
the pore water pressure equal to zero, this procedure also results in theEquation 28 relationship (-y' -yt).
The resulting pressures acting along the back of the wall are equal tothe sum of (I) the thrust of the soil skeleton as a result of its unit weightand (2) the thrust of the soil skeleton as a result of the surcharge. Thepore water pressure acting on the wall is equal to zero, with horizontal equi-potential lines and the total head equal to the elevation head within thedrained backfill. In this case, the effective weight is equal to the totalweight. 0 a is computed using Equation 21, inclined at an angle 6 from thenormal to the back of the wall and equal to the sum of the pressures shown inFigure 3.6. KA is computed using Equation 16, and QA is computed i'sing Equa-tion 17. Downward vertical steady state seepage in a backfill results innearly the same earth pressures as are computed in the case of a dry backfill,
In ba• '-fills where there is a lateral component to the seepage force orthe gradients -ary throughout the backfill, the trial wedge procedure, inconjunction wittz ) flow net, must be used to compute PA and UA. Spacial vari-ations in u with (.:.: .nt elevation will alter the location of the criticalslip surface from the vw,9ie given in Equation 17. The trial wedge procedureis also required to find tht, values for PA and UA when point loads or loads offinite width are placed on top of the backfill. An example using the trialwedge procedure for a retaining wall similar to that shown in Figure 3.6 butwith a vertical drain along the back of the wall is described it Section 3.4.
3.3.4 Coulomb Theory - Passive Earth Pressures
The forces and stresses corresponding to the passive states of stressare shown in Figure 3.4 for a wall with a face inclined at angle +6 from ver-tical, and retaining a frictional backfill inclined at angle +0. The effect'of surcharge and groundwater pressures are not included in this figure. Todevelop passive earth pressures, the wall moves towards the backfill, with theresulting displacements sufficient to mobilize fully the shear resistancealong the linear slip plane. Coulomb's theory allows for a shear force alongthe back of the walls that is due to the upward movement of the backfill asthe wall moves towards the backfill. The passive earth pressure force PF isr.omputed using Equation 13 and oriented at an angle 6 to the normal along theback of the wall at a height equal to H/3 above the heel of the wall, as shownin Figure 3.4. The shear component of Pp acts downward on the soil wedge dueto the upward movement of the soil wedge along the face of the wall. This isthe reverse of the situation for the shear component of PA. Kr is equal to
for frictional backfills. The passive earth pressure, op, along the back ofthe wall at depth z is computed using Equation 15 and oriented at an angle 6to the normal along the back of the wall. The variation in up is assumedlinear with depth for a dry backfill, as shown in Figure 3.4. The planar slipsurface extends upwards from the heel of the wall through Ohe backfill and isinclined at an angle aF, from horizontal, ap is equal to
This procedure is illustrated in example 4 at the end of this chaptet.
3.3.4.1 Accuracy of Coulomb's Theory for Passive Earth Pressure Coefficients
Equations 29 and 30 provide reasonable estimates for Kp and the orienita-tion of the slip plane, as., so long as 6 is restricted to values which areless than 0/2. Coulomb's relationship overestimates the value for Kp when 6is greater than 0/2. The large shear component of Pp introduces significantcurvature in the failure surface. The Coulomb procedure, however, restrictsthe theoretical slip surface to a plane. When 6 is greater than 0/2, thevalue for Kp must be computed using a method of analysis which uses a curvedfailure surface to obtained valid values. Section 3.5 presents a graphicaltabulation of Kp values obtained by using a log spiral failure surface. Fig-ures 3.7 and 3.8 show the variation in the values for Kp with friction angle,computed using Coulomb's equation for Kp based on a planer failure surfaceversus a log spiral failure surface analysis.
3.4 Earth Pressures Computed Using the Trial Wedge Procedure
The trial wedge procedure of analysis is used to calculate the earthpressure forces acting on walls when the backfill supports point loads orloads of finite width or when there is seepage within the backfill. The pro-cedure involves the solution of the equations of equilibrium for a series oftrial wedges within the backfill for the resulting earth pressure force on theback of the wall. When applying this procedure to active earth pressure prob-lems, the shear strength along the trial slip plane is assumed to be fullymobilized. The active earth pressure force is equal to the largest value forthe earth pressure force acting on the wall obtained from the series of trialwedge solutions. The steps involved in the trial wedge procedure aredescribed using the retaining wall problem shown in Figure 3.9, a problemoriginally solved by Terzaghi (1943) and described by Lambe and Whitman(1969). A 20 feet high wall retains a saturated sand backfill with 4 equal to30 degrees and 6 equal to 30 degrees. The backfill is drained by a verticalgravel drain along the back of the wall, with weep holes along its base. Inthis problem, a heavy rainfall is presumed to have resulted in steady stateseepage within the backfill. The solution for the active earth pressure force
36
12 _ 1 1 II i I
1 - - - - - - - - - --CL10,
SLEGEND /
LZ 9 S 7I--61 COULOM8'S. -b/2010
-.. LOG-SPIRAL.-, - _ -2
3 -4
-- 4
a-
2 -- 5 L I- - I ! i !
20 2,5 30 3,5 40ANGLE OF INTERNAL FRICTl0N,@IN DEGREES
Figure 3.7 Coulomb and log-spiral passiveearth pressure coefficients with 6 =42 -
vertical wall and level backfill
90
, 80_z
"00_0:-4
'- LEGEND
60_ -- COULOdB'S. B -.z•----LOG-SPRIA .-
50 I5I _a. ,,oI30
Io -- -- "'"
20 25 30 35 40ANGLE OF INTERNAL FRICTION, e, IN DEGREES
Figure 3.8 Coulomb and log-spiral passive
earth pressure coefficients with 6 = 012vertical wall and level backfill
37
GRAVEL DRAWN
SATURATED SAND
Y V 131.6 pcI•1 I Jl ,-3Do
THE RETAINING WALL AND DRAIN
Lstr- ,• FLO# UNE -
LS5 F7 :
LS .; f EGIPOTENTIA- •- LNE
Psn
- n\
,LS
FLOW NET FOR STEADY RAINFALL
INTERVALPOINT hp AL (h)o0.*AL
"U.2 0.91.5 1.7 3.8
10 2. 7.23 3.9'•' 4 .8' 23 IIO,
3.1' 15.4
3.8 19.06 4.97 5.5' 23.1
8.51 14.96 0 94.3 FT2
PORE WATER FORCE FOR a. 450 U.,1 ..- 94.3 X 62.4 - 5890 LB/FT OF WALL
PA 10,200 LB/FT WALL10,000
P (LB/FT)
26321) LBfl T 11 0~~. ~ X BFI "a- 4',J••0 /
a~45 450
FORCE EQUILIBRIUM FOR CASE a - 45* PLOT OF P VERSUS a
After Terzoghi (1943) ond Lombe and
Whitmon (1969).
Figure 3.9 Example of trial wedge procedure
38
on the back of the wall using the trial wedge procedure, is outlined in thefollowing eight steps.
(1) Determine the variation in pore water pressures within the backfill. Inthis example the flow net for steady state seepage is constructed graphicallyand is shown in Figure 3.9.
(2) Assume an inclination for the trial slip surface, a, defining the soilwedge to be analyzed.
(3) Assume sufficient displacement so the shear strength of the sand is fullymobilized along the plane of slip, resulting in active earth pressures. Forthis condition, the shear force, T, required for equilibrium along the base ofthe soil wedge is equal to the ultimate shear strength force along the slipsurface.
T = N tano (31)
(4) Calculate the total weight of the soil within the trial wedge, W.
(5) Calculate the variation in pore water pressure along the trial slip sur-face. Using the flow net, the pore water pressure is computed at a point byfirst solving for h., using Equation 24, and then computing u usingEquation 25. An example of the distribution in u along the trial slipsurface for a - 45 degrees is shown in Figure 3.9.
(6) Calculate the pore water pressure force , U acting normal to thetrial slip surface, inclined at angle a to the horizontal. Ust~t,-.= is theresultant of the pore water pressures calculated in step (5).
(7) Analyze the trial wedge for the corresponding effective earth pressureforce, P, acting at an angle 6 30 degrees to the normal to the back of thewall. Using the equations of equilibrium (XF, = 0 and XFy = 0), the resultingequation for the unknown force P is equal to
P = (W - Ustticccose) tan(a - q) + U.tatic-* sina (32)
sin6 tan(cr - 40 cos6
Note that because of the presence of the free flowing drain along the back ofthe wall in which the total head equals tie elevation head, the pore waterpressures are equal to zero along the back of the wall.
(8) Repeat steps 2 through 7 for other trial slip surfaces until the largestvalue for P is computed, as shown in Figure 3.9. The slip surface that maxi-mizes the value for P corresponds to the critical slip surface, aA = a and
PA = P. In this case, aA = 45 degrees, and PA = 10,200 pounds per foot ofwall and acts at 6 = 30 degrees from che normal to the back of the wall.
Hydrostatic Water Pressures:
Consider the possibility is that the drain shown in Figure 3.9 does notfunction as intended and hydrostatic pore water pressures develop along theback of the wall as shown in Figure 3. 10. For each slip surface analyzed
39
\HDPOSTATIC GRAIEL DRANWATER TABLE (INEFFECTIVE)
SATURATED SAND pY = 131.6 pc •,
= 300*
RETAINING WALL WITH HYDROSTATICWATER TABLE AT TOP OF BACKFILL
FOR HYDROSTATIC WATER TABLE
U Static = 1/2 YW H 2
= 1/2(62.4)(20)2 = 12.480 LB/FT OF WALL
NORMAL TO SLIP PLANE q. 54.3
(Refer to Figure A.2)1
UstaticQ = U stotic * sin -54.V
UstoticC" = 15,368 LB/FT OF WALL
PLOT OF P VERSUS C.
4500 P= 4113 LB/FT
4000
3500
3000
S2500
J 2000
1500
1000
500 -- CA - 54.3"
80 75 70 65 60 55 50 45 40 35 30a , DEGREES
Figure 3.10 Example of trail wedge procedure, hydrostaticwater table
40
using the trial wedge method the effective force P, acting at angle 6 to thenormal for the wall, is given in section A.2 of Appendix A as
P= [ W - Ust.ti,_acosa ]tan( a - 4) ) (A-21)
cos6 + sin6tan( a - 0' )
The hydrostatic water pressure forces acting normal to the slip surface andnormal to the back of the wall are UstaticG and Ustatic, respectively, and arecomputed following the procedures described in section A.2.1 and A.2.2 ofAppendix A. Otherwise, the solution of the trial wedge analysis to computethe active earth pressure force follows the same eight sueps describedpreviously.
Using the trial wedge procedure for the problem shown in Figure 3.10,the wedge that maximizes the value for P corresponds to the rritical slipsurface, QA - 54.34 degrees, and PA - 4,113 pounds per foot of wall which actsat 6 - 30 degrees from the normal to the back of the wall. Although PA forthe ineffective drain case (Figure 3.10) is 6,087 pounds per foot less thanfor the effective drain case (Figure 3.9), the total horizontal design loadfor the ineffective drain is larger by 7,208 pounds per foot of wall comparedto the effective drain case due to the contribution of the water pressureforce (Utti. = 12,480 pounds per foot of wall).
A closed form solution exists for this example, as PA may be calculatedusing Equation 7, with KA computed using the Coulomb Equation 16. The corre-sponding critical slip surface aA is given in Equation 17.
3.5 Active and Passive Earth Pressure Coefficients from Log Spiral Procedure
A logarithmic spiral failure surface may be used to determine the activeand passive pressures against retaining structures when interface frictionacts along the back of the wall.
Values for the active and passive earth pressure coefficients arepresented in Figures 3.11 and 3.12 and Table 3. Figure 3.11 provides valuesfor KA and Kp for walls with inclined faces retaining horizontal backfills.Figure 3.12 provides values for KA and Kp for walls with vertical facesretaining horizontal or inclined backfills. These figures and Table 3 wereassembled from tables of KA and Kp values given in Caquot and Kerisel (1948).Kerisel and Absi (1990) have also assembled tables of KA and Kp values basedon a log-spiral failure surface. The sign convention for the angles are shownin the insert figures in Figures 3.11 and 3.12. Note that the sign conventionfor 6 is determined by the orientation of the shear stress acting on the wedgeof the soil. 6 is positive when the shear is acting upward on the soil wedge,the usual case for active pressures, and negative if the shear acts downwardon the soil mass, the usual case for passive pressures. The values for KA andKp from these figures and this table are accurate for all values of 6 lessthan or equal to 4.
These procedures are illustrated in examples 5 and 6 at the end of thischapter.
41
u ~ ~ 010 EML.*3,zQO~4a. -Qk -0, OL A A
25 .912 -I) .71 La -2.7 -0
208 306 .2626ANGL OFS M NERA FRCTON DGR
Fiue .IIAtiean asieeat pesuecofi/etswt
wall~ ~~~ trcin-oIngwl
I42
90.0 A44.
REDUCTION FACTOR (R) OF Kp SOD ":.:.50/4FOR VARIOUS RATIOS OF -3/# 70D
FigureE 3425* Active an asie e0rth prsur000i etswtwal fPcin-sOpigbOfl
-Kps(KpFR1/#-I) 00 0'03
U10 Go -a -T 0 (*n O C4U -~o (ýoý'o! r-.rm oo 0, ' n ,, n.no(n e'4.-4 m'fl£'"1 0 " D oI c'J4 c.4 en-4 m-4r'D
c14.-.4 CN.4.4 f£cn£e" ,-4r
a, 001, m OA.. C4lo m O0%C! - 'nrl 4 c,4 n .- 4a! -! n r!- o 0I4o
-44 -4 s 44 In"c4
c4 c-40 0% 0 Go'J 1.40 r r.
m lC~ <*4r a. r-.o n' u,4 no D co' M'
4.In - Ul C% f-n c4u~ m~r Go
m c4 .-4 kn m.4f"
IAn~ N( 04r- U'rn t- '41 0
ýo mN4- 1.lzýe' Ou1. Ln w N r- n4 m*~4 ý4 4 4 4 -4 n' - ' .1
41 -o " _ _.1 a, 'a a,~a 'CS 1. 0. 0 -0 n C a, +
4. . 4~S 1-4 0 0S4 en Or M~ r-j In 4 a -.
,-4 -4 14 J 4 -4 -44 41.4 ( N 1-4 ý r-n,4
00
-4 -4 -4 4 C1 C4 -4 - -4 I-4 -4j -ý 4 (n~
C>1 . u.C 0 Q >00 C D0 Cgo -4 -4a-
41 a,__ -4 -- -. 04 co f nI
44 e4 +4 - q - 4 ý1 - l. 0 0 0 00 00( ,0 0
J31 m IT_ m 4 , M 1 -_ rl I _ _c4
44. * l - -4 r-cn C). c14 40a, r-D .4 (4040 t rr- lo~
000 00 00 0 001 0 0 00 0 0 ,0 0C> 4
o -4-4 M ? 4 (N -40 -0 -4 14D . (=) r,4 M N o c
00C>0 000 00 0 00 00 0 0 0l DC 300C
41~~- - -4___ _ _ ___ _
* 44 4 --4'. 1.( 4 (51 44' -40 4 1S(N'
I-.0
444
3.6 Surface Loadings
There are three approaches used to approximate the additional lateralearth pressures on walls due to surface loadings; (i) the wedge method ofanalysis, (2) elastic solutions, and (3) finite element analyses.
Trial wedge analyses, as described in Section 3.4, may be performed toaccount for uniform and irregular surface load distributions for those wallswhose movements satisfy the criteria listed in Table 1. The wedge analysisdescribed in Section 3.4 is modified by including that portion of the surfaceloading between the back of the wall and the intersection of the trial slipsurface and the backfill surface in the force equilibrium calculation for eachwedge analyzed. The resulting relationship for a vertical wall retaining apartially submerged backfill (for a hydrostatic water table) is given insection A.2.8 of Appendix A. The difficult part of the problem is to deter-mine the point of action of this force along the back of the wall. The pointof action of the resulting earth pressure force for an infinitely long lineload parallel to the wall may be computed using the simplified procedure de-scribed in Article 31 of Terzaghi and Peck (1967).
Elastic solutions of the type shown in Figure 3.13 can be used to calcu-late the increase in the horizontal earth pressure, Oa, using either a solu-tion for a point load, a line load or a strip load acting on the surface of anelastic mass, i.e. the soil backfill. Most applications of elastic solutionsfor s~irface loadings to earth retaining structures assume the wall to be un-yielding (i.e. zero movement horizontally) and zero shear stress induced alongthe soil to wall interface (Clough and Duncan 1991). To account for the zerowall movements along the soil to wall interface, the computed value for a,using elastic theory is doubled. This is equivalent to applying an imaginaryload of equal magnitude equidistant from the soil to wall interface so as tocancel the deflections at the interface as shown in Figure 3.14. Experimentsby Spangit-r (1938) and Terzaghi (1954) have validated this procedure ofdoubling the a, values computed using the Boussinesq solution for point loads.
The finite element method of analysis has been applied to a variety ofearth retaining structures and used to calculate stresses and movements forproblems involving a wide variety of boundary and loading conditions. Somekey aspects of the application of the finite element method in the analysis ofU-frame locks, gravity walls, and basement walls are summarized in Ebeling(1990).
45
m > 0.4
r- 4p E m 2 n -L w L -- M2 - n2) 2
O m 0.4 q - 2S... []....n - s c o s 2
or F 0 .20n O-Na]
a. Line load (factor of two included) b. Strip loadfrom Terzoghi (1954)
,AT I l- IT 'N
q F-I ,
R2 2z RVi *bo ]
c. Romp load d. Triangular load
from Dowkins (1991)
NOTES:(1) FOR FIGURES (c) AND (d) THE ANGLES nL AND 0 ARE
EXPRESSED IN UNITS OF RADIANS.(2) NEGATIVE PRESSURES MAY BE COMPUTED
AT SHALLOW DEPTHS (Z).
Figure 3.13 Theory of elasticity equations for pressures on wall due to
surcharge loads
46
P (FORCE) U - 0
UN-YIELDINGHOIOGENIOUJS WALL
ELASTIC SOIL (tu O0
__7/xW
a. Model of un-yielding wall problem
P uP
ALONG THIS UNE.THE HORIZONTALDEFLECTIONS DUE TO THE REALAND IMAGINARY LOADS CANCEL (U - 0.THE HORIZONTAL STRESSES ARETWICE AS HIGH AS FOR ASINGLE LOAD.
REAL LOAD IMAGINARY LOAD
b. Two point loads on on elastic half-space
AFTER CLOUGH AND DUNCAN (1991)
Figure 3.14 Use of an imaginary load to enforce azero-displacement condition at the soil-structure
interface
47
CHAPTER 3 - EXAMPLES
Contents
Example Problems I through 6.
Commentary
The following examples illustrate the proceduresdescribed in Chapter 3. The results of the computa-tions shown are rounded for ease of checking calcula-tions and not to the appropriate number of significant
figures. Additionally, the values assigned to vari-ables in these problems were selected for ease ofcomputations.
48
Example No. 1 Reference Section: 3.2.1
For a wall of height H - 20 ft retaining a dry level cohesionless backfillwith 3" - 30 degrees and 8 - 0 degrees, compute KA, CA, and PA.
'EMEWT - RANKIME11WA4,_ WEIZE
• ORY COH*SIONLESS -11 aACKFLL
." / •'t " 120 PO:| .
* -' - 30•
S. .. . "•/'( l PA :1
• .i'"•aA hPA
KA = tanZ(45° - 30'/2) (by eq 5)
KA - 1/3
i i(120 pcf)(20 ft) 2 (by eq 7)
PA - 8,000 lb per ft of wall
kA = 450 + 30'/2 (by eq 6)
aA = 600 from the horizontal
hpA = H/3 = 6.67 ft
49
Example No. 2 Reference Section: 3.2.2
For a wall of height H - 20 ft retaining a dry level cohesionless backfill
with 4" = 30 degrees and 6 = 0 degrees, compute Kp, ap, and Pp.
I•E•ENS•- A$SSAI *EDGE
Y, - 120 PO "
S' 30° *" •
-= tan2 (45° + 300/2) (by eq 11)
K- - 3.0
Pp = 3.0 • 1(120 pcf)(20') 2 (by eq 13)
Pp = 72,000 lb per ft of wall
rp = 450 - 30'/2 (by eq 12)
ap - 30* from the horizontal
hvp = H/3 - 6.67 ft
50
Example No. 3 Reference Section: 3.3.1
For a wall of height H - 20' retaining a dry cohesionless backfill with
0' - 30 degrees, 6 = 3 degrees, • = 6 degrees, and 6 - 0 degrees, compute KA,
eA, and PA
MOVEMENrFS ,6
0 -- COJULOMBACTIVE
* . . ,, WE XE
z PA
K' - cos'(1-0-0)
Cos, (0) Cos(0*3)1 + s-in(30+3)sin(30-6)cos(3+0) cos (6-0)
KA = 0.3465
S= 0. 3465. (120 pcf)(20' (by eq 7)
PA - 8316 lb per ft of wall
ci =V[tan(30-6)][tan(30-6) + cot(30)] [I + tan(3)cot(30)]
cl = 1.0283
c2 -- I + Eftan(3)] . [tan(30-6) + cot(30)]]
t2 = 1.11411
A 30[-a(30 - 6) + 1.0283] (by eq 17)
AA - 57.6' from the horizontal
) I
Example No. 4 Reference Section: 3.3.4
For wall of height H - 20 ft retaining a dry cohesionless backfill with 0' -
For the Example No. 3 problem of a wall retaining a dry cohesionless backfillwith j" - 30 degrees, 6 - +3 degrees, P - +6 degrees, and 0 - 0 degrees.compute KA using the log spiral procedu.e of Figure 3.12. Compare this valuewith the KA value computed in Example No. 3 using the Coulomb relationship.
6/1 - +0.1 and P/4 - +0.2
KA - 0.35 from Figure 3.12 with p/4 +0.2 and using the curve for 6 =
This value for KA agrees with the value computed using Coulomb's theory foractive earth pressures in Example No. 3 ( KA - 0.3465).
53
Example No. 6 Reference Section: 3.4
For the Example No. 4 problem of a wall retaining a dry cohesionless backfill
with ý = 30 degrees, 6 = -3 degrees*, 8 = +6 degrees, and 0 - 0 degrees, com-pute Kp. Compare this value with the Kp value computed in Example No. 4.
6/rh -0.1 and fi/= +0.2
R (for 6/0 = -0.1) 0.52 and Kp (for 6/0 = +0.2) = 8 from Figure 3.12
The value for Kp is nearly the same as the value computed using Coulomb's the-
ory for passive earth pressures in Example No. 4 (K4 = 4.0196) because 6 < 0/2(Section 3.3.4.1). The resultant force vector Pp acts in the same direction
as shown in the Example No. 4 figure.
. Note the difference in sign for 6 in the passive earth pressure solution
iusing the Figure 3.12 log spiral solution procedure compared to that used inthe Coulomb's solution, with sign convention as shown in Figure 3.4.
Okabe (1926) and Mononobe and Matsuo (1929) extended Coulomb's theory ofstatic active and passive earth pressures to include the effects of dynamicearth pressures on rv=Liiing walls. The Mononobe-Okahe theory incorporatesthe effect of earthquakes through the use of a constant horizontal accelera-tion in units of g, ah = kh'g, and a constant vertical acceleration in unitsof g, a, = k,'g, acting on the soil mass comprising Coulomb's active wedge (orpassive wedge) within the backfill, as shown in Figure 4.1. The term kh isthe fraction of horizontal acceleration, kv is the fraction of vertical accel-eration, and g is the acceleration of gravity (1.0 g - 32.174 ft/sec/sec =980.665 cm/sec/sec). In Figure 4.1, positive a, values act downward, and pos-itive ah values act to the left. The acceleration of the mass in the direc-tions of positive horizontal and positive vertical accelerations results inthe inertial forces kh'W and k,,W, as shown in Figure 4.1, where W is theweight of the soil wedge. These inertial forces act opposite to the directionin which the mass is accelerating. This type of analysis is described as apseudostatic method of analysis, where the effect of the earthquake is modeledby an additional set of static forces, kh'W and k,-W.
The Mononobe-Okabe theory assumes that the wall movements are sufficientto fully mobilize the shear resistance along the backfill wedge, as is thecase for Coulomb's active and passive earth pressure theories. To develop thedynamic active earth pressure force, PA, the wall movements are away from thebackfill, and for the passive dynamic earth pressure force, PpE, the wallmovements are towards the backfill. Dynamic tests on model retaining wallsindicate that the required movements to develop the dynamic active earth pres-sure force are on the order of those movements required to develop the staticactive earth pressure force, as discussed in Section 2.2.2.
The Mononobe-Okabe theory gives the net static and dynamic force. Forpositive kh > 0, PA is larger than the static PA, and PPE is less than thestatic Pp.
4.2 Dynamic Active Earth Pressure Force
The Mononobe-Okabe relationship for PA for dry backfills, given byWhitman and Christian (1990), is equal to
1
PAE = KA ' [yt(I - k,)]H2 (33)
and acts at an angle 6 from the normal to the back of the wall of height H.The dynamic active earth pressure coefficient, KAE, is equal to
55
p
a
Sli
Mesier ation
Wkth
~h
WI -kV%
a. Mononobe-Okabe (active) wedge
p
p-p
PC
From EM 1110-2-2502Figure 4.1 Driving and resisting seismic wedges, no saturation
The seismic inertia angle represents the angle through which the re-sultant of the gravity force and the inertial forces is rotated fromvertical. In the case of a vertical wall (9 = 0) retaining a horizon-tal backfill (• - 0), Equation 34 simplifies to
Figures 4.2 and 4.3 give charts from which values of KAE may be readfor certain combinations of parameters.
The planar slip surface extends upwards from the heel of the wallthrough the backfill and is inclined at an angle aAE from horizontal.cAE is given by Zarrabi (1978) to be equal to
Figure 4.3 Variation in KAE'(?OS 6 with kh, and 13
58
Figures 4.4 and 4.5 give aA as a function of 0 for several values of 0 forvertical walls retaining level backfills.
A limited number of dynamic model retaining wall tests by Sherif andFang (1983) and Ichihara and Matsuzawa (1973) on dry sands show 6 to rangefrom 4/2 to 24/3, depending upon the magnitude of acceleration.
These procedures are illustrated in examples 7 and 8 at the end of thischapter.
The validity of the Mononobe-Okabe theory has been demonstrated by theshaking table tests described in Section 2.2.1. These tests were conducted atfrequencies much less than the fundamental frequency of the backfill, so thataccelerations were essentially constant throughout the backfill. Figure 4.6gives a comparison between predicted and measured values of the seismic activepressure coefficient KA.
An alternative method for determining the value of KAE using tabulatedearth pressures was developed by Dr. 1. Arango in a personal communication, asdescribed by Seed and Whitman (1970). Dr. Arango recognized that by rotatinga soil wedge with a planar slip surface through the seismic inertia angle, theresultant vector, representing vectorial sums of W, khW and k,-W, becomes ver-tical, and the dynamic problem becomes equivalent to the static problem, asshown in Figure 4.7. The seismic active pressure force is given by
PAE = [KA(,B,O).FA -]. 1 [-Y(l - k') ]H 2 (38)
where
H = actual height of the wall*= +
6* = O + 4
and
F s = s( + 4)) (39)cosO cos2 0
Sis computed using Equation 35. Values of FA are also given as a functionof 4) and 6 in Figure 4.8. KA(#*,O*) is determined from the Coulomb static KA
values by Equation 16. An alternative procedure is to approximate KA(,f*,8*)
by using the static KA values that were tabulated by Caquot and Kerisel (1948)or Kerisel and Absi (1990) as given in Table 3. The product of KA(#*,6*)times FAE is equal to KA.
These procedures are illustrate in examples 9 and 10 at the end of thischapter.
59
30,
0, 2~0 0, 25. 30.
Figure 4.4 Variation in CtPE with for 6equal to vertical wall and level
backfill
f0.
400
Figure 4.5 Variation in aAE with ' for 6equal to zero degrees, vertical wall and
level backfill
60
0.8
DENSE OTTAWA SAND
Yovg - 1.66 g/cc
(q'- 40.90) Y 1.559 g/cc0.7
0.6 // 1.699
//
/
1.653@,
E 0.4
1.6670
BY MONONOBE-OKABE
0.3 .0
1.64 1.703y1.651
1.654o0.2 oe
LEGEND0o EXPERIMENTALS
AT WALL ROTATION Y/H - 0.0010.1
0 0.1 0.2 0.3 0. 4 0.5 0.6
HORIZONTAL ACCELERATION COEFFICIENT, %/g
From Sherif and Fong (1983).
Figure 4.6 Variation in dynamic active horizontal earthpressure coefficient with peak horizontal acceleration
61
H
aAE A
Dynamic Problem
'er
P
Equivalent Static Problem
Figure 4.7 Equivalent static formulation of the Mononobe-Okabe active dynamic earth pressure problem
62
020
0 4
'. , 1'. 15 2*25 -0
N5 .. ... .
Fiur 4. Vaue offato____ordeerintinf_
"0° 0° 150 200 250 0
Figure 4.8 Values of factor F• for determination of K•
4.2.1 Vertical Position of PA along Back of Wall
The Mononobe-Okabe analysis procedure does not provide a means for cal-culating the point of action of the resulting force. Analytical studies byPrakash and Basavanna (1969) and tests on model walls retaining dry sands(Sherif, Ishibashi, and Lee 1982; Sherif and Fang 1984a; Sherif and Fang1984b; and Ishibashi and Fang 1987) have shown that the position of PA alongthe back of the retaining wall depends upon the amount of wall movement andthe mode in which these movements occur. These limited test results indicatethat the vertical position of PA ranges from 0.4 to 0.55 times the height ofthe wall, as measured from the base of the wall. PA acts at a higher posi-tion along the back of the wall than the static active earth pressure forcedue to the concentration of soil mass comprising the sliding wedge above mid-wall height (Figure 4.1). With the static force component of PA acting below
mid-wall height and the inertia force component of PA actitig above mid-wallheight, the vertical position of the resultant force, PA, will depend upon
the magnitude of the accelerations applied to the mass comprising soil wedge.
63
This was shown to be the case in the Prakish and Basavanna (1969) evaluationof the moment equilibrium of a Mononobe-Okabe wedge. The results of theiranalyses are summarized in Figure 4.9.
4.2.2 Simplified Procedure for Dynamic Active Earth Pressures
Seed and Whitman (1970) presented a simplified procedure for computingthe dynamic active earth pressure on a vertical wall retaining dry backfill.They considered the group of structures consisting of a vertical wall (0 - 0)retaining a granular horizontal backfill (P - 0) with 0 equal to 35 degrees,6 = 0/2 and k, equal to zero. PA is defined as the sum of the initialstatic active earth pressure force (Equation 7) and the dynamic active earthpressure force increment,
PA =PA + APAE (40)
where
AP = AKE • YtH 2 (41)
The dynamic active earth pressure coefficient is equal to
KA = KA + AKAE (42)
and
AKAE = . kh. (43)
Using this simplified procedure, KA is computed using Equation 16, and AKA iscomputed using Equation 43. All forces act at an angle 6 from the normal tothe back of a wall, as shown in Figure 4.10. PA acts at a height equal to H/3above the heel of the wall, and APA acts at a height equal to 0.6 H. PA actsat a height, Y, which ranges from H/3 to 0.6 H, depending upon the value ofkh.
PA'( ) + APAE'(0. 6 H) (44)
PAE
The results of instrumented shake table tests conducted on model wallsretaining dense sands show APAE acts at a height of between 0.43H and 0.58H,depending upon the mode of wall movement that occurs during shaking. Theheight of the model walls used in the shake table tests, as summarized inMatsuzawa, Ishibashi, and Kawamura (1985), were 2.5 and 4 feet.
64
0.6
0.5
H
0.4
0.3 L0.1 0.2 0.3 0.4
Kh
i3csec on Moment iEquil;brium
for -p
Kv - 0
Adopted from F-rokaksh and [3osovonno (1969).
Af ttr 1'xik;h and PiH ,i;~v~ta ',
() I'A
N.A
PA. - PA PA
P. (0.6H
AEA
Figure 4.10 Static active earth pressure force and incremental dynamicactive earth pressure force for dry backfill
Seed and Whitman (1970) approximate the value for ax as equal to •,where 0 equals 35 degrees. Thus, for a wall retaining a dry granular backfillof height H, the theoretical active failure wedge would intersect the top ofthe backfill at a distance equal to 1.5 times H, as measured from the top ofthe wall (tan 35' = 1/1.5).
This procedure is illustrated in example 11 at the end ot this chapter.
4.2.3 Limiting Value for Horizontal Acceleration
Richards and Elms (1979) show that Equations 34 and 36 are limited tocases where (0 - P3) is greater than or equal to 0. Substituting (0 - f/) equalto 0 into Equation 37 results in .AF equal to the slope of the backfill(/P), which is the stability problem for an infinite slope. Zarrabi (1978)shows that this limiting value for ip corresponds to a limiting value for kh,
which is equal to
kh" = (I - k,) tan(ý -. ). (45)
When kh is equal to kh*, the shear strength along the failure surface is fullymobilized, and the backfill wedge verges on instability. Values of kh arealso shown in Figure 4.11.
This procedure is illustrated in examples 12 and 1I at the end of thischapter.
4.3 Effect of Submergence of the Backfill on the Monotiobe-()kabe Mt'thod ofAna l ys• is
The Mononobe-Okabe re lationshiip:; for "AE, KAE, and 4' will differ fromthose expressed in Equations 33, 34, and 3'), respectively, when water ispresent in the backfill. Spat i l variat ions in pore water pressurt, wit hconstant elevation in the backtill will alter the location of the criticalslip surface and thus the va"ue of P'AE, sifmil-ar to the cast, -f PA that was
66
0.45CTV
=- K'~/ (ACCELERATION
INERTIA FORCEDOWN'PWARD)
0.3Ky0
KV /,N 2 (ACCELERATION
INERTIA FORCE
K INW
0 11010 200 30
Figure. 4.11 Limiiting values for horizontal acceleration equals kh g
67
discussed in Section 3.3.3. In addition, the pore water pressures mayincrease above their steady state values in response to the shear strainsinduced within the saturated portion of the backfill during earthquake shak-ing, as discussed in Tokimatsu and Yoshimi (1983), Tokimatsu and Seed (1987),Seed and Harder (1990), and Marcuson, Hynes, and Franklin (1990). The trialwedge procedure of analysis is used to locate the critical slip surface withinthe backfill and to compute PA, following the steps described in Section 3.4and including the excess pore water pressures due to earthquake shaking in theanalysis are described in Appendix A. In some situations, such as the case ofa hydrostatic water table within the backfill or the case of excess pore waterpressures equal to a constant fraction of the pre-earthquake effective over-burden pressures throughout the backfill (ru = constant), modified Mononobe-Okabe relationships may be used to compute PAE.
4.3.1 Submerged Backfill with No Excess Pore Pressures
In this section it is assumed that shaking causes no associated buildupof excess pore pressure. The most complete study of this case appears inMatsuzawa, Ishibashi, and Kawamura (1985), Ishibashi, Matsuzawa, and Kawamura(1985), and Ishibashi and Madi (1990). They suggest two limiting conditionsfor design: (a) soils of low permeability - say k < 1 x 10-3 cm/sec where porewater moves with the mineral skeleton; and (b) soils of high permeability -
say k > 1 cm/sec, where pore water can move independently of the mineralskeleton. Matsuzawa, Ishibashi, and Kawamura (1985) also suggest a parameterthat can be used to interpolate between these limiting cases. However, under-standing of case (b) and the interpolation parameter is still very incomplete.
Restrained water case: Here Matsuzawa Ishibaski, and Kawamura (1985)make the assumption that pore pressures do not change as a result of horizon-tal accelerations. Considering a Coulomb wedge and subtracting the staticpore pressures, there is a horizontal inertia force proportional to Y,. kh anda vertical force proportional to Vb. Thus, in the absence of vertical accel-erations, the equivalent seismic angle is:
tan-IYt'~k (46)'yb
and the equivalent horizontal seismic coefficient is:
'ie1 ft = (47)
Using kht1 in the Mononobe-Okabe theory together with a unit weight yh willgive PIE, to which the static water pressures must be added.
If vertical accelerations are present, Matsuzawa, Ishibashi, andKawamura (1985) recommend using:
68
= tan-1 b ] (48)
This is equivalent to assuming that vertical accelerations do affect porepressures, and then it is not strictly correct to use the Mononobe-Okabetheory. However, the error in evaluating total thrust is small.
This procedure is illustrated in example 14 at the end of this chapter.
Free water case: It is difficult to come up with a completely logicalset of assumptions for this case. Matsuzawa, Ishibaski, and Kawamura (1985)suggest that the total active thrust is made up of:
(I) A thrust from the mineral skeleton, computed using:
=d GS (49)
and
0.]2 = tan] (50)
where G. is the specific gravity of the solids. A unit weight of 7b is usedin the equation for P,.
(2) The hydrodynamic water pressure force for the free water within the back-fill, Pwd, is given by the Westergaard (1931) relationship (Appendix B)
7 .d _ .7 (51)
and acts at 0.4 H above the base of the wall.
The total force behind the wall would also include the hydrostatic water pres-sure. This procedure is not totally consistent, since the effect of theincreased pore pressures is ignored in the computation of the thrust from themineral skeleton as is the effect of vertical acceleration upon pore pressure.
This procedure is illustrated in example 15 at the end of this chapter.
4.3.2 Submerged Backfill with Excess Pore Pressure
Excess pore pressures generated by cyclic shaking can be represented byr- - Au/a,' , where Au is the excess pore pressure and a,' is the initial
69
vertical stress, While there is no rigorous approach for adapting theMononobe-Okabe solution, the following approaches are suggested.
Restrained water case: Ignoring vertical accelerations, the effective"-nit weight of soil becomes:
7.3 = -Yb(l ~ (52)
while the effective unit weight of water is
1Yw3 = 7Yw + 7b r.r (53)
The thrust from the soil skeleton, PAE, is computed using
k,. f-Lk (54)793
and
03= tan1 Ikh.3] (55)
together with a unit weight from Equation 52. The effective unit weight ofwater, Equation 53, is used to compute the "static" pore pressure. The effectof vertical acceleration may be accounted for by inserting (1-k,,) in thedenominator of Equation 55.
As ru approaches unity, 7.3 -> 0 and 1,,3 = yso that the fully-lique-fied soil is a heavy fluid. It would now be logical to add a dynamic porepressure computed using Equations 51 and 53.
This procedure is illustrated in example 16 at the end of this chapter.
Alternate Procedure:
An alternative approach is to use a reduced effective stress frictionangle in which the effects of the excess pore water pressures are approximatedwithin the analysis using a simplified shear strength relationship. In aneffective stress analysis, the shear resistance on a potential failure surfaceis reduced by reducing the effective normal stress on this plane by the amountof excess residual pore water pressure, assuming the effective friction angleis unaffected by the cyclic loading. This is equivalent to using the initial,static effective normal stress and a modified effective friction angle, O.q'
where
tanolq = (1 - r.) tano' (56)
as shown in Figure 4.12. In the case of r. equal to a constant within thefully submerged backfill, the Use Of Oeq in Equations 34 and 38 for KAE andKA(fi*, 0*) approximating the effects of these excess pore water pressures
70
MOHR'S CIRCLE "c'- AT CONSOLIDATION BEFORE CYCLIC LOADINGMOHR'S CIRCLE "" - AFTER CYCLIC LOADING
%3- o-3" - AU
Au- ,f---- --ton qeý - (1-ru) ton 4
V) rff /
Sfo MOHR' S CIRCLE 'a'
BI-
O-f L{ Cc01 NORMAL STRESS. 0-
For the static stress path to failure from point A to point C
FSstatic = f:tan=rfc T
fc
For the cyclic stress path to failure from point A' to point B'
FS, y ffa ,atan, _, ofctanoeqyclic -. fc "[ Tfc
Figure 4.12 Modified effective friction angle
within the analysis. Using kha1, Oh., (Equations 47 and 46 in Section 4.3.1)and 0eq in the Mononobe-Okabe theory together with a unit weight 7b will give
Calculations by the authors of this report showed that reducing theeffective scress friction angle of ghe soil so as to account for the excesspore water pressures when computing a value for PA is not exact. Comparisonsbetween the exact value of PAE, computed using 1.3, khe3, Oh.3 in the Mononobe-Okabe theory, and the value computed using the 0,q procedure shows thisappr ximation to overpredict the value of PAE. The magnitude error in thecomputed value of PA increases with increasing values of r. and increaseswith decreasing values of kh. The error is largest for the kh equal to 0case.
71
This procedure is illustrated in example 17 at the end of this chapter.
Free water case: The thrust from the mineral skeleton may be estimated using:
Yd k (57)
783
where
l+w
To this thrust are added the dynamic Westergaard water pressure (computedusing -y,,) and a "static" water pressure computed using -lw3 from Equation 53.
This procedure is illustrated in example 18 at the end of this chapter.
4.3.3 Partial Submergence
Situations with partial submergence may be handled by weighing unitweights based on the volume of soil in the failure wedge above and below the
ph-eatic surface, as shown in Figure 4.13.
This procedure is illustrated in example 19 at the end of this chapter.
4.4 Dynamic Passive Earth Pressures
The trial wedge procedure of analysis may be used to find the orienta-tion of the critical slip surface that minimizes the value of the earth pres-sure force acting on the wall for the passive earth pressure problem shown inFigure 4.1b. This minimum earth pressure force corresponds to the dynamicpassive earth pressure force, PPE. The orientation of the inertial forces khWand kv-W that minimize the value of PpE is directed away from the wall andupwards (Figure 4.ib). This corresponds to the case where the soil wedge isaccelerating towards the wall (positive ah values) and downwards (positiveav values).
The Mononobe-Okabe relationship for PPE for dry backfill, given byWhitman and Christian (1990), is equal to
FEI- k))12 (58)
and acts at an angle 6 from the normal to the back of the wall of height H.The dynamic passive earth pressure coefficient, KpE, is equal to
PE = cos2 (0 - 0 - 6)
cOS • cos 2 B cos (I- B " 6) 1 - sin ( + 6) sin (4 - + )-) (59)C~ .os (6 +OS + Cos +cos (/3-9)
72
ton(90"- 4) 2 - I
AREA 2 hi hA2
or
1~hiAREA 1 h =
I h
SArea = Area + Area,2
Area 1 /21, h, lh)Area 1/2 1h
Y,* Area 1 + Y2* Area 2Ye = Area
= Areare l .+ Area ý-"
S(h), .2-(h)
NOTES:(1) EXACT SOLUTION WHEN r." 0.(2) APPROXIMATE SOLUTION WHEN r. > 0.
Figure 4.13 Effective unit weight for partiallysubmerged backfills
In the case of a vertical wall (0 - 0) retaining a horizontal backfill()6 - 0), Equation 59 simplifies to
YE = cos2(0 - 0)
coso cos(4 + 6) 1- sin(o + 5) sin( -
[1 cos(6 + 0)
The planar slip surface extends upwards from the heel of the wall through the
backfill and is inclined at an angle aCP from the horizontal, aPE is equal to
Figures 4.14 and 4.15 give CpE as a function of 1 for several values of 4.
This procedure is illustrated in example 20 and 21 at the end of thischapter.
The Mononobe-Okabe equation assumes a planar failure surface, which onlyapproximates the actual curved slip surface. Mononobe-Okabe's relationshipoverpredicts the values for KPE and the error increases with increasing valuesfor 6 and 0.
Rotating the passive soil wedge with a planar slip surface through theseismic inertia angle, the resultant vector, representing vectorial sums of W,khW, and kvW, becomes vertical, and the dynamic passive earth pressure forceproblem becomes equivalent to the static problem, as shown in Figure 4.16.
1 is computed using Equation 35. Values of FpE are also given as a functionof 0 and 6 in Figure 4.17. Kp(,*,O*) is determined from the Coulomb static Kpvalues by Equation 29. The Coulomb formulation assumes a planar failure sur-face which approximates the actual curved failure surface. The planar failuresurface assumption introduces errors in determination of Kp and the errorincreases with increasing values of 6. The error in slip surface results inan overprediction of Kp. Thus the equivalent static formulation will be inerror since the product of Kp(•*,O*) times FPE is equal to KPE. An alternateprocedure is to approximate Kp(p*,6*) by using the static Kp values tabulatedby Caquot and Kerisel (1948) or Kerisel and Absi (1990). Calculations showKpE values by the alternate procedure are smaller than KpE values by Mononobe-Okabe.
This procedure is illustrated in examples 22 and 23 at the end of thischapter.
74
4S' i
40. r- 7-III
f Figure 4.14 Variation aZP. with 0'5o for 6 equal to 0/2, vertical wall25. • _ _ _ _ and level backfill
-V
30
20*0
______ 1'
II
0. 25 !
*5'*
30'i~i
75E
Figure 4.15 Variation in QPE withfor 6 equal to zero degrees, 0
vertical wall and level backfill
S.
'0.3'1
75
Dynamic Problem
Eauivalent Static Problem
Figure 4.16 Equivalent static formulation of the Mononobe-Okabe passive dynamic earth pressure problcm
This procedure is illustrated in the procedures outlined in Section 4.3.The procedures are used to account for the effect of submergence of the back-fill in computing the value of PPE. For example, in t! - restrained water caseof a fully submerged backfill, an effective unit equal to lb is assigned tothe backfill for the case of ru - 0 or Equation 52 with ru > 0. KPE orKp(o*,O*) and FPE are computed using an equivalent seismic inertia angle usingEquation 48 for the case of ru - 0 or Equation 55 with ru > 0.
This procedure is illustrated in example 24 at the end of this chapter.
4.4.1 Simplified Procedure for Dynamic Passive Earth Pressures
Towhata and Islam (1987) recommended a simplified approach for computingthe dynamic passive earth pressure force that is similar to the Seed andWhitman (1970) procedure for the dynamic active earth pressure force. Theyalso considered the group of structures consisting of a vertical wall (0 = 0)
76
1.6
1.5
1.4 ,_ _ ,,
1.,3 ___ __ ___...._ _
S1.2 ......... _ _ _
1.o0.09
08 5oo 10 200 250 305
Figure 4.17 Values of factor FpE
77
retaining a granular horizontal backfill (6 = 0) with 0 equal to 35 degrees,6 equal to 0, and k, equal to zero. Equation 65 is presented as developed byTowhata and Islam, while Equations 64, 66, and 67 have been modified by theauthors of this report. PPE is defined as
PPE = PP - APPE (64)
where the reduction in the static passive earth pressure value Pp due toearthquake shaking is given by
APPE lYtH2*AKPE (65)
for a dry granular backfill. The dynamic passive earth pressurecoefficient is equal to
KPE = Kp - AKPE (66)
and
AKPE = 17kh (67)
Using this simplified procedure, Kp is computed using Equation 11(Rankine), and AKPE is computed using Equation 67. The incremental dynamicforce APPE acts counter to the direction of Pp, reducing the contribution ofthe static passive pressure force to PPE . The resulting forces Pp (Equa-tion 13) and APpF (Equation 65) act normal to the back of a wall.
This procedure is illustrated in example 25 at the end of this chapter.
The simplified procedure was developed for vertical walls retaininghorizontal backfills with 6 = 0. This simplified procedure should not beapplied to dynamic passive earth pressure problems involving values of 6 > 0,due to the magnitude of the error involved.
4.5 Effect of Vertical Accelerations on the Values for the Dynamic Active andPassive Earth Pressures
In a pseudo-static analysis the horizontal and vertical accelerations ofthe soil mass during an earthquake are accounted for by applying equivalentinertial forces kh'W and kv'W to the soil wedge, which act counter to thedirection of the accelerating soil wedges, as shown in Figure 4.1. A positivehorizontal acceleration value increases the value of PAE and decreases thevalue of PPE- The vertical component of acceleration impacts the computedvalues of both PA and PPE and KA and KPE.
Upward accelerations (-kvg) result in smaller values of KA and largervalues of PA as compared to the KA and PA values when k, is set equal tozero. Upward accelerations (-kvg) increase the value of PA due to the con-tribution of the term (I - k) in Equation 33. This trend is reversed when
78
the vertical acceleration acts downward (+kvg). Seed and Whitman (1970) andChang and Chen (1982) showed that the change in the KA value varied with boththe value of kv and kh. Calculations with k, ranging from 1/2 to 2/3 of the khvalue show that the difference between the computed values of KE with anonzero kv value and kv equal to zero is less than 10 percent. Seed andWhitman (1970) concluded that for typical gravity retaining wall design prob-lems, vertical accelerations can be ignored when computing KA. The k, valuehas a greater impact on the computed value of PPE than on the value of PA.
Chang and Chen (1982) show that the change in the KpE value varies withboth the value of k, and kh. The difference between the values of KPE with anonzero kv value and kv set equal to zero increases with increasing magnitudesof both kv and kh. This difference can easily be greater than 10 percent. Ingeneral, vertical accelerations acting downward (+kv-g) will decrease the KFE
and PPE values from the corresponding KPE and PPE values for which k, is setequal to zero. The trend is reversed when the vertical acceleration actsupward (-k.,g). When PpE acts as a stabilizing force for a structure, verticalaccelerations should be considered in the computations of the value for PpE.An example is the soil region below the dredge level and in front of ananchored sheet pile wall (refer to the design example in Section C.2 ofAppendix C).
4.6 Cases with Surface Loadings
There are two approaches used to approximate the additional lateralearth pressures on walls due to surface loadings; (1) the wedge method ofanalysis and (2) finite element analyses.
In the case of a uniform surcharge q,, the value of the dynamic activeearth pressure force is computed using the modified Mononobe-Okabe relation-ships listed in Figure 4.18 and Equation 34 (or Equation 36 tor a verticalwall retaining a horizontal backfill) for KA. The point of application of PAalong the back of the wall is computed using the procedure outlined in Fig-ures 4.19 and Figure 4.20. In this approximate procedure, the surcharge q, isreplaced by the addition of a layer of soil of height h. equal to q,/y7. Theresulting problem is analyzed by adapting the Seed and Whitman's simplifiedprocedure (of section 4.2.2) to the problem of a uniform surcharge loading asoutlined in Figure 4.20.
This procedure is illustrated in example 26 at the end of this chapter.
Pseudo-static trial wedge analyses may be performed to accountapproximately for both uniformly and non-uniformly distributed surfaceloadings, as described in Section A.2 of Appendix A for dynamic active earthpressure problems. These analyses may be performed on walls whose movementssatisfy the criteria listed in Table 1. Such analyses will give the totalthrust against a wall. The effects of surface loading is included within thewedge analysis by including that portion of the surface loading between theback of the wall and the intersection of the slip surface and the backfill_urface in the force equilibrium calculation for each wedge analyzed, asdescribed in Section 3.6 for the static problem. The effect of the earthquakeis modeled in the pseudo-static trial wedge analysis by an additional set ofstatic forces, khW, k, W, kh Ws, and kv'Ws, where W is equal to the weight ofthe soil contained within the trial wedge and W, is equal to the weight ofsurcharge contained within the region located above the trial wedge as shownin Figure A.3 for the active earth pressure problem. The difficult part of
79
"LAEAR SUP SURFACE
*0
0, - kV4g
PI= KA 2 2q. r Cos 0 1 1 * [y(I k) y1~2
AZ 4 1 Y'H [Cos (P-0)1 2
for a vertical wall (8 0) retaining a horizontal backfill (fi -0) becomes
,P KA I. 2 1 ' -1~ }(1 - jIH
II It HI III 2I I
These relationships are exact when the critical sliding surface is planar,as discussed in Chang and Chen (1982).
Figure 4.18 Mononobe-Okabe active wedge relationships including surchargeloading
the pseudo-static analysis is to determine the point of action of this forcealong the back of the wall (refer to Appendix A).
Two-dimensional finite element analyses may be used to estimate thedynamic forces against walls as a result of surface loadings. See Appendix Dfor a discussion of available methods.
80
-AOOIAMONK& SOIL EOUIVALEffr TO 00
A SURCHARGE. q(s
H
Hw
a. Surcharge
CrT*P- KA.Yt ha 3 O
O-W.T KAIYh + - H&t
/ YI - -Y,(H - HrF- - H~w
b. Effective horizontalearth pressure - b- 0 degrees
Figure 4.19 Static active earth pressure force includingsurcharge (Continued)
81
E V2~ (oq"- H.O).( -r H)
E 2 -E2
E 4 -E 3I,). (..)
II
c. Equivalent forces
PA E, E2 E3 , E4
PA 7
1"
YPA E 1 /3(H - Hw) Hw] E 2[1/2(H - H.) H ] -E [1/3H w] E 4 {1/2H w]
d. Resultant effective force and point of application
Figure 4.19 (Concluded)
82
Q04
04-4
14)
CL)
I-, Cu
ca
0h
a* bO
ww
"-4
w >>
41
(12
U14A 0. 0
,4
83-
CHAPTER 4 - EXAMPLES
Contents
Example Problems 7 through 26.
Commentary
The following examples illustrate the proceduresdescribed in Chapter 4. The results of the computa-tions shown are rounded for ease of checkingcalculations and not to the appropriate number ofsignificant figures. Additionally, the valuesassigned to variables in these problems were selectedfor ease of computations.
84
Example No. 7 Rrforence Section: 4 2
For a wall of height H - 20 ft retaining a dry cohesionless backfill with0' - 30 degrees, 6 - 3 degrees, P - 6 degrees, 0 - 0 degrees, kh = 0.1 (accel-
eration kh'g away from the wall and inertia force kh'W towards the wall) andk, - 0.067 (acceleration kv-g acting downward and inertia force k' W acting
upward), compute KAE, PAE, and aAE.
-t 621 120 pcf
= cos 2 (306.12)1 (byeq
cos (6.12) cosZ(O) cos (6.123) [:sin(30÷3) sin(30-6.12-6)I• + cos(3+6. 12) cos(6')...
The values for KAE and PAE by Equations 34 and 33, respectively, are equal tothe values for the product [KA(fl*,6*)-FAE] and PA (Equatici 38).
91
Example No. 10 Reference Section: 4.2
For the example 9 problem, compute the increase in magnitude for the dynamic
active earth pressure force above the static actjvw earth pressure value,
APAE.
KA = [ (3) si_(35)sin (35-5) (2 (by eq 16)
cos 2 (0) cos (0) 1 + cos () cos (5)
KA - 0.2842
S= 0.2842 (120 pcf) ( 2 0')2 (by eq 7)
PA- 6,821 lb per ft of wall
PA- 11,008 lb per ft of wall (from example 9)
APAE AEP - PA
APAE 11,008 - 6,821
APAE 4,187 lb per ft of wall
Summary
The dynamic active earth pressure force is 61 percent greater than the staticactive earth pressure force for the example 9 problem.
92
Example No. 11 Reference Section: 4.2.1
For a wall of height H - 20 ft retaining a dry cohesionless backfill with 0' -35 degrees, 6 - 17.5 degrees (- 0/2), f - 0 degrees, 0 - 0 degrees, kh = 0.2(acceleration khg away from the wall and inertia force kh'W towards the wall)and k, - 0, compute KAE, PAE, and its point of action at elevation Y along theback of the wall using the simplified procedure for dynamic active earthpressures.
KA = 120 pcf (PAEA
PA = 35,04 b
SA 0cos (35) 2. (by eq 16)
V lCos (17. 5) cos (0)
KA - 0.246
S= 0.246 • (120 pcf) (20')' (by eq 7)
PA = 5,904 lb per ft of wall, acting at6.67 ft (H/3) above the base of the wall.
KAE 3 . 0.2 (by eq 43)
AKA - 0. 915
APAE = 0.15 *1(120 pcf) (201')2 (by eq 41)
APAE - 3,600 lb per ft of wall, acting at 12 ft (0.6 H) above thebase of the wall.
=A 0. 246 + 0.1 (by eq 42)
-PAE= 5,904 + 3,600 (by eq 40)
PAE - 9, 504 lb per ft of wall
93
Example No. 11 (Continued) Reference Section: 4.2.1
5904 (20 + 360U (0.6 " 20) (by eq 44)
9504
Y - 8.69 ft (0.43 H) above the base of the wall
94
Example No. 12 Reference Section: 4.2.3
For a wall retaining a dry cohesionless backfill with 0' - 35 degrees, 6 - 0degrees, P - 15 degrees, 0 - 0 degrees, and k, - - kh/ 2 (acceleration k, gacting upward and inertia force k.,W acting downward), compute kh*, 0, aAE, KA,and PA.
Introducing k, - - kh*/2 and rearranging, Equation 45 becomes
2 tan(- 0)2 - tan( -(
For (0 - P) - 20 degrees,
kh* - 0.44494
and k, - - 0.22247
Note that the use of Figure 4.11 results in the same value for kh*.
By Equation 35, 0 - 20 degreesBy Equation 37, cA - 15 degreesBy Equation 34, KA - 1.05
95
Example No. 13 Reference Section: 4.2.3
Repeat example 12 with k, - + kh/2 (acceleration k,'g acting downward andinertia force k,'W acting upward).
Introducing k, - + kh*/2 and rearranging, Equation 45 becomes
Examples 12 and 13 show that for the limiting case of (6 - f) equal to 4, themagnitude of kh* is dependent upon the orientation of the vertical inertiaforce. Both analyses result in the same values for 0, KA, and a. For theselimiting cases the slip plane is orientated parallel to the slope of the back-fill, aE - P. Additionally, when the inertia force k,.W acts downward (exam-ple 12) in conjunction with the weight of the backfill wedge, the valuecomputed for PA is 44 percent greater than the value for PA when k,'W actsupward (example 13) due to the term (1 - k,) in Equation 33.
96
Example No. 14 Reference Section: 4.3.1
For a wall of height H - 20 ft retaining a submerged cohesionless backfill
with 0' - 35 degrees, 6 - 17.5 degrees (- 0/2), fi - 0 degrees, 0 - 0 degrees,
kh - 0.2 (acceleration kh'g away from the wall and inertia force khW towards
the wall) and k, - 0, compute the earth and water pressure forces acting on
the wall for the case of restrained water within the backfill. Assume a
hydrostatic water table within the backfill and r, - 0.
qY = 12,0 POcf-E
46- 350 'saic
Hydrostatic Water Pressure Force
U~tatj. - 1/2 (62.4 pcf) (20')2
Ustatic = 12,480 lb per ft of wall acting
at Yst - 201/3 - 6.67ft.
Dynamic Earth Pressure Force
= tan-[J120 0.2 (by eq 46)120 62].4
22.62 degrees
kl = 2 ]. 2 = 2.08 - 0.2 0.417 (by eq 47)
Method 1 (KA by Mononobe-Okabe, KA by Coulomb)
KA= cos 2 (35 - 22.62)
cos (22.62) cos (22.62+17.5) [ [ sin (35 + 17.5) sin (35 - 22.62) (by eqL cos(17.5 + 22.62) 36)
KAE = 0.624
97
Example No. 14 (Continued) Reference Section: 4.3.1
Example No. 14 (Continued) Reference Section: 4.3.1
APA - 3606 lb per ft of wall, acting at12 ft (0.6 H) above the base of the wall.
From Method I calculations,
P- 2,834 lb per ft of wall acting at6.67 ft above the base of the wall.
PA- 2,834 + 3,606 - 6,440 lb per ft of wall (by eq 40)
2834 20 + 3606 (0.6 " 20) (by eq 44)(6440)
Y - 9.65 ft (0.48 H)
Summar:y
The simplified procedure of analysis underestimates the PAE value com-puted using the Mononobe-Okabe relationship by 10 percent due to the accuracyof the simplified relationship for large kh., values (refer to the discussionon page 134 of Seed and Whitman 1970).
Static pore water pressures must be added for both methods.
99
Example No. 15 Reference Section: 4.3.1
For a wall of height H - 20 ft retaining a submerged cohesionless backfillwith 0' - 35 degrees, 6 - 17.5 degrees (- 0/2), P - 0 degrees, 6 - 0 degrees,kh = 0.2 (acceleration kh'g away from the wall and inertia force khW towardsthe wall) and k, - 0, compute the earth and water pressure forces acting onthe wall for the case of free water within the backfill. Assume a hydrostaticwater table within the backfill and ru - 0.
1 20 pcf p \
4, 350 wd "
G- 2.65 Ustatic
Hydrostatic Water Pressure Force
Ustatic = 1/2 (62.4 pcf) (20')2
Ustatic = 12,480 lb per ft of wall,acting at Yut - 20'/3 - 6.67 ft
Hydrodynamic Water Pressure Force
=7 0. 2 (62.4 pcf) (201)2 (by eq 51)
Pwd - 2,912 lb per ft of wall, acting at
Ypwd = 0.4 20' = 8 ft
Dynamic Earth Pressure Force
khe 2 = 2.65 0.2 (by eq 49)k Te =265-7 I
khez - 0.32
0.2 = tan- i ] (by eq 50)
0.2 - 17.74 degrees
100
Example No. 15 (Continued) Reference Section: 4.3.1
KAE = cos 2 (35 - 17.74)
cos (17.74) cos (17.74 + 17.5) [+ [ sin (35 + 17.5) sin (35 - 17.74) (by eqcos (17.5 t 17.74) 36)
A -- 0.4965
(adaptedfrom
PA = 0.4965 [ 1(120 pcf - 62.4 pcf) (I - 0)] (20/)2 eq 33)
P- - 5,720 lb per ft of wall
(PA). - PA (cos 6) - 5,455 lb per ft of wall
Determine Point of Application of PA
From the Method 1 calculations in Example 14,
KA - 0.246 and PA - 2,834 lb per ft of wall.
PAE PA + APA (eq 40)
APAE - PAE - PA
APAE - 5,720 - 2,834 - 2,886 lb per ft of wall, acting at 12 ft(0.6 H) above the base of the wall.
2,834 20i + 2,886 (0.6 " 20)Y = 2,8
5,720
Y - 9.4 ft (0.47 H)
Sumrmary
For the restrained water case (Example 14, Method 1), the total forceacting normal to the wall - PA(cos6) + Ustatic
- 6,855 + 12,480
- 19,335 lb per ft of wall.
For the free water case (Example 15), the total force acting on thewall - PAE(cos6) + Pwd + UStti, - 5,455 + 2,912 + 12,480 - 20,847 lb per ft ofwall
For this dynamic problem, the free water analysis results in an8 percent larger total dynamic earth pressure force acting normal to the wall.
101
Example N). 16 Reference Section: 4.3.2
For a wall of heigiit H = 20 ft retaining a submerged cohesionless backfill
with 0' - 35 degrees, 6 = 17.5 degrees (= 0/2), fi - 0 degrees, 6 - 0 degrees,kh = 0.2 (acceleration kh'g away from the wall and inertia force kh'W towards
the wall) and k, = 0, compute the earth and water pressure forces acting on
the wall for the case of restrained water within the backfill. Assume a
hydrostatic water table within the backfill and r, = 0.3.
4 ' - 3 5 0 K AAY
,Z4 +U T
Hydrostatic Water Pressure Force
Linear pressure distribution with depth.
Ustatic = 1 62.4 pcf (20')22
Ustatic = 12,480 lb per ft of wall
20'
acting at Yu= 3 - 6.67 ft above the base
Excess Pore Water Pressure Force
Linear pressure distribution with depth for r. = constant.
PAE - 8,331 lb per ft of wall(PAE)x - PAE(cos6) - 7,921 Ib per ft of wall
Determine Point of Application of PAE
KA cos 2 (35)[ ( z (by eq 16)
cos 2 (0) cos (17.5) [ + sin (35 + 17.5) sin (35)cos (17.5) cos (0)
KA - 0.246
(adapted from-0.246- * (40.32 pcf) (20' )1 eq 7)
103
Example No. 16 (Continued) Reterence Section: 4.3.2
PA 1,984 lb p*r ft of wall, acting at 6.67 ft above the base of the wall.
APAE = PAE - PA (solre eq 40 for APx)
APAE 8,331 - 1,984 - 6,347 lb per ft of wall, acting at 12 ft (0.6 H) above
the base of the wall.
1,984 201 + 6,347 (0.6 . 20-)8,331
Y - 10.7 ft (0.54 - H)
Summary
Excess pore water pressures within the submerged portion of the backfilli-creasec both the effective earth pressures and the total earth and waterpressures acting along the back of the wall.
PE increased by 16 percent, from a value equal to 7,188 lb per ft ofwall for the case of cu - 0 (Method 1, example 14), to a value equal to 8,331lb per ft of wall for the case of ru = 0.3 (example 16).
The total force acting normal to the wall for tie case of ru equal to 0(Method 1, example 14) = PA (cos 6) + Ustat:, = 6,855 + 12,480 = 19,355 lb perft of wall.
The total force acting normal to the wall for the case of r. equal to0.3 (example 16) = PAE (cos 6) + Ustatic + Ushear = 7,921 + 12,480 + 3,456
23,857 lb per ft of wall.
The total force acting normal to the back of the w. Il increased by 23percent from the case of :u equal to 0, in the case of r,, equal to 0.3.
104
Example No. 17 Reference Section: 4.3.2
Repeat Example 16 using the reduced effective stress friction angle procedureto account for excess pore water pressures within the backfill and usingr, - 0.3.
P't"- 120 pO f I
-. 350 UHAý
USTATIC Y.04 Y." Y
Hydrostatic Water Pressure Force
From Example 16,
Ust.tic - 12,480 lb per ft of wall, acting at
Y,,=6.67 ft_
Excess Pore Water Pressure Force
From Example 16,
Ushear -3,456 lb per ft of wall acting at
Yush 6.67 ft -- due to r, = constant.
Dynamic Earth Pressure Force
tanogq = (I - 0.3) tan 350 (by eq 56)
O•q = 26. 11 degrees
09, = tan-1 ( 0.2 (by eq 46)
0.1 :22.62 degrees
105
Example No. 17 (Continued) Reference Section: 4.3.2
2, = 120 • 0.22(120- 6 (by eq 47)
SkheI = 0.417
K-E cos 2 (26.1 - 22.62)
cos (22.62) cos (22.62 + 17.5) +t sin (26.1 + 17.5) sin (26.1 - 22 .62) (bycos (17.5 + 22.62) eq
KA= 0.928 36)
(adapted from
P = 0.928 • [(120 - 62.4) (1 - 0)] (20' )2 eq 33)
PAE - 10,690 lb per ft of wall
(PAE) -= PA(cosb) - 10,196 lb per ft of wall
Summary
The value of PAE computed using the reduced effective friction angle is28 percent larger than the value of PAE computed in example 16.
106
Example No. 18 Reference Section: 4-3.2
For a wall of height H = 20 ft retaining a submerged cohesionless backfillwith (water content = 15%) 4" 35 degrees, 6 - 17.5 degrees (- 0/2), ' - 0degrees, 9 = 0 degrees, kh = 0.2 (acceleration khg away from the wall andinertia force kh-W towards the wall), and k, - 0, compute the earth and waterpressure forces acting on the wall for the case of free water within thebackfill. Assume a hydrostatic water table within the backfill and r, - 0.3.
Yt"120 pcf AE
Gs 2.65 USHA
Hydrostatic Water Pressure Force
Ustatic I (62.4 pcf) (20' )2
Ustatic = 12,480 lb per ft of wall, acting at20'
Yust = 3 = 6.67 ft
Excess Pore Water Pressure Force
Ushear = 1 [(120 pcf - 62.4 pcf). 0.3] * (20' )2
Ushear = 3,456 lb per ft of wall, acting at
Yush = 6-67 ft [-H_ above the base of the wall with
H,, = H and r, = constant.
Hydrodynamic Water Pressure Force
Pwd =7. (0.2) - (62.4 pcf) (20")2 (by eq 51)121
Pwd = 2,912 lb per ft of wall, acting at
Ypwd 0.4 - 20' - 8 ft
107
Example No. 18 (Continued) Reference Section: 4.3.2
cos (27.38) cos (27.38 + 17.5) [1 + I sin (35 + 17.5) sin (35 - 27.38)cos (17.5 + 27.38) (by eq
36)K= - 0. 8136
PAE= 0.8136 1 1 [(40.32 pcf) (I - 0)] (20' )2 (adapted from eq 33)-7
P - 6,561 lb per ft of wall
(PA)x - PA (cos6) - 6,257 lb per ft of wall
108
Example No. 18 (Continued) Reference Section: 4.3.2
Determine Point of Application of PA
From example 16,
KA 0.246
PA = 1,984 lb per ft of wall, acting at
6.67 ft above the base of the wall.
APA - PAE - PA (solve eq 40 for APE)
APA - 6,561 - 1,984 - 4,577 lb per ft of wall, acting at 12 ft(0.6H) above the base of the wall.
1,984 [21 + 4,577 (0.6 • 20')6,561
Y = 10.4 ft (0.52 H) above the base of the wall
Summary
For the restrained water case (example 16), the total force actingnormal to the wall
- PA (cos6) + Ustatic + Ushear
- 7,921 + 12,480 + 3,456
- 23,857 lb per ft of wall
For the free water case (example 18), the total force acting normal tothe wall
- PE (cos6) + Ustatic + Ushear + Pwd
- 6,257 + 12,480 + 3,456 + 2,912
- 25,105 lb per ft of wall
For this problem, the free water analysis results in a 5 percent larger totaldynamic earth pressure force acting normal to the wall, as compared againstthe restrained water case.
109
Example No. 19 Reference Section: 4.3.3
For a wall of height H = 20 ft retaining a partially submerged cohesionlessbackfill with (Hw - 12 ft) with 0' - 35 degrees, 6 - 17.5 degrees (- 0/2), f -0 degrees, 0 - 0 degrees, kh - 0.2 (acceleration kh'g away from the wall andinertia force kh-W towards the wall) and k, = 0, compute the earth and waterpressure forces acting on the wall for the case of restrained water within thebackfill. Assume a hydrostatic water table within the backfill and ru - 0.1.
Example No. 19 (Continued) Reference Section: 4.3.3
KA = cos 2 (35 - 14.11) 12
cos (14.11) cos (14.11 + 17.5) + sin (35 + 17.5) sin (35 - 14.11) (adaptedL cos (17.5 + 14.11) from
eq 36)
KAE - 0.4254
PAE = (0.4254) (1/2) [95.45 pcf (I - 0)] (20' )Z (adapted from
PAE = 8,121 lb per ft of wall eq 33)
Determine Point of Application of PAE
From example 16,
KA = 0.246
Determine PA and the point of application.
Find the vertical effective stresses slightly above the water table (ay)+wTslightly below the water table (a!,)-wl and at the bottom of the wall (Gý )BOT.
(-Y WUT"
/'i 120Pcf(H HW) (H-Hi,)
#WT
USRU STAIC 1 VH--.; - 51.8 pcf
TOTAL STRESS PWP EFFECTIVE STRESS
112
Example No. 19 (Continued) Reference SEction: 4.3.3
Vertical Total and Effective Stresses Slightly Above G.W.T.
a, - -,(H - H.) = (120 pcf) (20' - 12') = 960 psf
u = 1
.tatic + Ushear = 0
(ay)*' = ay - u = 960 psf
Vertical Total and Effective Stresses Slightly Below G.W.T.
Determine the horizontal active effective stresses slightly above the water
table (a.+W), slightly below the water table (aa-wT), and at the bottom of the
wall (a. BOT)
TOP"o~ -o0
Xtt. 120 pcf 'a
o-. WT -..9 9
1 Hw Y' " 51.8 pcf f
13 SOT YPA
113
Example No. 19 (Continued) Reference Section: 4.3.3
ca = KA (ay)÷WT = (0.246) (960 psf)IWT
aea = 236.2 psf
Oa = KA(ar)-wT= (0.246) (864 psf)-WT
Ga = 212.5 psf
BOT K(y) (0.246) (1,485.6 psf)
aBOT 365. 5 psf
Break the effective stress distribution diagram into rectangles and trianglesto find the magnitude of the resultant force and its point of application.
APAE = 3,708 lb per ft of wall, acting at 12ft (0.6H) above the base of the wall.
y = PA (YPA) + APAE (0.6H)
PAE
y (4,413) (7.44') + (3,708) (0.6) (20')8121
Y = 9.52 ft (0.4811) above the base of the wall.
115
Example No. 20 Reference Section: 4,4
For a wall of height H - 20 ft retaining a dry cohesionless backfill with0' - 30 degrees, 6 = 3 degrees, P = 6 degrees, 9 = 0 degrees, kh = 0.1(acceleration kh-g towards the wall and inertia force khW away tromn the wall)and k, - 0.067 (acceleration k,-g acting downward and inertia force k,1 W actingupward), compute KpE, PpE, and cpPE.
Examples 20 and 21 show that when the inertial force k, W acts downward(example 21) the computed value for PPE is 15 percent larger than PpE for thecase when k, W acts upward (example 20).
I 11
Example No. 22 Reference Section: 4.4
For a wall of height H - 20 ft retaining a dry cohesionless backfill with0' - 35 degrees, 6 - 0 degrees, P - 0 degrees, 8 = 5 degrees, k, - 0.3(acceleration kh-g towards the wall and inertia for-e kW away from the wall),and k, - -0.12 (acceleration k,-g acting upward and inertia force k, W actingdownward) compute KpE, PpE, and QPE.
MOVEMENT"Y•'- 120 pcf
. 00 .
H -2ULNEAR? SLIP PLANE
Method 1 (KpE. by Mononobe Okabe)
Stan-1 .SL-a"- (-0.123 ] (by eq 35)
4 - 15.00°
KpE = cos 2 (35 - 15 + 5)
cos (15) cos 2 (5) cos (15 - 5 + 0) - sin (35 + 0) sin (35 - 15 + 0)cos (0 + 15 - 5) cos (0 - 5)
The values for KPE and PpE computed using Mononobe - Okabe (by Equations 58 and59) are 12 percent larger than the values for EKp (p*, 80) • FPE) and PPE byEquation 62.
121
Example No. 23 Reference Section. 4.4
For a wall of height H - 20 ft (,6 - 0 degrees, 8 - 5 degrees) retaining a drycohesionless backfill with 0' - 35 degrees, 6 - 4, compute the value of PFEfor the case of kh = 0.3 (acceleration kh.g towards the wall and inertia forcekh*W away from the wall), and k, = -0.12 (acceleration k,.g acting upward andinertia force k,.W acting downward). Note that when using the log-spiralsolutions, 6 is set equal to -35 degrees (for Table 3 and Kp(#*, 8*). Calcu-late the magnitude error in the Mononobe-Okabe solution for the value of PFE(KPE by Equation 59 with 6 - 35 degrees) versus the value of PPE determinedusing the equivalent static formulation.
MOVEMENT
S.l• -5 0•v, 120 pcf
H 0. 5" 35-
= tan- 1 - -0.3 (by eq 35)
4= 15.000
Method 1 (Equivalent Static Formulation with Kp by Log-Spiral Method)
cos (15) cos 2 (5) cos (15 - 5 + 35) - sin (35 + 35) sin (35 - 15 + 0)cos (35 + 15 - 5) cos (0 - 5) I
KpE = 11.507 (by eq 59)
PPE = 11.507 (1/2) [(120 pcf) (I - (-0.12))] (20' )2 (by eq 58)
PPE = 309,308 lb per ft of wall
Summary
The Mononobe-Okabe procedure over predicts the value for PPE by 63 percent.
The accuracy of the Mononobe-Okabe solution decreases with increasing values
of 6.
123
Example No. 24 Reference Section: 4.4
For a wall of height H - 20 ft retaining a submerged cohesionless backfillwith 0 - 35 degrees, 6 - 17.5 degrees (= 0/2), fi = 0 degrees, 9 - 0 degrees,kh - 0.2 (acceleration kh • g towards the wall and inertia force kh . W awayfrom the wall), and k, - 0, compute the passive earth pressure force and waterpressure forces acting on the wall for the case of restrained water within thebackfill. Assume a hydrostatic water table within the backfill and r,, 0.3.
a
- 120 pcfS•q- 35=
P PE H -He,, 2V'
U SHEARy _V9 ,• . T STATIC
Hydrostatic Water Pressure Force
Ustatic = 1/2(62.4 pcf)(20) 2
Usttic = 12,480 lb per ft of wall, acting at Yu,,= 20' 6.67 ft
cos (30.75) cos2 (0) cos (30.75 - 0 + 17.5) _ sin (35 + 17.5) sin (35 - 30.75 - 0)cos (17.5 + 30.75 - 0) cos (0 - 0)
KPE = 3.518 (by eq 59)
PpE = 3.518 (1/2) (,40.32 pcf [1 - 0]) (20")1 (adapted from eq 58)
PPE - 28,369 lb per ft of wall
125
Example No. 25 Reference Section: 4.4.1
For a wall of height H = 20 ft retaining a dry cohesionless backfill with 4" -
35 degrees, 6 = 0 degrees, - 0 degrees, 0 - 0 degrees, kh - 0.2(acceleration kh • g towards the wall and inertia force kh W away from thewall), and k, - 0, compute the value for PpE using the simplified procedurefor dynamic passive earth pressures.
-120 pcf
V.. 3504 H2V P PE/V v. ='v - 0°V
Since 6 = 0, the Rankine equation gives the same result as the Coulomb&quation.
Kr- tan2 (45 + 35/2) (by eq 11)
Kp =3.69
Pp 3.69 (1/2) (120 pcf) (20')2 (by eq 13)
Pp -88,560 lb per ft of wall, acting at 6.67 ft (1/3 H) above the base of
the wall
AKpE 17/8 (0.2) (by eq 67)
AKpE 0.425
APpE = 1/2 (120 pcf) (20")2 (0.425) (by eq 65)
APpE - 10,200 lb per ft of wall, acting at 13.33' (2/3 H) above the base ofthe wall.
PPE = 88,560 - 10,200 (by eqn 64)
PPE = 78,360 lb Fer ft of wall
Summary
The value of PpE computed using the simplified procedure agrees with thevalue computed using the Mononobe-Okabe relationship (calculations not shown).
The simplified procedure is limited to values of 6 = 0, vertical wallsand level backfills.
126
Example No. 26 Reference Section: 4.6
For a wall of height H - 20 ft retaining a submerged cohesionless backfillwith surcharge q. - 500 psf, 0' = 35 degrees, 6 = 17.5 degrees (= 0/2), 8 = 0degrees, 9 = 0 degrees, kh = 0.1 (acceleration kh g towards the wall andinertia force kh - W away from the wall), and k, = 0, compute the active earthpressure force and water pressure forces acting on the wall for the case ofrestrained water within the backfill. Assume a hydrostatic water table withinthe backfill and ru = 0.1.
is_ _ _ _ __ _ _ _ _ __ _ _ _ _ qQ. 500 psf
-0. 120 pcf U SHEA H } W 235' " 350 U STATIC Y
Y Y ft.
Hydrostatic Water Pressure Force
Ustatic = 1/2 (62.4 pcf) (20') 2
Usttic = 12,480 lb per ft of wall
Yu5 t - 20'/3 = 6.67' (H,/3) above the base of the wall.
Excess Pore Water Pressure Force
Linear pressure distribution with depth for ru = constant.Lop
cos 2 (0) cos (0 + 17.5) 1+ sin (35 + 17.5) sin (35 - 0)cos (17.5 + 0) cos (0 - 0) (by eq 16)
KA = 0. 2461
Determine PA and the point of application.
Find the vertical effective stress at the ground surface.
129
Example No. 26 (Continued) Reference Section: 4.6
cy % = 500 psf
Ustatic = 0top
Ushear Ushear = 50 psf
U U 8tati + Ushear = 0 + 50 psf = 50 psf
(ay)tOP co, - u = 500 psf - 50 psf = 450 psf
Find the vertical effective stress at the base of the wall.
(0 ,)bot = (ay)toP + y . H. = 450 psf + (51.84 pcf) (20')
(u,)b"t - 1,487 psf
Determine the horizontal active effective stress at the ground surface (actoP),and at the bottom of the wall ( 0abot)
top"
ca KA(ay)t0 P = (0.2461) (450 psf)
yat -- 110.8 psf
a bot = KA (y)bOt (0.2461) (1,487 psf)
aot 366 psf
Break the trapezoidal effective stress distribution diagram into a rectangleand a triangle to find the magnitude of the resultant force and its point ofapplication.
y = YE = PA (YIA) + APAE (YapAd - (4768) (8.22') + (3520) (14.5')PAE 8,288
Y = 10.89 ft (0.54 H) above the base of the wall
131
CHAPTER 5 EARTH PRESSURES ON WALLS KI&T*TNING NONYIELDING BACKFILLS
5.1 Introduction
This part of the report describes two procedures that are used t) com-pute the dynamic earth pressures acting along the back of walls retainingnonyielding backfills due to earthquake shaking. In practical terms, a wallretaining a nonyielding backfill is one that does riot develop Qhe limitingdynamic active or passive earth pressures because sufficient wall movements donot occur and the shear strength of the backfill is not fully mobilized - wallmovements that are less than one-fourth Lo one-half ot Table 1 wall movementvalues. Because of this, earth retaining structures such as massive concretegravity retaining walls founded on firm rock or U-frame locks and dry docksare sometimes referred to as structures retaining "nonyielding" backfills inthe literature. Two procedures for analyzing such cases are a simplifiedanalytical procedure due to Wood (1973) and a complete soil-structure interac-tion analysis using the finite element methocd (see Appendix D).
5.2 Wood's Solution
Wood (1973) analyzed the response of a wall retaining nonyielding hick-fill to dynamic excitation assuming the soil backfill to be an elasticmaterial. He provided normal mode solutions for the case of both a uniformmodulus and a modulus varying linearly with depth. Since Lhese solutions areslowly convergent for practical problems Wood (1973) presented approximateprocedures based on findings from the normal mode solut ions. Wood showed thata static elastic solution for ai uniform 1g horizontal body force gave veryaccurate results for the pressures, forces, and moments on the wall underharmonic excitatic- Df frequency f (cyclic frequency) when dynamic amplifica-tion effects were negligible. This occurs whe.n W = f/fi is less than about0.5 where f is the frequency of notion and f, = V5/4H is the cyclic frequencyof the first shear mode of the Lackfill considered as a semi-infinite layer ofdepth H. The limiting W depends on the value of V. and the geometry of 'theelastic backfill but the value 0 < 0,5 covers many practical cases.
In cases of wide backfills, the lateral seismic force against the wallwhen N < 0.5 is given by
Fsr = -y H2 - kh (68)
acting at a height of 0.63 H above the back of the wall.
The normal stress distributions along the back of the wall were shown tobe a function of (i) Poisson's ratio, u, and (2) the lateral extent of theela',tic medium behind the w.ill, expressed in terms of the ratio of the width-f the elastic backfill divided by the height of the wall, L/H (see Fig-ure 5.1). Two examples of the variation in the values for the normalizedhorizontal stresses with normalized elevations above the base of the wall are.,,own in Figure 5.2. A L/H value equal to I corresponds to a narrow backfillplaced within rigid containment and a L/H value equal to 10 corresponds to abackfill of great width. The horizontal stresses at any elevation Y along theback of the wall, (j, are normalized by the product of t"H in this figure.
1 33
Y HOMJOGEN'OUS ELASTCJ SOILj (P1LM1E 5TRAJN
cry- 0 •DISPIJEMENT X . 0rxy = 0 .. • 0
O/5PLK4EUENT X -0
H RIGID
WALL
DISPLACEMENT X -0
/ RIGID BOUNDARY /xL
Figure 5.1 Model of elastic backfill behind a rigid wall
The resulting distributions for the horizontal stresses are parabolic, -.itfilarger values computed along the upper half of the wall, as compared to thevalues computed along the lower half. In addition, the results show a, to belarger for wide elastic backfills, as compared to those values computed fornarrow elastic backfills. Figure 5.3 shows the corresponding resultant hori-zontal force, Far, along the back of the rigid wall and the correspondingseismic moment about the base of the rigid wall, Msr, as a function of u andL/H. Figure 5.3 presents the resultant force and moment in terms of theirdimensionless values. Fs, acts at a height
yr .Msr (69)Ysr =- 'r6
The stresses shown in Figure 5,2 and the forces and moments shown in Fig-ure 5.3 result from the application of a l-g static horizontal body force.The values for a. and Fsr corresponding to other constant horizontal accelera-tion values are computed by multiplying the a, value from Figure 5.2 and theFsr value from Figure 5.3 by the ratio of the new acceleration valuecoefficient, kh.
Shaking table tests by Yong (1985) using dry sand backfill and one-haltmeter high walls have confirmed the applicability of Wood's simplifiedprocedure when the predominant frequency of shaking is significantly less thanthe fundamental frequency of the backfill. The measured forces exceeded by afactor of 2 to 3 those predicted by the Mononobe-Okabe theory. The tests
DIMENSIONESS NORMAL STRESS r o/-YH DIMENSIONESS NORMAL STRESS ao/lYH
Figure 5.2 Pressure distributions on smooth rigid wall for 1-g statichorizontal body force
clearly showed the limitations of Woods simplified procedure when this condi-tion is not met. If the dynamic response of the backfill amplifies theaccelerations at the level of the base of the backfill,the assumption of con-stant acceleration is not met and much greater earth pressures can result.
Woods (1973) has given two approximate procedures for estimating seismicsoil pressures against walls retaining nonyielding backfills when dynamiceffects are important; typically when U > 0.5. In cle procedure the dynamicresponse is represented by a number of low frequency modes together with apseudomode called a rigid body mode :o represent the combined effects of thehigher modes.
The other procedure is based on the use of an equivalent two mode systemwith frequencies and damping ratios predefined to provide the best fit of thefull dynamic modal solution.
Effective use of these procedures requires at least a broad understand-ing of Wood's general approach to the dynamic response of unyielding retainingstructures. Therefore, the reader is referred to Wood (1973) for details onhow to implement the approximate dynamic procedure.
Wood's simplified procedures do not account for: (1) vertical accelera-tions, (2) the typical increase of modulus with depth in the backfill, (3) the
135
1.2
U00.5-.
0
tn 0.6
E-
0.4
0.2
0
0.8
0.6...-
I-~o
00.3I
: 0.4
3-
z 0.2
0
0 2 4 6 3• 10
L/H
FROM WOOD (1973)
Figure 5.3 Resultant force and resultant
moment on smooth rigid wall for 1-gstatic horizontal body force
136
influence of structures or other loads on the surface of the backfill, (4) thephased response at any given time for the accelerations and the dynamic earthpressures with elevation along the back of the wall, and (5) the effect of thereduced soil stiffness with the level of shaking induced in both the soilbackfill and soil foundation. These and many other factors are addressed inthe procedures used to simulate the dynamic response of earth retaining struc-tures by a complete soil-structure interaction analysis.
137
CHAPTER 6 ANALYSIS AND DI.EicN EXAMPLES FOR GRAVITY WALLS RETAINING YIELDINGBACKFILLS
6.1 Introduction
Gravity walls generally are designed assuming that some permanent dis-placement will occur during the design seismic event. This assumption isimplicit in procedures using a seismic coefficient significantly less than theacceleration coefficient corresponding to the design event. Newer methods,such as the displacement controlled approach developed by Richards and Elms(1979) explicitly consider such permanent displacements. If permanent dis-placements greater than about I inch per 20 foot height of wall (<iH = 0.004,see Table 1) are not permissible, the analyses described in Chapter 8 shouldbe used.
The procedures described in this chapter quantify the effect of earth-quakes on the backfill by means of inertial forces acting on the soil masscomprising the sliding wedge within the backfill using the Mononobe-Okaberelationships for dynamic active and passive earth pressures. Where signifi-cant permanent displacements do occur, it is appropriate to use the Mononobe-Okabe theory to evaluate static and dynamic earth pressures. As discussed inChapter 4, there is ample evidence that this theory is correct for dry sand"-ckfills, although supporting evidence is very weak in the case of submergedbackfills. With gravity walls, the dynamic increments of earth pressure gen-erally are small compared to the inertia force on the wall itself and changesin water pressure on the poolside of the wall. Hence the exact values fordynamic earth pressures usually are not crucial The procedures outlined inthis chapter assume that all dynamic forces act simultaneously in the worstpossible direction. This assumption is likely conservative (Whitman 1990;Anderson, Whitman, and Germaine 1987; Al Homound 1990), but is retainedpending more complete studies of case histories from earthquakes.
Dynamic finite element analyses seldom are suitable for use duringdesign of gravity walls, but will prove very useful for further research intoissues such as the phasing of the various earth and water pressures actingupon a wall. When such studies are made, the wall should be modeled as mov-able in response to the forces acting upon it, and not as a rigid, nondisplac-ing wall.
The Mononobe-Okabe theory for computing PAE and PpE is described in Chap-ter 4. The presence of water within the backfill affects not only the staticpressures acting on the wall, as discussed in Chapter 3, but also the dynamicpressures. During an earthquake, the saturated portion of the backfill thatis below the water table may experience the development of additional porewater pressures due to the shear strains that occur within the backfill duringearthquake shaking. These excess pore water pressures reduce the effectivestresses within the backfill, resulting in both a reduction in the strength ofthe soil and adding to the destabilizing forces which act along the back ofthe wall. The magnitude of the excess pore water pressures generated withinthe soil during an earthquake can range from zero to the extreme case of pres-sures that are equal to the pre-earthquake vertical effective stresses, astate that corresponds to the liquefaction of the, backfill. For those wallsthat have a pool of water in front of the wall, the earthquake shaking resultsin hydrodynamic pressures acting along the submerged portion at the front ofthe wall. The Westergaard procedure is used for computiltg the hydrodynamic
130
water pressures, which are superimposed on the static water pressure distribu-tion along the front of the wall. The hydrodynamic pressure force acts todestabilize the wall and acts counter to the direction of the static waterpressure force.
The seismic stability analysis of rigid walls that undergo movementsduring earthquakes is categorized as one of four types of analyses, as shownin Figure 6.1 and as listed in Table 4. These categories include rigid wallsretaining dry backfills (Case 1), and three categories for rigid walls retain-ing submerged backfills, depending upon the magnitude of excess pore waterpressures that are generated during the earthquake. They range from the caseof no excess pore water pressures (Case 2) to the extreme case which corre-sponds to the complete liquefaction of the backfill (Case 4) and the interme-diate case between the two (Case 3). In Figure 6.1, Ustatic corresponds to thesteady state pore water pressure force acting along the back of the wall andthe water pressure force when a pool exists in front of the wall. Usher cor-responds to the excess pore water pressure force acting along the back of thewall when excess pore water pressures are generated within the submerged por-tion of the backfill during the earthquake. HFinrtia corresponds to thehydrodynamic water pressure force of a liquefied backfill. Procedures fordetermining the potential for liquefaction within the submerged backfill orthe potential for the development of excess pore water pressures are discussedin Seed and Harder (1990) and Marcuson, Hynes, and Franklin (1990).
Experience gained with the evaluation of the stability and safety ofexisting Case 1 walls subjected to earthquake shaking over the last 20 yearshave established the validity of both the conventional equilibrium method ofanalysis and the displacement controlled approach for dry backfills. However,most of the case histories reported in the literature are for walls retainingsubmerged backfills that had liquified during earthquakes. The proceduresoutlined in this section for the analysis of the stability of the Case 2through Case 4 retaining walls are proposed extensions of the procedures usedfor the analysis of walls retaining dry backfill.
The design of gravity walls generally begins with design for staticloadings. Then the wall is checked for adequacy during the design seismicevent, using the procedures described in the following sections. Adequacy forpost-seismic conditions should also be checked, considering the effect ofresidual lateral earth pressures and any excess pore pressures as discussed inChapter 2.
6.2 Procedure Based upon Preselected Seismic Coefficient
The force equilibrium method of analysis expresses the safety and sta-bility of an earth retaining structure subjected to static and/or dynamicearth and water forces in terms of (1) the factor of safety against slidingalong the base of the wall, (2) the ability of the wall to resist the earthand water forces acting to overturn the wall, and (3) the factor of safetyagainst a bearing capacity failure or crushing of the concrete or rock at thetoe in the case of a rock foundation. The ability of the retaining wall toresist the overturning forces is expressed in terms of the portion of the wallbase remaining in contact with the foundation or, equivalently, the base arearemaining in compression (Headquarters, Department of the Army EM 1110-2-2502,Ebeling et.al. 1990; Ebeling et al. 1992). Recommended minimum static and
140
r p
AMA
CASE 1: Dry BockFill
AN U pnrd/
N*N
CASE 2: Submerged BackFill, No Excess PoreWater Pressures Due to Earthquake.
CASE 1: Dumry edBacack, xcssllr
PAE
CASE 2: Submerged BackFill, Excess PoreWater Pressures Due To Earthquake.
14 1U Iila
CASE 4:Liquefied BackFill.
Figure 6.1 Rigid walls retaining backfills whichundergo movements during earthquakes
141
Table 4 Section Numbers That Outline Each of the Two Design Proceduresfor Yielding Walls for the Four Categories of Retainig Walls
dynamic factors of safety and minimum base contact areas are listed inTable 5. Post-earthquake settlements should also be checked.
6.2.1 Stability of Rigid Walls Retaining Dry Backfills which Undergo Movementsduring Earthquakes
The force equilibrium procedure for evaluating the stability and safetyof rigid walls retaining dry backfills, of the type shown in Figure 6.2, isdescribed in Seed and Whitman (1970). This analysis, described as Case 1 inFigure 6.1, is an extension of traditional force equilibrium procedure that isused in the evaluation of the stability and safety of rigid walls under staticloadings. The rigid wall is presumed to have undergone sufficient movementsso that the active dynamic earth pressure force develops along the back of thewall. The eight steps in the stability analysis of the displaced rigid wallshown in Figure 6.2 are as follows:
(1) Select the kh value to be used in the analysis; see Section 1.4 ofChapter 1.
(2) Select the k, values to be used in the analysis; see Section 1.4.3 ofChapter 1.
Seed and Whitman (1970) found that for typical gravity earth retainingwall design problems with no toe fill in front of the wall, PA values varied
142
Table 5 Minimum Factors of Safety When Usingthe Preselected Seismic Coefficient Method
of Analysis
From U.S. Army Corps of Engineers EM 1110-2-2502 (1989)
Factor of Safety Factor of SafetyFailure Mode Static Earthquake
Sliding 1.5 1.1 - 1.2
BE/B 100% 75%(50%-Rock)
Bearing* 3 >2
*Check for settlements, including differential settlements.
by less than 10 percent (as discussed in Section 4.5). In other casesvertical accelerations can contribute to the forces attempting to destabilizethe wall (e.g. slender walls). In general, k, values other than zero would beincluded in the analysis when vertical accelerations impact wall stability.
(3) Compute the dynamic active pressure force using the Mononobe-Okabe rela-tionships as described in Chapter 4. PA is computed using equation number33, with KAE given by Equation 34 and acting at the height as given in Fig-ure 4.7. For a vertical wall retaining a horizontal backfill, PA may be com-puted directly or defined in terms of the static force PA and the incrementalinertial force APA. PA is computed using Equation 7 with KA given by Equa-tion 16, using the Seed and Whitman's simplified procedure,and APA is com-puted using Equation 41 with AKA given by Equation 43. PA is equal to thesum of these two forces (Equation 40) with a point of action, Y, given byEquation 44, as shown in Figure 4.8. For most engineered granular backfills,6 equal to 0/2 is a reasonable value. Table 2 provides a list of ultimatefriction angles for a variety of dissimilar materials that may interface withone another.
(4) Compute the weight of the wall W and point of application, and using theforce PA and its point of application as determined in step 3, solve for theunknown forces N and T which act along the base of the wall using the horizon-tal and vertical force equilibrium equations.
The force W is computed per lineal foot of wall by multiplying the unitcross-sectional area of the wall by a representative value for the unit weightof the section. The resultant force acts at the center of mass for the crosssection.
The total normal force between the wall and the foundation is equal to
N W + (PAO)v (70)
143
Aovemert$
N.
Slip Occurs
(a) Slip planes
//A K h'9
B (b) Effective Accelerations
P..
W K A1W •Kh
W "..
TOE TNN N
Nt T TaVK.9NE • -XN (c) Forces On Gravity Wall
Be
Figure 6.2 Rigid walls retaining dry backfill which undergo movementsduring earthquakes (case 1 in Figure 6.1)
where
W = weight of the wall(PAE)y = the vertical component of PA.
144
The point of action of the force N, XN is computed by summing momentsabout the toe of the wall
Xw, Yw = center of mass for the wall, as measured from the toe of thewall and the base of the wall, respectively.
The horizontal force T is the shear force required for slidingequilibrium of the wall and is equal to
T = (PAR) X + W*kh (72)
where
W'kh = horizontal inertia force of the wall.
(5) Compute the factor of safety against sliding, F,.
ultimate shear force (73)shear force required for equilibrium
The ultimate shear force along the base, TuLt, is given by
Tut = N. tan6b (74)
where
6 b - the base interface friction angle.
(6) Compare the computed factor of safety against sliding to the requiredfactor of safety. Many retaining walls are designed using static active earthpressures with a factor of safety of 1.5 against sliding along the base. Fortemporary loading cases, such as earthquakes, the minimum required factor ofsafety is equal to 1.1 or 1.2 (Table 5). For a ductile wall to foundationinterface, as the value of F, approaches the minimum required value, the mag-nitude of the translation of the structure will increase as the value of F,decreases (Newmark 1965). For a bonded interface, the displacements will besmall until the bond is ruptured (at F, - 1.0) and a brittle failure results.
145
(7) The overturning criterion is expressed in terms of the percentage of basecontact area Be/B, where Be is the width of the area of effective base con-tact. Assuming that the bearing pressure varies linearly between the base ofthe wall and the foundation, the normal stress is a maximum at the toe (qqmax) and a minimum at. the inner edge (q 0 0) as shown in Figure 6.3.
B, = 3xN (75)
An alternative assumption regarding base pressure distribution and contactarea was suggested by Meyerhof (1953). Meyerhof assumed a uniform distribu-tion of pressure along the base, resulting in an effective base contact equalto
B 2.xN. (76)
Meyerhof's pressure distribution has been used widely for foundations on soiland is most. appropriate for foundation materials that exhibit ductile mecha-nisms of failure. The assumption is less appropriate for brittle materials.
Many retaining walls are designed using static active earth pressures-iv, o..ll contact along the base, Be/B ( or B',/B), equal to 100 percent. For
temporary loading cases, such as earthquakes, this criteria is relaxed to aminimum value of 75 percent, 50 percent for rock foundations (Table 5).
(8) For those structures founded on rock, the factor of safety against bearingcapacity failure, or crushing of the concrete or the rock at the toe, can beexpressed as
Fb = quIt (77)qmax
where q,,,, is the ultimate bearing capacity or compressive strength of theconcrete or the rock at the toe and 9max is the maximum bearing pressure atthe toe. For brittle materials like unconfined concrete, the ultimate bearingcapacity is equtl to the compressive strength of the material. Building codesare commonly used to obtain values for the allowable bearing stress on rock,qal1 . Alternately, a large factor of safety is applied to the unconfined com-pressive strength of intact samples. The maximum bearing pressure q... isrestricted to an allowable bearing capacity qail. For ductile foundationmaterials that undergo plastic failure, the ultimate bearing capacity isgreater than the compressive strength of the material, excluding those founda-
ti,)n materials exhibiting a loss in shear resistance due to earthquake- induceddeformations or due to the development of residual excess pore water pres-sures. In these cases, a conventional bearing capacity evaluation isconducted to establish the post-earthquake stability of the structure.
In stability analyses in which the vertical accelerations are consid-ered, the force acting downward through the center of mass of the wall thatrepresents the weight of the wall, W, in Figure 6.2, is replaced by the force
1 4 G
6* -3XN
2Niq mox 23XN
N
e
N
a. Linear base pressure distribution
e "2XN
q maq -, 2 XN
L MalI N X
..,_
B e/
b. Uniform base pressure distribution
Figure 6.3 Linear and uniform base pressuredistributions
W(l k) acting downward. The first term in Equations 70 and 71, W and WXWare replaced by W(l-kQ) and W (1-k,)4X, respcctively The direction in whichthe vertical inertia force, k+W, acts is counter to the direction assigned tn
the effective vertical acceleration, kvg. Vertical accelerations will alsoaffect AhP values for PAV (Equation 33) and KAE (Equa tion 34), as described in
14/
Section 4.2. The stability should be checked for the possibility of k, actingin either direction.
This procedure is illustrated in example 27 at the end of this chapter.
6.2.2 Stability of Rigid Walls Retaining Submerged Bacl.fills which UndergoMovements During Earthquakes - No Excess Pore Water Pressures
The presence of water within the backfill an"' in front of the wallresults in additional static and dynamic forces acting on the wall and altersthe distribution of forces within the active soil wedge developing behind thewall. This section describes the first of three proposed force equilibriumprocedures used in the evaluation of the stability and safety of rigid wallsretaining submerged or partially submerged backfills and including a pool ofwater in front of the wall, as shown in Figure 6.4. This analysis, describedas Case 2 in Figure 6.1, assumes that no excess pore water pressures are gen-erated within the submerged portion of the backfill or within the foundationduring earthquake shaking. The evaluation of the potential for the generationof excess pore water pressures during the shaking of the submerged soilregions is determined using the procedure described in Seed and Harder (1990)or Marcuson, Hynes, and Franklin (1990). The rigid wall is presumed to have-undergone sufficient movements so that the active dynamic earth pressure forcedevelops along the back of the wall. Many of the details regarding the pro-cedures used in the eight steps of the stability analysis of walls retainingdry backfills (Section 6.2.1) are similar to those procedures used for sub-merged backfills, and the explanations for these common steps are not repeatedin this section. The eight steps in the stability analysis of the displacedrigid wall retaining submerged backfill as shown in Figure 6.4 are as tollows:
(1) Select the kh value to be used in the analysis; see Section 1.4 ofChapter 1.
(2) Consider kv, as discussed in Section 1.4.3.
(3) Compute PA using the procedure described in Section 4.3. tJstati, is deter-mined from the steady state flow net for the problem. By definition, onlysteady state pore water pressures exist within the submerged backfill andfoundation of a Case 2 retaining structure (r ,- 0). In the restrained watercase of a fully submerged soil wedge with a hydiostatic water table, PAE iscomputed (Equations 33 and 38) using an effective unit weight equal to 1b.
KA (Equation 34) or KA(fl*,O*) (Equation 38) are computed using an equivalenthorizontal acceleration, khel, and an equivalent seismic inertia angle, 0,given by Equation 47 and 48. In the case of a partially submerged backfill,this simplified procedure will provide approximate results by increasing thevalue assigned to the effective unit weight based upon the proportion of thesoil wedge that is above and below the water table. A more refined analysismay be conducted using the trial wedge procedure (Section 3.4) for the forcesshown in Figure 6.4. For most engineered granular backfills, 6 equal to f/2is a reasonable value (Table 2).
(4) Compute the weight of the wall W and point of application, and using theforce PA and the point of application as determined in step 3, solve for theunknown forces N' and T which act along the base of t-he wall using the hori-zontal and vertical force equilibrium equations.
148
The force W is computed per lineal toot of w;0 l by multiplying tOh tillit,cross-sectional area of the wall by a representative ,:.alue for the unit we-ightof the section. The resultant force acts at the ceitter of mass for theý crossection.
The effective normal force between the wall and the toundat ion is (-qkLilto
N' = W + (PAE)y - Ub (78)
where
W = weight of the wall
(Pu)y = the vertical component of PAE
Ub = resultant pore water pressure force along the base of thi.wall
The point of action of the force N', XN, is computed by summing momentsabout the toe of the wall
x_ + MPA - Ustat.i(Yus,) - Ub(Xh) + Mpo}l
N
where
MW = W(Xw) - W(kh)YW
MpAE = (PAE)Y (XpAE) - (PAE)X (YPAE)
MPool = Upool(Yup) - Uinertia(Yud)
(PAE)X = PA cos( 6 + 0 )
(PAE)y = PAE sin( 6 + 8 )
XpAE = B - (YpAE) tan 0
YPAE = Y
Yut = point of action of U 5taic (from flow net)
Yp = point of action of Upo., (ý Hp/3)
Yi = point of action of Uinertia (see Appendix B)
Yb - point of action of Ub (from flow net)
Xw, Yw - center of mass for the wall, as measured trom the toeof the wall and the base of the wall, respectivelv.
1,49
P
PAE W~
QAAEH H W UF
A
(a.) Backfill
B
Xw
P 8X
PAE
W ~
I Itoo
Ubl N4 T
4X N
(b.) Wall and Pool
Figure 6.4 Rigid wall retaining submerged backfill which undergomovements during no excess pore water pressures (Case 2 in
Figure 6.1)
150
The horizontal force T is the shear force required for equilibrium ofthe wall and is equal to
Ustatic ý resultant steady state pore water pressure force alongthe back of the wall.
UPOOI= resultant hydrostatic water pressure force for the pool
Uinert-a - hydrodynamic water pressure force for the pool directed awayfrom the wall (see Appendix B).
(5) Compute the factor of safety against sliding, F,, using Equation 73. Theultimate shear force along the base, Tuf,, is given by
Tu=t = Nl.tantb (81)
where
6b = the effective base interface friction angle.
(6) Compare the computed factor of safety against sliding to the requiredfactor of safety of 1.1 or 1.2 for temporary loading cases (Table 5).
(7) The stability against overturning is expressed in terms of the base areain compression, Be. B. is computed by either Equation 75 or 76, as describedin Section 6.2.1. Many retaining walls are designed using static active earthpressures with full contact along the base, Be/B ( or B' 8 /B), equal to100 percent. For temporary loading cases, such as earthquakes, this criteriais relaxed to a minimum value of 75 percent, 50 percent for rock foundations(Table 5).
(8) Check the stability of the wall against a bearing capacity failure, asdiscussed in step 8 of Section 6.2.1.
6.2.3 Stability of Rigid Walls Retaining Submerged Backfills which UndergoMovements During Earthquakes - Excess Pore Water Pressures
This section describes the second of three proposed force equilibriumprocedures for evaluating the stability and safety of rigid walls retainingsubmerged or partially submerged backfills and including a pool of water infront of the wall, as showa in Figure 6.5. This analysis, described as Case 3in Figure 6.1, assumes that excess pore water pressures, in addition to thesteady state pore water pressures, are generated within the submerged portionof the backfill or within the foundation during earthquake shaking. The mag-nitude and distribution of these excess pore water pressures depend upon sev-eral factors including the magnitude of the earthquake, the distance from the
151
site to the fault generating the earthquake and the properties of the sub-merged soils. The evaluation of the magnitude of the residual excess porewater pressures within the submerged soil regions due to earthquake shaking isdetermined using the procedure described in Seed and Harder (1990) orMarcuson, Hynes, and Franklin (1990). The rigid wall is presumed to haveundergone sufficient movements so that the active dynamic earth pressure forcedevelops along the back of the wall. Many of the details regarding theprocedures used in the nine steps of the stability analysis are common to theCase 1 and Case 2 analyses. The nine steps in the stability analysis ofFigure 6.5 displaced rigid wall retaining a submerged backfill with excesspore water pressures within the soil regions are as follows:
(1) Select the kh value to be used in the analysis; see Section 1.4 ofChapter 1.
(2) Consider kv, as discussed in Section 1.4.3.
(3) Compute PAE using the procedure described in Section 4.3. The total porewater pressures existing near the end of earthquake shaking are equal to thesum of the steady state pore water pressures and the residual excess porewater pressures. Ustatic is determined from the steady state flow net for theproblem. The post-earthquake residual excess pore water pressures areidentified as Ushear and AU, respectively, in Figure 6.5 and are determinedusing the procedures described in Seed and Harder (1990) or Marcuson, Hynes,and Franklin (1990). In the restrained water case of a fvlly submerged soilwedge with a hydrostatic water table and r, equal to the average value withiinthe backfill, PAE is computed (Equations 33 and 38) using an effective unitweight (Equation 52). KAE (Equation 34) or KA(O,O*N) (Equation 38) is computedusing an equivalent horizontal acceleration, khe3, and an equivalent seismicinertia angle, 'P•e3, given by Equations 54 and 55.
An alternative approach is to compute PAE using an effective unit weightequal to Ib and a modified effective friction angle, 0, (Equation 56). KAE
(Equation 34) or KA(3*,,*) (Equation 38) are computed using an equivalenthorizontal acceleration, khel, and an equivalent seismic inertia angle, 0,,given by Equations 47 and 48.
In the case of a partially submerged backfill, either of the simplifitedprocedures providt-3 for approximate results by increasing the value assigne.dto the effective unit weight based upon the proportion of the soil wedge thatis above and below the water table. A more refined analysis may be conductedusing the trial wedge procedure (Section 3.4) for the forccs shown in Figure6.5. For most engineered granular backfills, 6 equal to 0/2 is a reasonablevalue (Table 2).
(4) Compute the weight of the wall W and corresponding point of application,with the forces determined in step 3 and their points of application, solvefor the unknown forces N' and T which act along the base of the wall using thehorizontal and vertical force equilibrium equations.
152
P
7 WW.Kh
H H~ W7 UŽJU j WPA
V U "•"
&U Ub N'I-- O X# ,U~
/ 't (AE Y Yh
(a.) Backfill
- -IN - h
(b. Wal adtool
Figure 6.5 Rigid wall retaining submerged backfill which undergomovements during earthquakes, including excess pore water
pressures (Case 3 in Figure 6.1)
The force W is computed per lineal foot of wall by multiplying the unit
cross-sectional area of the wall by a representative value for the unit weight
of the section. The resultant force acts at the center of mass for the crosssection.
153
The effective normal force between the wall and the foundation is equalto
N' = W + (PAE)Y - Ub - AU (82)
where
AU = resultant excess pore water pressure force along the baseof the wall
The point of action of the force N', XN, is computed by summing momentsabout the toe of the wall
x= M, +M + Mp - A U(XDU) - Ub(X.),) + Moi (83)N'
where
Mw = W(Xw) - W(kb)Yw
MPAE = (PAE)Y (XPA.) - (PAE)X (YPAE)
Moo0 1 = Upoo 1 (Yup) - Uinertia(Yud)
Mpwp = -Ustatic(Yust) - Ushear(Yush)
and
(PAE)X = PAE COS( 6 + 0 )
(PAE)Y = PA sin( 6 + 0 )
XpAE = B - (YpAE) tan 0
YPAE = Y
Yush = point of action of Ushear
XDU = point of action of AU
The horizontal force T is the shear force required for equilibrium ofthe wall and is equal to
Ushear = resultant excess pore water pressure force along the back ofthe wall.
Procedures for the computation of values for Ushear, Yush, AU, and XDU are dis-cussed in Seed and Harder (1990) or Marcuson, Hynes, and Franklin (1990).
(5) Compute the factor of safety against sliding, F5, using Equation 73. Theultimate shear force along the base, T,,t, is given by Equation 81.
(6) Compare the computed factor of safety against sliding to the requiredfactor of safety of 1.1 or 1.2 for temporary loading cases (Table 5).
154
(7) The stability against overturning is expressed in terms of the base areain compression, Be. Be is computed by either Equation 75 or 76, as describedin Section 6.2.1. Many retaining walls are designed using static active earthpressures with full contact along the base, B./B ( or B'e/B), equal to100 percent. For temporary loading cases, such as earthquakes, this criteriais relaxed to a minimum value of 75 percent, 50 percent for rock foundations(Table 5).
(8) Check the stability of the wall against a bearing capacity failure, asdiscussed in step 8 of Section 6.2.1.
(9) Additional stability considerations for the retaining wall are discussedin Chapter 2. Some of the factors to be considered are the potential forstrength loss within looser foundation materials and the post-earthquakeredistribution of excess pore water pressures. Post-earthquake stability ofthe wall and post-earthquake settlements should also be considered.
This procedure is illustrated in example 28 at the end of this chapter.
6.2.4 Stability of Rigid Walls Retaining Submerged Backfills which UndergoMovements During Earthquakes - Liquified Backfill
This section describes the force equilibrium procedure used in the eval-uation of the stability and safety of displaced rigid walls retaining sub-merged or partially submerged backfills and including a pool of water in frontof the wall, as shown in Figure 6.6. This analysis, described as Case 4 inFigure 6.1, assumes that the submerged portion of the backfill has liquified(ru = 100%) during the earthquake and that excess pore water pressures (r" <100%) are generated within the foundation during earthquake shaking. Theevaluation of the liquefaction potential for the backfill and the magnitude ofthe residual excess pore water pressures within the foundation are determinedusing the procedure described in Seed and Harder (1990) or Marcuson, Hynes,and Franklin (1990). Many of the details regarding the procedures used in thenine steps of the stability analysis are common to the previously describedanalyses. The steps in the stability analysis of Figure 6.6 displaced rigidwall retaining a liquified backfill with excess pore water pressures withinthe soil foundation are as follows:
(1) Select the kL value to be used in the analysis; see Section 1.4 ofChapter 1.
(2) Consider k,, as discussed in Section 1.4.3.
(3) Compute the forces acting along the back of the wall,=I H (5
HFstatic = tH (85)
identified as HFt.atic and HFinertia in Figure 6.6. Upon liquefaction of thebackfill during the earthquake, the earth pressure forces acting along theback of the wall are equivalent to a heavy fluid with a density equal to thetotal unit weight of the backfill, -yr. The inertial force of the heavy fluidduring shaking is approximated using the Westergaard procedure (Appendix B)for the inertia force of a fluid as acting at 0.4H above the base of thewall.
(4) Compute the weight of the wall W and corresponding point of applicationwith the forces determined in step 3 and their points of application; solvefor the unknown forces N' and T which act along the base of the wall using thehorizontal and vertical force equilibrium equations.
The force W is computed per lineal foot of wall by multiplying the unitcross-sectional area of the wall by a representative value for the unit weightof the section. The resultant force acts at the center of mass for the crosssection.
The effective normal force between the wall and the foundation is equalto
N' = W - Ub - AU (87)
156
The point of action of the force N', XN, is computed by summing momentsabout the toe of the wall
XN= MW + MHF - AU(Xu) - Ub(X b) + (88)N'
where
Mw = W(Xw) - W(kh)Y1
M• = -HF.tatj,:(Yu) - HFinertia (Yi)
MPol = UPOo| (Y.p) - Uinýtt (YW)
andYms - point of action of HFttiC - H/3)
Yj - point of action of HFinertia ( - 0.4H)
In the case where excess pore water pressures are generated within the founda-tion, the steady state flow net is used to compute the steady state pore waterpressure force U~b along the base of the wall, and the excess pore water pres-sure force AU is computed using the procedure described in Seed and Harder(1990) or Marcuson, Hynes, and Franklin (1990). The horizontal force T is theshear force required for equilibrium of the wall and is equal to
(5) Compute the factor of safety against sliding, F,, using Equation 73. Theultimate shear force along the base, Tult, is given by Equation 81.
(6) Compare the computed factor of safety against sliding to the requiredfactor of safety of 1.1 or 1.2 for temporary loading cases (Table 5).
(7) The stability against overturning is expressed in terms of the base areain compression, B.. B. is computed by either Equation 75 or 76, as describedin Section 6.2.1. Many retaining walls are designed using static active earthpressures with full contact along the base, Ba/B ( or B'./B), equal to100 percent. For temporary loading cases, such as earthquakes, this criteriais relaxed to a minimum value of 75 percent, 50 percent for rock foundations(Table 5).
(8) Check the stability of the wall against a bearing capacity failure, asdiscussed in step 8 of Section 6.2.1.
(9) Additional stability considerations for the retaining wall are discussedin Chapter 2. Some of the factors to be considered are the potential forstrength loss within looser foundation materials and the post-earthquakeredistribution of excess pore water pressures. Post-earthquake stability ofthe wall and post-earthquake settlements should also be considered.
157
6.3 Displacement Controlled Approach
The displacement controlled approach incorporates wall movements explic-itly in the stability analysis of earth retaining structures. It is, ineffect, a procedure for choosing a seismic coefficient based upon explicitchoice of an allowable permanent displacement. Having seleect-d the seismiccoefficient, the usual stability analysis against sliding is performed,including use of the Mononobe-Okabe equations. No safety factor is applied tothe required weight of wall evaluated by this approach; the appropriate levelof safety is incorporated into the step used to calculate the horizontal seis-mic coefficient. This procedure of analysis represents an alternative to theconventional equilibrium method of analysis which expresses the stability of arigid wall in terms of a preselected factor of safety against sliding alongits base, as described in Section 6.2.
The analytical procedure that was developed by Richards and Elms (1979)recognizes that for some limiting value of horizontal acceleration, identifiedas N*'g in Figure 6.7, the horizontal inertia force acting on a retaining wallwith no toe fill will exceed the shear resistance provided by the foundationalong the interface between the base of the wall and the foundation. Thisimplies that although the soil base may be accelerating horizontally at valuesgreater than N*-g, the wall will be sliding along the base under the action ofthe horizontal inertial force that corresponds to the horizontal accelerationN" g. This results in the movement of the soil base relative to the movementot r:he wall and vice-versa. The relative movement commences at the point int-ime designated as point a in Figure 6.8 and continues until the velocity ofthe base is equal to the velocity of the wall, designated as time point b inthis same figure. The velocity of the soil base is equal to the integral overtime of the soil acceleration, and the velocity of the wall between timepoints a and b is equal to the integral of the wall acceleration, which is aconstant N*g. The relative velocity of the wall, vr, is equal to theintegral of the difference between the base acceleration and the constant wallacceleration N*-g between time points a and b, as shown in Figure 6.8. Therelative displacement of the wall is equal to the integral of the relativevelocity of the wall, which occurs between the two points in time labeled aaind b in Figure 6.8. Additional relative displacements occur for the wallhetw•en the two laLt-er points in time labeled c and d in Figure 6.8, with the
residual relative wall displacements, dr, equal to the cumulative relativedisplacements computed during the entire time of earthquake shaking.
This problem was first studied in detail by Newmark (1965) using thesliding block on a sloping plane analogy, with procedural refinements contri-buted by Franklin and Chang (1977), Wong (1982), Whitman and Liao (1985),Ambraseys and Menu (1988) and others. Makdisi and Seed (1978) and Idriss(1985, Figure 47), proposed relationships based on a modification to thleNewmark pecrmanent displacement procedure to allow tor the dynamic response ofembankments. The approach has been reasonably well validated for the case ofwall retaining dry backfills. The major problem is; the selection ot a suit-able friction angle. This is particularly troublesome when the peak frictionangle is significantly greater than the residual friction angle. It is con-servatiwfy to use the residual friction angle, and this shtound he the usualpractice _
158j
FAILURE SUP OCCURS WHEN ah l) ,>PWEDGE MOVEMENT
N-
BACKFILL N.
SLIP'Ch k11 OCRS LIMITING ACCELERATIONS
From Whitman (1990)
Figure 6.7 Gravity retaining wall and failure wedgetreated as a sliding block
W I
WALL a
ABSOLUTE RELATIVE b-c dVELOCITY VELOCITY
ON --, 'X 0; i0 - Vr 0--1 d
..' 'ABSOLUT RELATIVE bcDISLCMN
0 x tO y r 0
Froa alRELATIVEiDISPLACEMENT
bb
b b
SOILACCELERATION
WALL -ACCELERATION . .__ --------
t It t
From Elms and Richards (1900)
Figure 6.8 Incremental displacement
The Richards and Elms procedure was developed using a sliding blockanalogy to calculate the magnitude of wall displacements in ,liding duringearthquake shaking. Whitman and Liao improved this procedure by usingstatistical methods to address the several sources of imncertainty in the
159
displacement controlled procedure. However, the reader is cautioned againstrelying solely upon this simplified procedure for waterfront structureslocated within severe seismic environments or epicentral regions, structureswith significant deformations, or critical structures. It does not includewall displacements due to post-earthquake settlements or due to creepdisplacements. The method has not yet been extended to take into accounttilting of walls; this matter is discussed by Whitman (1990).
Among the uncertainties are the effects of vertical and transverseaccelerations, including their influence upon the passive stabilizing forcefor walls with toe fill. Results of studies by Sharama (1989), as describedby Elms and Richards (1990), indicate that the effect of the verticaiacceleration component is negligible. Other research as described by Whitman(1979) indicated that the effect of vertical acceleration can be to increasethe total displacement by 50 to 100 percent for N*/A > 0.6. Whitman and Liao(1985) determined that the detrimental effects of vertical accelerations onwall stability were offset by consideration of other variables. Sharama(1989), as reported by Elms and Richards (1990), determined that transverseaccelerations oriented along the length of the wall contribute to wall dis-placement. Sliding block displacements must always increase due to transverseaccelerations. Displacement increases of 70 percent or higher for N*/A valuesbetween 0.5 and 0.9 were found. These additional displacements are based onanalysis of a wall with no transverse support other than base friction. Amore sophisticated analysis is required to investigate, or to consider threeffects of k, (or vertical acceleration) in the deformations of watertrontstructures.
The stabilizing force for sliding resistance may be less than the fullpassive earth pressure force because of insufficient wall displacements. Aconservative evaluation of this resistance should be used.
The displacement controlled procedure for the analysis of earth retain-ing structures is categorized as one of four types of analyses, as was donefor the conventional equilibrium method of analysis. These categories, thatare shown in Figure 6.1, include rigid walls retaining dry backfills (Case 1)and three categories for rigid walls retaining submerged backfills, dependingupon the magnitude of excess pore water pressures that are generated duringthe earthquake. They range from the case of no excess pore water pressures(Case 2) to the extreme case which corresponds to the complete liquefaction ofthe backfill (Case 4) and the intermediate case between the two (Case 3).This proposed procedure for submerged backfills is not applied to the case ofliquified backfills due to the complexity of the post-earthquake behaviorwithin the soil regions. In addition, the steps in the application of thedisplacement controlled approach to the design of a new wall are distinguishedfrom the steps in the application of the displacement controlled approach tothe analysis of an existing wall. Table 4 identifies the appropriate Chapter6 section that describes either the design of a new wall or the analysis of anexisting wall for the first three Figure 6.1 categories of displacement con-trolled analyses.
6.3.1 Displacement Controlled Design Procedure for a Wall Retaining DryBackfill
This section describes the application of the displacement controlled
approach to the design of a wall retaining dry backfill identified as Case i
160
it Figure 6.1. Richards and Elms (1979) first applied this analysis procedureto walls that retain dry backfill. The eight steps in the design of the earth
retaining structure shown in Figure 6.9 are as follows:
(I) Decide upon the value for the permanent relative displacement dr that isacceptable for the wall. For most walls, displacements on the order ofseveral inches would be acceptable. The value for dr must be consistent withthe dynamic active earth pressure used in step 5 during the design of tile wall
(see the discussions in Sections 6.1 and 2.2.2).
(2) Select the site specific average peak horizontal acceleration, A g, andthe site specific average peak horizontal velocity, V, within ýhe soil back-fill comprising the dynamic active wedge attd the retrining structure. Refer
to the discussion in Section 1.4 of Chapter 1.
(3) In typical earth retaining wall design problems, by 1.r itman and Liao dis-placement controlled procedure, k, = 0.
(4) Calculate the maximum transmissible acceleration, N*g, coefficient N'using the Whitman and Liao (1985) relationship
N -= A * 10.66 - 1 .n Vdr -(A -g)(9
where
A-g = base acceleration in units of in/sec 2
V is expressed in units cf inches per secondd, is expressed in units of inches
g - 386 in/sec2
According to Whitman and Liao, this relationship for the maximum tranLmissinleacceleration coefficient, N*, ,onsures that there will be 95 percent confidencethat the prescribed allowable permanent displacement will not be exceededduring an earthquake for the assigned A and V values. Equation 90 was derivedusing 14 earthquake records. All but two of the records were for earthb,,,akes
with magnitudes between 6.3 and 6./. For severe seismic environments, struc-tures located in epicentral regions, significant deformations, or criticalstructures, additional calculations should be made using other relationships
(see Section 6.2).
(5) Compute the value for the dynamic active earth pressure force. PA usingthe Mononobe-Okabe relationship described in Section 4.2, or jor verticalwalls and level backfills, in terms of PA and APA using the simplifiedMononobe-Okabe procedure described in Section 4.2.2. When using the relation-ships for 0, KAE, AKA, and aA, N* is substituted for kh, and k, is set equalto zero. Additional comments regarding these calcailations are giver, in stepin Section 6.2.1.
(6) Compute the required weight of wall. Horizontal force equilibrium
requires that the shear stress required for equilibrium, T, (Equation 72) beequal to the ultimate shear force along the base of the wall , T,,t (Equa-tion 74). Setting Equation 72 equal to Equation 74, and introducing tilcnormal force N (Equation 70) and solving for W results in the relationship
161
Movements
(a) Slip planes
NN'-9
'NN
Ov "0 "
• B p (b) Limiting Acceleration
W'N" • W'N
X, "-..
P EO AE //<AE
S I NpO c u s 'N %
N_• " v = 9 N T"(c) Forces p l, a re ,ity Wall
1N
oN*
Fa• ' 6.(b),. c in • ;av~y w l L o im iting Ac.eleroaio i o
xe.. a ~ ~~~ __ _ _ __ _ _ __ _ _
W (|AF)X (PAE)Y (91)
tan6j, - N'
where
(PAE)X = PA cos( 6 + 0 )
(PAE)y = PAE sin( 6 + 0 )
(7) No factor of safety needs to be applied to the wall weight W computed instep 6 when using Equation 90 (FSw = 1.0).
(8) Proportion the geometry of the wall so that the overturning criterion issatisfied. This is expressed in te-ms of the percentage of base contact areaBe/B, where B, is the width of the area of effective base contact, asdescribed in stt:) 7 in Section 6.2.1. For a giv,-.n trial geometry, the pointof action of the niormal force along the base, xN, is computed using Equationnumber 71, followed 1by the computation of the value of B, using either Equa-tion 75 or 76, depending upon the foundation material. This B, valuc- is thnllcompared to the minimum Be value, which is equal to 75 percent ot the basewidth B for earthquake loading conditions (50 percent for rock foundations).
This procedure is illustrated in example 29 at the end of this chapter.
6.3.2 Analysis of Earthquake Induced Displacements for a Wall Retaining Dry
Backfill
This section describes the analysis of the earthquake induced displace-ments of an existing wall retaining dry backfill, identified as Case 1 in
Figure 6.1. The four steps in the analysis of the earth retaining structure
shown in Figure 6,9 are as follows:
(1) Determine the value for the average site specific peak horizontal acceler-
ation, A-g, and the value for the average peak horizontal velocity, V, at thesite. Refer to the discussion in step 2 of Section 6.3.1.
(2) In typical earth retaining wall design problems by tVhittman and Liao dis-
placement controlled procedure, k-, - 0.
(3) Compute the value for the maximum trra.ismissible. acceLeration, N' g, coot -
ficient N*. An iterativw method coi; i sting of then followizqig five st-,psused to determine the value for N'.
(3-A) Using the assume-'d value for N', computtc tihe Va luc- for the
dynamic cti',e earth pressure force e AE u-inig Iit her thbe Mononobe,-Okabe relationship described in Section 4.2 or in te rms of PA and
AFPAF assuming the s:iimplifietd Monoiob,-Okabe proc dnir- d. scribed in
Sýct-ion 14.2.2 applies;. Whien c;istijg th, re lat ioiis;hij:; for U), KA•:
AK•F, and (CAE, N* is fsibst i at ed tor kj, and k, is sset eqlial to
zero. Addi tional comment!; regarding ths;e (,,a -lculations are given
in s;t (p .3 i.n Sect ion 6.)..I.
1 61,
(3-B) Calculate the value of the shear force required for equilib-rium along the base of the wall, T, using Equation 72.
(3-C) Calculate the value for the normal force, N, usingEquation 70.
(3-D) Calculate the value for the ultimate shear torce along thebase of the wall, Tut, using Equation 74.
(3-E) If the value for T is not equal to the value for T,,,adjust the value used for N* and repeat steps 3-A through step 3-Duntil T - Tult. The resulting value for N* is equal to the limitacceleration.
(4) Calculate the permanent relative displacement dr using the Whitman andLiao (1985) relationship
dr .(A'g) exp (92)
where
N*'g = maximum transmissible acceleration in unit:, of in/sec2
Ag = base acceleration in units of in/sec 2
V is expressed in units of inches per seconddr is expressed in units of inchesg = 386 in/sec 2 .
The value of dr must be consistent with those movwmients that are requi red todevelop the dynamic active earth pressure (used in step 3-A). Refer to thediscussion in Section 2.2.2. The actual earthquake induced di splacemk.nit wi 11be of the same relative magnitude as the computed dr value.
This procedure is illustrated in example 30 at- the end of this chapter.
6.3.3 Displacement Controlled Design Procedure for a Wall Retaining SubmtrgedBackfill - No Excess Pore Water Pressures
The displacement controlled approach was originally formulated by Rich-ards and Elms (1979) for gravity walls retaining drv hackfills. This; sectionuutlines a proposed procedure for extending this method of analysis to prob-lems involving walls retaining submerged backfills that do not develop excesspore water pressures during earthquake shaking, the Case 2 structure ofFigure 6.1. A pool of water is also present in front of the retaining wall.The same procedures that were described in the conventional force equilibriummethod of analysis to compute the effective earth pre-ssures (['HA) and bothsteady state pore water pressure forces, tJat and U b, and residual excetsswater pressure forces, Ushar and AU, acting on the wall. are used in the dis-placement controlled design approach. Thte procedure used to evalkuate theliquefaction potential within the backfill and founda tion and the magnitude' of
the residual excess pore water pressures after shakinpg arte, dhs!;ribld in Seedand Harder (1990) or Marcuson, Hynes, and Franklin (1) '1(1 .
1664
This section describes the application of the displacement controlledapproach to the design of a wall retaining submerged backfill, identified asCase 2 in Figure 6.1. No excess pore water pressures result from earthquakeshaking. There are eight steps in the design of the earth retaining structureshown in Figure 6.4. The first four steps are the same as those listed inSection 6.3.1, with the first being the selection of the value for the perm;i-nent relative displacement d, that is acceptable for the wall.
For steps (1) through (4), see Section 6.3.1.
(5) Compute the value for the effective dynamic active earth pressure tor't.PAE using the procedure described in step 3 of Section 6.2.2. When using the-relationships for 0, KA, and aAE, N* is substituted for kh, and k, is setequal to zero (a more sophisticated analysis is required to consider k,).
(6) Compute the required weight of wall. Horizontal force equilibrium re-quires that the shear stress required for equilibrium, T, (Equation 80) bt-equal to the ultimate shear force along the base of the wall, T,, (Equatioll81). Setting Equation 80 equal to Equatimn 81, and introducing tie effec' t i'enormal force N' (Equation 78) and solving for W results in the relationrship
(PAE)X - (PAE)y(tan 6 b) - Ustali, - Up0 01 1 3inerta . , b
tan~b - N"
where
(P - P)= cos( 6 + 8 )
(PA)Y = PAE sin( 6 + 0 )
(7) No factor of safety needs to be applied to the wall weight W compurted instep 6 when using Equation 90 (FSw = 1.0).
(8) Proportion the geometry of the wall so that the overturning criterion is;satisfied. This is expressed in terms of the percentage of base contact areaBe/B, where Be is the width of the area of effective base contact, asdescribed in step 7 in Section 6.2.2. For a given trial geometry, the pointof action of the effective normal force along the base, xN., is computed usingr,Equation 79, followed by the computation of the value for B. using eitherEquation 75 or 76, depending upon the foundation material. This B, value isthen compared to the minimum B. value, equal to 75 percent of the base width Bfor earthquake loading conditions.
With no residual excess pore water pressures generated within the hack-fill nor the soil foundation during earthquake shaking, there is no redistri-bution of excess pore water pressures after the earthquake. This implics thi.itthe wall displacements are due entirely to inertial effects during the earth-quake (and not due to any post earthquake consolidation), Additional wallmovements would occur should the foundation soils exhibit cretp behavior i!:discussed in Seed (1987) and Wlhitman (1985). Creep displacemeints are rIot
included in this procedure.
6.3.4 Analysis of Earthquake Induced Displ acements for a Wall Re-taining :oub-merged Backfill - No Excess Pore Water Pressuires
165
This section describes the proposed procedure for the analysis of theearthquake induced displacements of an existing waill retaining submerged back-fill, identified as Case 2 in Figure 6.1. No excess pore water pressures aregenerated within thy backfill and the foundation during earthquake shaking.The four steps in the analysis of Figure 6.4 retaining wall are as follows:
For steps (1) and (2), see Section 6.3.2.
(3) Compute the value for the maximum transmissible acceleration, N* g, coef-ficient N*. An iterative method consisting of the following five steps areused to determine the value for N*.
(3-A) Using the assumed value for N*, compute the value for thedynamic active earth pressure force PA using the proceduredescribed in step 3-A of Section 6.2.2. When using the relation-ships for 0,e KA, AKAE, and cAE, N* is substituted for kl, -md kv
is set equal to zero.
(3-B) Calculate the value the shear force requires for equilibriumalong the base of the wall, T, using Equation 80.
(3-C) Calculate the value for the effective normal force. N,using Equation 78.
(3-D) Calculate the value for the ultimate shear force along thebase of the wall, Tult, using Equation 81.
(3-E) If the value for T is not equal to the value [or Tult,adjust the value used for N* and repeat steps 3-A through 3-Duntil T = Tul,. The resulting value for N ' is equal to the limitacceleration.
(4) Calculate the permanent relative displacement (1, using Equation 92. Thevalue of dr must be consistent with those movements t hat are required todevelop the dynamic active earth pressure (used in step 3-A), as described inSection 2.2.2. The commentary following step 8 in Section 6.3,3 also appliesin this case.
6.3.5 Displacement Controlled Design Proccdure for a Wall Retaining SubmergedBackfill - Excess Pore Water Pressures
This section describes the application of the proposed displacementcontrolled approach to the design of a wall retaining a submerged backfillthat develops excess pore water pressures withhit, the- backfill or within thefoundation during earthquake shaking, the Case 3 structure of Figure 6.1. Apool of water is also present in front of the retainin:g wýill. There are ninesteps in the design of the earth retaining structoure shown in Figure 6.5. Thefirst four steps are the same as those listed in Sect ion 6. 3.1. with the firstbeing the selection of the value for the permanent relat:ive displacement drthat is acceptable for the wall.
For steps (1) through (4) see Section 6.3.1
( ') Compute the value for the effect.ive dynamic act ivye eairth pre s.sure torce
PAE u1;ing the procridure described in ste p I of Section 6,2.3. When using the
160
relationships for Oe2, KA, and aAE, N* is substituted for kl,, and k' is setequal to zero (a more sophisticated analysis is required to consider k,).
(6) Compute the required weight of wall. Horizontal force equilibriumrequires that the shear stress required for equilibrium, T, (Equation 84) beequal to the ultimate shear force along the base of the wall, Tult(Equation 81). Setting Equation 84 equal to Equation 81, and introducing theeffective normal force N' (Equation 82) and solving for W results in therelatiorýhip
W = (PAE)X - (PAE)y(tan6b) + Ustatic + Ushear - Up001 + Uinortia + Ub + AU (94)tan6 b - N"
where
(PAE)X = PIE cos( 6 + 0 )
(PAE)Y = PAE sin( 6 + 0 )
(7) No factor of safety needs to be applied to the wall weight W computed instep 6 when using Equation 90 (FSw = 1.0).
(8) Proportion the geometry of the wall so that the overturning criterion issatisfied. This is expressed in terms of the percentage of base contact areaBe/B, where Be is the width of the area of effective base contact, asdescribed in step 7 in Section 6.2.2. For a given trial geometry, the pointof action of the effective normal force along the base, XN,, is computed usingEquation 83, followed by the computation of the value for Be using eitherEquation 75 or 76, depending upon the foundation material. This Be value isthen compared to the minimum Be value, which is equal to 75 percent of thebase width B for earthquake loading conditions.
(9) Compute the additional wall movements that occur as a result of thedissipation of the residual excess pore water pressures. In this problem,residual excess pore water pressures are generated during earthquake shakingwithin the backfill and/or the soil foundation, resulting in a redistributionof excess pore water pressures after the earthquake. The design wall dis-placement selected in step 1 results from the inertial forces acting duringthe earthquake and do not include the post earthquake settlements.
The cautions expressed regarding wall stability during the dissipationof these excess pore water pressures as expressed in step 9 of Section 6.2.3remain applicable.
This procedure is illustrated in Example 31 at the end of this chapter.
6.3.6 Analysis of Earthquake Induced Displacements for a Wall Retaining Sub-merged Backfill - Excess Pore Water Pressures
This section describes the proposed procedure for the analysis of theearthquake induced displacements of an existing wall retaining a submergedbackfill that develops excess pore water press;ures within the backfill orwithin the foundation during earthquake shaking, the Case, i st-ructure of
167
Figure 6.1. A pool of water is also present in front of the retaining wall.The five steps in the analysis of Figure 6.4 retaining wall are as follows:
For steps (1) and (2) see Section 6.3.2.
(3) Compute the value for the maximum transmissible acceleration, N*g, coef-ficient N*. An iterative method consisting of the following five steps areused to determine the value for N*.
(3-A) Using the assumed value for N', compute the value for thedynamic active earth pressure force PAE using the proceduredescribed in step 3 of Section 6.2.3. When using the relation-ships for 0e2, KAE, AKAE, and aAE, N* is substituted for kh, and k,is set equal to zero.
(3-B) Calculate the value the shear force requires for equilibriumalong the base of the wall, T, using Equation 84.
(3-C) Calculate the value for the effective normal force, N',using Equation 81.
(3-D) Calculate the value for the ultimate shear force along threbase of the wall, Tult, using Equation 81.
(3-E) If the value for T is not equal to the value for Tut,adjust the value used for N* and repeat steps 3-A through step 3-Duntil T = Tlt. The resulting value for N* is equal to the limitacceleration.
(4) Calculate the permanent relative displacement dr using Equation 92.
(5) Compute the additional settlements that occur during the dissipation ofthe excess pore water pressures and add these computed values to the lateraldisplacement value calculated in step 4. Note that this value of displacemHetdoes not include any creep displacements that may occur within the foundationsoils. The resulting displacements must be consistent with those movementsthat are required to develop the dynamic active earth pressure (used instep 3-A), as described in Section 2.2.2.
The commentary included in step 9 of Section 6.2.3 also applies in thiscase.
168
CHAPTER 6 - EXAMPLES
Contents
Example Problems 27 through 31.
Commentary
The following examples illustrate the proceduresdescribed in Chapter 6. The results of the computa-tions shown are rounded for ease of checking calcula-tions and not to the appropriate number of significainrfigures. Additionally, the wall geometry and valuesfor the material properties were selected for ease of
computations.
169
Example No. 27 Reference Section: 6.2.1
For a wall of height H 40 ft and base width B = 32 ft founded oiv rock and
retaining a dry dense sand backfill, determine if the wall satisfies thestability criterion listed in Table 5 for a peak horizontal site acceleration
equal to 0.3 g. Assume the contact surface between the wall and the founda-tion rock to be entirely frictional (no bond).
DENSE SAND BACKFILL
120 pcf H.=H4V
' 350
ROCK
Step 1
Determine Seismic Coefficient kh
ah= 0.3 g
kh = 0.2
Step 2
Determine Seismic Coefficient kV
k,- 0.
Step 3
Determine P from Mononobe-Okabe relationships
tan- 0.-2 ] (by ,, V)
-' 11. 31
6 = 21.8'
1/0
Example No. 27 (Continued) Reference Section: 6.2.1
For structures with borderline values of F•, %e/B or Fb, vertical accelerarionis Imu.tvbe considered to correctly evaluate wall stability.
177
Exarmp[le No. 28 Reference Section: 6.2
For a wall of height H - 20 ft and base width B = 20 ft founded on "weathered" rock:nd retaining a partially submerged cohe-,iconless backfill (H. = 12 ft), determine ifthe wall satisfies the stability criterion listed in Table 5 for a peak horizontalsite acceleration equal to 0.3 g. Assume the contact surface between the wall andthe foundation rock to act as a granular material (i.e. with rio bond), ru is equal to0.1.
-t 120 pcf
17r. 15 VIHW 1Z I.
_____ ____ ____ ____ ___ OE.~- -i---
WEATHERED ROCK B 20'
Step I
Determine the seismic coefficient kh
ah = 0.3 g
kh = 0.2
Step 2
Determine seismic coefficient kv
k, = o.
Step 3
Determine PAE from the Mononobe-Okabe relationships.
PAE = 8,121 lb per ft of wall (see Example 19)
YPAE = Y = 9.52 ft (0.49 H) above the base of the wall (see Example 19)
(PAE)x, = 8,121 cos (17.5)
( PAE) = 7,745 lb per ft of wall
(IAE)y = 8,121 sin (17.5)
(PAE)y =2,442 lb per ft of wall
XAE = B = 20 it
178
Example No. 28 (Continued) Reference Section: 6.2.3
Determine hydrostatic water pressure force
Ustatic = 4,493 lb per ft of wall (see Example 19)
Yut = 4 ft (see Example 19)
Assume 80 percent of the base in compression (Be - 16 ft) with full upliftpressures acting along 4 ft (B - Be) of the wall to rock interface.
Example No. 28 (Continued) Reference Section: 6.2.3
Determine the excess pore water pressure force along the back of the wall
Ushear = 1,567 lb per ft of wall (see ex 19)
Yush = 5.47 ft above the base of the wall (see ex 19)
Determine the pore water pressure force along the base of the wall
Assuming redistribution of excess pore water pressure within the backfillalong the interface between the base of the wall and the foundation, the pres-sure distribution will be distributed as discussed for Ub.
Determine the width of the area of effective base contact (B")
B, = 3 (5.53') (by eq 75)
Be = 16.59'
B,. 16.59' = 0.83 > 0.5 req'd ". o.k.
Calculations show Be/B = 83 percent as compared to the initially assumed valueof 80 percent. If the calculated Be value differed sufficiently from theassumed value, it would be necessary to recompute the uplift pressure dis-tribution and repeat the analysis.
Step 8
Determine the factors of safety against bearing capacity failure or crushingof the concrete and the rock at the toe of the wall.
Compute q max
qmax = (2/3) (N'/XN') = 2/3 51476] = 6,206
Theck Fb for concrete
Assume for concrete:
q.it = 576,000 lb per ft of wall (see ex 27)
182
Example No. 28 (Continued) Reference Section: 6.2.3
Fb)oncete qult 576,000 = 92.8 (by eq 77)
Value for Fb for concrete is adequate,
Check Fb for rock
Calculations omitted.
183
Example No. 29 Reference Section: 6.1.1
Design a rectangular wall of height H = 20 ft to be founded on "weathered"rock and retaining a dense sand backfill for a peak average hori:zont-al siteacceleration equal to 0.3 g and a peak average velocity equal to 12 in/sec.Assume the contact surface between the wall and the foundation rock to act asa granular material (i.e. with no bond). Use the displacement controlleddesign procedure for a wall retaining a dry backfill.
rt- 120 pcf
0 - 35"H eff i " - 8.75"
_____T TOE
WEATHERED ROCK LB'72
Step 1 Decide upon a value for dr
Minimum value for dr, To achieve active earth pressures behind a 20 ft highwall retaining a dense sand backfill, the minimum wall displacement equals0.24 inch (Y/H = 0.001 from Table 1).
Specify a maximum allowable wall displacement dr equal to 0.5 inch (use theWhitman and Liao method).
Step 2
A,g= 0.3 g
A.g = 0.3.(386 in/sec/sec) = 116 in/sec/sec
A = 0.3
V = 12 in/sec
Step 3
k,=0
184
Example No. 29 (Continued) Reference Section: 6.3.1
Ste_ 4
N = (0.3), 0.66 - i in [(0.5 2in)/(16 in/sec))2
to " " .7 (12 in/see )2 (by eqI :46)
N= 0.227, = 0.76]
Step 5
kh - N" - 0.227
k,-0
Use the simplified Mononobe-Okabe procedure, described in Section 4.2.2.
AKAE - 3/4 (0.227) = 0.170 (by•eq 4
ApE = (0.170) (1/2) (120 pcf) (20' )2 = 4,080 lb per ft of wall
YAE = 0.6H = 0.6 (20') = 12 ft above from the base of the wall
KA = cos 2 (35 - 0)
cos2 (0) cos (0 + 8.75) F sin (35 + 8.75) sin (35 - 0)
cos (8.7/5 + 0) cos (0 - )T
(by eq 16)
KA = 0.2544
P^ = (0.2544) (1/2) (120 pcf)(20') (by eq 7)
PA = 6,106 lb per ft of wall
P, = 6,106 + 4,080 (by eq 40)
PA = 10,186 lb per ft of wall
185
Example No. 29 (Continued) Reference Section: 6.3.1
= 6,106 (20'/3) + 4,080 (0.6) (20')ý10,186
(by eq 44)
Y -YpAE = 8.80 ft above the base of the wall
step 6
Compute the required weight of the wall.
(PA)x = 10,186 cos (8.75 + 0) = 10,068 lb per ft of wall
(PAE)y = 10,186 sin (8.75 + 0) = 1,550 lb per ft of wall
B =. 6.u15 ft.= = . B5 = = 5 (from Table 5)B a ctu al B2 5 "B r q 'd
Check Fb
Compute qmax
qmax 2/3 (39,050/2.05) = 12,700 lb/ft (see Figure 6.3)
187
Example No. 29 (Continued) Reference Section: 6.3.1
Check Fb for concrete
Assume for concrete:
quIt= 576,000 lb/ft (see ex 27)
(Fb)concrete =qult 576,000 45 (by eq I)
Value for Fb for concrete is adequate.
Check Fb for rock
Calculations omitted.
Summary
Overturning stability governs the design of the gravity wall (refer tostep 7). It would be more efficient to make a gravity wall thitirier at topthan at the base. Doing so lowers the center of gravity and. hence the :;,ismicoverturning moment. A T-wall may be mrere economical for structures of thisheight. In contrast with gravity walls, the addition of reinforced concreteto the toe of the T-wall increases the overturning resistance with arelatively minor increase in mass (and cost) of the structure.
188
Example No. 30 Reference Sectioni: 6.1.2
Compute the value of dr (Equation 92) for a rectangular wall of height H ,2 ()ft and width equal to 12.5 ft to be founded on "weathered" rock and retaininga dense sand backfill for a peak average horizontal site acceleration equal to0.3 g and a peak average velocity equal to 12 in/sec. Assume active eartlhpressure forces acting along the back of the wall and the contact surfacebetween the wall and the foundation rock to act as a granular material (i.e.with no bond).
Yt"120 pcf0'- 35"
H.20" " - 8.75"
" TOEI
WEATHERED ROCK 2 8.12.5'L
Step 1
A.g - 0.3g
A'g - 0.3 (386 in/sec2 ) = 116 in/sec2
A =0.3
V = 12 in/sec
Step 2
k•-0
Stev 3
N= 0.227 (from example 29)
Step 3-A
PE= 10,186 lb per ft of wall (see ex
189
Example No. 30 (Continued) Reference Section: 6.3.2
YpAE = 8.80 ft above the base ot the wall (see Ex 29)
Step 3-B
T = 10,068 + (37,500) (0.227) (by eq 72)
T = 18,581 lb per ft of wall
Step 3-C
N = 37,500 + 1.550 ýby eq 70)
N = 39,050 lb per ft of wall
sLL-p 3-D
6 b = 29- (from Table 2)
Tuit = 39,050 tan (29) (by eq 74)
Tu,, = 21,646 lb per ft of wall
Step 3-E
Adjust the value used for N*
F. Tt - 21,646 - 1.165T 18,58
kh = N* - (N*)oid (F,) - (0.227) (1.165)
kh = N* = 0.264
Step 3-A 2nd Iteration
AK, = 3/4 (0.264) -0.198 (by eq 43)
190
Example No. 30 (Continued) Reference Section: 6.3.2
Ap . = (0.198) (1/2) (120 pcf) (20' )2(by eq 41)
APAE = 4,752 lb per ft of wall
P^ = 6,106 lb per ft of wall (see ex 29)
PA= 6,106 + 4,752(by eq 40)
PA = 10,858 lb per ft of wall
(PAE)X = 10,858 cos (8.75 + 0) = 10,732 lb per ft of wall
(PAE)y = 10,858 sin (8.75 + 0) = 1,652 lb per ft of wall
Step 3-B 2nd Iteration
T = 10,732 + 37,500 (0.264) (by eq 72)T = 20,632 lb per ft of wall
Step 3-C 2nd Iteration
N = 37,500 + 1,652(by eq 70)
N = 39,152 lb per ft of wall
Step 3-D 2nd Iteration
6b= 29' (see ex 29)
TULT = 39,152 tan (29) - 21,702 lb per ft of wall (by eq 74)
Step 3-E 2nd Iteration
Adjust the value used for Nh
191
Example No. 30 (Continued) Reference Section: 6.3.2
F TULT - 21,702 = 1.05 1.1
k, = N= (N')oId (F.) = (0.264) (1.1) = 0.290
Step 3-A 3rd Iteration
AKAE = 3/4 (0.290) = 0.218 (by eq 43)
APAz = (0.218) (1/2) (120 pcf) (20' )' = 5,232 lb per ft of wall (by eq 41)
PA = 6,106 lb per ft of wall (see ex 29)
'AE = 6,106 + 5,232 = 11,338 lb per ft of wall (by eq 40)
(PA)x = 11,338 cos (8.75 + 0) 11,i206 lb per ft of wall
(PAE)y = 11,338 sin (8.75 + 0) - 1,725 lb per ft of wall
Step 3-B 3rd Iteration
T = 11,206 + 37,500 (0.290) (by eq 72)
T = 22,081 lb per ft of wall
Step 3-C 3rd .Iteration
N - 37,500 + 1,725 = 39,225 lb per ft of wall (by eq 70)
Step 3-D 3rd Iteration
6b = 29° (see ex 29)
192
Example No. 30 (Continued) Reference Section: 6.3.2
TULT = 39,225 tan (29) = 21,743 lb per ft of wall (by eq 74)
Step 3-E 3rd Iteration
Adjust the value used for N*
F - TULT - 21, 743 0.985
Assume TULT T since F, is less than 2 percent from a value of 1.0 and use
The calculated earthquake induced displacement (approximately 1/10 inch)
is less than 1/4 inch displacement, the minimum value that is required todevelop active earth pressures in a dense sand backfill of 20 ft height (referto Example 29). The computed dr value is less than this required minimumvalue due to the fact that to satisfy the stability criterion against over-turning, the required width of the gravity wall was increased. The additionalconcrete mass increased the shear resistance along the base of the wall andthus reduced the magnitude of wall displacement for the design earthquakeload.
Since the computed displacement of the rectangular gravity wall is lessthan that minimum value required to develop active earth pressures for the
design earthquake by a factor of four, the procedures discussed in Chapter 5
193
Example No. 30 (Continued) Reference Section: 6.3.2
(walls retaining nonyielding backfills) would be used to compute the dynamic
earth pressure acting on the gravity wall. In general, the dynamic earth
pressures for "nonyielding backfills" are two to three times larger than the
dynamic active earth pressure force. Analysis and design of walls retaining
nonyielding backfills are discussed in Chapter 8.
If the wall had been made thinner at the top than at the base, as dis-
cussed in the summary to Example 29, then the necessity to design the wall to
retain a nonyielding backfill might be avoided.
19/4
Example No. 31 Reference Section: 6.3.5
Design a rectangular wall of height H 20 ft to be founded on "sound" rockand retaining a dense sand backfill for a peak average horizontal site accel-eration equal to 0.275 g and a peak average velocity equal to 10 in./sec.Assume active earth pressure forces acting along the back of the wall and thecontact surface between the wall and the foundation rock acts as a granularmaterial (ie. with no bond). Use the displacement controlled designprocedure for a wall retaining a submerged backfill, with dr = 0.5 inches andru= 0.1.
-Yt- 12 0 pcf
H.2a IF 08 - .1..EHw 12t POOL
ROCK
Step 1
Specify a maximum allowable wall displacement dr equal to 0.5 inch.
Example No. 31 (Continued) Reference Section: 6.3.5
kh. - 0. 251 (see ex 19)
(PAE). - 8,121 cos 17.5' (see ex 19)
(PAE),, - 7,745 lb per ft of wall (see ex 19)
XpAE - B
(PAE)y - 8,121 sin 17.5* (see ex 19)
(PAE)y - 2,442 lb per ft of wall (see ex 19)
YpAE - Y - 9.52 ft (0.49 H) above the base of the wall (see ex 19)
Determine hydrostatic water pressure force
U.t.tic 4,493 lb per ft of wall (see ex 19)
Y,,,t 4 ft from the base of the wall (see ex 19)
Assume full hydrostatic pressure beneath the base of L:,e wall.
Ub (H.) (I.) (B) - (12' ) (62.4 pcf) B
Ub 748.8 B
X.b B/2 - 0.5 B
Determine the excess pore water pressure force alony- the back of the wall.
Ush.., - 1,567 lb per ft of wall (see ex 19)
YUh - 5.47 ft (see ex 19)
Determine the excess Pore water Rressure force along the base of the wall
Assume B /B - 0.5
Assume the excess pore water pressure generated in the backfill during earth-quake shaking will propagate under the wall at a constant value in the baseseparation zone (B - B.). The pore water pressure in the base undercompression (B.) will linearly decrease from the maximum value to zero at thetoe of the wall.
196
Example No. 31 (Continued) Reference Section: 6.3.5
bH
t"H20'T b H.
SHw - 1 Hp •
*SHEA 165.1
197
Example No. 31 (Continued) Reference Section: 6.3.5
MpAE = 2,442 B - 7,745 (9.52') = (2,442 B - 73,732)
MP0o1 = 4,493 (4') - 1,048 (4.8') = 12,942
Ub (Xub) = (748.8 B) (0.5 B) = 374.4 B 2
AU(XDIJ) = (123.83 B) (0.6111 B) = 75.7 B 2
N' = (H) (B) ( + 2,442 - 748.8 B - 123.83 B
N' = (20') (150 pcf) B + 2,442 - 872.6B = (2,127.4 B + 2,442)
Solution continues on following page.
Summary
The width of the retaining wall cannot be directly determined becausethe resultant pore water pressure forces (both hydrostatic and excess) alongthe base of the wall vary as a function of the base width. Pressure distri-bution diagrams, for a specified value of the ratio Be/B, are expressed as afunction of the width of wall B for both hydrostatic and excess porepressures.
The design procedure is based on determining the weight of wall (usingEquation 94) which will satisfy base shear requirements. Values of N' and XN'are next calculated. The value of XN. defines the value of Be. Be/B is usedto express the stability of the wall against overturning. If the value ofB./B is sufficient and consistent with the assumed uplift pressures used inthe calculations, then base shear would have controlled the design width. ifBe/B is not acceptable (as in this example) then overturning controls the de-sign width which must be increased such that the minimum value for B./B issatisfied.
201
Examnpl e No. 31 (Cont:inued) Rv t ei dcv Sec lull
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CHAPTER 7 ANALYSIS AND DESIGN OF' ANCUORED SitEET PILE WALLS
7. 1 Introduction
This section describes the procedures for evail at ing tkhte stabi lity andsafety of anchored sheet pile walls during earthquakes. Anchored sheet pilewalls are comprised of interconnected flexible sheet piles that form a contin-uous and permanent waterfront structure. The free earth support method isused to determine the required depth of sheet pile penetration below thedredge level and the force the anchor must resist so that excessive sheet pilewall movements do not occur during earthquake shaking. The forces acting onboth the sheet pile wall and anchor during the earthquake include the staticand dynamic earth pressure forces, the static and hydrodynamic nool waterpressure forces and the steady state and residual excess pore water pressureforces within the submerged backfill and foundation soils. Because anchoredwalls are flexible and because it is difficult to prevent some permanent dis-placement during a major seismic event, it is appropriate to use active andpassive earth pressure theories to evaluate dynamic as well as static earthpressures. The Mononobe-Okabe theory is used to evaluate the dynamic earthpressures.
There have been very few documented cases of waterfront anchored wallsthat have survived earthquakes or of walls that. have tailed for reasons otherthan liquefaction. Hence uncertainty remains concerning the procedures out-lined in this chapter and the difficulty of ensuring adequacy of anchoredsheet pile walls during strong earthquake shaking (e.g. one rough index isseismic coefficients above 0.2).
One of the few seismic design procedures for anchored sheet pile wallsis the Japanese Code, which is summarized in Section 7.2.1. Using the obser-vations regarding the performance of anchored sheet pile walls during earth-quake shaking (summarized in Section 7.2), the following improvements overpast practice are recommended:
(1) Anchors must be placed further away from the wall.
(2) Larger seismic coefficients are required. They arte to be assignedwith consideration of the seismotectonic structures as well. as thecharacteristics of soil and structural features comprising the wall, theanchorage and its foundation.
(3) There is a limitation upon the build-up of excess pore pressures inbackfill.
The procedures outlined in this chapter are to be viewed as interimguidance, an improvement over past practice. An anchored sheet pile wall is acomplex structure and its performance (e.g. displacements) during earthquakeshaking depends upon the interactions between the many components )f thestructural system (e.g. sheet pile wall, backfill, soil below dredge level,foundation, and anchorage), which impact overall wall performance. Theseismic design of anchored sheet pile walls using the procedures described inthis chapter requires considerable jiudgement during the course, of design by anearthquake engineer experienced in the problems associated with the seismicdesign of anchored sheet pile walls.
203
As a general design principle, anchored sheet pile walls sited inseismic environments should be founded in dense and dilative cohesionlesssoils with no silt or clay size particles. The proposed design procedurepresume this to be the case. Strength parameters are to be assigned inaccordance with the criteria in Section 2.3. Additionally, the designprocedure is limited to the case where excess pore water pressures are lessthan 30 percent of the initial vertical effective stress (see Section 1.3,Chapter 1).
7.2 Background
Agbabian Associates (1980) summarize the performance of anchored sheetpile walls at 26 harbors during earthquakes in Japan, the United States, andSouth America. Their survey indicates that the catastrophic failures of sheetpile walls are due to the large-scale liquefaction of the backfill and/or thefoundation, including the foundation soil located in front of the sheet pilewall and below the dredge level. For those structures that underwent exces-sive movements but did not suffer a catastrophic failure, there was little orno evidence of damage due to the vibrations of structures themselves. Forthose walls whose backfill and foundation soils did not liquify but didexhibit excessive wall moments during the earthquake, the survey identifiedthe source of these excessive sheet pile wall movements as (1) the soil infront of the sheet pile wall and below the dredge level moved outward (toefailure), (2) the anchor block moved towards the pool (anchor failure), or(3) the entire soil mass comprising the sheet pile structure and the anchorblock moved as one towards the pool (block movement). These three potentialfailure modes within the backfill and the foundation soils are idealized inFigure 2.1, along with the two potential structural failure modes duringearthquake shaking of anchored sheet pile walls. The report identified anumber of factors which may contribute to the excessive wall movements,including (1) a reduction in soil strength due to the generation of excesspore water pressures within the submerged soils during the earthquake shaking,(2) the action of the inertial forces due to the acceleration of the soilmasses in front and behind the sheet pile wall and the anchor block, and(3) the hydrodynamic water pressures along the front of the wall during theearthquake.
The Japanese Ports and Harbors commissioned a study by Kitajima andUwabe (1979) to summarize the performance of 110 quay walls during variousearthquakes that occurred in Japan during the past several decades. Thissurvey included a tally of both damaged and undamaged waterfront structuresand the dates on which the earthquakes occurred. Most of these waterfrontstructures were anchored bulkheads, according to Gazetas, Dakoulas, andDennehy (1990). In their survey, Kitajima and Uwabe were able to identify thedesign procedure that was used for 45 of the bulkheads. This is identified asthe Japanese code. Their survey showed that (1) the percentage of damagedbulkheads was greater than 50 percent, including those designed using theJapanese design procedure and (2) the percentage of bulkhead failures did notdiminish with time. These two observations indicate that even the more re-cently enacted Japanese code is not adequate. To understand the poor perfor-mance of anchored sheet pile walls during earthquakes, it is useful to reviewthe Japanese code that was used in the design of the most recent sheet pilewalls that were included in the Kitajima and Uwabe survey.
204
7.2.1 Summary of the Japanese Code for Design of' Anchored Sheet Pile Walls
Most of the case histories regarding the performance of anchored sheetpile walls during earthquakes that were included in the Agbabian Associates(1980) and the Kitajima and Uwabe (1979) surveys are for Japanese waterfrontstructures. To understand the performance of these Japanese waterfront struc-tures, it is useful to review the Japanese design procedures that were usedfor the most recently constructed waterfront structures included in the sur-veys. The Japanese code for the design of anchored sheet pile walls as de-scribed by Gazetas, Dakoulas, and Dennehy (1990) consists of the followingfive steps:
(1) Estimate the required sheet pile embedment depth using the freeearth support method, with the factor of safety that is applied to theshear strength of the soil reduced from 1.5 for static loadings to 1.2for dynamic loadings. The effect of the earthquake is incorporated inthe analysis through the inertial forces acting on the active and pas-sive soil wedges by using the Mononobe-Okabe method to compute PaE and
PEr..
(2) The horizontal seismic coefficient, kh, used in the Mononobe-)kAberelationships for PA and PPE is a product of three factors: a regionalseismicity factor (0.10 ± 0.05), a factor reflecting the subsoil condi -tions (I ± 0.2), and a factor reflecting the importance of the stru'ture(1 ± 0.5).
(3) Design the tie rod using a tension force value computed on theassumption that the sheet pile is a simple beam supported at the drecOgeline and by the tie rod connection. Allowable stress in the tie rodsteel is increased from 40 percent of the yield stress in a design totstatic loadings to 60 percent of the yield stress in the design fordynamic loadings.
(4) Design the sheet pile section. Compute tme maximum bending moment,referred to as the free earth support moment, in the sheet pile usingthe simple beam of step 3. In granular soils Rowe's procedure is usedto account for flexure of the sheet pile below the dredge level. Areduction of 40 to 50 percent in the free earth support moment value isnot unusual. Allowable stress in the sheet pile steel is increased from60 percent of the yield stress in a design for static loadings to90 percent of the yield stress in the design for dynamic loadings.
(5) Design the anchor using the tie rod force of step 2 increased hv afactor equal to 2.5 in the design for both static and dynainmic loadingsand assume the slip plane for the active wedge starts at the dredgeline.
From the modes of failure observed in the Kitajima and Uwabe study of anchoredsheet pile walls that were designed using the Japanese code, (Cazetas, [).Ikoulasand Dennehy (1990) identified the following as the primary deficiencies in theJapanese code procedure:
(i) The values for the seismic coefficients, k, and kh, used in thieMononobe-Okabe relationships for PAE and PPE are not determined from asite response analysis but are specified within the ,Iapaiiese code (k,.
205
0, and kh is within a narrow range of values for most of the waterfrontstructures involved in the study).
(2) The resistance provided by the anchor is over estimated because thecode allows the anchor to be placed too close to the sheet pile wallsuch that the passive wedge that develops in front of the anchor inter-feres with the active wedge developing within the backfill behind thesheet pile wall.
(3) The code does not account for the earthquake induced excess porewater pressures within the submerged soils and the corresponding reduc-tion in the shear strength for the submerged soil regions, nor the ex-cess water pressure forces and hydrodynamic forces acting on the sheetpile structure.
Gazetas, Dakoulas, and Dennehy (1990) listed only one of the failures ofthe sheet pile walls designed using the Japanese Code as a general flexuralfailure. In this case, the structural failure was attributed to corrosion ofthe steel at the dredge level.
Each of these deficiencies is addressed in the steps used in the design ofanchored sheet pile walls using the free earth support method of analysis asdescribed in Section 7.4.
7.2.2 Displacements of Anchored Sheet Piles during Earthquakes
In the Kitajima and Uwabe (1979) survey of damage to anchored sheet pilewalls during earthquakes, the level of damage to the waterfront structure wasshown to be a function of the movement of the top of the sheet pile during theearthquake. Kitajima and Uwabe (1979) categorized the damage as one of fivelevels as given in Table 6 and reported in Gazetas, Dakoulas, and Dennehy(1990). Their survey shows that for sheet pile wall displacements of 10 cm(4 inches) or less, there was little or no damage to the Japanese waterfrontstructures as a result of the earthquake shaking. Conversely, the level ofdamage to the waterfront structure increased in proportion to the magnitude ofthe displacements above 10 cm (4 inches). Using the information on theanchored sheet pile walls survey reported in Kitajima and Uwabe (1979) andusing simplified theories and the free earth support method of analysis,Gazetas, Dakoulas, and Dennehy (1990) showed that the post-earthquake dis-placements at the top of the sheet pile wall correlated to (i) the depth ofsheet pile embedment below the dredge level and (2) the distance between theanchor and the sheet pile.
Two anchored bulkheads were in place in the harbor of San Antonio,Chile, during the very large earthquake of 1985. A peak horizontal accelera-tion of about 0.6g was recorded within 2 km of the site. One experienced apermanent displacement of nearly a meter, and use of the quay was severelyrestricted. There was evidence of liquefaction or at least poor compaction ofthe backfill, and tie rods may not have been preloaded. The second bulkheaddeveloped a permanent displacement of 15 cm, but the quay remained functionalafter the earthquake. This bulkhead had been designed using the Japaneseprocedure with a seismic coefficient of 0.15, but details concerning compac-tion of the backfill are unknown.
206
Table 6 Qualitative and Quantitative Description ofthe Reported Degrees of Damage
DEGREE CF PERMANENT DISPLACEMENTDAMAGE AT TOP OF SHEETPILE
__,,I DESCRIPTION OF DAMAGE
C INCHES
0 No damage <2 <1
Negilble damage to thewall Itself; noticeble
damage to related structures 10 4(I.e. concrete aoron)
II 2 NNotcable damage to wall 30 12
General shape of anchored3 sheetpile preserved, but 60 24
significantly damaged
4 Complete destruction, no 120 48recognizable shape of wall
" A + Damaged Criteria Grouping by Gazetas, Dakouls, and Denneby (1990)."- B + Damage criteria Grouping by Kltajlima and Uwake (1978).
7.3 Design of Anchored Sheet Pile Walls - Static Loadings
In the design of anchored sheet pile walls for static earth pressure andwater pressure loads, the free earth support method or any other suitablemethod may be used to determine the required depth of sheet pile embedmentbelow the dredge level and the magnitude of the design anchor force requiredto restrict the wall movements to acceptable levels. The interrelationshipbetween the changes in earth pressures, the corresponding changes in the sheetpile displacements, and the changes in the distribution of bending momentsalong the sheet pile make the free earth support method of analysis an attrac-tive design tool, as discussed in Section 7.4. Rowe's (1952) free earth sup-port method of analysis assumes that the sheet pile wall moves away from thebackfill and displaces the foundation soils that are below the dredge leveland in front of the wall, as shown in Figure 7.1. These assumed displacementsare sufficient to fully mobili -, the shear resistance within the backfill andfoundation, resulting in active earth pressures along the back of the sheetpile wall and passive earth pressures within the foundation in front of thesheet pile wall, as shown in Figure 7.1.
207
FrAILR IJFC-TAC[Z
-ApAE" ,A
PVA PA
P AE
AFTER ELMS AN RICHARDS (1990g.
Figure 7.1 Decrease in failure surface slope of the active andpassive sliding wedges with increasing lateral accelerations
To begin the analysis, a factor of safety equal to 1.5 is applied to theshear strength of the soil comprising the passive block in front of the sheetpile wall, while active earth pressures are presumed behind the sheet pilewall (factor of safety on shear strength of the backfill = 1.0). Equilibriumof the moments for the active earth pressure distribution and the factoredpassive earth pressure distribution about the anchor results in the minimumrequired depth of sheet pile penetration. Horizontal equilibrium of theactive earth pressure distribution and the factored passive pressure earthdistribution results in the computation of the equilibrium anchor force. Thedistribution of moments along the sheet pile is then computed using the earthpressure distributions and the equilibrium anchor force.
Rowe's (1952) model studies showed that because of flexure in the sheetpile below the dredge level, the free earth support analysis predicts largermoments than those developing under working loads. According to Rowe's work,the maximum moment to be used in the design of the sheet pile wall is equal tothe maximum moment corresponding to the free earth support analysis times acorrection factor; rd, where
rd - the moment reduction factor due to flexure below thedredge level, as developed by Rowe. rd is typicallyless than 1.0. Values for rd are given in Figure 7.2.The value of rd is a function of the flexibility of thesheet pile and the type and characteristics of thefoundation soil below the dredge level.
The value of the correction factor is a value less than or equal to one,dependent upon (1) the flexibility of the sheet pile and (2) the type and
208
1.0 .....
• % •--_-PENETRATD3/IEI lid EDAIMl C12UP,•T AMD WOMPAC
LEGENDM F0 - MAXIMUN POSITIVE MOMENT IN SHEETING COMPUTED BY FREE EARTH SUPPORT METHODM m:•1 -MAXIMUM POSITIVE MOMENT FOR DESIGN OF SHEETING
(H.D) 4 E - SHEETING MODULUS OF ELASTICITY. PSIP - FLEXIBLITY NUMBER- I- SHEETING MOMENT OF INERTIA, 0 PER RUNNING
El FOOT OF WALLNOTES FO FWL
1. M O.•M IS OTAJNED BY SUCCESSIVE TRIALS OF SHEETING SIZE UNTIL MAX. BENDING
STRESS IN SHEETING EQUALS ALLOWABLE BENDING STREAA.2. NO REDUCTION IN MrES IS PERMITTED FOR PENETRATION IN FINE GRAINED SOLS OR LOOSE
OR VERY LOOSE COARSE GRAINED SOILS3. FLEXIB.TY NUMBER IS COMPUTED ON THE BASIS OF LUBRICATED INTERLOCKS.
FROM NAVFAC DM7.2
Figure 7.2 Reduction in bending moments in anchored bulkhead from wallflexibility
209
characteristics of the foundation soil. The entire moment diagram is altereddue to incorrect earth pressure assumptions, idealized in Figure 7.3.
The corresponding design load, sheet pile displacements shown inFigure 7.3 reflect the flexure that occurs below the dredge level. In sandfoundations the flexure below the dredge level increases with increasing den-sity for the foundation sand. These reduced outward displacements along thebottom of the sheet pile explain why the free earth support method over-predicts the required design moment values for flexible sheet pile structures.Note that the point of contraflexure is now above the tip of the sheet pile inthe case of the design loads.
For those anchored walls in which the water table within the backfilldiffers from the elevation of the pool, the differences in the water pressuresare incorporated in the analysis. Terzaghi (1954) describes a simplifiedprocedure used to analyze the case of unbalanced water pressures and steadystate seepage. The distributions for the unbalanced water pressures along thesheet pile for the case of no seepage and for the case of steady state seepageare shown in Figure 7.4. In an effective stress analysis of frictional soilsare computed within these two regions, and the effective unit weights (Equa-tion 27) are used to compute the active and passive earth pressures along thesheet pile wall using the simplified relationship of the type described inSection 3.3.3. The seepage force acts downward behind the sheet pile, in-creasing the effective unit weight and the active earth pressures, and actsupward in front of the sheet pile, decreasing the effective unit weight withsteady state seepage, and the passive earth pressures. For the case of noflow, the buoyant unit weights are assigned to the frictional soils below thewater table to compute the active and passive earth pressures using the sim-plified relationships of the type described in Section 3.3.2.
Various important load Pnd material factors in common practice are asfollows: The allowable stress in the sheet pile is usually restricted tobetween 50 percent and 65 percent of the yield stress of the steel (60 percentin the Japanese Code). The allowable stress (gross area) in the tie rod steelis usually between 40 and 60 percent of the yield stress, and the tie rodforce is designed using the equilibrium anchor force increased by a factorequal to 1.3. The anchor is designed using the equilibrium anchor force in-creased by a factor equal to between 2.0 and 2.5.
This design procedure for static loadings is extended to dynamic prob-lems in the following sections.
7.4 Design of AihchoL.=d Stheet Pile Walls for Earthquake Loadings
The first step is to check for the possibility of excess pore pressuresor liquefaction (see Seed and Harder (1990) or Marcuson, Hynes, and Franklin1990). The presence or absence of these phenomenon will have a major influ-ence on design. The potential for excessive deformations is to be considered(see National Research Council, 1985).
The proposed design procedure quantifies the effect of earthquake shak-ing in the free earth support analysis of anchored sheet pile walls throughtre use of inertial forces within the backfill, the soil below the dredgelevel in front of the sheet pile wall and the hydrodynamic water pressureforce in the pool in front of the wall. These inertial forces are
210
0=1
a WkW:~
w 'Uui-. to.
IjAlwL
,JJ 0
V)
ku uJ
IlI-
15
~.0
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z
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0w
.0
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UU
NO FLOW FLOW NET
HU HU
_W_ BOTTOM OF WALL ' BOTTOM OF WALL
(o). NET WATER PRESSURE (b). NET WATER PRESSUREWITHOUT SEEPAGE WITH SEEPAGE
FROM TERZAGHI (1954) AND DAWKINS (1991)
Figure 7.4 Two distributions for unbalanced water pressures
superimposed on the static forces along the sheet pile wall. Certain adjust-
ments are made to the load and material factors, as is detailed in the follow-
ing sections, when earthquake loads are included in the analysis.
An important design consideration is the placement of the anchor. It
should be located far enough from the wall such that the active wedge from thewall (starting at the bottom of the wall) and the passive wedge from the
212
anchor do not intersect. The inertial forces due to the acceleration of thesoil mass have the effect of decreasing the slope of the active and passivesoil wedge failure surfaces, as shown in Figure 7.1 and described in Chap-ter 4. The slope angles aAE and aPE for the slip planes decrease (the slipplanes become flatter) as the acceleration levels increase in value.
When the horizontal accelerations are directed towards the backfill(+kh'g), the incremental increases in the earth pressure forces above the sta-tic earth pressure forces, denoted as APA and APpE in Figure 7.1, are directedaway from the backfill. This has the effect of increasing the driving forcebehind the sheet pile wall and decreasing the stabilizing force in front ofthe sheet pile wall. The effect of increased accelerations on the distribu-tion of moments are twofold, (1) increased values for the maximum momentwithin the sheet pile and (2) a lowering of the elevation of the point ofconflexure along the sheet pile (refer to Figure 7.3 for definition). Theanchored sheet pile wall model tests in dry sands by Kurata, Arai, and Yokoi(1965), Steedman and Zeng (1988) and Kitajima and Uwabe (1979) have confirmedthis interrelationship, as shown in Figure 7.5. This type of sheet pileresponse shows that as the value for acceleration increases, the point ofconflexure moves down the pile, and the response of the sheet pile (describedin terms of sheet pile displacements, earth pressures along the sheet pile anddistribution of moments within the sheet pile) will approach those of the freeearth support. This increase in the value of the maximum moment and the move-ment of the point of contraflexure towards the bottom of the sheet pile withincreasing acceleration reflects the development of a fully active stressstate within the soil that is located below the dredge level and behind thesheet pile wall. Thus, th. value for Rowe's moment reduction factor that isapplied to the moment distribution corresponding to the free earth supportmethod will increase in value, approaching the value of one, with increasingvalues for accelerations. This effect is not taken into account directly inthe design. However, it is indirectly considered if the moment equilibriumrequirement of the free earth method requires a greater depth of embedmentwhen earthquake loadings are included.
Another factor affecting the orientation of the failure planes and thusthe corresponding values for the dynamic earth pressure forces is the distri-bution of total pore water pressures within the backfill and foundation. Thetotal pore water pressure is a combination of the steady state seepage and anyexcess pore water pressures resulting from earthquake induced shear strainswithin the submerged soils.
The proposed procedures for the seismic stability analysis of anchoredsheet pile walls that undergo movements during earthquakes are categorized asone of three types of analyses, depending upon the magnitude of excess porewater pressures generated during the earthquake (Figure 7.6). They range fromthe case of no excess pore water pressures (Case 1) to the extreme case cor-responding to the complete liqu-faction of the backfill (Case 3) and theintermediate case of residual excess pore water pressures within the backfilland/or the soil in front of the sheet pile (Case 2).
In Figure 7.6, Ut~ticjb corresponds to the steady state pore water pres-sure force along the back of the sheet pile wall, Ustaict the steady statepore water pressure force along the front toe of the wall and Up,,, the hydro-static water pressure force exerted by the pool along the front of the wall.In the case of balanced water pressures, the sum of Ustaticb is equal to UPool
Figure 7.5 Measured distributions of bendingmoment in three model tests on anchored
bulkhead
214
SU ,static -b Ustac ".• • . ,,- ,'
-IN
NN
NtPPE
CASE 1: Submerged BockFill, No Excess PoreWater Pressures Due to Earthquakes.
IN.
%N
Ný UUlft~-
CAE1: Submeorged - baklNoExcer Pret
S• ~SWItc - toP PE
CASE 2: Submerged BackFill, Excess PoreWater Pressures Due To Earthquake.
HF sNtic - 0. U
HFlr~rtjG -indrU t a
-b U ,•.
N Ushw -1
P PE
CASE 3: Liquefied BackFill.
Figure 7.6 Anchored sheet pile walls retaining backfills which
undergo movements during earthquakes
215
and Ustatic..t Uinertia corresponds to the hydrodynamic water pressure forcealong the front of the wall due to earthquake shaking of the pool. Jn,,-band Ushear-t correspond to the excess pore water pressure force acting alongthe back of the wall and along the front of the wall (Case 2). In the case ota liquified backfill, HFsQaic and HFinerta-b are equal to the equivalent heavyfluid hydrostatic pressure of the liquified backfill and the inertia force dueto the acceleration of a liquified backfill.
An anchored sheet pile wall cannot be designed to retain a liquifiedbackfill and foundation, and hence Case 3 is only of academic interest. Siteimprovement techniques (the National Research Council 1985) or the use ofalternative structures should be investigated in this situation. A procedurefor determining the potential for liquefaction within the submerged backfillor the potential for the development of excess pore water pressures is dis-cussed in numerous articles, including the National Research Council (1985),Seed, Tokimatsu, Harder, and Chung (1985), Seed and Harder (1990) or Marcuson,Hynes, and Franklin (1990). The design procedure (Section 7.4.2) is limitedto the case where excess pore water pressures are less than 30 percent of theinitial vertical effective stress.
Flexure of the Sheet Pile Wall Below the Dredge Level:
Justification of the use of Rowe's moment reduction factor values,obtained from static tests (Rowe 1952) on dynamic problems, is empirical. Thedamage surveys of anchored sheet pile walls that failed due to earthquakeshaking listed one sheet pile wall that exhibited a general flexural failure(Section 7.2.1). The structural failure of this wall, designed using theJapanese Code, was attributed to corrosion at the dredge level. The JapaneseCode uses the Rowe's reduction factor values to reduce the maximum free earthsupport moment in the design of the sheet pile section, thus relying onflexure of the sheet pile wall below the dredge level during earthquake shak-ing. Flexure of the sheet pile below the dredge level is caused by severalfactors, including the depth of penetration and flexural stiffness of thesheet pile wall and the strength and compressibility of the soil (Rowe 1952,1956, and 1957, Tschebotarioff 1973). In Rowe's procedure, the dependence ofthe value of rd on the soil type incorporates the dependence of the level ofmoment reduction on the compressibility and strength of the soil as well isthe magnitude and distribution of sheet pile displacements below the dredtelevel.
The ability of the system to develop flexure below the dredge levelduring earthquake shaking must be carefully evaluated prior to application ofRowe's moment reduction factor or any portion of the reduction factor. Thisis especially true when analyzing the seismic stability of an existing sheetpile wall founded in a contractive soil. A sheet pile wall founded in dhsegranular soils is far more likely to develop flexure below the dredge levelduring earthquake shaking than one founded in loose soils. I)ense soils thatdilate during shearing are far less susceptible to large displacements duringearthquake shaking than are loose soils (Seed, 1987 and Seed, Tokimarsu,Harder, and Chung, 1985). Loose soils contract during shearing and are sus-ceptible to large displacements and even flow failures caused by earthquakeshaking (National Research Council, 1985, and Whitman, 1985). As a generaldesign principle, anchored sheet pile walls sited in seismic environmentsshould be founded in dense and dilative cohesionless soils with no silt orclay site particles.
216
7.4.1 Design of Anchored Sheet Pile Walls - No Excess Pore Water Pressures
The presence of water within the backfill and in front of the shee-t piltewall results in additional static and dynamic forces acting on the wall andalters the distribution of forces within the active and passive soil wedgesdeveloping behind and in front of the sheet pile wall. This section describesthe first of two proposed design procedures using the free earth support.method to design anchored sheet pile walls retaining submerged or partiallysubmerged backfills and including a pool of water in front of the sheet pilewall, as shown in Figure 7.7. This analysis, described as Case 1 in Fig-ure 7.6, assumes that no excess pore water pressures are generated within thesubmerged portion of the backfill or within the foundation during earthquakeshaking. The evaluation of the potential for the generation of excess porewater pressures during the shaking of the submerged soil regions is determinedusing the procedure described in the National Research Council (1985), Seed,Tokimatsu, Harder, and Chung (1985), Seed and Harder (1990) or Marcuson,Hynes, and Franklin (1990). Stability of the structure against blockmovements, as depicted in Figure 2.1, should also be checked during the courseof the analysis. The ten stages of the analyses in the design of anchoredwalls for seismic loadings using the free earth support method of analysis arelabeled A through J in Table 7. Appendix C contains a worked example. The13 steps in the design of the anchored sheet pile wall retaining submergedbackfill as shown in Figure 7.7 are as follows:
(1) Perform a static loading design of the anchored sheet pile wall using the-free earth support method of analysis, as described in Section 7.3. or anyother suitable method of analysis.
(2) Select the kh value to be used in the analysis; see Section 1.4 ofChapter i.*
(3) Consider k, as discussed in Section 1.4.3.
(4) Compute PAE using the procedure described in Section 4.3 and with theshear strength of the backfill fully mobilized. PAE acts at a'1n angle 1 to tlhnormal to the back of the wall. The pore pressure force [It.._b is determinedfrom the steady state flow net for the problem. By definition, only steadystate pore water pressures exist within the submerged backfill and founditionof a Case 1 anchored sheet pile wall (ru = 0). in the reLstrained water caseof a fully submerged soil wedge with a hydrostatic water table, PAE is com-puted (Equations 33 and 38) using an effective unit weight equal. to the buoy-ant unit weight. KAE (Equation 34) or KA(fl,O*) (Equation 38) is computedusing an equivalent horizontal acceleration, kl,,, and an equivalent seismitcinertia angle, 4 e1, given by Equations 47 and 46 (Section 4.3. 1).
The values for seismic coefficients are to he es alished by the sveijsnicdesign team for the project considering the seismotectonic .'r'ucture-s wi tin ithe region, or as specified by the design agency. The earthquak -idicddisplacements for the anchored sheet pile wa ll are dependelit upoU Tlulflle rotfactors, including how conservatively the strengths, seismic coefficients(or accelerations), and factors of safety have bf-en a-ssirigt'd , is well ý]!a the,compressibility and density of the soils, and the displacement aIt theanchorage.
211
iN
-AE U I ____
NU staft -b --
Figure 7.7 Anchored sheet pile wall with no excess pore water pressure
due to earthquake shaking (Case 1).
Table 7 Ten Stages of the Analyses in the Design of Anchored Walls for
Seismic Loadings
Section 7.4.1
Stage of Analysis Design Steps Description
A Evaluate potential for liquefaction or
excessive deformations.
B 1 Static design: Provides initial depth
of penetration for seismic analysis.
C 2, 3 Determine the average site specific
acceleration for wall design.
D 4, 5 Compute dynamic earth pressure forces
and water pressure forces.
E 6 Sum the moments due to the driving
forces and the resisting forces about
the tie rod elevation,
F 4-6 Alter the depth of penetration and
repeat steps 4 and 6 until momentequilibrium is achieved. The minimum
depth of embedment has been computed
when moment equilibrium is satisfied.
G 7 Sum horizontal forces to compute the
tie rod force (per foot of wall).
H 8, 9 Compute the maximum bending moment,
apply Rowe's moment reduction factor
and size the flexible wall (if
applicable).
10 Size the tie rods and select their
_ _.. .. ... ..__ _ _ _ _ _ spac in g ._
11-13 Design and site the anchorage.
218
In the case of a partially submerged backfill, this simplifiedprocedure will provide approximate results by increasing the valueassigned to the effective unit weight, 7e, based upon the proportion ofthe soil wedge that is above and below the water table (see Figure 4.13in Section 4.3.3). PAE is computed (Equations 33 and 38) with 1t replacedby Ie. KAE (Equation 34) or KA(fl ,O*) (Equation 38) is computed using anequivalent horizontal acceleration, kh.l, and an equivalent seismicinertia angle, 0.1, given by Equations and 46 in Section 4.3.1 with lb
replaced by Ie. A more refined analysis may be conducted using the trialwedge procedure (Appendix A) for the forces shown in Figure 7.7.
To compute the point of action of PAE in the case of a partially sub-merged backfill, redefine PAE in terms of the static force, PA, and thedynamic active earth pressure increment, APAE, using Equation 40. Thisprocedure is demonstrated in Figure 7.8. First compute KA and the staticeffective earth pressure distribution along the back of sheet pile wallusing one of the procedures described in Chapter 3. PA is equal to the
resultant force for this static effective stress distribution along the
back of the wall, which also provides for the point of action for PA.Solve for the force APAE as equal to the difference between PAE and PA-
Asstume that APAE acts at a height equal to 0.6H above the base of the
sheet pile. Compute the point of action of force PAE using Equation 44and correcting this relationship for the new locations along the back of
the sheet pile for the forces PA and APAE (refer to Example 19).
(5) Compute PPE acting in front of the sheet pile using the proceduredescribed in Section 4.4 (Chapter 4) and using a factor of safety, FSp.
applied to both the shear strength of the soil and the effective angle offriction along the interface. 6 equal to 0'/2 (Section 3.3.1) is a
reasonable value for dense frictional soils. In a static free earth sup-port method of analysis, FSp is set equal to 1.5, and in a dynamic earth
pressure analysis, the minimum value assigned to FSp is 1.2. U isdetermined from the steady state flow net for the problem. By defini-
tion, only steady state pore water pressures exist within the submerged
backfill and foundation of a Case I anchored sheet pile
= tanO' (95)FSp
and
tan6t = tan6 (96)FSp
wall (r,, = 0) . In the restrained water case of a fully submerged soil
wedge with a hydrostatic water table, PPE is computed (Equations 58 and62) using an effective unit weight equal to the buoyant unit weight. For
low to moderate levels of earthquake shaking, assume that PPE acts at a
height equal to approximately 1/3 of the height of the soil in front of
the sheet pile wall and at an angle 6t, to the normal to the face of the
21.9
H WY DREDGE LEVEL
Pr
(a.) Mononobe - OKabe Earth Pressure Forces PA andP .
PAE COS b PA .c a
DREDGE LEVEL
0.6 H
(b.) Ho,.1z-,taI Force Components. of Pa and PPE"
Figure 7.8 Static and inertial horizontal force components of the
Mononobe-Okabe earth pressure forces
220
wall.* KpE (Equation 59) or Kp(fi',O*) (Equation 62) is computed using anequivalent horizontal acceleration, khel, and an equivalent seismic inertiaangle, 0'e1, given by Equations 47 and 46. In the case of a steady stateseepage, this simplified procedure will provide approximate results bydecreasing the value assigned to the effective unit weight (Equation 27)according to the magnitude of the upward seepage gradient (Equation 26).
Equation 59 for KPE is restricted to cases where the value of 0 (Equa-tion 95) is greater than 0.1 (Equation 46). This limiting case may occur incases of high accelerations and/or low shear strengths. One contributingfactor is the submergence of the soil in front of the anchored wall, whichapproximately doubles the value of the equivalent seismic inertia angle overthe corresponding dry soil case.
(6) To determine the minimum required depth of sheet pile penetration, theclockwise and counterclockwise moments of the resultant earth pressure forct'sand resultant water pressure forces about Figure 7.7 anchor are computed asfollows:
Counterclockwise Moment = PAECOS 6 b'(Ya - YAE) Ut,,-- B'(Ya - Y1 b)
(97)
+ Uinertia* (Ya - Y')
and
Clockwise Moment = - pool. (Ya - Yup) - PPE'Cs 6 t (Ya - YPE)
(98)
- Ustatic-t* (Ya - Yut)
* In a static design by the free earth support method of analysis, a
triangular earth pressure is assumed along the front of the wall, with theresulting force Pp assigned to the lower third point. Experience has shownthat reasonable static designs resulted when the appropriate strengthparameters and adequate factors of safety were used in conjuncticn with thissimplified assumption. A similar approach is used in the dynamic dtsigii.The point of application of PPE may move downward from its static point ofapplication for anchored sheet pile walls as the value for kh increases.However, no satisfactory procedure was found for computing the point ofapplication of PPE for this structure. In the interim, the assumption of PpEacting at approximately 1/3 of the height of the soil in front of the wallis restricted to low to moderate levels of earthquake shaking (e.g. onerough index is kh < 0.1) and with conservative assumptions regarding allparameters used in the analysis. For higher levels of shaking and lessconservative assumptions for parameters, a larger value for FSp than 1.2and/or a lower point of application would be assigned.
221
where
6b = effective angle of friction along the backfill to sheet pile wallinterface
6, = effective angle of friction along the toe foundation to sheet pilewall interface
Ustatih-b - resultant steady state pore water pressure force along theback of the wall
,static-t = resultant steady state pore water pressure force below thedredge level along the front of the wall
up,,, = resultant hydrostatic water pressure force for the poolUinertia = hydrodynamic water pressure force for the pool, directed away
from the wall (see Appendix B)Y== distance from the base of sheet pile to the anchorYAE =distance from the base of sheet pile to PAE
Yub distance from the base of sheet pile to Ustatic-b (from a flow net)Y= distance from the base of sheet pile to U inertia (see Appendix B)Y= distance from the base of sheet pile to Upaoo
YPE = distance from the base of sheet pile to PPEY= distance from the base of sheet pile to Uta[.Ic-t (from a flow net).
The value for the Clockwise Moment about Figure 7.7 anchor is compared to thev;ilue for the Counterclockwise Moment, resulting in the following threepossibilities:
(6a) If the value of the Clockwise Moment is equal to the value of theCounterclockwise Moment, the sheet pile wall is in moment equilibrium,and the depth of penetration below the dredge level is correct for theapplied forces.
(6b) If the value of the Clockwise Moment is greater than the value ofthe Counterclockwise Moment, the trial sheet pile embedment depth belowthe dredge level is too deep and should be reduced.
(6c) If the value of the Clockwise Moment is less than the value of theCounterclockwise Moment, the trial sheet pile embedment depth below thedredge level is shallow and should be increased.
Note that the sheet pile wall is in moment equilibrium for only one depth ofsheet pile penetration within the foundation. For those trial sheet pilepenetration depths in which moment equilibrium is not achieved, a new trialdepth of sheet pile penetration is assumed, and step 4 through step 6 arerepeated.
(7) Once the required depth of sheet pile penetration is determined in step 6,the equilibrium anchor force per foot width of wall, TFES, is computed usingthe equations for horizontal force equilibrium.
In some situations the value for TFES computed in a seismic analysis canbe several times the value computed in the static analysis due to the effect.of the inertial forces acting on both the active and passive soil wedges andthe pool of water. Large atnchor forces per foot: width of wall will impact
222
both the selection of the type of anchorage, anchor geometry and the number ofrows and spacing of tie rods along the wall (see steps 10 through 12).
(8) The distribution of the moments within the sheet pile is computed from theexternal earth pressures along the front and back of the sheet pile and fromthe anchor force. To accomplish this, the earth pressure forces shown inFigure 7.7 must be converted to equivalent earth pressures distributions. Oneapproach for doing this is to separate PAE into its static and incrementaldynamic components and correspnndinc points of action, as dis, ,,s:;,-,i 'n step 4and shown in Figures 7.8 and 7.9. Figure 7.10 is used to define the variationin horizontal stress with depth for the dynamic earth pressure force incrementAPAE. At a given elevation, an imaginary section is made through the sheetpile, as shown in Figure 7.10, and the internal shear force V and internalbending moment M are represented. The internal shear force V is equal to thesum of earth pressures and water pressures and TFEs acting on the free bodydiagram of the sheet pile above Section A-A'. The internal bending moment Mis equal to moment of the earth pressures, water pressures about Section A-A'.The maximum bending moment within the sheet pile is denoted as MFES. Thevalue for MFES is determined by calculating the internal bending moment at theelevation at which the shear is equal to zero.
(9) The design moment for the sheet pile, Mdeslgn, is equal to
Mdesign = MFES " rd (100)
where MFES is the value of the maximum moment calculated using the Free EarthSupport Method, and rd is the Rowe's moment reduction factor discussed inSection 7.3. Using the currently available moment reduction curve shown inFigure 7.2, the value of correction factor will change from the static caseonly if the depth of penetration or the flexural stiffness, El, of the wallchanges in order to meet moment equilibrium requirements for seismic loadings.The ability of the system to develop flexure below the dredge level duringearthquake shaking must be carefully evaluated prior to application of Rowe'smoment reduction factor or any portion thereof. This aspect of the design isdiscussed in Section 7.4.
In a static design, the allowable stress in the sheet pile is usuallyrestricted to between 50 and 65 percent of the yield strength. Higher allow-able stresses may be considered for use in the de• ign for dynamic earth pres-sures, given the short duration of loading during earthquakes. The allowablestresses for earthquake loading may be increased 33 percent above the valuespecified for static loading. This corresponds to an allowable stress in thesheet pile restricted to between 67 and 87 percent of the yield strength. Theeffects of corrosion should be considered during the course of wall design forstatic and seismic loadings.
(10) In a static design, the design tie rod force per foot width of wall,Tdesign, is equal to
Tdosgn - 1 .3-TFES (101)
23
of horizon0tal tese
coreP odntoPA
A E
T-; H
Oh6 H
4 Al
Figure 7.9 Distributionsof horizontal stressescorresponding to APAE
SHE PIILE
STFE WATER PRESSRA I
A + +
SDREDGE I LEVEL
\ I F-- I --- I
L --- -- -- ---
DYNAMIC HYDROSTATIC STATIC DYNAMC HYDROSTATIC14C R E M EN T AL W T RA T VACTIVE WATER ACTIVE PASSIVE WATER
Figure 7.10 Horizontal pressure components and anchor force acting onsheet pile wall
224
and the allowable stress in the tie rods is usually restricted to between 40and 60 percent of the yield strength. The factor 1.3 is also recommended forearthquake loading conditions. The Japanese code restricts the allowablestresses to 60 percent of the yield strength for earthquake loading (see thediscussion at the end of step 9). The value of 60 uercent is recommended.The effects of corrosion should be considered during t.hl ':ourse of wall designlfor both static and seismic loadings.
(11) The design of the anchorage for seismic loadings follows the approachthat is proposed for the design of the flexible wall and differs from the-appioachL used when desigiii.,g fur static luiJd.iags. !i t lite a, • f :ý;tati'Lloads, the ultimate force (per foot width of wall) which the anchor is to bedesigned, Tult-a, is equal to
Tult-a = 2 .5*TFES (102)
and the static earth pressure forces PA and Pp on the front and back of theanchor block are computed using the ultimate shear strength with 6 - 0' forslender anchorage (refer to discussion in Section C.1.9 of Appendix C or toDismuke (1991). The proposed design procedure for seismic loadingsdescribed in steps 12 and 13. Seismic loads usually control the anchoragedesign.
(12) For those waterfront structures in which the anchor consists of a plateor a concrete block, a major contribution to the forces resisting the pullingforce Tuita is provided by the formation of a passive soil wedge in front ofthe block, as shown in Figure 7.11a. In a seismic analysis, T•,,,-, is setequal to TFES. The Mononobe-Okabe equations 33 and 58 are used to compute thedynamic active earth pressure force, PAE, and the dynamic passive earth pres-sure force, PPE, acting on the anchor block during earthquake shaking(Figure 7.11b). PAE is computed with the shear strength of the backfill fullymobilized and 6 - 0' for slender anchorage and 6 _< 0/2 for mass concreteanchorage (Section C.I.9 of Appendix C). PPE is computed using a factor ofsafety FSp applied to the shear strength of the soil (Equation 95) and theeffective angle of friction along the interface (Equation 96). At a minimum,FSp is set equal to a value between 1.2 and 1.5, depending on the allowabledisplacement and on how conservatively the strengths and seismic coefficientshave been assigned. In general and with all parameters constant, the largerthe factor of safety, the smaller the anchorage displacement due to earthquakeshaking.
Water pressure forces are not included along the sides of the blockbecause most anchor blocks are constructed on or just above the water table,as idealized in this figure. If the water table extends above the base of theblock, these forces are to be included in the analysis.
The size of the block is proportioned such that
225
- P P PE-A J'7
APAE.-A PA ---- 00
V V(L IIE- a. FE-A
AUA
(a) Forces On Anchor Block.
""V A E -A I N K h
PFE-A17INotc: UA - 0 for Anchor Bose
AA UA N"A Above Water Table as Shown.
(b) Slip Plane And Dynamic Forces.
Figure 7.11 Dynamic forces acting on an anchor block (for 6 = 0°)
When the magnitude of computed anchor block forces prohibit the use of
shallow anchor blocks, alternative anchorage systems are to be investigated.These include the use of multiple tie rods and anchorage, A-frame anchors,sheet pile anchorage, soil or rock anchors and tension H-piles. Discussionsof anchorage are readily available in numerous textbooks and sheet pile designmanuals, including the USS Steel Sheet Piling Design Manual (1969),Dismuke (1991), McMahon (1986) and U. S. Army Corps of Engineers ManualEM 1110-2-2906 (Headquarters, Department of the Army 1991).
By definition, no excess pore water pressures are generated within thebackfill (AUA = 0) for the Case I anchored sheet pile walls. UA is equal tothe resultant steady state pore water pressure force along the base of theanchor. The orientation of a linear failure plane in front of the anchorblock, cpe, in Figure 7.11a is approximated using Equation 61.
(13) The anchor block is to be located a sufficient distance behind the sheetpile wall so that the active failure surface behind the sheet pile wall does
226
not intersect the passive failure surface developing in front of the anchorduring earthquake shaking. The required minimum distance between the back ofthe sheet pile and the anchor block increases with increasing values ofacceleration, as shown in Figure 7.1. The orientation of the active slipsurface behind the sheet pile wall, aAE, is calculated in step 4, and theorientation of the passive slip surface in front of the anchor block, aFE, iscalculated in step 12.
7.4.2 Design of Anchored Sheet Pile Walls - Excess Pore Water Pressures
This section describes the proposed procedure, using the free earth sup-port method to design anchored qheet pile walls retaining submerged or par-tially submerged backfills and including a pool of water in front of the sheetpile wall, as shown in Figure 7.12. This analysis, described as Case 2 inFigure 7.6, assumes that excess pore water pressures are generated within thesubmerged portion of the backfill or within the foundation during earthquakeshaking. The magnitude and distribution of these excess pore water pressuresdepend upon several factors, including the magnitude of the earthquake, thedistance from the site to the fault generating the earthquake and the proper-ties of the submerged soils. The evaluation of the magnitude of these excesspore water pressures is estimated using the procedure described in Seed andHarder (1990) or Marcuson, Hynes, and Franklin (1990). This design procedureis limited to the case where excess pore water pressures are less than30 percent of the initial vertical effective stress. Stability of thestructure against block movements, as depicted in Figure 2.1, should also bechecked during the course of the analysis. Many of the details regarding theprocedures used are common to the Case I analysis. The 14 steps in the designof the anchored sheet pile wall retaining submerged backfill as shown inFigure 7.12 are as follows:
(i) Perform a static loading design of the anchored sheet pile wall using thefree earth support method of analysis, as described in Section 7.3, or anyother suitable method of analysis.
(2) Select the kh value to be used in the analysis; see -ection 1.4 ofChapter l.*
(3) Consider k•, as discussed in Section 1.4.3.
(4) ComIpute PAE using tile procedure described in Section 4.3 and with theshear strength of the backfill fully mobilized. PAE acts at an angle 6 to thenormal to the back of the wall. The pore pressure force Usttlc-b is determinedfrom the steady state flow net for the problem. The post-earthquake residualexcess pore water pressures are identified as U1shear in Figure 7.12 and aredetermined using the procedures described in Seed and Harder (1990) or
The values for seismic coefficients are to be established by the seismicdesign team for the Vroject considering the seismotectonic structures withinthe region, or as specified by the design agency. The earthquake-induceddisplacements for the anchored sheet pile wall are dependent upon numerousfactors, including how conservatively the strengths, seismic coefficients(or accelerations), and factors of safety have been assigned, as well as thecompressibility and density of the soils, and the displacement at theancho rage.
227
N UU ster - -% ,-" U//WHO
NUIk - _A78
Figure 7.12 Anchored sheet pile wall with excess pore water preisuresgenerated during earthquake shaking (Case 2)
Marcuson, Hynes, and Franklin (1990). In the restrained water case of a fullysubmerged soil wedge with a hydrostatic water table, PA is computed (Equa-tions 33 and 38) using an effective unit weight equal to the buoyant unitweight. KA (Equation 34) or KA(#*,O*) (Equation 38) is computed using ;IIIequivalent horizontal acceleration, khe3, and an equivalent seismic inertiaangle, 0. 3 , given by equations 55 and 54 (Section 4.3.2). An alternative ap-proach is to use a modified effective friction angle, ieq (Equation 56), withru equal to the average value within the backfill.
In the case of a partially •ubmerged backfill, this simplified procedurewill provide approximate results by increasing the value assigned to theeffective unit weight, I., based upon the proportion of the soil wedge that isabove and below the water table (see Figure 4.13 in Section 4.3.3). PA iscomputed (Equations 33 and 38) with It replaced by I.. The unit weightassigned to the soil below the water table is given by Equation 52 when usingthe procedure described in Figure 4.13 to compute the value of je- KA (Equa-tion 34) or KA(P*,O*) (Equation 38) is computed using an equivalent horizontalacceleration, kh.3, and an equivalent seismic inertia angle, 0e3, given byEquations 54 and 55 in Section 4.3.2 with Ye3 replaced by I, For this case,the excess residual pore water pressures are superimposed upon the hydrostaticpore water pressures.
To compute the point of action of PA in the case of a partially sub-merged backfill, redefine PA in terms of the static force, PA, and thedynamic active earth pressure increment, APE, as described in step 4 ofSection 7.4.1.
(5) Compute PPE acting in front of the sheet pile using the proceduredescribed in Section 4.4 of Chapter 4 and apply a factor of safety FSp equalto 1.2 to both the shear strength of the soil and the effective angle of fric-tion along the interface. Refer to step 5 of Section 7.4.1. The pore pres-sure force Ustatic-t is determined from the steady state flow net for theproblem. In the restrained water case of a fully suhmerged soil wedge with ahydrostatic water table, PPE is computed (Equations 58 and 62) with -y, re-placed by the effective unit weight of soil below the water table, Yj(Equation 52 in Section 4.3.2). An average ru value is used within the soilin front of the wall. KPE (Equation 59) or Kp(/3*,O*) (Equation 62) is computed
228
using an equivalent horizontal acceleration, khe 3 , and an equivalent seismicinertia angle, 0.3, given by Equations 54 and 55 in Section 4.3.2. 1r) thecase of a steady state seepage, this simplified procedure will provide approx-imate results by decreasing the value assigned to the cffective unit weight(Equation 27) according to the magnitude of the upward seepage gradient (Equa-tion 26). For low to moderate levels of earthquake shaking, assume that PPEacts at a height equal to approximately 1/3 of the height of the soil in frontof the sheet pile wall and at an angle 6, to the normal to the face of thewall.*
(6) To determine the required depth of sheet pile penetration, the clockwiseand counterclockwise moments of the resultant earth pressure forces and resul-tant water pressure forces about Figure 7.12 anchor are computed as follows:
Counterclockwise Moment PAEcos6b. (Ya - YA.) + Ustatij-b" (Y8 - Yub)
(105)
+ Ushear-*" (Ya - Yutaub) + Uinertia* (Ya - Y)
and
Clockwise Moment = - Upool (Ya - Yu,) - PPE' Cs 6t" (Ya - YPE)
(106)
- Ustatic-t (Ya - Yut) - Ushear-t (Ya - Yutaut)
where
Ushear-b = resultant excess pore water pressure force along the back ofthe wall
Ushear-t = resultant excess pore water pressure force below the dredgelevel along the front of the wall
Yutaub = distance from the base of sheet pile to Ushear-b
Yutaut = distance from the base of sheet pile to 4shear-t
Values for Yutaub, Ushear-b, Yutaut and Ushear-t are computed using the proceduredescribed in Seed and Harder (1990) or Marcuson, Hynes, and Franklin (1990).
* In a static design by the free earth support method of analysis, atriangular earth pressure is assumed along the front of the wall, with theresulting force Pp assigned to the lower third point. Experience has shownthat reasonable static designs resulted when the appropriate strengthparameters and adequate factors of safety were used in conjunction with thissimplified assumption. A similar approach is used in the dynamic design.The point of application of PpE may move downward from its static point ofapplication for anchored sheet pile walls as the value for kh increases.However, no satisfactory procedure was found for computing the point ofapplication of PPE for this structure. In the interim, the assumption of PPEacting at approximately 1/3 of the height of the soil in front of the wallis restricted to low to moderate levels of earthquake shaking (e.g. onerough index is kh < 0.1) and with conservative assumptions regarding allparameters used in the analysis. For higher levels of shaking and lessconservative assumptions for parameters, a larger value for FSp than 1.2and/or a lower point of application would be assigned.
229
The value for the Clockwise Moment is compared to the value for the Counter-clockwise Moment, resulting in one of three possibilities listed in steps 6athrough step 6c in Section 7.4.1. The sheet pile wall is in moment equili-briuin for only one depth of sheet pile penetration within the foundation Forthose trial sheet pile penetration depths in which moment equilibrium is niotachieved, a new trial depth of sheet pile penetration is assumed, and step 4through step 6 is repeated.
(7) Once the required depth of sheet pile penetration is determined in step 6,the equilibrium anchor force per foot width of wall, TFES, is computed usingthe equations for horizontal force equilibrium.
TFES = PPEcOS6t + Ustatic-t + Ushear-t + Up0oo
(107 )
- Uinert1ia PAE Cos 6b - Ustatlc-b - Ulnear-b
Additional commentary is provided in step 7 of Section 7.4.1.
(8) The distribution of the moments within the sheet pile, MFES, is computedusing the procedure described in step 8 of Section 7.4.1.
(9) The computation of the design moment for the sheet pile, Md~sign, isdescribed in step 9 of Section 7.4.1.
(10) The design tie rod force, Taesign, is computed using the proceduredescribed in step 10 of Section 7.4.1.
(11) The design of the anchor block for seismic loadings differs from theapproach used when designing for static loadings. The reader is referred tothe discussion in step 11 of Section 7.4.1.
(12) For those waterfront strurtrres in which the anchor consists of slenderanchorage or mass concrete anchorage, a major contribution to the forces re-sisting the pulling force Tu 1a a is provided by the formation of a passive soilwedge in front of the block, as shown in Figure 7.11a. The procedure de-scribed in step 12 of Section 7.4.1 is used to compute PO, APE, and UPE(Figure 7.11b). The size of the block is proportioned using Equation 103relationship, where N' is equal to
The excess pore water pressure force along the base of the block is equal toAIA (see Seed and Harder (1990) or Marcuson, Hynes, and Franklin (1990)).
An alternative procedure for incorporating residual excess pore water pres-sures in the analysis is by using r. and an equivalent angle of interfacefriction along the base of block, 6A.
tan6A q = (I - r) tan16A (109)
In this case, the value for N' in Equation 103 is given by
210
N' = W(l - k1) - UA. - IPE-sinb, + PA.sin6j (110)
Reducing the effective stress friction angle along the soil to concrete inter-face so as to account for the excess pore water pressures is not an exjctprocedure (see discussion in Section 4.3.2).
(13) The required minimum distance between the back of the sheet pilo :nd theanchor block is computed following the procedure described in step I .,Section 7.4.1.
(14) The residual excess pore water pressures within the submerged backfilland foundation will be redistributed after earthquake shaking has ended, Thepost earthquake static stability (kh and k, equal to zero) of any earthretaining structure should be evaluated during the redistribution of theexcess pore water pressures within the soil regions (see discussions in the-National Research Council 1985 or Seed 1987).
7.5 Use of Finite Element Analyses
Finite element analyses should be considered only if: (a) the cos;timplications of the simplified design procedures indicate that more detailedstudy is warranted, (b) it is necessary to evaluate permanent displacemenLt-sthat might result from the design seismic event, or (c) there is concern aboutthe influence of surface loadings. It is particularly difficult to model wellthe various features of an anchored wall, especially when there is concernabout excess pore pressures. One example of a detailed analysis of an actualfailure is given by Iai and Kameoka (1991).
231
CHAPTER 8 ANALYSIS AND DESIGN OF WALLS RETAINING NONYIELDING BACKFILLS
8.1 Introduction
This chapter applies to design problems in which the allowable movementof a wall is small - less than one-fourth to one-half of Table I wall movementvalues. Typical situations include the walls of U-shaped structures such asdry docks, walls of basements, and the lateral walls of underground struc-tures. Under these conditions it may be inappropriate to base design uponearth pressures computed using the Mononobe-Okabe theory. which assumes thatactive stress conditions are achievrtd. Hence, earth pressures generallyshould be computed using the theory set forth in Chapter 5.
Design criteria for such situations will involve permissible combinedstatic plus dynamic bending stresses within the wall. In many cases it may benecessary to ensure that such moments do not cause yielding of the materialcomposing the wall. If the wall is free-standing, then avoidance of slidingor overturning will be design criteria.
In many cases it may be appropriate to use Wood's simplified theory tocompute the dynamic increment of stresses. In this case, a key decision willbe the choice of the horizontal acceleration coefficient kh. Important con-siderations are:
* If displacement of the wall is not permissible, the assigned peak
acceleration coefficient should be used. Use of a seismic coefficient lessthan the peak acceleration coefficient implies that some displacement of thebackfill is acceptable during the design earthquake event.
* The acceleration at ground surface should be used to define kh. Thisis a conservative assumption. If the peak acceleration varies significantlyover the height of the backfill, which may often be the situation when thehigh side walls of dry docks are involved, consideration should be givento the use of dynamic finite element studies (see Appendix D).
Use of finite element studies should also be considered when there are impor-tant surface loadings. In many cases an elastic analysis using soil moduliand damping adjusted for expected levels of strain will suffice.
There iay be cases in which it is overly conservative to design struc-tures using lateral pressures from the theory for walls retaining nonyieldingbackfills. If the structure is founded upon soil with the same stiffness asthe backfill (see Figure 8.1), the structure itself will experience movementsthat may be sufficient to develop active stress conditions. Finite elementstudies, and measurements as large scale field models in Taiwan (Chanig et al,1990), have shown this to be the case. However, in such situations, it wouldseem that larger, passive-type stresses should develop on the opposite wall.If there are large cost implications for design using stresses computed assum-ing nonyielding backfills, finite element studies should again be considered.
It liquefaction is of concern, methods for evaluating residual porepressures may be found in Seed and Harder (1990) or Marcuson, Hynes, andFranklin (1990). In principle it is possible to design walls to resist thepressures from fully liquefied soil, including Westergaard's dynamic pressure
233
Concrde tonolith
, ~V
H Soil BackFill Pool Of Water Hp
Ro xw~dto
(o)
W _.___.,. W'd
U0
XL
(b)
Figure 8.1 Simplified procedure for dynamic analysisof a wall retaining nonyielding backfill
increment based upon the total unit weight of the soil. However, in such asituation the lateral pressures on a wall can be very high. Unless there arestructures (including cranes) adjacent to the wall, it might be possible toallow values of ru in excess of 40 percent. If so, a check should be made forpost-seismic stability, uýing the residual strength of the backfill soil.
8.2 An Example
The application of the simplified procedure to the dynamic analysis isdemonstrated for a wall retaining nonyielding backfill founded on rock asshown in Figure 8.1a. A pool of water is included in front, of the wall inthis problem. The forces acting along the back, front, and base of the wallinclude both static and dynamic incremental forces (Figure 8.1b). Withnegligible wall movements, the value for the static effective earth pressure,
234
Ph, corresponds to at-rest earth pressures. For gravity earth retainingstructures founded on rock, K. usually ranges in value from 0.45 tor com-pacted granular backfills to 0.55 for uncompacted granular backfills (Duncan,Clough, and Ebeling 1990). Ustaic and Ub are determined from the steady stateflow net for the problem. Up.., is equal to the hydrostatic water pressureforce along the front of the wall. Uinertia is the hydrodynamic water pressureforce for the pool computed using the Westergaard procedure that is describedin Appendix B. Given the horizontal base acceleration value, kh g, thedynamic earth pressure force Fsr is computed using Equation 68, acting at Yrequal to 0.63-H above the base of the wall. The horizontal force T is the
shear force required for equilibrium of the wall and is equal to
T = Ph + Fsr - W-kh , Ustatic - Upool + Uinertia. (ii)
The effective normal force between the base of the wall and the rock founda-tion is equal to
NI = W - Ub. 012)
The ultimate shear force along the base, Tult, is given by
Tult = N'tan6b (113)
where
6b - the effective base interface friction angle.
The factor of safety against sliding along the base, F., is given by
ultimate shear force (114)shear force required for equilibrium
and compared to the minimum value of 1.1 or 1.2 for temporary loading cases.The point of action of the force N', xN,, is computed by stuming moments aboutthe toe of the wall.
YPh - point of action of Ph. YPh - 0.4'H for a completely dryor completely submerged backfill with a hydrostatic water table (Duncan,Clough, and Ebeling 1990)
The overturning criterion is expressed in terms of the percentage ofbase contact area B./B, where B. is the width of the area of effective basecontact. Assuming that the bearing pressure varies linearly between the baseof the wall and the foundation, the normal stress is a maximum at the toe(q = qmax) and a minimum at the inner edge (q = 0) as shown in Figure 8.2.
B, 3-x, (116)
ZNN'
Linear Sase Pressure Distribution
N"
Uniform Bose Pressure Distribution
Figure 8.2 Linear and uniform base pressuredistributions
216
An alternative assumption regarding base pressure distribution and contactarea was suggested by Meyerhof (1953). Meyerhof assumed a uniform distribu-tion of pressure along the base, resulting in the effective base contact equalto
B (117)
Meyerhof's pressure distribution has been used widely for foundations on soil,and is most appropriate for foundation materials that exhibit ductile mechan-isms of failure. The assumption is less appropriate for brittle materials.
Many retaining walls are designed using static active earth pressureswith full contact along the base, B./B ( or B'e/B), equal to 100 percent. Fortemporary loading cases, such as earthquakes, this criteria is relaxed to aminimum value of 75 percent (50 percent for rock foundations, Table 5).
For those structures founded on rock, the factor of safety against bear-ing capacity failure, or crushing of the concrete or the rock at the toe canbe expressed as
FL, (118)qmax
where qult is the ultimate bearing capacity or compressive strength of theconcrete or the rock at the toe, and qmax is the maximum bearing pressure atthe toe. For brittle materials like unconfined concrete, the ultimate bearingcapacity is equal to the compressive strength of the material. Building codesare commonly used to obtain values for the allowable bearing stress on rock,qai1- Alternately, a large factor of safety is applied to the unconfined com-pressive strength of intact samples. The maximum bearing pressure qmax isrestricted to an allowable bearing capacity qal,. For ductile foundationmaterials that undergo plastic failure, the ultimate bearing capacity is larg-er than the compressive strength of the material, excluding those foundationmaterials exhibiting a loss in shear resistance due to earthquake induceddeformations or due to the development of residual excess pore water pres-sures. In these cases, a conventional bearing capacity evaluation is con-ducted to establish the post-earthquake stability of the structure.
In those stability analyses where the vertical accelerations are con-sidered, the force acting downward through the center of mass of the wall thatrepresents the weight of the wall, W, in Figure 8.1, is replaced by the force(l-kv)W acting downward. The first term in equations 112 and 115, W and W-x,are replaced by (l-kv)W and (l-k,)-W'xw, respectively. The direction in whichthe vertical inertia force, kvW, acts is counter to the direction assigned tothe effective vertical acceleration, kv'g. A kv-W force acting upward destabi-lizes the wall, while a kv'W acting downward increases the stability of thewall.
This procedure is illustrated in example 32 at the end of this chapter.
237
CHAPTER 8 - EXAMPLE
Content
Example Problem 32.
Commentary
The following example illustrates the proceduresdescribed in Chapter 8. The results of the computa-tions shown are rounded for ease of checking calcula-tions and not to the appropriate number of significantfigures. Additionally, the wall geometry and valuesfor the maturial properties were selected for ease ofcomputations.
238
Example No. 32 Reference Section: 8.2
Design an "nonyielding" rectangular wall (i.e. no wall displacements) ofheight H - 20 ft to be founded on rock and retaining a dense sand backfill fora peak horizontal acceleration at t>. ground surface equal to 0.2 g. Assume africtional contact surface between The wall and the foundation rock (i.e. withno bond).
Ko- 0.45 Yt -120 pcf
- 35* .
H, - 12' .. .r,- 0 A - 0.29
RROK
Determine the horizontal acceleration
For Wood's procedure:
kh = A = 0.2 (where A is peak groundsurface acceleration)
Determine the vertical acceleration
k= 0
Determine Ph (at rest horizontal effective earth pressure) and the point of
application.
Find the vertical effective stresses at the ground surface (U,)TOe, at thewater table (ac)T, and at the base of the wall (a.)`T.
Vertical Effective Stresses at the Top of the Wall
Determine the horizontal at rest effective stress at the top of the wall Ch,
at the water table chT, and at the bottom of the wall Ch
TOP 0
(N = K,, =)wT 0.45 (960 psf) = 432 psf
OT = K(a)BOT = 0.45 (1,651.2 psf) = 743 psf
Break the stress distribution diagram into rectangles and triangles to findthe magnitude of the resultant force (Ph) and its point of application (YPh).
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. 1936 (Apr). "A Fundamental Fallacy in Earth Pressure Calcula-tions," Boston Society of Civil Engineers, pp. 71-88.
1943. "Theoretical Soil Mechanics", John Wiley & Sons, New York.
1954. "Anchored Bulkheads," Transactions of the American Societyof Civil Engineers, Vol. 119, pp. 1243-1324.
Terzaghi, K., and Peck, R. 1967. "Soil Mechanics in Engineering Practice,"Second Edition, John Wiley & Sons, Inc., New York.
Tokimatsu, A. M., and Seed, H. B. 1987 (Aug). "Evaluation of Settlements InSands Due to Earthquake Shaking," ASCE, Journal of the Geotechnical Division,Vol. 113, No. 8, pp. 861-878.
Tokiinatsu, K., and Yoshimi, Y. 1983. "Emperical Correlation of Soil Lique-faction Based on SPT N-Value and Fines Content," Soils and Foundations,Vol. 23, No. 4, pp 56-74.
Towhata, I., and Islam, S. 1987 (Dec.). "Prediction of Lateral Movement ofAnchored Bulkheads Induced by Seismic Liquefaction," Soils and Foundations,Vol. 27, No. 4, pp. 137-147.
Tschehotarioff, G. P. 1973. Foundations, Retaining and Earth Structures,McGraw-Hill, Second Edition, 642 p.
USS Steel Sheet Piling Design Manual 1969, 132p.
Vasquez-Herrera, A., and Dobry, R. 1988. "The Behavior of Undrained Contrac-tive Sand and Its Effect on Seismic Liquefaction Flow Failures of EarthStructures," Rensselaer Polytechnic Institute, Troy, NY.
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255
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256
APPENDIX A: COMPUTATION OF THE DYNAMIC ACTIVE AND PASSIVE EARTH PRESSUREFORCES FOR PARTIALLY SUBMERGED BACKFILLS USING THE WEDGEMETHOD
A.1 Introduction
This appendix describes the derivation of the dynamic active and passiveearth pressure forces for partially submerged backfills using the wedgemethod. The effect of earthquakes is incorporated through the use of aconstant horizontal acceleration, ah - kh-g, and a constant verticalacceleration, a, - kv'g, acting on the soil mass comprising the active wedge(or passive wedge) within the backfill, as shown in Figure A.1 (and
Figure A.3).
P
a AE
aA MOVEMENTS
I
u top u topW-hk v static shear
shw u W static shear
Acee totu oorubo
*sai- /YuOTAI WAT R TAB OSTTEN/ r .OSITsai COINSUMPOER NT AKF
"h h I- \ /)_. NDUCEDU u' , Io ...... ...... C O M P O N E •N T l
Acceleraorlon
•0v kv .9
•HYDROSTATIC WATER TABL.Eo r uCONSTANT WITHIN SUBMERGED BACKFILL
Figure A.1 Dynamic active wedge analysis with excess pore water pressures
Al
The earth and water pressure forces acting on the wedge are derived forthe case of restrained water within the backfill and a hydrostatic watertable. Any increase in the pore water pressures above their steady statevalues in response to the shear strains induced within the saturated portionof the backfill during earthquake shaking is reflected in a value of ru, > 0.A constant r0 value is used throughout the submerged portion of the backfillin this derivation.
A.2 Active Earth Pressures
Figure A.1 represents a free body diagram for the derivation whichfollows. The base of the wedge is the trial planar slip surface representingthe active failure plane, which is inclined at angle alpha to the horizontal.The top of the wedge is bounded by a horizontal ground surface, and a verticalface along the interface between the backfill and the retaining wall.
The weight of the wedge acts at the center of mass and is computed as
W It H 2 1(A-1)
W = • •t tan(A
The three forces acting along the planar slip surface are represented byan effective normal force N', a shear force T and the pore water pressureforce. Assuming a cohesionless backfill and full mobilization of shearresistance along the slip surface, the shear force may be computed utilizingthe Mohr-Coulomb failure criteria as
T = N' L (A-2)
The total pore water pressures acting along the submerged faces of the soilwedge are described in terms of the steady state pore water pressure componentand the excess pore water pressure component attributed to earthquake shaking.
A.2.1 Calculation of Water Pressure Forces for a Hydrostatic Water Table
The pore water pressure at the ground water table (Figure A.2) is
top = 0 (A-3)static
For a hydrostatic water table the pore water pressure distribution is linfearwith depth, and at the bottom of the wedge is computed as
bot (A-4)U static = •W H.
A2
H H~ Of W W TAI(U , , , , , U!S!
STAT - C L
U *COS ( I'CSTATIC - /I
USTATIC-OL %UbatL
STATIC
XFh -0 U sin Co - U STATICSTATIC -a
Figure A.2 Equilibrium of horizontal hydrostatic water
pressure forces acting on backfill wedge
A.2.2 Static Water Pressure Forces Acting on the Wedge
The static pore pressure distribution immediately behind the wall istriangular and the resultant force may be calculated as
U _i y j2 (A-5)
The static pore pressure force acting along the planar slip surface is alsotriangular and the resultant force may be computed as
1 H2 1 (A-6)I- H sincn
A.2.3 Excess Pore Water Pressures Due to Earthquake Shaking with Constant r"
Excess pore water pressures due to earthquake shaking are computed
assuming the restrained water case as described in section 4.3.2. With r,constant throughout the submerged portion of the backfill the pressuredistribution is linear. The excess pore water pressure at the ground watertable is computed as
u shar *Y (H - H.) ru (A-7)
top
Note that when the water table is below the surface of the backfill u shear> .The excess pore water pressure at the bottom of the wedge is computed as
hot (A-8 )Ushear [ -t (H - H,) , ( - Hy,) H,-8r
A3
The total pore water pressures are equal to the sum of the hydrostatic porewater pressures plus the excess pore water pressures.
A.2.4 Excess Pore Water Pressure Forces Acting on the Wedge.
The resultant excess pore water pressure force of a trapezoidal pressuredistribution acting normal to the back of the wall is equal to
1[ top bot(A9
U shear = U [uhear + U h ]r H (A-9)
The resultant excess pore water pressure force of the trapezoidal pressuredistribution acting normal to the planar 'slip surface is equal to
1[top hot ea 1 (A-10)U sh1ar-* shear + U shear
A.2.5 Equilibrium of Vertical Forces
Equilibrium of vertical forces acting on the Figure A.1 soil wedgeresults in the relationship
- P sinS + W( 1 - k ) - T sina - N'cosa
- (Ustatic_ +Ushear-a )Cosa * 0 (A-11)
Introducing Equation A-2 into A-11 results in
- P sinS + W( 1 - kv ) - N'tan'"sina
- N' cosa - (Ustatic.a + Ushear-a )Cosa = 0 (A-12)
and solving for the normal effective force, N', becomes
N' -- P sinS +W (W-kg)tano'sina + cosa tanO'sina + cosa
Cosa (A-13)
Us'÷U° tan)'sina + cosa
A4
A.2.6 Equilibrium of Forces in the Horizonal Direction
Equilibrium of horizontal forces acting on the Figure A.1 soil wedge
results in the relationship
Pcos6 - N' sina ( Uztaic-cp Ushe.r.. )sirta
+ Tcosa - Wkh, ( Uttatc + Ushear ) - 0 (A-14)
Substituting Equation A-2 into A-14, and with the horizontal components ofwater pressure forces of equal magnitude and opposite direction, (refer to
Figure A.2), Equation A-14 simplifies to
Pcos6 - N'sina + N' tano'cosa - Wkh = 0 (A-1,-
Combining the N' terms results in
Pcos6 - N' ( sinfl - tano'cosa ) - Wkh = 0 (A-16)
Multiplying Equation A-13 (for N') by [ - ( sina - tan6'cosa )Iand simplifying becomes
- N ( - tano'cosa + sina )+ Psin6tan( a
- W( I - k )tan( a - 1" )
+ ( ustatic-a + Uhear-a )cosatan( a - ) kA 1 ,
Substituting Equation A-17 into A-16 gives
Pcos6 + Psin6 tan( a -
-W( 1 - k, )tan( a - ' )
( U _ + U~hear_• )cosatan( a - ' ) -Wk ) (A-
Combining terms results in
P cos6 + sin6tan( of - €" )
W[ ( I - k, ) ta n a - ' ) j kh '1
A5
Solving for the resultant force P which acts at angle 6
P CONSTANTAl - CONSTANTA2 (A-20)
cos6 + sin6tan(c -)
where
CONSTANTAI =W[( I -k, )tan(a- ' ) +k 1 J
and
CONSTANTA2 = (Ustatic-a + Ushear-a) cosatan( a -
The dynamic active earth pressure force PAE is equal to the maximum value of Pfor the trial wedges analyzed and aA = C for this critical wedge, as discussedin Section 3.4 and shown in Figure 3.10.
A.2.7 Surcharge Loading
The presence of an additional mass located on top of the backfill duringearthquake shaking can increase the magnitude of the dynamic active earthpressures acting on the wall. The effects of an additional surcharge massidealized in Figure A.3, or a surcharge loading idealized in Figure A-4, maybe incorporated within the dynamic active wedge analysis of Section A-2.6 byexpanding Equation A-20. For each slip surface analyzed, that portion of thesurcharge loading contained within the wedge is included within the equationsof equilibrium of forces acting on the wedge. When the surcharge isrepresented as a uniform pressure distribution q(, that port ion of thesurcharge loading contained above the wedge is replaced by an equivalent torceW, acting at its center of mass. The uniformly distributed surcharge pressure,q, shown in Figure A.4 is replaced by the equivalent force (per foot of wall)
Ws = qs le (A-21)
where
Ie = i ((H/tana) - x) for 1q -, I (refer to Figure A-4),
otherwise
The variable 1. rI-present.; lhe tfetective lengt-h of tht. surch.1 i Fge load
Equation A-20 becomes
P = CONSTANTAI CONSTANTA-: C'ONSTANTA2 (A-22)
coS'6 * sin tan( a, - ' )
AO
P
PAE -- - -
I= [ 14ton a
0 al1 (AE (90-. • a)-
W, "k'
Figure A,3 Dynamic active wedge analysis including a surchargeloading
whe re
CONSTANT~si= Ws [ ( 1 - kv )tan( a - •" ) k•
W.
[ ~and CONSTANTAl and CONSTANTA2 are computed as in Section A.2.6 forEquation A-20.
i For surcharge loadings of finite length, a wide range of slip surfacesI must be investigated to ensure that the maximum value for P is calculatzed and
equal to PAE, corresponding to the critical slip surface a• as shown inFigures A.3 and A.4.
A.2.8 Static Active Wedge Analysis
In the case of a static wedge analysis with k•. = kh = 0ser_ ,Equation A-2O simplifies to
P_ [ W - IJs~ t•cosa ltan( a - '" ) (A-23)
tOSb + sin6tan'( a'•
with a restricted to values of a ", , since P 0 .
A7
! !
P/
0 aI a A 6o-a) H a x
i ,o
Figure A.4 Dynamic active wedge analysis including a surcharge
loading
PA =P and aA a for the static critical wedge as well. Fo r asurcharge loading, Equation A-22 simplifies to
[ W ' ý4s - UstaticacosO' I tan( a*-• (A-24)
cos6 + sin6tan( a •
where W., is computed using Equation A-21.
4.3 Passive Earth Pressures
Figure A.5 represents a free body diagram that is used in the derivationof the wedge procedure for computing the value of the dynamic passive earthpressure force PPE. The base of the wedge represents the trial planar slit)
surface and is inclined at angle a to the horizontal. The top of the wedge isdefined by a horizontal ground surface, and the vertical face is located along
the interface between the backfill and the retaining wall.
The weight of the wedge acts at the center of mass and is• computed usingEquation A-1. The three forces acting along the planar slip surface are th|enormal force N" , the shear force T, and the pore water pressure force. Theshear force T shown in Figure A.3 for the passive case acts opposite to the
shear force shown in Figure A.1 for the active case,. Assuming a cohen.,ionloss
A8
p MOVEMENTS
I I_ . H. ton--- a
d PE Iu top u top.O Wo-k
shear stot;c
+P-- + + + W'hto Hw
U shear Ustati
u bat bat o topshear static shear
AI OO statit-
HYHDROTAICTATRTICLCOMPONENT WIHI SUMRGDBAKtL
sha freARmOay e cmue tlzn h orCuon falue critri
SINDUCED ' iICOMPONENT •\ %gu bat Yio
Theao t s o t he k hs
\ Ground
watr resurs lusth exes wterprssuesdu to artquaeo shaing
* HYDROSTATIC WATER TABLE Aclrto
Swr fCONSTANT WITHIN SUBMERGED BACKFILL arev c k Vou
Figure A.5 Dynamic passive wedge analysis with excess pore waterpressures
backfill and full mobilization of shear resistance along the slip surface, the
shear force may be computed utilizing the Mohr-Coulomb failure criteria asgiven by Equation A-2.
A.3.1 Calculation of Water Pressure Forces for a Hydrostatic Water Table
The total water pressure forces are equal to the sum of the steady state
water pressures plus the excess water pressures due to earthquake shaking.
Steady state water pressure forces for a hydrostatic water table are computed
using the procedures described in Sections A.2.1 and A.2.2. Excess pore water
pressures due to earthquake loads with constant ru throughout the submerged
portion of the backfill are computed using the procedures described in
Sections A.2.3 and A.2.4.
A9
A.3.2 Equilibrium of Vertical Forces
Equilibrium of vertical forces acting on Figure A.3 wedge results in therelationship
P sin6 + W( I - k, ) + T sina - N' cosa
- ( Ustatica + Ushear-or ) Cosa = 0 (A-25)
Introducing Equation A-2 into A-25 results in
P sin6 + W( 1 - k, ) + N' tanO' sina
- N'cosa - ( Urtatic-a + Usheara )C Sa = 0 (A-26)
and solving for the normal effective force becomes
N, P sins W ( 1- k )- tan•'sincr + cosa - taný' sinc + cosa
us + Ushear -) Cosa (A-27)Usttic= Us~ar= - tanO' sina + cosa
A.3.3 Equilibrium of Forces in the lHorizonal Direction
Equilibrium of horizontal forces acting on Figure A,5 soil wedge resultsin the relationship
Pcosb - N' sina - ( U.t.t.. Ush+ &,r.a )sitck
- Tcosa + Wkh + ( Urt.t,: + Ushear ) = 0 (A-28)
Stst-it-utitg Equation A-2 into A-28 and with the horizontal coimiponents of Hitiwater pressure force.: of equal magnitude and opposite direction (refer toFigure A.2), Equation A-28 simplifies to
Pcos6 - N'sina - N' tanO'cosa + lW=k 0 (A-29)
combining the N' terms results in
Pcos6 - N' ( sina * tano' cosa ) + Wkh = 0 (A-30)
AIO
Multiplying Equation A-27 (for N') by - ( sin + tanO' cosa ) ]and simplifying becomes
Solving for the resultant force P which acts at angle 6
p= CONSTANTpI - CONSTANT. 2 (A-34)
cosS - sin6tan( a + d" )
where
CONSTANTp, = W[ ( I - k, )tan( a ÷' ) + k I
and
CONSTANTp 2 = (Ustatic- + U shear-_) cos+tan ( a +
The dynamic passive earth pressure force PPE is equal to the minimum value ofP for the trial weeges analyzed and arp = a for this critical wedge.
All
A.3.4 Surcharge Loading
The presence of an additional mass located on top of the backfill duringearthquake shaking can decrease the magnitude of the dynamic passive earthpressures acting on the wall. The effects of an additional surcharge massidealized in Figure A.6, or a surcharge loading idealyzed in Figure A.7, maybe incorporated within the dynamic passive wedge analysis of Section A.3.3 byexpanding Equation A-34. For each slip surface analized, that portion of the
P
tH, on a
PPE IW
x
0 a1lciE (90*i)
W - ..
W w / U
-
C,'g/
GROUNDACCELERATION a,• k,'g
Figure A.6 Dynamic passive wedge analysis including a surcharge load
surcharge loading contained within the wedge is included within the equationsof equilibrium of forces acting on the wedge, When the surcharge isrepresented as a uniform pressure distribution q5, that portion of thesurcharge loading contained above the wedge is replaced by an equivalent forceW. acting at its center of mass. The uniformly distributed surcharge pressureq, shown in Figure A.7 is replaced by the equivalent force (per foot of wall)Ws, computed using Equation A-21 in Section A.2.7. Figure A.7 surchargepressure q, is equivalent to Figure A.6 case of a surcharge of weight W5, andEquation A-34 becomes
p CONSTANTp1 + CONSTANTpsl - CONSTANTP 2 (A- 35)cosS - sin&tan( a + 4" )
A12
H - tonia
a/ aPE (90 --a) Q
UWfkfl / T1f
,Q
W " k k,/
G N T NP s Ws r l-k,, ) a a ' ) -k
USXu U st0 •; -CL
GROUND k*ACCELERATION a "#g
Figure A.7 Dynamic passive wedge analysis including a surcharge load
where
CONSTANTpsI = WS[ ( 1 k, )tai-( cz + •')-kh
and CONSTANTpt and CONSTANTp 2 are computed as in Section A.3.3 for Equation
A-34.
For surcharge loads of finite length, a wide range of slip surfaces mustbe investigated to ensure that the minimum value for P is calculated and equalto PPE, corresponding to the critical slip surface OPE as shown in Figures A.6and A.7.
A.3.5 Static Passive Wedge Analysis
Note that for static problems with k, = kh = ~shear-o = 0 Equation A-34simplifies to
S[W - Ustatic- cosa I tan( a + 4' )cosS - sin6tan ( a + •' )
A13
with at restricted to values of a > 0 and 6 < 0/2.
Pp = P and ap = a for this critical wedge. For a surcharge loading, EquationA-35 simplifies to
p W + Ws - ustatic_,cosa J tan( a +" ) (A-37)
cos6 - sinStan ( a + •" )
A I /1
APPENDIX B: THE WESTERGAARD PROCEDURE FOR COMPUTING HYDRODYNAMIC WATERPRESSURES ALONG VERTICAL WALLS DURING EARTHQUAKES
This section describes the Westergaard procedure for computing the mag-nitude of the hydrodynamic water pressures along rigid vertical walls luringearthquake shaking. The solution developed by Westergaard (1931) is for tilecase of a semi-infinite long water reservoir retained by a concrete dam andsubjected to a horizontal earthquake motion. The fundamental period of theconcrete dam is assumed to be much smaller than the fundamental period of theearthquake so that the acceleration for the massive structure is approximatedas the accelefat.ion of the earthquake motion along the rigid base. Thisallows the problem of a very stiff concrete dam to be simplified to the caseof a rigid vertical face moving at the same horizontal acceleration as thebase horizontal acceleration. Using the equations of elasticity of a solid todescribe the propagation of sounds in liquids (waves which propagate withoutshear distortions) and with the water considered to be compressible, a solu-tion to the equation of motion of the water was developed for a harmonicmotion appliedi along the base of the reservoir, This solution ignores theeffects of surface waves and is valid only when the period of the harmonicexcitation is greater than the fundamental natural period of the reservoir
(Chopra 1967). The fundamental period for the reservoir, Tw. is equal to
T, 411) (B-I)C
where the velocity of sound in water, C, is given by
C =FK (B-2)
and the mass density of water, p, is given by
-YW (B- 3)g
With the bulk modulus of elasticity of water. K, equal to 4.32 X 10W lb perft 2 , the unit weight for water, yw, equal to 62.4 lb per ft'• and the accelera-tion due to gravity, g, equal to 32.17 ft per sec 2 , C is equal to 4,720 ft persec. For example, with a depth of pool of water, tip, equal to 25 ft, Tw isequal to 0.02 seconds (47 Hz) by Equation B-i.
The re.,ilting relationship for hydrodynamic pressure on the face of thedam is a function Of the horizontal seismic coefficient, kh, the depth ofwater, Y., the total depth of the pool of water, Hp, the fundamental period ofthe earthquake , and the compress ih i I i ty of the water ,". The hydrodynamic
pressure is opposite in phase to the base acceleration and for positive baseacrceleratio ns the hydrodyn;amic press:lire is a ten1;ile, Westergaard proposedthe following approximate solution for the hydrodynamic water pressure distri-bution: a parabolic dlynamnic p ressure distribution, P•,, described by the
relat ionship
BI.
7w (B3-4)P.d =-g Yh0. F--Y (-4
The resultant dynamic water pressure force, Pwd, is equal to7
P.d = (B-5)
acting at an elevation equal to 0.4 Hp above the base of the pool as shown inFigure B.1. This dynamic force does not include the hydrostatic water pres-sure force acting along the face of the dam.
MRgtD VERTCAL FACE
arfold bm"Kh" g
HYDROSTATIC WATER PRESSURES + HYDRODYNAMIC WATER PRESSURES
(Westergoord Procedure)
Figure B.1 Hydrostatic and Westergaard hydrodynamic water pressuresacting along vertical wall during earthquakes
B.1 The Westergaard Added Mass Procedure
A complete dynamic analysis of a structure LAiaL is in :uitact with apool of water requires that the hydrodynamic effects be accounted for duringthe dynamic analysis. This requires that the pool of water must be incor-porated within the idealized model for the problem. Most dynamic finite ele-ment computer code formulations that are used for soil-structure interactionanalyses do not include a fluid element in their catalog of elements. TheWestergaard added mass procedure is one method that is used to incorporate thehydrodynamic effects in the analysis for computer .odes without a fluid ele-ment formulation. With the hydrodynamic water pressure on the vertical faceof a rigid structure opposite in phase to the ground acceleration, thesehydrodynamic pressures are equivalent to the inertia force of an added massmoving with the dam (Chopra 1967). The Westergaard (1931) added water massprocedure adds an additional water mass to the mass matrix along the frontface of the structure. For pools that are wider than three times the depth ofthe pool, this additional mass of water is enveloped within the parabolicpressure distribution given by Equation B-4 and the front: of the wall. Thisprocedure is applicable when the period of harmonic excitation (i.e. theearthquake) is greater than the fundamental natural period of the earthquake(Chopra 1967), which is the case for shallow pools.
B2
APPENDIX C: DESIGN EXAMPLE FOR AN ANCHORED SHEET PILE WALL
The calculations involved in the design of Figure C.1 aincLored ý;Iieetpile wall and its anchorage is described in this appenclix tor bot-h styatic :iud
seismic loadings using the procedures described in Chapter 7. Assume kh --0.2, k, - 0.1 and no excess pore water pressures are generated during earth-quake shaking (r, = 0). The results of the computations shown are rounded forease of checking calculations and not to the appropriate number of signitic,-ritfigures.
SHEET PILE WALL
- TIE ROD H1-z- 7'
HTI" -3
ANCHOR BLOCK
DENSE SAND HpOOL - 20'
-'t - 120 pcf DREDGE LEVEL
q5'- 35 degrees -,
ru -O D-
Figure C.A Anchored sheet pile wall designproblem
Section C.A describes the design of anchored sheet pile wall for staticloading and Section C.2 the design for earthquake loading.
Section C.A Design of An Anchored Sheet Pile Wall for Static Loading
This section describes the design of Figure C.I anichored shtI,.- pile wallfor static loads using the free earth support method ot analysis;
C.1.1 Active Earth Pressures Coefficients KA
Factor of Safety on shear strength ý 1.0
Assume 6 = 0
S= 17.5 degrees
Ci
By Equation 16, KA - 0.246
Say KA - 0.25
KA'cos 6 - 0.24
C.1.2 "Factored" Passive Earth Pressure Coefficient K
Factor of Safety on shear strength - 1.5
By Equation 95,
tan 0' tan 35'
Ot 25 degrees
By Equation 96 with 6 - 0/2 and 6 - 17.50,
tanbt = tan(17.5)
6t = 11.9
say St = 12 degrees
Stand 6 = 0.5
Using the Log-spiral solutions in Figure 3.11 for K, with
6/,A = -0.5, R, = 0.808
KP (b/l = -1.0,0 = 25 degrees) = 4.4
K, (8/0 = -0,5) = 0.808 • 4.4 = 3.56
KP cos&t = 3.56 ,cosl2° = 3.48
C.1.3 Depth of Penetration
Table C.1 summarizes the horizontal force components acting on FigureC.2 sheet pile wall and are expressed in terms of the generalized dimensions
HTI, HTZ, Hpoo,, and D. The horizontal force components and their moment aboutthe elevation of the tie rod are summarized in Tables C.2 and G.3. The tor•e•and moments are expressed in terms of the unknown depth of penetration, 0.
C2
m m 1111111I
T F E S E 1 H T, -7 "___________~_____'______ HT2 , 3'
E3 Hpoo - 20'
5 DREDGE LEVEL
E4 f
E 5 -- PTOE D
ACTIVE "FACTORED"PRESUES PASSIVE
PRESSURES PRESSURE
Figure C.2 Horizontal earth pressurecomponents in free earth support
design
Table C.1 Horizontal Force Components
Horizontal DistanceForce to
Designation Horizontal Force Tie Rod
KAcos. 7,Yt (HT1 + HT2)2 4(2 HT2 - HT )
E2 KAcos6. 1 t(T T)H.1H2 Po
EK C +P 1) 21
1E 4O -D
KAcosS '[jYt(H'TI H7-~2) -y *l HpoolJ.D
KAcos6 -b (D) 2 H'2 + Hpool + 2 D
PTOE Wcos6t .y (D) 2 Rrz + HP.., + 2 D
7
Table C.2 Moments About Tie Rod Due to Active Earth Pr&:ssures
Distance Moment AboutHorizontal Horizontal to Tie Rod
Force Force Tie Rod -CCW Moment +'w.-Designation (lb per ft wall) (ft) (ft-lb per ft wal1l)
E, 1,440 -0.33 -475
E2 5,760 13 74,880
E3 2,765 16.33 45,149
E4 564.5D 23 + I D 12,9835 D + 282.1 D2
E5 6.91 (D)2 23 + 2 D 159,0 D2 + 4.6 D3S-3
MActive = 4.6 D01 + 441.3 D + 12,983.5 +119,554
Table C.3 Moments About Tie Rod Due to Passive Earth Pressures
Distance MomenC AboutHorizontal Horizontal to Tie Rod
Force Force Tie Rod -CCW Moment +'vv-Designation (lb per ft wall) (ft) (ft-lb per ft wall)
PTOE 100.2 (D)2 23 + 2 D -66,8 D: - 2,304.6 D2
Mpassive - -66.8 D3 2,3104.6 D2
Equilibrium of moments about the elevation of the tie rod(CCVI moment +'ve) requires
E4
tip rod 0
0 = MAtive + Mpassive
0 = -62.2 D3 - 1,863.3 D2 + 12,983.5 D + 119,554
(:4
From the calculations summarized in Table C.4, D - 10.02 tt for calculatiotipurposes (D - 10 ft for construction).
Table C.4 Calculation of the Depth of Penetration
Trial D Moment Imbalance(ft) (ft-lb per ft wall) Comment
9 40,134 shallow
10 859 shallow
10.1 -3,473 deep
10.02 -1 exact
C.1.4 Tie Rod Force TFES
Horizontal force equilibrium (refer to Figure C.2).
ZFh = 0
E, + E2 + E3 + E 4 + E5 - PTOE - TFES = 0
From the calculations summarized in Table C.5,
16,315 - 10,060 - TFES - 0
TFES - 6,255 lb per ft of wall
Table C.5 Horizontal Force Components for D = 10 Feet
Horizontal HorizontalForce Force
Designation (lb per ft wall)
E• 1,440
E2 5,760
E3 2,765
E4 5,656
E5 694
PTOE 10,060
C5
C. 1. 5 Maximum Moment MFES
The maximum value of moment, MFES, occurs at thre v1-v;'t ion of zero sl"'r.i
within the sheet pile. First, determine the elevat ion of zero shear tind io-icompute the moment internal to the sheet, pile by comput ing tie moment s of thiearth pressures and water pressures about the elevation of Ohe tie rod rtelex
to Step 8 discussion in Section 7.4.1). This usually occurs at an elevat ionabove the dredge level. By modifying the relationships given in Table CI.,the equilibrium of horizontal forces at a depth, y, below the water table if;
expressed as
El + E2x + E3x - TFES = 0
1,440 + 288.y + 6.ql2 .y 2 - 6,255 - 0
6.912-y 2 + 288y - 4.815 = 0
- -(288) t 1(288)' - 4(6,912)(-4815)
2(6.912)
y = 12.79 ft below the water table
From the calculations summarized in Table C.6, the maximum momelnti ix e-ocl tothe sheet pile at y = 12.79 ft below the ,;at•r- table is equal to MFE- -' 47,10'.
ft-lb per ft of wall.
Table C.6 Moment Internal to the Sheet Pile at y = 12.79 Feet Below the WaterTable and About the Elevation of the Tie Rod
Horizontal Horizontal Lever
Force Force Arm Moment
Designation (lb per ft wall) (ft) (ft-lb per ft wall)
_ E.l 1,440 -0.33 -475
E2X 3,683.5 3 + 1 (12 79) 34,607
E3x 1,130.7 3 + 2 (12.79) 1.033
M EIS Y 7, 165 ft - lb per f t wall
(:6
Retined Procedure for Computing MFES:
The computed value for maximum moment- MFE:, e a l to 41/(% ft -lb pe-r ft
of wall is greater than would he ohba ined usingg the US Army Corps of Engin.-,ers(Corps) design procedure for static loading, w; dse;eriltdi i n thlt, U.S. ArmyEngineers Mainual EM l110-2-2504 (Headtuairt~ers, l)ep.irtnwet of thb, Army 1992).The Corps design procedure is a refinement to the procedurt, described in tlssection with the value for the maximum moment. MFF. ('-omputed" ,: ing a dup h ofpenetration with FSp ill Equat. ions 95 and 96 set equal to uni tv. The Corpsprocedure avoids compounding factors of safety in the stc-Iect ion of tie .ht-*wtpile section. The value specified for depth of penertration for shte-t pilewall construction would be unchanged, equal to 10 ft in this example
(Section C. 1.4).
Section C.1.6 Design Moment Mdesn
The design moment, Misgz, is obtained through apptlicatiot' of Rowe's
moment reductioni procedure that is out lined in Figure 7 2.
"H = 1T1 + HT 2 ' Hii 0 4 1)
H 7 + 3 ± 20 + 10 40 ft (480_24 in.
E - 3 x l0 psiH"
Flexibility number, =E
where
I moment. of inertia per ft of wall
P (480.24 in.)'(3(0 x 10' ps i).-I
1, 773 .0I
The values of Mdj.,,n are given in Table C.7 for four sheet pilesect-ionls.
Table C. 7 Design Momentt for Sheet Pile Wa l1 i n Dense Sand
I P) M1,.,,, • ,
Se, t ioln (in. pe," ft (in.Z/lb per r, (ft-l per
Designa•tion of wall) ft of wall) (Figure, 7.2) ft of wall)
PZ22 84.4 21.0 0.45 21,224
22/ 184.2 9.62 0.68 32,072
P 45 .2 . 0 47,165
FPZ40 490.8 3.61 [ I.0 47. 165
C 7
where Mdesign = rd .MFES (by eq 100)
Section C.l.7 Selection of the Sheet Pile Section
In this design example of sheet pile walls for static loadings, assumethe maximum allowable stress within the sheet pile is restricted to
7allowable = 0.65- cyield
for ASTM A328 steel sheet piling,
ayileid = 39,000 psi
Ualowable = 0.65 39,000 psi = 25,000 psi
The allowable bending moment (Table C.8), Mallowable, is given by
Mallowable = S 'allowabie per ft run of wall
where
S - section modulus (in. 3 per ft run of wall)
Table C.8 Allowable Bending Moment for Four ASTM A328 Grade Sheet PileSections (Oallowable - 0.65 • ayield)
Section S Mal towable
Designation (in. 3 per ft of wall) (ft-kips per ft run of wall)
PZ22 18.1 38
PZ27 30.2 64
PZ35 48.5 102
PZ40 60.7 128
Comparison of the design moment values (Mdesitn in Table C.7) to theallowable bending moments (Mallawable in Table C.8) indicates that all four pilesections would be adequate. The lightest section, PZ22, would be selected forthis design based upon static loading. Corrosion must also be addressed dur-ing the course of the sheet pile wall design. Additionally, the deflection ofthe anchored sheet pile wall would be checked (Dawkins 1991).
C.1.8 Design Tie Rod
Tdsign = - . 3 TFES
TFES - 6,255 lb per ft of wall (from Section C.1.4)
Tdsign = 8,132 lb per ft of wall
C8
Assume
(a) 6 ft spacing of anchors
(b) Uyieid = 36,000 psi
0 allowable = 0.4 4yield (40 t of yield)
Minimum area of rod 6 ft - 8,132 lb per ft of wall0.4 -36,000 psi
Gross Area = 3.39 in. 2
Diameter = { 4 * Area = 2.08 in.
C.1.9 Design Anchorage
Tuit-a = 2.5 "TFES (by eq 102)
with
TFES = 6,255 lb per ft of wall (from Section C.1.4/
Tuita = 15,638 lb per ft of wall
Details regarding the design of anchorage are provided in numerous ret-erences including Dismuke (1991) and the USS Steel Sheet Piling Design Manual(1969). If the overall height of the anchor, h, is not less than about 0.6
times the depth from the ground surface to the bottom of anchorage, designatedda in Figure C.3, the anchor behaves as if it extended to the ground surface.
ha > 0.6 .d.
The full angle of interface friction, 6, used in computing Kp can only bemobilized if the anchor has sufficient dead weight or, in general, isrestrained against upward movement (Dismuke 1991). For a slender anchor theultimate capacity for a continuous anchor is required to satisfy the
expression
Tuit-a < PP - PA
with 6 = 0 degrees (refer to Figure C.3).
C9
0N0 SFACE ANCHOR PLAT
do-I"ho
Figure C.3 Horizontal active and passiveearth pressure components acting on a
continuous slender anchor
For anchorage above the water table
T,,lt-a -< 't (h.)2 -( K )
For 6' = 35 degrees and 8 = 0 degrees,
S= 3.69 (by eq 11)
KA = 027 (by eq 5)
Tjitia -120 pcf (10')2 -(3 69 - 0.27)
15,638 lb per ft of wall < 20,520 Ib per ft run of continuous anchor
h_ > 0 6 10'
h > 6 ft.
Because the value of Tu.t-a is significantly less than the capacity ot acontinuous wall, a series of separate anchorages would be investigated (referto the procedures described in the USS Steel Sheet Piling Design manual,1969).
C 10
C.1.1O Site Anchorage
To be effective, the anchorage must be located such that the potentialactive failure zone behind the sheet pile wall and the potential passivefailure zone in front of the anchorage does not intersect. Design criteiiafor deadman anchorage is shown in Figure C.4. The use of the estimated
point of zero moment in the wall ( at -iD) accounts for the increase
depth of penetration due to the use of FSp = 1.5 used in the calculation ofthe passive earth pressure force provided by the soil below the dredge level(Duncan 1985).
ANCHOR , HBLOCK _T2 ", H
ANCHOR BLOCK DREDGE LEVELSHOULD BE SIT EDBEYOND) THIS LINE'S 0
ESTIMATED POINT OF ¼DZERO MOMENT ABOVE 4
THE eASE OF THE WALL
(a) Simplified procedure for siting anchor block
,I ..- ]' " " • "'HTI- HV
ANCHOR 5 °-12) " HTzBLOCK'_ 445 NOH HP00,
ANCHOR BLOCK DREDGE LEVELSHOULD BE S/TED %BEYOND THIS UNE (45" */2)
ESTIMATED POINT OF .I !IO O
ZERO MOMENT ABOVETHE BASE OF THE WALL
(b) Simplified procedure for sit;ng a continuous anchor wall
From NAVFAC DM /.2
Figure CQ4 Design criteria for deadman anucho;rage
CIl
C.2 Design of an Anchored Sheet Pile Wall for Seismic Loading
This section describes the calculations iwlvoived in the desigo of Fig -
ure C.1 anchored sheet pile wall tor earthquake loading using the fre, ,.arth
support method of analysis (13 steps) described irt Sec-tion 7.4 1 with L- 1 .
C.2.1 Static Design (Step 1)
The static loading design of Figure C.I anchored sheet pile wall is described
in Section G.l. The calculated depth of penetration D equals 10.02 ft
(Section C.1.3).
C.2.2 Horizontal Seismic Coefficient, kh (Step 2)
= 0.2
C.2.3 Vertical Seismic Coefficient, k, (Step 3)
k, = +0.1, 0 and -0.1
according to Section 1.4.3. This appendix contains (detailS; regarlding rt ,
for k, ý +0.1 only due to the length of the calculationis involved.
C.2.4 Depth of Penetration (Steps 4 to 6)
The depth of penetration, D, equal to 10 ft was tound not to he !t;h-lt'
under earthquake loading. The required minimum depth ot pene-trattion is beks
determined by the trial and error procedure of first assuwiilg i valut- tot I)
and checking if moment eqo•ilibrium of the earth and watr pres:urt, iorcts
about the elevation of the tie rod is satisfied (steps 4 through 6).
This iterative procedure results in a minimum required depth .f pene-
tration equal to 20.24 ft. The calculations involved in Steps 4 through 6 . e
summarized in the following par.g'-aphs for the case of 1) set equal to
20.24 ft.
C12
ill
Effective Unit Weight for the Partially Submerged Backfill
According to Figure 4.13,
(hZ (h - + - 1,f
with D = 20.24 ft
h, = 40.24 ft
h = 50.24 ft
'Ye = [5 ]2 (120 pcf - 62.4 pcf) + - (0.24 120
-y = 79.97 pcf
Equivalent Horizontal Seismic Coefficient, khel, for the jackfill
For the restrained water case with ru - 0
kh-1 = - • kb (adapted from eq 40)le
khel 120 pcf . 0.2 = 0.3001khl=79.97T pcf
Seismic Inertia Angle, 0el' for the Backfill
tan-1 khOI (adapted from tq 48)
tan- , 0.3901]
0e. = 18.44'
Dynamic Active Earth Pressure, PAE
with •" = 35°*, 6 - 0/2 - 17.5k and Ve. = 18.44', KAE 0.512 (by eq 30)
* Strength parameters to be assigo~ed in dccordance with the cciteria in
Section 2.3.
C13
PA KAE -l Ik) H2 (adapte~d from eq 3",)
PAE = 0.512 1 {79 .97 psf 1( - 0.1)] (50.24')2
PAE = 46, 506 lb pcr ft of wall
(F&E), = PAE c Cos6 = 44 354 lb per ft cf wall
Horizontal Static Active Earth Pressure Component of PAE
With a hydrostatic water table and r, = 0, the horizontal static activeearth pressure force components of PAE are computed using the relationships inTable C.i.
With •" = 350 and 6 = 0/2 - 17.50,
KA = 0.246 (by eq 16)
KA cos 6 = 0.235
Above the water table -t = 120 ocf is used to calculate the effectiveoverburden pressu-e while below the water table -y' = -, - y- (z 57 6 pcf) isused to calculate the effective overburden pressure with -,, = 0. The result-ing values for the five horizontal static force components E, through E" of PAEare given in Table C.9 (forces shown in Figure C.2).
Table C.9. Five Horizontal Static Active Earth Pressure Force Componentsof PA with D = 20.24 feet
Horizontal Force Horizontal Force Distance to PileDesignation (lb per ft wall) Tip (ft)
Equilibrium of Moments About The Elevation of the Tie Rod
SMCC = ( A) (HT2 + Hp.., + D - YpAE)
+ PwdO (HT2 + Hpoo 1 - 0.4"pool)
= 44,354 .(3' + 20' + 20.24' - 23.87")
+ 2,912 -(3Y + 20' -8')
= 859,137 + 43,680
y - D for illustration purposes only. See discussion in footnore toPE 3
step 5.
C17
Mccw -= 902,817 ft. - lb per ft. of wall
MCw = -(PPE)x • (HT2 + 1Voo, + D - YpE)
= -24,740. (3' + 20' + 20.24' - 6.75' )
EMcw = -902,763 ft-lb per ft of wall
Moment Imbalance - Zbccw + ZMcw
= 54 ft-lb per ft of wall
Small moment imbalance value so D - 20.24 ft for the case of kh = 0.2 and k,+0.1.
The two additional cases of k, = and k, = -0.1 are summarized inTable C.10. The required minimum depth of penetration is equal to 20.24 ft
(20.5 ft for construction).
Table C.10 Summary of Depth of Penetration Calculations
D D
Case kh k, (ft) Dstat,
Static 0 0 10.02 1.0
Dynamic 0.2 -0.1 14.88 1.5
Dynamic 0.2 0 17.1 1.7
Dynamic 0.2 . +0.1 20.24 2.0
C.2.5 Tie Rod Force TFES (Step 7)
Horizontal force equilibrium for the case of D = 20.24 ft with kh 0.2 and k,= +0.1,
ZFh = 0
results in
TFES = (PAE)x + Pwd - (PP) (adapted from eq 99)
for a hydrostatic water table with r, = 0.
TFES = 44,354 + 2,912 - 24,740
TFES = 22,526 lb per ft of wall.
The two additional cases of k, = 0 and k, = -0.1 are summarized inTable C.11. The anchorage is designed using TFEs = 22,526 lb per ft of wall.
C18
Table C 11 Tie Rod For'-' TFES
D TFES TFESCase kh k, (ft) lb per ft
of wall (TFES) sttI
Static 0 0 10.02 6,255 1
Dynamic 0.2 -0.1 14.88 20,819 3.3
Dynamic 0.2 0 17.1 21,368 3.4
Dynamic 0.2 +0.1 20.24 22,526 3.6
C.2.6 Maximum Moment MFES (Step 8)
The maximum value of moment internal to the sheet pile wall, MFEs,occurs at the elevation of zero shear within the sheet pile. First determinethe elevation of zero shear and then compute the moment of earth and waterpressure forces about the tie rod (refer to Figure 7.10).
Above the dredge level, at elevation y below the hydrostatic water table
(PAE + Pwd - TFES = 0
with
(PAE)x = (PA)X + (APAE)x
(PA)x above the dredge level (refer to Figure C.2)
(PA). = E1 + Ezy + E3y
(PA) ý= 1,410 + 282 y + 6.768 YZ
With (APAE)x equal to 20,638 lb per ft of wall, the equivalent stressdistribution is given in Figure C.5 (adapted from Figure 7.9).
1(APA),." (1 oP + *Y) • (10' + y)
- (657.3 + 559.2 - 9 .80 7 y) - (10 + y)
APAE = -4.9035 yZ + 559.215 y + 6,082.5
CGI
GROUND SURFACE rOPt
a PAE = 20.638 Ib Y
H 50.24'"
0.6 H = 30./4"
PILE TIP 1
Utop= 1.6 (-"--PAE)= 657.3 psf
/'y = 559.2 - 9.807 ,,y
O-bot = 0.4 H = 164.3 psf
Figure C.5 Distributions of horizontal stresses corresponding to LPAE
* From Figure C.5 pressure distribution for y - l'.32 tt
The maximum moment internal to the sheet pile at y -15. P 2 tt blow the' W.tet
table is equal to MFES - 150,580 ft-lb per tt of wall.
Section C.2.7 Design Moment Mdesxgn (Step 9)
The design moment, Mdesain, is obtained through applicat ion of Rowt.'s(1952) moment reduction procedure that is outlined in Figure 7.2. The, ;i•ilitvof the system to develop flexure below the dredge lovel durinjg ,,artI-quatkeshaking must be carefully evaluated prior to applicatiot ot Rowe'sti momenttreduction factor or any portion of the reduction factor (reter to thte ittro-ductory discussion of Section 7.4).
H - HTl + HTz + Hpt,I, D
H - 7' + 3' + 20' + 20.24' - 50.24 ft - (602.88 in.)
E - 30 x 10 Psi
Flexibility number, p -11
Tl
whereI - moment of inertia ver ft of wall
(602.88 in) 4
(30 x TOs psi) . I
4,403.54I
The values of Md.i,5 , are given in Table C. 13 for 'ourI sIne't pi It' :,Q iot(T
C21
Table C.13 Design Moment for Sheet Pile Wall in Dense Sand
I p Mdes,&nSection (in." per ft (in. 2 /lb per rd (ftt-lb per
Designation of wall) ft of wall) (Figure 7.2) ft ot wall)
PZ22 84.4 52,2 0.38 57,220
PZ27 184.2 23.9 0.46 69,267
PZ35 361.2 12.2 0.58 87,336
PZ40 490.8 9.0 0.74 111,429
where Mdesign = rd • MFES (by eq 100)
In this design example, the maximum allowable stress within the sheetpile for seismic loadings is restricted to
OaLlowable = (1.33) 0 ,65 0yield 0 O. 87 'Oyeld
for ASTM A328 steel sheet piling,
ayield = 39,000 psi
'a7lcwable = 0.87 39,000 psi = 34,000 psi
The allowable bending moment, MaIlowable, is given by
Mallowable = S ' 0'allowable per ft run of wall
where
S = section modulus (in. 3 per ft run of wall)
Comparison of the design moment values (Mdesign in Table C.13) to the allowablebending moments (Mallowable in Table C.14) indicates that the pile section wouldbe upgraded from PZ22 to PZ27 due to seismic considerations. Corrosion mustalso be addressed during the course of sheet pile wall design.
Table C.14. Allowable Bending Moment for Four ASTM A328 Grade Sheet PileSECTIONS (al, wable = 0.87 ) . ...d)
S Mallowable
Section Designation (in.' per ft of wall) (ft-kips per ft of wall)
PZ22 18.1 51.3
PZ727 30.2 95.6
PZ35 48.5 137.4
PZ40 60,7 172.0
(722
C.2.8 Design Tie Rods (Step 10)
For seismic loadings
Tdesign = 1. 3 • TFES (by eq 101)
with TFES - 22,526 lb per ft of wall
Tdsign - 29,284 lb per ft of wall
Assume
(a) 6 ft spacing of tie rods
(b) ayleld = 36,000 psi
aallowable = O. 6
vyield (60% of yield)
Minimum area of rod = 6 ft. - 29,284 lb per ft of wall0.6 - 36,000 psi
Gross Area = 8.13 in. 2
Minimum Diameter = 4*Area = 3.22 inches
Table C.15 summarizes the required geometry of tie rod for the four load
cases.
Table C.15 Required Geometry of Tie Rod*
Tdesign Rod
D aallowable (lb per ft of Area DiameterCase kh k, (ft) oyield wall) (in. 2 ) (in.)
Static 0 0 10.02 0.4 8,132 3.30 2.08
Dynamic 0.2 -0.1 14.88 0.6 27,065 7.52 3.09
Dynamic 0.2 0 17.1 0.6 27,778 7.72 3.13
Dynamic 0. 2 +0.1 20.24 0.6 29,284 8.13 3.22
Calculated for the case of
(a) 6 ft spacing of tie rods(b) oyeld 36,000 psi
(c) Tdsign = 1.3 • TFES
C23
Comparison of the minimum diameter of tie rod (Table C.15) required for seis-mic loading to the diameter required for static loading indicates that for a6 ft spacing, the diameter of the tie rods (ayi 8 ld - 36,000 psi) would beupgraded from 2.08 in. to 3.22 in.
C.2.9 Design of Anchorage (Step 11)
For seismic loadings
Tult, = TFEs (refer to discussion in step 11)
In the case of kh = 0.2 and k, - +0.1,
Tuit-.a = 22,526 lb per ft of wall
The dynamic forces acting on the continuous anchor wall are shown inrigure C.6.
K GROUND SURFACE
ha do/.5' We kh
"r W. (I-k) W • r•
Figure C.6 Seismic design problem for acontinuous anchor block
C.2.10 Size Anchor Wall (Step 12)
Assume that a continuous concrete wall is selected to be the anchorage.The "factored" dynamic earth pressures that develop in front of the anchorwall provides nearly all of the lateral resistance to the pull force Tufta.
The anchor wall will be designed using 0. and 6t (Equations 95 and 96) due tothe magnitude of Tult-a for seismic loading (equal to 3.6 times the staticvalue). The required depth and width of anchor wall is best determined by thetrial and error procedure of first assuming a value for da and checking ifequilibrium of horizontal forces acting on the anchor (Equation 103) is
C24
satisfied (step 12). Once the value of d. is determined, equilibrium of thevertical forces acting on the anchor wall (Equation 104) will dictate theminimum value of wall width b.. Refer to Section C.1.9 in this appendix foradditional discussion of anchorage design.
This iterative procedure results in a minimum required depth ofanchorage equal to 11.5 ft and a minimum width of an-hor wall equal to 4.5 ftThe calculations involved in Step 12 are summarized in the followingparagraphs for da = 11. 5 ft and (ba).n = 4.5 ft in Figure C.b
Dynamic Active Earth Pressure Force PAE-A
For the case of da = 11.5 ft (the anchor submerged 1.5 ft below thewater table), the effective unit weight is equal to
.= 118.94 pcf
with h, = 1.5 ft and h = 11.5 ft in Figure 4.13.
The equivalent horizontal seismic coefficient khel is equal to 0.2018(obtained by substituting 1e for Ib in Equation 47). A value of khel equal to0.2 is used in the subsequent calculations.
For the case of kh.1 - 0.2 and k, = +0.1
0.1 = tan 1l -] (adapted from eq 48)
0.1 - 12. 529°
With q" - 35', 6 - 17.5° and 0e. = 12.529°
KA - 0.3987
andK*E .cos 6 = 0.38
KAE *sin 6 = 0.12
With da = 11.5 ft in Figure C.6.
(adapted trom
(PA-A). = KA .cos • [y( 1 - k ) (d )2 eq 33)
P 0.38 .1 [118.94 pcf (1 - 0.1) ] (11.5')2
(PA-A). -= 2,690 lb per ft of wall
C25
by a similar ,.alculation
(PAE-A)y - 849 lb per ft of wall
Dynami-ý Passive Earth Pressure Force PPE-A
With 0' = 35' and with FSp set equal to 1.2 in this example (see step 12
discussion regarding the relationship between anchorage displacement and FSp)
= 30.3' (by eq 95)
and 6 - 17.5',
6t = 14.7°
For 4 = 12.5290 (refer to PAE-A Lalculations), t = 30.3' and 6, = 14.70
The depth of the continuous anchor wall is governed by the equilibrium
of horizontal forces. Ignoring the contribution of the shear force along the
base of the wall, Equation 103Tufta = (PPEA), - (PAEA), - W"kh
For Figure C.6 concrete wall, the weight W per foot run of wall with d. -
11.5 ft and Icon, = 150 pcf is given by
C26
W = b .a da = 1,725 "b.
Introducing this relationship for W and
kh = 0.2
Tljlt-a 22,526 lb per ft of wall (ke , +0.1)
(PPE-A) = 27,818 lb per it ot wall
(PAE-Alx - 2,690 lb per ft of wall
into the modified equation of horizontal equilibrium results in a maximumvalue of ba equal to 7.5 ft for d. = 11.5 ft. Larger ba values would resultin exc-ssive horizontal inertia forces acting on the roncrete block, requiringrevisions of the previous calculations.
Mobilization of friction along interface between the front of the anchorwall and the passive wedge requires that the wall have sufficient dead weightto restrain against upward movement as it displaces the soil in front of thewall (Dismike 1991). The equation of equilibrium of vertical forces acting onthe wall is used to computte the minimum width of anchor wall. With N' setequal to zero, Equation 104 becomes
0 = W (1 - k1) - UA - (PPE-A) + (PAE-)y
w i th
W = 1,725 .ba
•0,.I
UA = 62.4 pcf "1.5' -b. = 93.6-ba
(PPE-A)y = 7,291 lb per ft of wall
(PE.z-A)y = 849 lb per ft of wall
the modified equation of vertical equilibrium results in a minimum value of baequal to 4.4 ft or (ba)mn = 4.5 ft.
Alternative Anchorage:
Other types of anchorages to be considered include slender anchorage,multiple tie rods and anchorage, A-frame anchors, sheet pile itachorage, soilor rock anchors and tension H-piles. Slender anchorage refers to a slenderwall designed using the procedure described in this section with 6 set equalto 0 degrees.
C.2.11 Site Anchorage (Step 13)
The anchor wall is to be located a sufficient dist.ance behind the sheetpile wall so that the dynamic active failure surface does not intersect thepassive failure developing i'i front of the anchor wall. Figure C.7 outlinesthe miiiirnum required distances for this design problem.
C27
Dynamic Active Wedge - Sheet Pile Wall
With 0' - 350, 6 - 17.50 and 0. - 18.44' (from Section C.2.4, Step 4)
aAE - 40.6950 (by eq 37)
XAE = 50.24' = 58'tan CAE
CONT INOUS LINEARANCHOR PLMNE SHET PiLE
GROUND SURFACE 7 WALLxTIE ROD K
LIE NEAR--E
SLIP PLANE DREDGE LEVEL"DY•NAMIC '
C * 20.24'
K-XPE A
Figure C.7 Simplified procedure for siti!.g a continuous
anchor wall
Dynamic Passive Wedge- Anchor Wall
with 01 = 30.30, 6t = 14.70 and
kei = 12.5290 (Section C.2.10, St-p 12)
opE 18.27' (by eq 61)
XPE = 11.5'tan aPE
Site Anchorage
Site concrete anchor wall at a distance of 93 ft behind the sheet pilewall (- xA + XPE),
C28
APPENDIX D: COMPUTER-BASED NUMERICAL ANALYSES
This appendix is a brief guide to issues; that must bh. faced whi.tI ,Iaki ga decision to utilize a computer-based rtumerical analysis'; ,id to thleliterature concerning such methods. As discussed in the maini body o0 this;report, there are circumstances in which analyses carried out by ;ome s;uchmethod may be appropriate during design of a waterfront structure-.
There exists a bewildering array of computer-based met hods a>ppliciz:il- toanalysis of the dynamic response of earthen mosses or soil-!;tructure sVtefmsTable D.1 presents a partial listing of some of the bettter-known methods.Most, but not all, such methods use a finite element formulatiioni, and hence.somewhat incorrectly are referred to collectively as tinite eltem--nt m•ethods;.Most methods were developed originally for applications other than waterfrontstructures - especially problems related to nuclear power plants andearthdams.
Some methods are relatively simple but approximate only one or twoaspects of soil behavior. Others, which can be quite complex titd ditti(:11luse, simulate a number of different features of soil behavior quit. well. Al]must be used with care and judgment. A key is to select a method no more-complex than is required for the problem at hand.
Table D.1 Partial Listing of Computer-Based Codes for Dynamic Anal';sisof Soil Systems
Reference Names of Code
Lysmer, Udaka, Tsai and Seed (1975) FLUSH
Earthquake Engineering Technology, Inc. (1983) SuperFLUSH
Hallquist (1982) DYNA2D)
Finn, Yogendrakumar, Yoshida, and Yoshida TARA(1986)
Provost (1981) DYNA-FLOW
Lee and Finn (1975, 1978) DESRA
Streeter, Wylie and Richart (1974) CHARSOIL
Provost (1989) DYNAID
Li (1990) SUMDES
Schnabel, Lysmer, and Seed (1972) SHAKE
Roth, Scott, and Cundall (1986) DSA(E
Zienkiewicz and Xie (1990) SWANDYNE-X
lai (see Tai and Kameoka 1991) ----
Earth Mechanics, Inc. of Fountai.n Valley, CA LIN()S
DI
D.1 Some Key References
For dynamic soil-structure interaction analysis related to heavybuildings resting on earth, a concise summary of the various proceduresavailable is reported in the 1979 ASCE report by the Ad Hoc Group on Soil-Structure Interaction of the Committee on Nuclear Structures and Materials ofthe Structural Division. While in many ways out-of-date, this is still auseful reference concerning basic principles.
Several different finite element formulations are described in detail inChapter 3, titled Geomechanics and written in part and edited by W, D. L. Finnin the Finite Element Handbook, edited by H. Kardestuncer. The scope and typeof laboratory and/or field testing program used to characterize the soil modelparameters will vary among the computer codes, as discussed by Finn, theCommittee on Earthquake Engineering of the National Research Council (1985),and others.
Whitman (1992) has suggested a scheme for categorizing the various typesof methods, and has discussed the ;tatus of validation of various methods bycomparison to observations during actual earthquakes or to results from modeltests.
D.2 Principal Issues
According to the guidelines set forth by the ASCE Ad Hoc Group on Soil-Structure Interaction of the Committee on Nuclear Structures and Materials ofthe Structural Division 1979 report on the "Analysis For Soil-StructureInteraction Effects For Nuclear Power Plants" and the ASCE Standard (1986), toperform a complete soil-structure interaction analysis the analyticalprocedure must (1) account for the variation of soil properties with depth,(2) give appropriate consideration to the material nonlinear behavior of soil,(3) consider the three-dimensional nature of the problem, (4) consider thecomplex nature of wave propagation which produced the ground motions, and(5) consider possible interaction with neighboring structures.
The reference to a "complete" analysis results from the existence of twodistinguishable aspects of soil-structure interaction: (1) the relativemotion of the foundation of the structure with respect to the surrounding soilas a result of the inertial forces in the structure being transmitted to thecompliant soil foundation and backfill and/or (2) the inability of the stifferstructural foundation and walls to conform to the distortions of the soilgenerated by the ground motion. The former is referred to as inertialinteraction and the latter is referred to as kinematic interaction. Bothfeatures co-exist in most actual problems. However, several an,-tlyticalprocedures available to perform the soil-structure interaction analysis ofearth retaining structures take advantages of this separation of behavior intheir numerical formulation.
Specific feature that must be accounted for in some problems includesoftening the soil stiffness during shaking, the material and geometricaldamping and the separation of portions of the backfill from the structure,followed by recontact or "slap," that can occur during shaking. It may benecessary to use special interface elements at boundaries between soil andstructure. It also may be necessary to model the actual process ofconstruction as accurately as possible.
D2
D.2.1 Total versus Effective Stress Analysis
Effective stress analyses explicitly predict and take into account theeffects of excess pore pressures caused by the cyclic shearing of soil duringearthquake shaking. The generation of significant excess pore pressurescauses the stiffness of soil to degrade and may lead to a nearly-total loss ofshear strength. TARA, DYNAFLOW, DESRA, and DSAGE are examples of effectivestress analyses. As a general rule, such analyses should h, used ifsignificant excess pore pressures are anticipated.
Total stress analyses do not explicitly account for the effects ofexcess pore pressures, although some may consider this effect indirectly byadjusting stiffness for the anticipated intensity of cyclic shear strains.FLUSH and SHAKE are examples of total stress analyses. Tot-at stress analysesare appropriate when cohesionless soils are dry or very coarse, with mostcohesive soils, and for problems such as analyzing lateral earth pressurescaused by surface loadings.
D.2.2 Modeling Nonlinear Behavior
Using an effective stress analysis accounts partially, but not fully,for the nonlinear behavior of soils. In addition, it is necessary to considerthe effect of shear strain upon stiffness at a given effective stress.
As somewhat of an oversimplification, three ways of introducing suchnon-linearity have been utilized. (1) By using a linear analysis in whichshear modulus is linked, via an iterative procedure, to a measure of cyclicshear strain during shaking, FLUSH -and SHAKE are examples ef this approach.(2) By introducing a nonlinear stress-strain law, such as a hyperbolicbackbone curve together with Masing rules for strain reversals, DESRA andTARA are examples. (3) By utilizing conepts and principles from the theoryof plasticity. DYNAFLOW is an example of this approach.
It is not really possible to say that. one way is bett or than another.
All involve some degree of approximation. The choice involves a trade-offbetween accuracy and convenience/cost, and perhaps the availability of a code.
D.2.3 Time versus Frequency Domain Analysis
Problems involving nonlinear material behavior can be solved in either(1) the time domain or (2) the frequency domain by using equivalent linearmaterial property approximations for the nonlinear material(s). The one-dimensional computer programs DESRA, CHARSOIL, DYNAID, and SUMDES and the two-dimensional programs TARA, DYNA-FLOW, and DYNA2D are examples of the timedomain procedure. The one-dimensional computer program SHAKE and the two-dimensional programs FLUSH and SuperFLUSH are examples of the frequency domainsolutions.
Frequency-domain techniques formerly favored owing to greatercomputational efficiency. However, the growth in theI power of relativelyinexpensive computers has diminished this advantage.
I) ý
D.2.4 I-D versus 2-D versus 3-D
Today it is, in principle possible to model the three-dimensionalaspects of soil response problems, but seldom is the effort justified. Inmany cases the responses of a soil profile can be modeled satisfactorily usingone-dimensional programs such as SHAKE, CHARSOIL, or DESRA. For most problemsinvolving retaining structures, a 2-D analysis (such as TARA, DYNAFLOW,DYNA2D, or DSAGE) will be necessary. The code FLUSH approximates some aspectsof 3-D response.
D.2.5 Nature of Input Ground Motion
Typically analyses use the idealization that the patterns of groundmotion are simple mechanisms; the most common procedures u'e verticallypropagating shear waves or dilatation waves. While it is possible to considermore general forms of input with horizontally viaveling waves, seldom willsuch an effort he warranted for waterfront structures.
D.2.6 Effect of Free Water
Consider the proLlem of a complete soil-structure interaction analysisof the earth retoining structure .hown in Figure D.la. The finite elementmesh used to model this problem includes the retaining structure, the soilbackfill and the pool of water in front of the wall, as shown in Figure D.lb.Th- mass and stiffness effects are included within the analysis for both thestructure and the soil backfill by incorporating these regimes within thefinite element mesh that is used to model the problem. Most computer codes donot include within their formulation a water element among their catalog offinite elements, so the Westergaard (1931) added water mass procedure is usedto account for the effect of the hydrodynamic water pressures on the dynamicresponse of the retaining wall (see Appendix B). One computer code that doesinclude a fluid element within its catalog of elements is SuperFLUSH.
D.3 A Final Perspective
The preparation time for developing the finite element mesh, assigningmaterial properties, selecting the ground motion, performing the analysis, andinterpreting the computed results is much greater than the time required forperforming a simplified analysis. However, the information provided by adynamic finite element analysis is much more complete and extensive. Thecomputed results include: the variation in computed accelerations with timeand the variation in computed dynamic normal and shear stresses with timethroughout the wall and the soil regime(s). Thus, a complete soil-structureinteraction analysis, when done properly, provides much more accurate anddetailed information regarding the dynamic behavior of the earth retainingstructure being analyzed.
In a complete soil-structure interaction analysis, the total earthpressures along the back of the wall at any time during the earthquake areequal to the sum of the computed dynamic earth pressures and the static earthand water pressures. At any elevation along the back of the wall, theeffective stress component (static + dynamic) of this total pressure isrestricted to range in values between the static active earth pressure valueand the static passive earth pressure value. Exceedence of these values mayoccur where in actuality separation may occur during earthquake shaking.
The potential for liquefaction within the submerged soils comprising thebackfill may be computed using the equivalent values for the induced shearstresses form the results of the complete soil-structure interaction analysis.The residual excess pore water pressures are then computed using the proceduredescribed in Seed and Harder (1990) or Marcuson, Hynes, and Franklin (1990).
D5
APPENDIX E: NOTATION
GREEK LETTER SYMBOLS
S(alpha) Inclination from horizontal of a planar slip surfaceextending upward through the backfill
"A (alpha) Inclination from horizontal of a planar slip surfaceextending upward through the backfill, static active case
"AE (alpha) Inclination from horizontal of a planar slip surfaceextending upward through the backfill, dynamic active case
ap (alpha) Inclination from horizontal of a planar slip surfaceextending upward through the backfill, static passive case
apE (alpha) Inclination from horizontal of a planar slip surfaceextending upward through the backfill, dynamic passive case
fi (beta) Inclination of backfill from horizontal
(beta) Inclination of backfill from horizontal, used in the
equivalent static procedure for computing KA and KpE
(delta) Effective angle of interface friction between the soil andthe structure
6b (delta) Effective angle of interface friction between the base of
the wall aud its foundation
Ah (delta) Change in total head
AKA (delta) Incremental dynamic active earth pressure coefficient
a'.t (sigma) Effective weight of backfill, excluding surcharge
7 (tau) Shear stress
rf (tau) Shear stress at failure
a (theta) Inclination of the back of wall to soil interface fromvertical
E2
0* (theta) Inclination of the back of the wall to soil interface from
vertical, used in the equivalent static procedure forcomputing KA and KpE
ROMAN LETTER SYMBOLS
A Maximum ground acceleration as a fraction of g (dimensionless)
ah Maximum horizontal ground acceleration, equal to khg
amax Maximum ground acceleration, equal to A'g
a, Maximum vertical ground acceleration, equal to k,-g
B Width of wall base
Be Effective base width of the wall in contact with the foundation
c Effective cohesion
cl Constant used to compute CA
c2 Constant used to compute CA
c 3 Constant used to compute ap
C4 Constant used to compute ap
C 1A Constant used to compute aA
C2A Constant used to compute a.A
C3PE Constant used to compute apE
C4PE Constant used to compute apE
dr Maximum displacement
FAE Factor used in the equivalent static procedure to compute KA
Fb Factor of safety against bearing capacity failure of a wall
FPE Factor used in the equivalent static procedure to compute KPE
Fr Factor of safety against sliding along the base of a wall
Fsr Lateral seismic force component by Woods procedure
FSp Factor of safety applied to both the shear strength of the soil and theeffective angle of friction along the interface when computing PPE fora sheet pile wall and the anchorage.
g Acceleration of gravity
H Height of wall
E3
h Total head
h. Elevation head
hp Pressure head
HFstatic Static component of heavy fluid force behind a wall retaining liquefiedbackfill
HFinertia Inertial component of heavy fluid force behind a wall retainingliquefied backfill during shaking
i Seepage gradient, equal to Ah/Al
KA Static active earth pressure coefficient
KAE Dynamic active earth pressure coefficient
Kh Horizontal earth pressure coefficient
kh Horizontal seismic coefficient as a fraction of g (dimensionless)
kh* Limiting value for the horizontal seismic coefficient as a fraction ofg (dimensionless)
kh. Equivalent horizontal seismic coefficient as a fraction of g(dimensionless)
khel Equivalent horizontal seismic coefficient as a fraction of g(dimensionless) for the restrained water case with r, = 0
khe2 Equivalent horizontal seismic coefficient as a fraction of g(dimensionless) for the free water case with ru, = 0
khe3 Equivalent horizontal seismic coefficient as a fraction of g(dimensionless) for the restrained water case with r, > 0
khe 4 Equivalent horizontal seismic coefficient as a fraction of g(dimensionless) for the free water case with ru > 0
KI, Static passive earth pressure coefficient
KPE Dynamic passive earth pressure coefficient
k, Vertical seismic coefficient as a fraction of g (dimensionless)
K. At-rest horizontal earth pressure coefficient
Mdesign Design moment for a sheet pile wall
MFES Maximum moment computed using the Free Earth Support method for a sheetpile wall
N Total normal force between the wall and the foundation
E4
N' Effective normal force between the wall and the foundation
N Maximum transmissible acceleration coefficient, as a fraction of g(dimensionless)
P Resultant earth pressure force acting on a wall
PA Active earth pressure force acting on a wall for static loading
PA Active earth pressure force acting on a wall for pseudo-static loading
PP Passive earth pressure force acting on a wall for static loading
PPE Passive earth pressure force acting on a wall for pseudo-static loading
Pwd Westergaard hydrodynamic water pressure force
q Vertical surcharge stress
qan1 allowable bearing pressure of rock
qmax maximum bearing pressure below toe of wall
qult ultimate bearing capacity or unconfined compressive strength ofconcrete
rd Moment reduction factor due to Rowe
r, Excess pore water pressure ratio, equal to 1u/0'v.•nitial
Su Undrained shear strength of soil
T Horizontal shear force along the base of the wall required forequilibrium
TdosiLn Design tie rod force for a sheet pile wall
TFES Tie rod force computed using the Free Earth Support method for a sheetpile wall
Tult Ultimate horizontal shear force along the base of the wall
Tulta Ultimate force for which the sheet pile wall anchorage is to bedesigned
Ub Resultant steady state pore water pressure force normal to the base ofthe wall
Uinertia Hydrodynamic water pressure force for the pool, directed away from thewall
UPOCI Resultant hydrostatic water pressure force for the pool
E5
Ushear Resultant excess pore water pressure force due to earthquake shakingacting normal to the backfill to wall interface
Ushear-b Resultant excess pore water pressure force due to earthquake shakingacting normal to the backfill to sheet pile wall interface
Ushear-t Resultant excess pore water pressure force due to earthquake shakingacting normal to the dredge level soil to sheet pile wall interface
Ushear-a Resultant excess pore water pressure force due to earthquake shakingacting normal to planar slip surface inclined at a from vertical
Ustatic Resultant steady state pore water pressure force acting normal to thebackfill to wall interface
Ustatic-b Resultant steady state pore water pressure force acting normal to thebackfill to sheet pile wall interface
Ustatict Resultant steady state pore water pressure force acting normal to thedredge level soil to sheet pile wall .,terface
Ustatic-a Resultant steady state pore water pressure force acting normal toplanar slip surface inclined at a from vertical
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WISS, JANNEY, ELSTNER, & ASSOC / DW PFEIFER, NORTHBROOK, ILWISWELL, INC. / SOUTHPORT, CTWOODWARD-CLYDE CONSULTANTS / R. CROSS, OAKLAND, CA; WEST REG, LIB,
OAKLAND, CA
DISTRIBUTION QUESTIONNAIREThe Naval Civil Engineering Laboratory is revising Its primary distribution lists.
SUBJECT CATEGORIES
1 SHORE FACILITIES 3D Alternate energy source (geothermal power, photovoltaic1A Construction methods and materials (including corrosion power systems, solar systems, wind systems, energy
control, coatings) storage systems)1B Waterfront structures (maintenance/deterioration control) 3E Site data and systems integration (energy resource data,1 C Utilities (including power conditioning) integrating energy systems)10 Explosives safety 3F EMCS design1 E Aviation Engineering Test Facilities 4 ENVIRONMENTAL PROTECTION1 F Fire prevention and control 4A Solid waste management1G Antenna technology 46 HazardousAoxic materials management1 H Structural analysis and design (including numerical and 4C Waterwaste management and sanitary engineering
computer techniques) 40 Oil pollution removal and recovery1J Protective construction (including hardened shelters, shock 4E Air pollution
and vibration studies) 4F Noise abatement1K Soil/rock mechanics 5 OCEAN ENGINEERING1L Airfields and pavements 5A Seafloor soils and foundations1M Physical security 5B Seafloor construction systems and operations (including2 ADVANCED BASE AND AMPHIBIOUS FACILITIES diver and manipulator tools)2A Base facilities (including shelters, power generation, water 5C Undersea structures and materials
supplies) 5D Anchors and moorings28 Expedient roads/airfields/bridges 5E Undersea power systems, electromechanical cables, an-2C Over-the-beach operations (including breakwaters, wave connectors
forces) 5F Pressure vessel facilities2D POL storage, transfer, and distribution 5G Physical environment (including site surveying)2E Polar engineering 5H Ocean-based concrete structures3 ENERGY/POWER GENERATION 5J Hyperbaric chambers3A Thermal conservation (thermal engineering of buildings, 5K Undersea cable dynamics
HVAC systems, energy loss measurement, power ARMY FEAPgeneration) BOG Shore Facilities
3B Controls and electrical conservation (electrical systems, NRG Energyenergy mondcng &id control systems) ENV Environmental/Natural Responses
3C Fuel flexibility (liquid fuels, coal utilization, energy from solid MGT Managementwaste) PRR Pavements/Railroads
TYPES OF DOCUMENTS
D - Techdata Sheets; R - Technical Reports and Technical Notes; G - NCEL Guides and Abstracts; I - Index to TOS. U - UserGuides; 0 None - remove my name
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DEPARTMENT OF THE NAVYNaval Civil Engineering Laboratory560 Laboratory DrivePort Hueneme CA 93043-4328
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We at NCEL want to provide you our customer the best possible reports but we need your help. Therefore, I ask youto please take the tim e from your busy schedule to fill out this questionnaire. Your response will assist us in providingthe best reports possible for our users. I wish to thank you in advance for your assistance. I assure you that theinformation you provide will help us to be more responsive to your future needs.
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improvements in the cost and or 7. The graphics, tables, and photo- ( ) ( ) ( ) (performance of my operation. graphs are well done.
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DEPARTMENT OF THE NAVYNaval Civil Engineering Laboratory560 Laboratory DrivePort Hueneme CA 93043-4328
NO POS'TAGEOfficial Business NECESSARYPenalty for Private Use, S300 IF MAILEDIN THEBUSINESS REPLY CARD UWDSAE
FIRST CLASS PERMIT NO. 12503 WASH D.C.
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COMMANDING OFFICERCODE L03560 LABORATORY DRIVENAVAL CIVIL ENGINEERING LABORATORYPORTHUENEME CA 93043-4328
