lateral force induced by rectangular surcharge loads on a...

Lateral Force Induced by Rectangular Surcharge Loadson a Cross-Anisotropic Backfill

Cheng-Der Wang1

Abstract: This article presents approximate but analytical-based solutions for computing the lateral force �force per unit length� andcentroid location induced by horizontal and vertical surcharge surface loads resting on a cross-anisotropic backfill. The surcharge loadingtypes include: point load, finite line load, and uniform rectangular area load. The planes of cross-anisotropy are assumed to be parallel tothe ground surface of the backfill. Although the presented solutions have never been proposed in existing literature, they can be derivedby integrating the lateral stress solutions recently addressed by the author. It is clear that the type and degree of geomaterial anisotropy,loading distances from the retaining wall, and loading types significantly influence the derived solutions. An example is given for practicalapplications to illustrate the type and degree of soil anisotropy, as well as the loading types on the lateral force and centroid location inthe isotropic/cross-anisotropic backfills caused by the horizontal and vertical uniform rectangular area loads. The results show that boththe lateral force and centroid location in a cross-anisotropic backfill are quite different from those in an isotropic one. The derivedsolutions can be added to other lateral pressures, such as earth or water pressure, which are necessary in the stability and structuralanalysis of a retaining wall. In addition, they can be utilized to simulate more realistic conditions than the surcharge strip loading ingeotechnical engineering for the backfill geomaterials are cross-anisotropic.

DOI: 10.1061/�ASCE�1090-0241�2007�133:10�1259�

CE Database subject headings: Lateral stress; Lateral loads; Vertical loads; Horizontal loads; Anisotropy; Backfills.

Introduction

Retaining structures usually carry earth pressure, water pressure,and frequently, surcharge pressure on their backfills. Surchargeloads might be anything from truck wheels, railway tracks, high-way pavements, or to the foundations of adjacent buildings. Theseloading types could be modeled by point loads, line loads, striploads, or area loads. It was found from a series of experimentsperformed at the Iowa Engineering Experiment Station �Spangler1936; Spangler 1938a; Spangler 1938b; Spangler and Mickle1956; Spangler and Handy 1982�, and using the theory of elastic-ity method �Boussinesq 1885; Mindlin 1936� that when surchargeloads on the backfill were close enough to the retaining wall,additional lateral stress was produced. Traditionally, backfill ma-terial has been assumed to be a homogeneous, linearly elastic, andisotropic continuum. Nevertheless, various investigators �i.e.,Michell 1900; Barden 1963; Pickering 1970; Gerrard and Wardle1973; Gibson 1974; Gazetas 1982; Wang 2003; Abelev and Lade2004� recognize the fact that many natural soils are deposited bygeological sedimentation over a long period of time; therefore,they are not generally isotropic materials, but rather anisotropic,

1Associate Professor, Dept. of Civil and Disaster PreventionEngineering, National United Univ., No. 1, Lien-Da, Kung-Ching-Li,Miao-Li, 360, Taiwan, ROC. E-mail: [email protected] [email protected]

Note. Discussion open until March 1, 2008. Separate discussions mustbe submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on October 10, 2005; approved on May 9, 2007. This paper ispart of the Journal of Geotechnical and Geoenvironmental Engineer-ing, Vol. 133, No. 10, October 1, 2007. ©ASCE, ISSN 1090-0241/2007/
10-1259–1276/$25.00.
JOURNAL OF GEOTECHNICAL AND GEO

since the properties in the horizontal and vertical planes are dif-ferent. That is, better results can be obtained by considering theanisotropic deformability. In this work, lateral force and its cen-troid location induced by three surcharge loads on a cross-anisotropic backfill are addressed.

Recently, Wang �2005; 2007� presented approximate butanalytical-based solutions for calculating the lateral stress �Wang2005�, lateral force, and centroid location �Wang 2007� due tohorizontal and vertical surcharge surface strip loads acting on across-anisotropic backfill. These surcharge loading types includeda horizontal/vertical point load, a horizontal/vertical infinite lineload, a horizontal/vertical uniform strip load, a horizontal/verticalupward linear-varying strip load, a horizontal/vertical upwardnonlinear-varying strip load, a horizontal/vertical downwardlinear-varying strip load, and a horizontal/vertical downwardnonlinear-varying strip load. However, in the case of a backfillresulting from an arbitrarily shaped load, it is well known that thearea could be divided into several regularly shaped subareas, suchas rectangles or triangles. Therefore, the strip loading solution forthe induced lateral stress, lateral force, and centroid locationmight not be suitable. For this reason, Wang �unpublished mate-rial, 2006� gave the lateral stress solution for a cross-anisotropicbackfill subjected to the aforementioned loadings, but distributedover a rectangular region. A part of Wang’s solutions �unpub-lished material, 2006�, lists loading Case A: A horizontal/verticalpoint load �P/Q, force�; Case B: A horizontal/vertical finite lineload �Pl /Ql, force per unit length�; Case C: A horizontal/verticaluniform rectangular area load �Ps /Qs, force per unit area� withrespect to the lateral stress solution is provided in Appendix I.Although the presentation of the lateral stress solution is clear andconcise, they cannot easily applied to calculate the lateral force�force per unit length�, since the computation of an irregular lat-
eral stress area behind a retaining wall is not as simple as esti-
ENVIRONMENTAL ENGINEERING © ASCE / OCTOBER 2007 / 1259

mating an earth or water pressure area. Hence, Jarquio �1981�derived the exact solution for determining the lateral force andcentroid location �measuring from the wall top� for unyieldingwalls owing to a vertical uniform strip load exerting on an isotro-pic backfill. Then, Steenfelt and Hansen �1983� suggested that theelastic solution based on Bousssinesq’s half-space �1885� wasonly reasonable for unyielding structures �Georgiadis and Anag-nostopoulos 1998�. Lately, Yildiz �2003� investigated the lateralpressures on unyielding walls due to surface strip loads by con-sidering the nonlinear stress-strain behavior of the soil using thecommercial finite element code, PLAXIS. Data obtained from thefinite element analyses were used to train neural networks inorder to acquire a solution to assess the lateral force and its pointof application. To the best of the author’s knowledge, no analyti-cal solutions have been proposed for the lateral force and centroidlocation generated by loading Cases A–C for a homogeneous,linearly elastic, and cross-anisotropic backfill. In deriving the pre-sented solutions, the retaining wall is taken to be vertical with thehorizontal backfill, meaning that the planes of cross-anisotropyare parallel to the horizontal ground surface. Moreover, two sim-plifying assumptions are made in this work: �1� the wall does notmove; and �2� the wall is perfectly smooth �there is no shearstress between the wall and the soil�. These two assumptionswould imply that the induced lateral stress on a retaining wall isthe same as the horizontal stress in an elastic half-space inducedby two loads of equal magnitude acting on the surface �Fang1991�. However, in a real situation, a large stress concentrationmight be developed around the lower corner of the retaining wallin contact with the backfill. In addition, the theory of elasticityutilized in this study does not consider the strength and the varia-tion of the stiffness of the soil with different stress states. Also,the assumption of a perfectly smooth wall is restrictive, andwould be limited to the applicability of the elasticity method inpractical applications. Nevertheless, as mentioned above, a seriesof experiments conducted at the Iowa Engineering ExperimentStation �Spangler 1936; Spangler 1938a; Spangler 1938b;Spangler and Mickle 1956; Spangler and Handy 1982� and byTerzaghi �1954�, confirmed the fact that doubling the horizontalstress in an elastic half-space could provide a good approximationto measured values of earth pressures on retaining walls �Fang1991�.

Another interesting and alternative approach by Constantinouand Gazetas �1986� by using a systematic relaxation-of-constraints technique of Hetenyi �1960, 1970�, a plane-strain so-lution was derived for stresses in an elastic orthotropic quarterplane loaded by a vertical line load located at a distance from theedge. Further, Constantinou and Greenleaf �1987� yielded the so-lutions of stresses in the same plane due to a horizontal line loadat a distance from the edge, a vertical uniform strip load at theedge, and at a distance from the edge. Their solutions were alsospecialized to the case of cross-anisotropic material with both realand complex-valued roots of the characteristic relation.

In this article, integrating the lateral stress of loading CasesA–C �Appendix I� with respect to z direction could obtain theclosed-form solutions for the lateral force and centroid locationinduced by the horizontal and vertical surcharge loads �point load,finite line load, and uniform rectangular area load� on a cross-anisotropic backfill. The proposed approximate but analytical-based solutions could not only complete the full analysis of aretaining wall structure with a cross-anisotropic backfill subjectedto a horizontal/vertical uniform rectangular load, but could alsobe extended to calculate the lateral force and centroid location
due to an arbitrarily shaped loaded area by superposition. These
1260 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGIN

solutions indicate that the type and degree of geomaterial aniso-tropy, loading distances from the retaining walls, dimensions ofthe loading area, and horizontal/vertical loading types deeply im-pact the induced lateral force and centroid location. An example isgiven for practical applications to illustrate the type and degree ofsoil anisotropy, and loading types on the lateral force and centroidlocation in the isotropic/cross-anisotropic backfills caused by ahorizontal/vertical uniform rectangular area load.

Lateral Stress Induced by a Horizontal/Vertical PointLoad, Finite Line Load, and Uniform RectangularArea Load

Point load solutions in exact closed-form have always played animportant role in applied mechanics. For the displacements andstresses in cross-anisotropic media subjected to a point load, ana-lytical solutions have been presented by numerous investigators�i.e., as shown in the classical textbook of Lekhnitskii �1963�;Liao and Wang 1998; Wang and Liao 1999�. In this article, thesolutions for lateral stress due to a horizontal/vertical surchargepoint load, finite line load, and uniform rectangular area load on across-anisotropic backfill were derived by Wang �unpublishedmaterial, 2006� by integrating the point load solution of Wang andLiao �1999�. A detailed deriving approach for solving the lateralstress, �h

p, subjected to a horizontal surface point load P, and avertical one, Q �loading Case A�, located at a horizontal distancein the x-axis from the retaining wall, a, and at a horizontal dis-tance in the y-axis from the retaining wall, c �x=a, y=c, z=0�,with height H, as depicted in Fig. 1�a�, can be referred to Wang�unpublished material, 2006�. In addition, integrating �h

p for asmooth, rigid retaining wall could produce the approximate butanalytical-based solutions for lateral stress, �h

l and �hu, subjected

to loading Cases B and C, respectively. A summary of the lateralstress solutions, including �h

p, �hl , and �h

u, is given in Appendix I.In Appendix I, ps1i-ps4i �Eqs. �1�–�4��, ds1i-ds4i �Eqs. �5�–�8��, andes1i-es4i �Eqs. �9�–�12�� i=1, 2, and 3� are defined as the stresselementary functions. Hence, the lateral force and centroid loca-tion solutions could be yielded by integrating the stress elemen-tary functions. The deriving procedures are as follows.

Lateral Force Induced by a Horizontal/Vertical PointLoad, Finite Line Load, and Uniform RectangularArea Load

Retaining walls support backfill earth pressure, water pressure,and often the surcharge pressure in the field of geotechnical en-gineering. Practically, it is easy to compute the resultant force andlocation of the resultant force induced by the earth and waterpressures. Nevertheless, from a previous study on lateral stress�Wang, unpublished material, 2006�, it seems to be difficult tocalculate an irregular lateral stress area as lateral stress solutionsare influenced by several factors, such as the type and degree ofgeomaterial anisotropy, loading distances from the retainingwalls, dimensions of the loading area, and different horizontal/vertical loading types. Hence, analytical solutions for lateral forceand its centroid location generated by the presented loading CasesA–C are necessary.

In this work, lateral force and centroid location could be di-
rectly obtained by integrating the stress elementary functions of
EERING © ASCE / OCTOBER 2007

each lateral stress solution �Appendix I�. For example, the com-plete solution of lateral force �force per unit length�, Ph

p, inducedby a horizontal/vertical point load, as shown in Fig. 1�a�, could bederived by integrating the stress elementary functions, ps1i-ps4i

�i=1, 2, and 3� �Eqs. �1�–�4� in Appendix I�, in the z directionbetween limits 0 and H �height of the retaining wall�. That is, theexpression of the lateral force, Ph

p, is the same as that of �hp,

except that ps1i-ps4i �i=1, 2, and 3� in Eqs. �1�–�4� should be

Fig. 1. Lateral force and centroid location induced by three types ofhorizontal/vertical surcharge loads on a cross-anisotropic backfill: �a�loading Case A: point load case; �b� loading Case B: finite line loadcase; and �c� loading Case C: uniform rectangular area load case

replaced by force integral functions, pt1i-pt4i �i=1, 2, and 3�,


respectively. The lateral force, Php, and related force integral func-

tions, pt1i-pt4i �i=1, 2, and 3� for loading Case A are presented inAppendix II. Similarly, solutions for the lateral force, Ph

l �finiteline load, Fig. 1�b�� and Ph

u �uniform rectangular area load, Fig.1�c��, and their related force integral functions for loading CasesB and C could be obtained. They are also presented in AppendixII. However, observing Appendix II, the force integral functions,et1i �Eq. �21��, et2i �Eq. �22��, et3i �Eq. �23��, and et4i �Eq. �24��i=1, 2, and 3� for loading Case C are all functions ofk1i=a /uiH, k2i= l /uiH, k3i=c /uiH, and k4i=w /uiH �i=1, 2, and 3�.Thus, if the five elastic engineering constants E, E�, �, ��, and G�are given, then the characteristic root ui �i=1, 2, and 3� can becalculated using the characteristic equation listed in Appendix I.Figs. 2–5 show the calculation charts of et1i �Eq. �21��, et2i �Eq.�22��, et3i �Eq. �23��, and et4i �Eq. �24�� for computing the lateralforce induced by loading Case C, respectively. In these figures, k1i

and k3i are equal to 0, that means the horizontal/vertical uniformrectangular area load is acting nearby the retaining structures�a=c=0�. Consequently, for the rest of variable nondimensionalfactors, k2i and k4i, in which k2i is in the range 0.01–10, and k4i isin the range 0.4–6. These four calculation charts could be utilizedto estimate the induced lateral force by a horizontal/vertical uni-form rectangular load in a conservative manner when computersor calculators are unavailable, but they are only suitable for theroot type of the characteristic equation �Appendix I� belonging toCase 1 �i.e., where u1 and u2 are two real distinct roots�. In otherwords, the proposed figures �Figs. 2–5� cannot be applied to cal-culate the lateral force for loading Case C when the root typebelongs to Case 2 and Case 3 �Appendix I�.

Centroid Location Induced by a Horizontal/VerticalPoint Load, Finite Line Load, and UniformRectangular Area Load

Since the location of resultant lateral force is significant for thestability and structural analysis of a retaining wall, the centroidlocation solution for loading Cases A–C are derived and presentedin Appendix III. For instance, the centroid location z̄h

p measuringfrom the top of a wall, induced by a horizontal point load�namely, the vertical-direction point force Q=0� could be ac-quired by multiplying z with �h

p �lateral stress resulting from ahorizontal point load, as shown in Appendix I�, and integratingwith the limits of z from 0 to H, then dividing by the yieldedlateral force, Ph

p �when Q=0, as expressed in Appendix II�. Thesame process holds for the other centroid location, z̄v

p, when thehorizontal-direction point force P=0. Repeatedly, in accordancewith Appendix I, the stress elementary functions, ps1i-ps4i �i=1, 2,and 3� �Eqs. �1�–�4�� in �h

p are integrated and limits are evaluated.They are then defined as centroid integral functions, pz1i-pz4i

�i=1, 2, and 3�. Hence, only centroid location solutions andrelated centroid integral functions for loading Cases A–C are pre-sented in Appendix III. According to Appendix III, another set ofcalculation charts could be drawn from the centroid integral func-tions, ez1i �Eq. �33��, ez2i �Eq. �34��, ez3i �Eq. �35��, and ez4i �Eq.�36�� i=1, 2, and 3� for computing the centroid location inducedby loading Case C. They are plotted, respectively, in Figs. 6–9.

Fig. 10 demonstrates a flow chart that illustrates the use of theeight calculation charts �et1i �Fig. 2�, et2i �Fig. 3�, et3i �Fig. 4�, et4i

�Fig. 5�, and ez1i �Fig. 6�, ez2i �Fig. 7�, ez3i �Fig. 8�, and ez4i �Fig.


9�� for computing the lateral force and centroid location inducedby the horizontal and vertical uniform surcharge rectangular arealoads �loading Case C�. Although these charts are given for fiveelastic engineering constants of a cross-anisotropic backfill be-longing to Case 1, they can be an alternative tool for supplyingresults with reasonable accuracy.

The use of the presented formulae could save users time incalculating the lateral force and centroid location owing to sur-

Fig. 2. Calculation charts of et1i �i=1, 2, and 3� from Eq. �21� forcomputing the lateral force induced by loading Case C



charge surface loads involving a horizontal/vertical point load,finite line load, and uniform rectangular load, resting on a cross-anisotropic backfill. Moreover, the derived solutions are identicalto Wang’s horizontal and vertical strip loading solutions �Wang2007� for a cross-anisotropic backfill in the case of c=0, andw=�. Additionally, they are in very good agreement with theisotropic solutions of Jarquio �1981� by using a limiting approachfor the vertical uniform strip loading case �c=0 and w=��.




Practical Applications

An example for computing the lateral force and centroid locationinduced by a horizontal/vertical uniform rectangular load is illus-trated in this section. The plane of a horizontal/vertical loadedarea EFGH with a uniform intensity Ps /Qs acting on the surfaceof an isotropic and cross-anisotropic backfill is shown in Fig. 11.Seven types of isotropic �Soil 1� and cross-anisotropic soils �Soils2–7� are considered as the constituted backfill geomaterials. Theinfluence of the degree of soil anisotropy, specified by the ratiosE /E�, � /��, and G /G� on the lateral force and centroid location is

Fig. 6. Calculation charts of ez1i �i=1, 2, and 3� from Eq. �33� forcomputing the centroid location induced by loading Case C



examined. Table 1 lists their elastic properties and root type. Theselected domains of anisotropic variation basically follow thesuggestion of Gazetas �1982� with E /E� ranging from 0.6 to 4 forclays, and as low as 0.2 for sands. Hence, they are hypotheticallyassumed that � /�� varying between 0.75–1.5, and E /E� andG /G� ranging between 0.15–1.5. The values of E and � adoptedin Table 1 are 50 MPa and 0.3, respectively.

Based on Appendixes II and III, a Mathematica program waswritten to compute the lateral force and centroid location due tothe proposed loading cases. However, in this section, a loaded




area as depicted in Fig. 11 is illustrated as a practical example. InFig. 11, a�horizontal and vertical uniform rectangular loads ap-plied at a horizontal distance in the x-axis from the retaining wall,and l�length of the rectangular load. The normalized lateralforce, Ph

u / Ps and Phu /Qs, and centroid location measuring from the

top of the retaining wall, zhu and zv

u, for Soils 1–7 are calculatedand plotted, respectively, in Figs. 12–15.

Fig. 12 presents the normalized lateral force Phu / Ps versus a/l

�from 0 to 5� for Soils 1, 2, and 3 �Fig. 12�a��, Soils 1, 4, and 5�Fig. 12�b��, and Soils 1, 6, and 7 �Fig. 12�c�� resulting from ahorizontal uniform rectangular loaded area of Fig. 11. The varia-tions of Ph

u / Ps for Soils 1–7 are within the 0–0.16 range, andexcept for Soils 2 and 6, they are slightly influenced by the typeand degree of geomaterial anisotropy.

Fig. 13 depicts the effect of a/l generated by a vertical uniformrectangular load �Fig. 11� on the normalized lateral force Ph

u /Qs

for Soils 1, 2, and 3 �Fig. 13�a��, Soils 1, 4, and 5 �Fig. 13�b��, andSoils 1, 6, and 7 �Fig. 13�c��. Clearly, the lateral force �Ph

u /Qs�,especially for Soils 2 �Fig. 13�a�� and 6 �Fig. 13�c��, is markedlyaffected by the type and degree of backfill anisotropy.

Fig. 14 exhibits the centroid location zhu �measuring from the

wall top�, for Soils 1–7 caused by a horizontal uniform rectangu-lar loaded area of Fig. 11. The variations of zh

u for Soils 1–7 arewithin −0.5–2.5. Again, they are slightly impacted by the type

Fig. 10. Flow chart for computing lateral force and centroid locationinduced by the horizontal and vertical uniform surcharge rectangulararea loads using the calculation charts

and degree of material anisotropy except for Soils 2 and 6. In


particular, within the smaller region of a/l �i.e., 0–0.5�, zhu could be

a negative value in the Soil 6 backfill.Fig. 15 displays the centroid location zv

u �measuring from thewall top�, for Soils 1–7 produced by a vertical uniform rectangu-lar load �Fig. 11�. In this figure, a significant phenomenon can befound in Fig. 15�c� that the ratio of G /G� �=0.15,1.5� do have agreat effect on zv

u. Also, within the smaller range of a/l �i.e.,0–0.5�, zv

u could be a negative value in the Soil 7 backfill.As the illustrative example demonstrates, the parametric study

results verify that the lateral force and centroid location inisotropic/cross-anisotropic backfills generated by a horizontal/vertical uniform rectangular load can be computed easily and cor-rectly by the proposed approximate but analytical-based solu-tions. In addition, Figs. 12–15 illuminate the effects of the soilanisotropy �Soils 2–7�, loading distance from the retaining wall�a�, dimension of the loading length �l�, and loading directions�horizontal and vertical� on the lateral force and centroid location.Hence, when computing the lateral force and centroid locationdue to the applied surcharge rectangular or irregularly shaped arealoads resting on a cross-anisotropic backfill, the anisotropic de-formability, loading distances and dimensions, and different load-ing types should be taken into account.

Conclusions

The closed-form solutions derived by Wang �unpublished mate-rial, 2006� for the lateral stress induced by a horizontal/vertical

Table 1. Elastic Properties and Root Type for the Isotropic/Cross-Anisotropic Soils

Soil type E /E� � /�� G /G� Root type

1. Isotropy 1 1 1 Equal

2. Cross-anisotropy 0.15 1 1 Distinct

3. Cross-anisotropy 1.5 1 1 Complex

4. Cross-anisotropy 1 0.75 1 Complex

5. Cross-anisotropy 1 1.5 1 Distinct

6. Cross-anisotropy 1 1 0.15 Complex

7. Cross-anisotropy 1 1 1.5 Distinct

Fig. 11. The plane of a horizontal/vertical uniform surcharge loadedarea EFGH acting on the surface of the isotropic/cross-anisotropicbackfills

Note: E=50 MPa and �=0.3 are adopted.


point load, finite line load, and uniform rectangular area load,exerting on a cross-anisotropic backfill can be expressed in termsof stress elementary functions. Integration of the stress elemen-tary functions, solutions of lateral force and centroid location’sintegral functions can be yielded. These integral functions can beapplied to construct the calculation charts for computing the lat-eral force and centroid location resulting from the horizontal/vertical uniform rectangular load when computers or calculatorsare unavailable. Nevertheless, these charts �Figs. 2–9� are onlysuitable for the root type of the characteristic equation belongingto Case 1 �i.e., where u1 and u2 are two real distinct roots�.

Fig. 12. Effect of a / l on lateral force caused by a horizontal uniformand 5; and �c� soils 1, 6, and 7

Based on the results of a parametric study by a practical ex-


ample, it is found that the lateral force and centroid location areboth intensely affected by the type and degree of soil anisotropy�Soils 1–7�, as well as different loading types �horizontal andvertical�. The calculation of the induced lateral force and centroidlocation by a horizontal/vertical point load, finite line load, anduniform rectangular load in an isotropic/cross-anisotropic backfillis fast and correct, since the presentation of the derived solutionsis clear and concise. These approximate but analytical-based so-lutions can be utilized to simulate more realistic conditions thanthe surcharge strip loading in geotechnical engineering for thebackfill geomaterials are cross-anisotropic. Additionally, the de-

ngular loaded area of Fig. 11 for: �a� soils 1, 2, and 3; �b� soils 1, 4,
recta
rived solutions can be added to other lateral pressures, such as


earth or water pressure, necessary for the stability and structuralanalysis of a retaining wall. The proposed formulae are identicalwith Wang’s horizontal and vertical uniform strip loading solu-tions �Wang 2007� for a cross-anisotropic backfill when c=0, andw=�. However, by limiting a procedure, they are also in goodagreement with Jarquio’s vertical uniform strip loading solutions�Jarquio 1981� for an isotropic backfill. Moreover, these solutionscan be extended to compute the lateral force and centroid locationsubjected to a horizontal/vertical uniform arbitrarily shaped areaload by superposition. Regarding the solutions of the lateral forceand centroid location resulting from a horizontal/vertical linear-varying and nonlinear-varying rectangular area loads can also be

Fig. 13. Effect of a / l on lateral force caused by a vertical uniform re5; and �c� soils 1, 6, and 7

explored. With these solutions, the lateral force and centroid lo-


cation owing to any conceivable irregular area and distributedload can be fully analyzed. The results of these investigations willbe addressed in forthcoming works.

Acknowledgments

The writer would like to thank the valuable comments of Editorand reviewers, and special thanks to the loving supports of hisfamily during this work.

lar loaded area of Fig. 11 for: �a� soils 1, 2, and 3; �b� soils 1, 4, and
ctangu

Fig. 14. Effect of a / l on centroid location caused by a horizontal uniform rectangular loaded area of Fig. 11 for: �a� soils 1, 2, and 3; �b� soils1, 4, and 5; and �c� soils 1, 6, and 7

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / OCTOBER 2007 / 1267

Fig. 15. Effect of a / l on centroid location caused by a vertical uniform rectangular loaded area of Fig. 11 for: �a� soils 1, 2, and 3; �b� soils 1,4, and 5; and �c� soils 1, 6, and 7

1268 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / OCTOBER 2007

Appendix I. Solutions of Lateral Stress due to Loading Cases A–C

Type ofloading Lateral stress solutions

Stress elementaryfunctions

Case A�h

p=2*P

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12* ps11−u2

2* ps12�−2A66� u1

m1+u1* ps21−

u2

m2+u2* ps22��+2u3* ps23�

+2*Q

2�

k�m2u1−m1u2�

u1−u2�A44�u1* ps31−u2* ps32�−2A66� 1

m1+u 1�ps31− ps41�−1

m2+u2�ps32− ps42��

Eqs. �1�–�4�

Case B�h

l =2*Pl

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*ds11−u2

2*ds12�−2A66� u1

m1+u1*ds21−

u2

m2+u2*ds22��+2u3*ds23�

+2*Ql

2�

k�m2u1−m1u2�

u1−u2�A44�u1*ds31−u2*ds32�−2A66� 1

m1+u 1�ds31−ds41�−1

m2+u2�ds32−ds42��

Eqs. �5�–�8�

Case C�h

u=2*Ps

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*es11−u2

2*es12�−2A66� u1

m1+u1*es21−

u2

m2+u2*es22��+2u3*es23�

+2*Qs

2�

k�m2u1−m1u2�

u1−u2�A44�u1*es31−u2*es32�−2A66� 1

m1+u 1�es31−es41�−1

m2+u2�es32−es42��

Eqs. �9�–�12�

where

ps1i =x + a

Ri3 �1�

ps2i =x + a

Ri3 −

3�x + a�Ri�Ri + zi�2 +

�x + a�3�3Ri + zi�Ri

3�Ri + zi�3 �2�

ps3i =zi

Ri3 �3�

ps4i =1

Ri�Ri + zi�−

�x + a�2�2Ri + zi�Ri

3�Ri + zi�2 �4�

ds1i = −x + a

��x + a�2 + zi2�� c

��x + a�2 + c2 + zi2

−c + l

��x + a�2 + �c + l�2 + zi2 �5�

ds2i = − �x + a� c��x + a�2 + c2 + zi2 − zi�2

��x + a�2 + c2�2��x + a�2 + c2 + zi2

−�c + l��x + a�2 + �c + l�2 + zi

2 − zi�2

��x + a�2 + �c + l�2�2��x + a�2 + �c + l�2 + zi2� �6�

ds3i = −zi

��x + a�2 + zi2�� c

��x + a�2 + c2 + zi2

−c + l

��x + a�2 + �c + l�2 + zi2 �7�

ds4i =c

�x + a�2 + c2 −c + l

�x + a�2 + �c + l�2 −zi

�x + a�2 + zi2 c�2�x + a�2 + c2 + zi

2�

��x + a�2 + c2��x + a�2 + c2 + zi2

−�c + l��2�x + a�2 + �c + l�2 + zi

2�

��x + a�2 + �c + l�2��x + a�2 + �c + l�2 + zi2�

�8�

es1i = ln��n1i + n2i�2 + n3i2 + 1 + n3i

�n1i2 + n3i

2 + 1 + n3i

� − ln��n1i + n2i�2 + �n3i + n4i�2 + 1 + �n3i + n4i��n1i

2 + �n3i + n4i�2 + 1 + �n3i + n4i�� 9�


es2i = − n3i 1

1 + �n1i2 + n3i

2 + 1−

1

1 + ��n1i + n2i�2 + n3i2 + 1

�+ �n3i + n4i� 1

1 + �n1i2 + �n3i + n4i�2 + 1

−1

1 + ��n1i + n2i�2 + �n3i + n4i�2 + 1� �10�

es3i = tan−1 n1in3i

�n1i2 + n3i

2 + 1− tan−1 �n1i + n2i�n3i

��n1i + n2i�2 + n3i2 + 1

− tan−1 n1i�n3i + n4i��n1i

2 + �n3i + n4i�2 + 1+ tan−1 �n1i + n2i��n3i + n4i�

��n1i + n2i�2 + �n3i + n4i�2 + 1�11�

es4i = es3i + tan−1 n3i

n1i− tan−1 n3i

n1i + n2i− tan−1 n3i + n4i

n1i+ tan−1 n3i + n4i

n1i + n2i+ tan−1 n1i

n3i�n1i

2 + n3i2 + 1

− tan−1 n1i + n2i

n3i��n1i + n2i�2 + n3i

2 + 1�12�

− tan−1 n1i

�n3i + n4i��n1i2 + �n3i + n4i�2 + 1

+ tan−1 n1i + n2i

�n3i + n4i��n1i + n2i�2 + �n3i + n4i�2 + 1

where Ri=��x+a�2+ �y+c�2+zi2, n1i= a � zi , n2i= l � zi , n3i= c � zi , n4i= w � zi , zi=uiz �i=1, 2, and 3�.

• The definition of a, l, c, w, P /Q �force�, Pl /Ql �force per unit length�, and Ps /Qs �force per unit area� can be referred to Fig. 1 and theNotation section.

• x, y denote the desired horizontal positions; and z�vertical distance from point to load in a Cartesian coordinate system.• Aij �i , j=1–6� denote the elastic moduli or elasticity constants of the backfill, in which A11=E�1− �E /E��2� / �1+��1−�

− �2E /E��2�, A13=E�� /1−�− �2E /E��2, A33=E��1−�� /1−�− �2E /E��2, A44=G�, A66=E /2�1+��.• E represents the Young’s modulus in the horizontal direction, E� represents the Young’s modulus in the vertical direction, � represents

the Poisson’s ratio for the effect of horizontal stress on complementary horizontal strain, �� represents the Poisson’s ratio for the effectof vertical stress on horizontal strain, G� represents the shear modulus in the vertical plane.

• u3=�A66�A44 , u1 and u2 denote the roots of the characteristic equation: u4−su2+ t=0, in which s= A11A33−A13�A13+2A44��A33A44 ,

t= A11�A33 , and it can be categorized into three cases: �1� Case 1. u1,2= ±��1 � 2 �s±��s2−4t�� has two real distinct roots whens2−4t�0, �2� Case 2. u1,2= ±�s /2 , ±�s /2 has double equal real roots when s2−4t=0 �i.e., complete isotropy�, �3� Case 3. u1

= 1 � 2��s+2�t�− i1 � 2��−s+2�t�=− i, u2=+ i has two complex conjugate roots �where �0� when s2−4t�0, and• k= �A13+A44��A33A44�u1

2−u22� , mi= �A13+A44�ui� �A33ui

2−A44� = �A11−A44ui2�� A13+A44�ui �i=1,2�

Appendix II. Solutions of Lateral Force due to Loading Cases A–C

Type ofloading Lateral force solutions

Force integralfunctions

Case APh

p=2*P

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12* pt11−u2

2* pt12�−2A66� u1

m1+u1* pt21−

u2

m2+u2* pt22��+2u3* pt23�

+2*Q

2�

k�m2u1−m1u2�

u1−u2�A44�u1* pt31−u2* pt32�−2A66� 1

m1+u 1�pt31− pt41�−1

m2+u2�pt32− pt42��

Eqs. �13�–�16�

Case BPh

l =2*Pl

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*dt11−u2

2*dt12�−2A66� u1

m1+u1*dt21−

u2

m2+u2*dt22��+2u3*dt23�

+2*Ql

2�

k�m2u1−m1u2�

u1−u2�A44�u1*dt31−u2*dt32�−2A66� 1

m1+u 1�dt31−dt41�−1

m2+u2�dt32−dt42��

Eqs. �17�–�20�

Case CPh

u=2*Ps

2� �k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*et11−u2

2*et12�−2A66� u1

m1+u1*et21−

u2

m2+u2*et22��+2u3*et23�*H

+2*Qs

2�

k�m2u1−m1u2�

u1−u2�A44�u1*et31−u2*et32�−2A66� 1

m1+u 1�et31−et41�−1

m2+u2�et32−et42��*H

Eq. �21� for Fig. 2Eq. �22� for Fig. 3Eq. �23� for Fig. 4Eq. �24� for Fig. 5

where

pt1i =�x + a�H

2 2 2 2 2�13�

��x + a� + �y + c� ��x + a� + �y + c� + �uiH�


pt2i = − �x + a�H ��x + a�2 − 3�y + c�2��x + a�2 + �y + c�2�3 �uiH − ��x + a�2 + �y + c�2 + �uiH�2� −

�y + c�2

��x + a�2 + �y + c�2�2��x + a�2 + �y + c�2 + �uiH�2� �14�

pt3i =1

ui 1

��x + a�2 + �y + c�2−

1��x + a�2 + �y + c�2 + �uiH�2� �15�

pt4i =1

ui��x + a�2 + �y + c�2�3/2− uiH��x + a�2 − �y + c�2��x + a�2 + �y + c�2

+ �y + c�2� +��x + a�2 + �y + c�2 + �uiH�2

ui��x + a�2 + �y + c�2� �x + a�2 − �y + c�2

�x + a�2 + �y + c�2

−�x + a�2

�x + a�2 + �y + c�2 + �uiH�2� �16�

dt1i = −1

ui tan−1 cuiH

�x + a��x + a�2 + c2 + �uiH�2− tan−1 �c + w�uiH

�x + a��x + a�2 + �c + w�2 + �uiH�2� �17�

dt2i = − �x + a�H c

��x + a�2 + c2��x + a�2 + c2 + �uiH�2 + uiH�−

c + w

��x + a�2 + �c + w�2��x + a�2 + �c + w�2 + �uiH�2 + uiH�� 18�

dt3i =1

ui�ln� c + ��x + a�2 + c2 + �uiH�2

c + ��x + a�2 + c2 � − ln� c + w + ��x + a�2 + �c + w�2 + �uiH�2

c + w + ��x + a�2 + �c + w�2 � �19�

dt4i =c

ui��x + a�2 + c2 uiH

��x + a�2 + c2+ 1 −

��x + a�2 + c2 + �uiH�2

��x + a�2 + c2 � −c + w

ui��x + a�2 + �c + w�2 uiH

��x + a�2 + �c + w�2+ 1

−��x + a�2 + �c + w�2 + �uiH�2

��x + a�2 + �c + w�2 � +1

ui�ln� c + ��x + a�2 + c2 + �uiH�2


c + w + ��x + a�2 + �c + w�2 � �20�

et1i = k1i�T1i − T2i� − �k1i + k2i��T3i − T4i� + L1i − L2i − k3i�L3i − L4i� + �k3i + k4i��L5i − L6i� �21�

et2i = −k3i

2 �k1i

2 + k3i2 + 1

k1i2 + k3i

2 −��k1i + k2i�2 + k3i

2 + 1

�k1i + k2i�2 + k3i2 � +

�k3i + k4i�2

�k1i2 + �k3i + k4i�2 + 1

k1i2 + �k3i + k4i�2 −

��k1i + k2i�2 + �k3i + k4i�2 + 1

�k1i + k2i�2 + �k3i + k4i�2 �−

�2k1i + k2i�k2ik4i�k1i2 �k1i + k2i�2 − k3i�k3i + k4i��k1i

2 + �k1i + k2i�2 + k3i2 + �k3i + k4i��2k3i + k4i��

2�k1i2 + k3i

2 ��k1i + k2i�2 + k3i2 ��k1i

2 + �k3i + k4i�2��k1i + k2i�2 + �k3i + k4i�2�−

k3i

2�L3i − L4i� +

�k3i + k4i�2

�L5i − L6i�

�22�

et3i = T5i − T6i − T7i + T8i − k1i�L7i − L8i� + �k1i + k2i��L9i − L10i� − k3i�L11i − L12i� + �k3i + k4i��L13i − L14i� �23�

et4i = − �T1i − T2i − T3i + T4i + T9i − T10i − T11i + T12i�

− k1i�L7i − L8i� + �k1i + k2i��L9i − L10i� �24�

where k1i= a � uiH , k2i= l � uiH , k3i= c � uiH , k4i= w � uiH �i=1, 2, and 3�; H denotes the height of the retaining wall; T1i-T12i and L1i-L14i

�i=1, 2, and 3� are expressed as

T1i = tan−1 k3i

k1i�k1i

2 + k3i2 + 1

T2i = tan−1 k3i + k4i

k1i�k1i

2 + �k3i + k4i�2 + 1

T3i = tan−1 k3i

2 2
�k1i + k2i��k1i + k2i� + k3i + 1


�k1i + k2i��k1i + k2i�2 + �k3i + k4i�2 + 1

T5i = tan−1 k1ik3i

�k1i2 + k3i

2 + 1

T6i = tan−1 k3i�k1i + k2i��k1i + k2i�2 + k3i

2 + 1

T7i = tan−1 k1i�k3i + k4i��k1i

2 + �k3i + k4i�2 + 1

T8i = tan−1 �k1i + k2i��k3i + k4i��k1i + k2i�2 + �k3i + k4i�2 + 1

T9i = tan−1 k1i

k3i


k3i

T11i = tan−1 k1i

k3i + k4i


k3i + k4i

L1i = ln��k1i + k2i�2 + k3i2 + 1 + k3i

�k1i2 + k3i

2 + 1 + k3i

�L2i = ln��k1i + k2i�2 + �k3i + k4i�2 + 1 + k3i + k4i

�k1i2 + �k3i + k4i�2 + 1 + k3i + k4i

�L3i = ln��k1i + k2i�2 + k3i

2

�k1i2 + k3i

2 �

L4i = ln��k1i + k2i�2 + k3i2 + 1 + 1

�k1i2 + k3i

2 + 1 + 1�

L5i = ln��k1i + k2i�2 + �k3i + k4i�2

�k1i2 + �k3i + k4i�2 �

L6i = ln��k1i + k2i�2 + �k3i + k4i�2 + 1 + 1

�k1i2 + �k3i + k4i�2 + 1 + 1

�L7i = ln��k1i

2 + k3i2 + 1 + k3i

�k1i2 + k3i

2 + k3i

�

L8i = ln��k1i2 + �k3i + k4i�2 + 1 + k3i + k4i

�k1i2 + �k3i + k4i�2 + k3i + k4i

�L9i = ln��k1i + k2i�2 + k3i

2 + 1 + k3i

��k1i + k2i�2 + k3i2 + k3i


��k1i + k2i�2 + �k3i + k4i�2 + k3i + k4i

�

L11i = ln��k1i2 + k3i

2 + 1 + k1i

�k1i2 + k3i

2 + k1i

�L12i = ln��k1i + k2i�2 + k3i

2 + 1 + k1i + k2i

��k1i + k2i�2 + k3i2 + k1i + k2i

�L13i = ln��k1i

2 + �k3i + k4i�2 + 1 + k1i

�k1i2 + �k3i + k4i�2 + k1i


��k1i + k2i�2 + �k3i + k4i�2 + k1i + k2i

�Appendix III. Solutions of Centroid Location due to Loading Cases A–C

Type ofloading Centroid location solutions �measuring from the top of the retaining wall�

Centroidintegral

functions

Case A

z̄hp=

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12* pz11−u2

2* pz12�−2A66� u1

m1+u1* pz21−

u2

m2+u2* pz22��+2u3* pz23

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12* pt11−u2

2* pt12�−2A66� u1

m1+u1* pt21−

u2

m2+u2* pt22��+2u3* pt23

�when Q=0�

Eqs. �25�–�28�

z̄vp=

A44�u1* pz31−u2* pz32�−2A66� 1

m1+u 1�pz31− pz41�−1

m2+u2�pz32− pz42��

A44�u1* pt31−u2* pt32�−2A66� 1

m1+u 1�pt31− pt41�−1

m2+u2�pt32− pt42��

�when P=0�

Case B

z̄hl =

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*dz11−u2

2*dz12�−2A66� u1

m1+u1*dz21−

u2

m2+u2*dz22��+2u3*dz23

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*dt11−u2

2*dt12�−2A66� u1

m1+u1*dt21−

u2

m2+u2*dt22��+2u3*dt23

�when Ql=0�

Eqs. �29�–�32�

z̄vl =

A44�u1*dz31−u2*dz32�−2A66� 1

m1+u 1�dz31−dz41�−1

m2+u2�dz32−dz42��

A44�u1*dt31−u2*dt32�−2A66� 1

m1+u 1�dt31−dt41�−1

m2+u2�dt32−dt42��

�when Pl=0�

Case C

z̄hu=

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*ez11−u2

2*ez12�−2A66� u1

m1+u1*ez21−

u2

m2+u2*ez22��+2u3*ez23

k�m2u1−m1u2�

m1m2�u1−u2� �A44�u12*et11−u2

2*et12�−2A66� u1

m1+u1*et21−

u2

m2+u2*et22��+2u3*et23

*H

�when Qs=0�

Eq. �33� for Fig. 6Eq. �34� for Fig. 7Eq. �35� for Fig. 8Eq. �36� for Fig. 9

z̄vu =

A44�u1*ez31−u2*ez32�−2A66� 1

m1+u 1�ez31−ez41�−1

m2+u2�ez32−ez42��

A44�u1*et31−u2*et32�−2A66� 1

m1+u 1�et31−et41�−1

m2+u2�et32−et42��

*H

�when Ps=0�

where

pz1i =x + a

ui2 1

��x + a�2 + �y + c�2−

1

��x + a�2 + �y + c�2 + �uiH�2� �25�


pz2i =�x + a�

3ui2��x + a�2 + �y + c�2� �x + a�2

��x + a�2 + �y + c�2−

2�uiH�3��x + a�2 − 3�y + c�2��x + a�2 + �y + c�2�2 + ��x + a�2 + �y + c�2 + �uiH�2

2�uiH�2��x + a�2 − 3�y + c�2��x + a�2 + �y + c�2�2

−3�y + c�2

��x + a�2 + �y + c�2 + �uiH�2�

−�x + a�2 − 3�y + c�2

�x + a�2 + �y + c�2 �� 26�

pz3i = −1

ui2 uiH

��x + a�2 + �y + c�2 + �uiH�2− ln� uiH + ��x + a�2 + �y + c�2 + �uiH�2

��x + a�2 + �y + c�2 �� 27�

pz4i =1

2ui2��x + a�2 + �y + c�2�− �uiH�2��x + a�2 − �y + c�2�

�x + a�2 + �y + c�2 + ��x + a�2 + �y + c�2 + �uiH�2uiH��x + a�2 − �y + c�2��x + a�2 + �y + c�2

−uiH�x + a�2

�x + a�2 + �y + c�2 + �uiH�2�� +1

2ui2 ln� uiH + ��x + a�2 + �y + c�2 + �uiH�2

��x + a�2 + �y + c�2 � �28�

dz1i =x + a

ui2 �ln� c + ��x + a�2 + c2 + �uiH�2


c + w + ��x + a�2 + �c + w�2 � �29�

dz2i =c�x + a�

3ui2��x + a�2 + c2 2�uiH�3

��x + a�2 + c2�3/2 − 1 +��x + a�2 + c2 − 2�uiH�2��x + a�2 + c2 + �uiH�2

��x + a�2 + c2�3/2 �−

�c + w��x + a�

3ui2��x + a�2 + �c + w�2 2�uiH�3

��x + a�2 + �c + w�2�3/2 − 1 +��x + a�2 + �c + w�2 − 2�uiH�2��x + a�2 + �c + w�2 + �uiH�2

��x + a�2 + �c + w�2�3/2 � �30�

dz3i =1

ui2 �x + a��tan−1 cuiH

�x + a��x + a�2 + c2 + �uiH�2− tan−1 �c + w�uiH

�x + a��x + a�2 + �c + w�2 + �uiH�2 − c ln� uiH + ��x + a�2 + c2 + �uiH�2

��x + a�2 + c2 �+ �c + w�ln� uiH + ��x + a�2 + �c + w�2 + �uiH�2

��x + a�2 + �c + w�2 �� 31�

dz4i = −cH��x + a�2 + c2 + �uiH�2

2ui��x + a�2 + c2�−

wH2��x + a�2 − c�c + w��2��x + a�2 + c2��x + a�2 + �c + w�2�

+�c + w�H��x + a�2 + �c + w�2 + �uiH�2

2ui��x + a�2 + �c + w�2�

+x + a

ui2 tan−1 cuiH

�x + a��x + a�2 + c2 + �uiH�2− tan−1 uiH�c + w�

�x + a��x + a�2 + �c + w�2 + �uiH�2� −1

2ui2 c ln� uiH + ��x + a�2 + c2 + �uiH�2

��x + a�2 + c2 �− �c + w�ln� uiH + ��x + a�2 + �c + w�2 + �uiH�2

��x + a�2 + �c + w�2 �� 32�

ez1i =1

2�k3i��k1i

2 + k3i2 − ��k1i + k2i�2 + k3i

2 − �k1i2 + k3i

2 + 1 + ��k1i + k2i�2 + k3i2 + 1� − �k3i + k4i��k1i

2 + �k3i + k4i�2 − ��k1i + k2i�2 + �k3i + k4i�2

− �k1i2 + �k3i + k4i�2 + 1 + ��k1i + k2i�2 + �k3i + k4i�2 + 1� + L1i − L2i − k1i

2 �L7i − L8i� + �k1i + k2i�2�L9i − L10i�� 33�

ez2i =1

3k3i �k1i2 + k3i

2 − ��k1i + k2i�2 + k3i2 −

�k1i2 + k3i

2 + 1�3/2

k1i2 + k3i

2 +��k1i + k2i�2 + k3i

2 + 1�3/2

�k1i + k2i�2 + k3i2 � − �k3i + k4i� �k1i

2 + �k3i + k4i�2

− ��k1i + k2i�2 + �k3i + k4i�2 −�k1i

2 + �k3i + k4i�2 + 1�3/2

k1i2 + �k3i + k4i�2 +

��k1i + k2i�2 + �k3i + k4i�2 + 1�3/2

�k1i + k2i�2 + �k3i + k4i�2 �−

�2k1i + k2i�k2ik4i�k1i2 �k1i + k2i�2 − k3i�k3i + k4i��k1i

2 + �k1i + k2i�2 + k3i2 + �k3i + k4i��2k3i + k4i��

�k2 + k2 ��k1i + k2i�2 + k2 ��k2 + �k3i + k4i�2��k1i + k2i�2 + �k3i + k4i�2� � �34�
1i 3i 3i 1i

ez3i =1

2�T5i − T6i − T7i + T8i − k1i

2 �T1i − T2i� + �k1i + k2i�2�T3i − T4i� − k3i2 �T13i − T14i� + �k3i + k4i�2�T15i − T16i� + 2k3i�k1iL15i

− �k1i + k2i�L16i� − 2�k3i + k4i��k1iL17i − �k1i + k2i�L18i�� 35�

ez4i =1

2�− �T1i − T2i − T3i + T4i + T9i − T10i − T11i + T12i�

− k1i2 �T1i − T2i� + �k1i + k2i�2�T3i − T4i� + k3i�k1iL15i

− �k1i + k2i�L16i� − �k3i + k4i��k1iL17i − �k1i + k2i�L18i�� 36�

where T13i-T16i, and L15i-L18i �i=1, 2, and 3� are expressed as

T13i = tan−1 k1i

k3i�k1i

2 + k3i2 + 1


k3i��k1i + k2i�2 + k3i

2 + 1

T15i = tan−1 k1i

�k3i + k4i��k1i2 + �k3i + k4i�2 + 1


�k3i + k4i��k1i + k2i�2 + �k3i + k4i�2 + 1

L15i = ln��k1i2 + k3i

2 + 1 + 1

�k1i2 + k3i

2 �L16i = ln��k1i + k2i�2 + k3i

2 + 1 + 1

��k1i + k2i�2 + k3i2 �

L17i = ln��k1i2 + �k3i + k4i�2 + 1 + 1

�k1i2 + �k3i + k4i�2 �

L = ln��k1i + k2i�2 + �k3i + k4i�2 + 1 + 1

18i � ��k1i + k2i�2 + �k3i + k4i�2 �
Notation
The following symbols are used in this paper:

Aij �i , j=1–6� � elastic moduli or elasticity constants;a � loads applied at a horizontal distance in

the x-axis from the wall;c � loads applied at a horizontal distance in

the y-axis from the wall;ds1i-ds4i � stress elementary functions for loading

Case B �Appendix I�;dt1i-dt4i � force integral functions for loading Case B

�Appendix II�;dz1i-dz4i � centroid integral functions for loading

Case B �Appendix III�;E ,E� ,� ,�� ,G� � elastic engineering constants of a

cross-anisotropic backfill;es1i-es4i � stress elementary functions for loading

Case C �Appendix I�;et1i-et4i � force integral functions for loading Case C

�Appendix II�;ez1i-ez4i � centroid integral functions for loading

Case C �Appendix III�;H � height of the retaining wall;i � complex number �=�−1�;


k � coefficients;k1i-k4i � functions of a, l, c, w, H, and ui �i=1, 2,

and 3�;l � length of the rectangular area load;

m1 ,m2 � coefficients;n1i-n4i � functions of a, l, c, w, z, and ui �i=1, 2,

and 3�;P � a horizontal point load �force�;Pl � a horizontal finite line load �force per unit

length�;Ps � a horizontal uniform rectangular area load

�force per unit area�;Ph

l � lateral force due to loading Case B�Appendix II�;

Php � lateral force due to loading Case A

�Appendix II�;Ph

u � lateral force due to loading Case C�Appendix II�;

ps1i-ps4i � stress elementary functions for loadingCase A �Appendix I�;

pt1i-pt4i � force integral functions for loading Case A�Appendix II�;

pz1i-pz4i � centroid integral functions for loadingCase A �Appendix III�;


Q � a vertical point load �force�;Ql � a vertical finite line load �force per unit

length�;Qs � a vertical uniform rectangular area load

�force per unit area�;s , t � coefficients;

u1 ,u2 ,u3 � roots of the characteristic equation;w � width of the rectangular area load;z̄h

l � centroid location due to loading Case Bwhen Ql=0 �Appendix III�;

z̄vl � centroid location due to loading Case B

when Pl=0 �Appendix III�;z̄h

p � centroid location due to loading Case Awhen Q=0 �Appendix III�;

z̄vp � centroid location due to loading Case A

when P=0 �Appendix III�;z̄h

u � centroid location due to loading Case Cwhen Qs=0 �Appendix III�;

z̄vu � centroid location due to loading Case C

when Ps=0 �Appendix III�;�h

l � lateral stress due to loading Case B�Appendix I�;

�hp � lateral stress due to loading Case A

�Appendix I�;�h

u � lateral stress due to loading Case C�Appendix I�.

References

Abelev, A. V., and Lade, P. V. �2004�. “Characterization of failure incross-anisotropic soils.” J. Eng. Mech., 130�5�, 599–606.

Barden, L. �1963�. “Stresses and displacements in a cross-anisotropicsoil.” Geotechnique, 13�3�, 198–210.

Boussinesq, J. �1885�. Application des potentiels à l’Étude de l’Équilibreet due mouvement des solides élastiques, Gauthier-Villars, Paris.

Constantinou, M. C., and Gazetas, G. �1986�. “Loading of anisotropicquarter plane.” J. Eng. Mech., 112�10�, 1021–1040.

Constantinou, M. C., and Greenleaf, R. S. �1987�. “Stress systems inorthotropic quarter plane.” J. Eng. Mech., 113�11�, 1720–1738.

Fang, H. Y. �1991�. Foundation engineering handbook, 2nd Ed., Chap-man & Hall, New York.

Gazetas, G. �1982�. “Stresses and displacements in cross-anisotropicsoils.” J. Geotech. Engrg. Div., 108�4�, 532–553.

Georgiadis, M., and Anagnostopoulos, C. �1998�. “Lateral pressure onsheet pile walls due to strip load.” J. Geotech. Geoenviron. Eng.,124�1�, 95–98.

Gerrard, C. M., and Wardle, L. J. �1973�. “Solutions for point loads and
generalized circular loads applied to a cross-anisotropic half space.”

CSIRO Technical Paper 13, 1–39.Gibson, R. E. �1974�. “The analytical method in soil mechanics.” Geo-

technique, 24�2�, 115–140.Hetenyi, M. �1960�. “A method of solution for the quarter-plane.” J. Appl.

Mech., 27�2�, 289–296.Hetenyi, M. �1970�. “A general solution for the elastic quarter space.” J.

Appl. Mech., 37�1�, 70–76.Jarquio, R. �1981�. “Total lateral surcharge pressure due to strip load.” J.

Geotech. Engrg. Div., 107�10�, 1424–1428.Lekhnitskii, S. G. �1963�. Theory of elasticity of an anisotropic elastic

body. Holden-Day, San Francisco.Liao, J. J., and Wang, C. D. �1998�. “Elastic solutions for a transversely

isotropic half-space subjected to a point load.” Int. J. Numer. Analyt.Meth. Geomech., 22�6�, 425–447.

Michell, J. H. �1900�. “The stress in an anisotropic elastic solid with aninfinite plane boundary.” Proc. London Math. Soc., 32, 247–258.

Mindlin, R. D. �1936�. “Pressure distributions on retaining walls.” Proc.,Int. Conf. Soil Mech. Found. Engng., Vol. III, 155–156.

Pickering, D. J. �1970�. “Anisotropic elastic parameters for soil.” Geo-technique, 20�3�, 271–276.

Spangler, M. G. �1936�. “The distribution of normal pressure on a retain-ing wall due to a concentrated surface load.” Proc., Int. Conf. SoilMech. Found. Engng., Vol. I, 200–207.

Spangler, M. G. �1938a�. “Lateral pressures on retaining walls caused bysuperimposed loads.” Proc., 18th Ann. Meet. High. Res. Board, 57–65.

Spangler, M. G. �1938b�. “Horizontal pressures on retaining walls due toconcentrated surface loads.” Bulletin 140, Iowa Engng. Exp. Sta.

Spangler, M. G., and Handy, R. L. �1982�. Soil engineering, 4th Ed.,Harper and Row, New York.

Spangler, M. G., and Mickle, J. L. �1956�. “Lateral pressure on retainingwalls due to backfill surface loads.” High. Res. Board, 141.

Steenfelt, J. S., and Hansen, B. �1983�. “Discussion of ‘Total lateral sur-charge pressure due to strip load.’” J. Geotech. Engrg., 109�2�, 271–273.

Terzaghi, K. �1954�. “Anchored buckheads.” Trans., ASCE, 119, 1243–1324.

Wang, C. D. �2003�. “Displacements and stresses due to vertical subsur-face loading for cross-anisotropic half-spaces.” Soils Found., 43�5�,40–52.

Wang, C. D. �2005�. “Lateral stress caused by horizontal and verticalsurcharge strip loads on a cross-anisotropic backfill.” Int. J. Numer.Analyt. Meth. Geomech., 29�14�, 1341–1361.

Wang, C. D. �2007�. “Lateral force and centroid location caused by hori-zontal and vertical surcharge strip loads on a cross-anisotropic back-fill.” Int. J. Numer. Analyt. Meth. Geomech., in press.

Wang, C. D., and Liao, J. J. �1999�. “Elastic solutions for a transverselyisotropic half-space subjected to buried asymmetric-loads.” Int. J.Numer. Analyt. Meth. Geomech., 23�2�, 115–139.

Yildiz, E. �2003�. Lateral pressures on rigid retaining walls: a neural
network approach, MS thesis, The Middle East Tech. Univ., Ankara.

lateral force induced by rectangular surcharge loads on a...

Documents