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**Increasing with Depth
**

W. Y. Shen1 and C. I. Teh2

Abstract: This article reports a variational solution and its spreadsheet calculation procedure for the analysis of laterally loaded piles in

a soil with stiffness increasing with depth. The aim of the paper is to provide solutions that can be used simply with recourse only to

spreadsheet calculation to solve the displacement and bending moment of laterally loaded piles, so that they can be easily applied in

practice as an alternative approach to analyze the response of laterally loaded piles.

DOI: 10.1061/共ASCE兲1090-0241共2004兲130:8共878兲

CE Database subject headings: Piles; Lateral displacement; Lateral loads; Soils; Stiffness; Spreadsheets.

Introduction Method of Analysis

**Laterally loaded piles are commonly used in engineering practice, Basic Variational Formulation
**

and a number of theoretical methods have been available for ana-

lyzing such piles. For example, the subgrade reaction approach by The variational approach for the analysis of laterally loaded pile

Barber 共1953兲 and Matlock and Reese 共1960兲, the p-y curve groups has been described in detail by Shen and Teh 共2002兲. For

method by Reese 共1977兲, and the elastic continuum approach by a single pile in a soil modeled using the subgrade reaction

Poulos and Davis 共1980兲, Zhang and Small 共2000兲, and Shen and method, the potential energy can be written as

Teh 共2002兲. These aforementioned methods, however, need com-

plex computer programs to perform fully numerical analysis, and

this makes them less accessible to practicing engineers in the

p ⫽U p ⫹

1

2 冕l

z k hz z ddz⫺ t H t ⫺

t

z t

M (1)

**routine design. In Eq. 共1兲, the first term U p equals elastic strain energy of the pile.
**

This article reports a variational solution for the analysis of The second term is the work done by the soil reaction pressure,

laterally loaded piles that can be used simply based on a spread- where z ⫽pile displacement; k hz ⫽soil modulus of subgrade re-

sheet calculation procedure using Microsoft Excel. The purpose of action; l⫽pile length; and d⫽pile diameter. The third and fourth

the paper is to provide solutions that are simple and efficient and terms are, respectively, the work done by the horizontal load H t

can be used as an alternative approach for the analysis of laterally and moment M t acting at the pile head, where t is the pile head

loaded piles at working load levels. The present solutions are displacement. The pile displacements z , which can be repre-

developed mainly based on the variational approach by Shen and sented by finite series as given by Shen and Teh 共2002兲, can be

Teh 共2002兲. The soil is modeled using a subgrade reaction method written as

and its stiffness can be increased with depth. It should be noted z⫽ 兵 Z 其 T兵  其 (2)

that solutions that are able to consider soil stiffness increasing

with depth are important to laterally loaded piles, since adopting where 兵 Z 其 ⫽vector related to only the depth coordinate z; and

this soil stiffness variation is an approximate way to take into 兵其⫽vector containing undetermined constants. Thus, with the

account the high strain levels in the soil near the ground surface application of the principle of minimum potential energy, Eq. 共1兲

can be reduced as

so that simple elastic analysis can provide at least a first estimate

of pile response at working load levels. t

冉 冊

冕

U p z t z

⫹ k hz z ddz⫽ H t⫹ Mt (3)

1

Research Fellow, NTU-PWD Geotechnical Research Centre, School  i l  i  i  i

of Civil & Environmental Engineering, Nanyang Technological Univ.,

Singapore. where  i ⫽constants in the vector 兵其. For the case of a pile

2

Associate Professor, NTU-PWD Geotechnical Research Centre, subjected to a horizontal load, Eq. 共3兲 can be reduced to a matrix

School of Civil & Environmental Engineering, Nanyang Technological equation given by

Univ., Singapore.

Note. Discussion open until January 1, 2005. Separate discussions 共 b k pH c ⫹ 关 k sH 兴 兲 兵  H 其 ⫽ 兵 H 其 (4a)

must be submitted for individual papers. To extend the closing date by For the case of a pile subjected to a moment, Eq. 共3兲 can be

one month, a written request must be filed with the ASCE Managing

reduced to a matrix equation given by

Editor. The manuscript for this technical note was submitted for review

and possible publication on October 15, 2002; approved on December 12, 共 b k pM c ⫹ 关 k sM 兴 兲 兵  M 其 ⫽ 兵 M 其 (4b)

2003. This technical note is part of the Journal of Geotechnical and

Geoenvironmental Engineering, Vol. 130, No. 8, August 1, 2004. where b k pH c and 关 k sH 兴 are the matrices reflecting the pile and soil

©ASCE, ISSN 1090-0241/2004/8-878 – 882/$18.00. stiffness under a horizontal load loading, respectively. b k pM c and

878 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004

Downloaded 08 Jul 2010 to 158.125.80.124. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org

5.0. and K r where h ⫺1 ⫺1 M 2 j and h M 22 are the coefficients in the inverted matrix ⫽E p I p /k hl dl 4 is the pile-soil relative stiffness. and the rest of the terms for i⭓3 or j⭓3. These quantities are described as follows..⫺ 2 sin z l .sin .ascelibrary. and ˆM‰ The row number i and column number j of the matrix 关 h H 兴 are typed in cells in a manner as shown in Table 1. respectively. 共4a兲 and 共4b兲.org . For convenience. Redistribution subject to ASCE license or copyright. . 共6a兲 and 共6b兲 is valid for i⬍3 or j ⬍3.125. cos .5 m. .. Using the menu command insert/name/define. ␣ is a coeffi- ⫽ 兵 0. k hl dl l l l pile head can be derived based on the zero-rotation condition. where E p ⫽pile 关 h M 兴 ⫺1 .. 再 冎 which can be expressed as 1 z z 共 2k⫺1 兲 z 兵 B M 其 T ⫽ ⫺1. 2 共 ␣⫺1 兲共 i⫺2 兲共 j⫺2 兲关 1⫺ 共 ⫺1 兲 i⫹ j⫺4 兴 the displacement and bending moment along the pile shaft that ⫹ (6e) are the key responses of practical interest can eventually be de.. 兵 H 其 ⫽ 兵 H t .80..0. d⫽0. the moment generated at the 兵 B H 其 T ⫽ 1. d.M t . For the problem of a fixed- 1 z z kz head pile under a horizontal load.sin . as h Mi j⫽ 冋 K r 共 i⫹ j⫺5 兲 4 4 ⫹8 共 ␣⫹1 兲 32 ⫹ ␣⫺1 共 i⫹ j⫺5 兲 2 2 册 z ⫽ 兵 B H 其 T 关 h H 兴 ⫺1 兵 H 其 ⫹ 兵 B M 其 T 关 h M 兴 ⫺1 兵 M 其 (5a) ⫹ 2 冉 ␣⫺1 1⫺ 共 ⫺1 兲 i⫹ j⫺5 1⫺ 共 ⫺1 兲 i⫺ j 共 i⫹ j⫺5 兲 2 2 ⫹ 共 i⫺ j 兲 2 2 冊 (6f) M z ⫽ 兵 B mH 其 T 关 h H 兴 ⫺1 兵 H 其 ⫹ 兵 B mM 其 T 关 h M 兴 ⫺1 兵 M 其 (5b) The first term in Eqs. the value of is 共 ⫺1 兲 i⫹ j⫺4 also typed in a cell.. . The displacement zH and moment M zH at depth z can be obtained..⫺cos . . l⫽10 m. k hl ⫽soil modulus of subgrade reaction at the pile toe. The coefficients in the rest of the rows pile head. . In the above Eqs. respectively. This size is adopted in the spreadsheet calcula- (6b) tion procedure. where the first term in Eqs.. K r is then calcu- lated in a cell 共see Table 1兲. 共6c兲 and 共6d兲 is valid for i⫽ j and the vectors reflecting the horizontal load and moment acting at the rest of the terms for i⫽ j. Visithttp://www. Both 关 h H 兴 and 关 h M 兴 are dimensionless and symmetrical ma- trices with their coefficients given as follows.⫺ . 2l 共 2k⫺1 兲 2 2 4 cos 共 2k⫺1 兲 z 2l 冎 ⫽ 兺 h ⫺1 j⫽1 M2j h ⫺1 ⫺1 M 22 in which k⫽number of terms used in the trigonometric function. 兵H其 and 兵M其 are the where the first term in Eqs..关 k sM 兴 are the matrices for a moment loading. and pi. 共5a兲 and 共5b兲.⫺cos k hl dl 2 l 2l 2l M t ⫽lH t (7) 再 兵 B mH 其 T ⫽K r l 0. ˆH‰..0 其 T cient defined as the ratio of soil modulus of subgrade reaction at 再 冎 the pile head to that at the pile toe. The coefficients in Spreadsheet Calculation Procedure the first row and first column are given by A spreadsheet calculation procedure using Microsoft Excel is de- 冋 册 veloped to solve the pile displacement and bending moment as i⫹ j⫺1⫹␣ ␣⫺ 共 ⫺1 兲 i⫹ j⫺3 described above by the present solutions.e. 冎 The coefficient k⫹2 2 4 再 z 兵 B mM 其 T ⫽K r 0.. as l.0. 共6a兲–共6f兲. and columns are given by Solution for Pile Response Estimate h Hi j ⫽ 冋 K r 共 i⫹ j⫺4 兲 4 4 ⫹8 共 ␣⫹1 兲 32 册 After solving the vectors 兵  H 其 and 兵  M 其 in Eqs. respectively. Taking E p ⫽2⫻107 kPa. 共6e兲 and 共6f兲 is valid for i⫽ j and the In Eqs. . 共 i⫹ j⫺4 兲 2 共 i⫺ j 兲 2 2 termined.124.0. Young’s modulus and I p ⫽second moment of area of the pile sec- tion. which is demonstrated h Hi j ⫽ ⫹ (6a) 共 i⫹ j⫺1 兲共 i⫹ j 兲 共 i⫹ j⫺3 兲 in the following through an example of a fixed-head pile sub- jected to a horizontal load.0 其 T and 兵M其 second term for i⫽ j. ⫺ (6c) the cells with the values of these above stored parameters are 共 i⫹ j⫺4 兲 selected and named. It should be mentioned that the present h Mi j⫽ 冋 i⫹ j⫺1⫹␣ 共 i⫹ j⫺1 兲共 i⫹ j 兲 ⫹ 册 2 共 ⫺1 兲 i⫹ j⫺2 ⫹ 4 共 ␣⫺1 兲 共 2i⫹2 j⫺7 兲 共 2i⫹2 j⫺7 兲 2 2 method converges quickly and the adoption of a size 10⫻10 for the matrices 关 h H 兴 and 关 h M 兴 is usually enough to give sufficiently accurate results. and k hl ⫽50 000 kN/m3 . k hl . .⫺k 2 2 sin kz l .. h Mi j⫽ 冋 i⫹ j⫺1⫹␣ 共 i⫹ j⫺1 兲共 i⫹ j 兲 ⫹ 册 2 共 ⫺1 兲 i⫹ j⫺3 ⫺ 4␣ 共 2i⫹2 j⫺9 兲 共 2i⫹2 j⫺9 兲 2 2 Calculation of †hH‡. ␣⫽0. ␣. The coefficients in the second row and second column are given by Basic Parameters The basic parameters for the pile and soil are typed in cells in a h Hi j ⫽ 冋 i⫹ j⫺1⫹␣ 共 i⫹ j⫺1 兲共 i⫹ j 兲 ⫹ 册 2 共 ␣⫺1 兲关 1⫺ 共 ⫺1 兲 i⫹ j⫺4 兴 共 i⫹ j⫺4 兲 3 3 manner as shown Table 1. i.. K r . †hM‡. The coefficients in 16共 ␣⫺1 兲共 ⫺1 兲 i⫹ j⫺3 ⫹ (6d) 关 h H 兴 can then be calculated by making use of these row 共 2i⫹2 j⫺9 兲 3 3 and column numbers when typing in cells the relevant JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004 / 879 Downloaded 08 Jul 2010 to 158..... . .

兵H其. respectively. It tained are shown in cells in Table 1.2. 共5a兲. Calculation of 兵 BH其 . the matrix 关 h M 兴 can be calculated in the involved in these equations with respect to row and column same way. 共6a兲. Similarly.125. 共5a兲 and 共5b兲 in a way similar to built-in spreadsheet functions for matrix operations. Eqs. 兵 B mH 其 .’’ ‘‘index.2 兲 ⫺1 兲 * l * H t (8) where ‘‘sum..org . The vector 兵M其. and 兵 B mM 其 can be calculated typing in a cell as 880 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004 Downloaded 08 Jul 2010 to 158. selected and named B H . based on Eq. Table 1. 共7兲. based on Eq.e. i. Visithttp://www. The entire matrices 关 h H 兴 . and 共6e兲. Note that for the coefficient calculation of the matrices 关 h H 兴 and 关 h M 兴 . selected and named h H .0 兲 )/ 共 index共 minverse共 h M 兲 . 兵 BmH其 . B mH . The coefficients in the matrices 关 h H 兴 and 关 h M 兴 ob- locations can be represented using the IF statement of Excel.80. 关 h M 兴 .’’ and ‘‘minverse’’ are Microsoft Excel’s based on their definition in Eqs. can select the cell with the coefficient calculated and then moving the be calculated by typing in the second row of the vector 兵M其 cursor to the other cells of the matrix 关 h H 兴 where the equation is 共see Table 1兲 as sum(index共 minverse共 h M 兲 . These a free-head pile. and 兵 BmM其 Calculation of Pile Displacement and Bending Moment Taking a value of z/l. the second row of the vector 兵M其 just needs to be vectors are. can be obtained by vectors 兵 B H 其 . 兵 BM其 . The vector 兵H其 can be easily should be noted that in this way. set as M t . the The pile displacement z . each equation only needs to obtained by typing in the horizontal load 共here assumed to be 100 be typed once to calculate one coefficient. 共6c兲. The rest of the kN兲 in a corresponding cell of the vector 兵H其 as shown in Table 1 coefficients can be obtained by pressing the right mouse key to and this cell is named H t . Spreadsheet Calculation of Pile Displacement and Bending Moment expressions. and naming it z/l.ascelibrary. B M . Redistribution subject to ASCE license or copyright.2. and 兵M其 are.2 共see Table 1兲. and M. say 0. The conditions applicable. 兵 B M 其 . H.124. and B mM . h M . respectively.

where I H and I M are. which is obtained as k hl ⫽187. The results presented by Barber correspond to the case of ␣⫽0 in the present solutions. the value k hl can be back-analyzed based on a estimating the response of laterally loaded piles merely by input. Comparison of displacement and bending moment profiles JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004 / 881 Downloaded 08 Jul 2010 to 158. Using the established spreadsheet cal- It should be pointed out that once the above spreadsheet cal. moment can also be easily determined if ‘‘Solver. tical purpose.H 兲兲 In the analysis. Solutions from the present method are compared in Fig. 1 with especially for the maximum bending moment. The pile bending moment at other depths can be easily obtained by just displacement and bending moment profiles are then calculated inputting corresponding values of z/l. the rotation factors due to horizontal load and moment.mmult共 minverse共 h H 兲 . 共5b兲. The computed displacement and bending moment distributions are plotted and compared with Comparison with Numerical Solutions the measured results in Fig. and this is useful for prac. based on Eq. mmult共  mH .25 m with a diameter d⫽0. 1 is presented in terms of a set of influence factors I H .80. The maximum bending and compared with the measured results.5 MN/m3 . 1. match with the measured pile head displacement by trying differ- ting relevant pile and soil parameters. 2.90 kN is selected for analysis. culation procedure in Table 1 and typing in the cells in the rel- culation procedure is established. Thus.125. 2 and agreement is generally good. Redistribution subject to ASCE license or copyright. and I F is the displacement factor due to horizontal load for a fixed-head pile. Visithttp://www.75 mm at the above-mentioned load level. I H and I M are. Comparison of displacement and rotations factors Fig. the soil modulus of subgrade reaction is assumed to increase with depth with the coefficient ␣ taken as ␣⫽0. respectively. The instrumented pile Pile IN1 at a working load level of Similarly.org .M 兲兲 (9b) their values at the working load level are determined through The obtained results are shown in Table 1. The above comparison demonstrates the accuracy of the present solutions for the analysis of laterally loaded piles in a soil with stiffness increasing with depth. and I F . Application to Field Pile Test The spreadsheet calculation procedure is used to calculate the displacement and bending moment profiles of one instrumented Fig. the displace- ment factors due to horizontal load and moment for a free-head pile.124.H 兲兲 pile as reported by Mohan and Shrivastava 共1971兲 out of a series of field tests on laterally loaded piles subjected to horizontal load- ⫹mmult共  M . ent values of k hl . is invoked 共not described the pile is l⫽5. The basic parameters for the pile are as follows. I H .M 兲兲 (9a) ing. the bending moment M z can be 4.ascelibrary. respectively. Similar agreement the numerical results by Barber 共1953兲 for piles in a soil with stiffness increasing with depth to validate the performance of the present method. the displacement and bending moment profiles can be calculated by inputting relevant values of z/l. The comparison in Fig.mmult共 minverse共 h M 兲 . rigidity E p I p ⫽320 kN m2 . Good agreement can be observed between the present solutions and those by Barber 共1953兲.mmult共 minverse共 h M 兲 . The length of timization routine in Microsoft Excel. it can be easily used later on for evant parameters.’’ a built-in op. I M . I M . and ⫹mmult共  mM . Displacement and back-analysis from the measured pile head displacement. mmult共  H .mmult共 minverse共 h H 兲 .1 m and a bending here兲. The pile was embedded into a obtained as layer of silty sand followed by a clay layer and had a pile head displacement about 8.

Tech. C.. C. and Reese.. S.’’ Geotechnique. widely used spreadsheet software. 共1977兲. 882 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004 Downloaded 08 Jul 2010 to 158. 共1971兲.’’ J. Y. New York.’’ Proc.. D. a subjected to horizontal and vertical loads. Found. Zhang. which is a References more reliable way compared to using empirical correlations with other soil properties. Pile foundation analysis and design.80.. Gleser. and Davis. Mohan. and Small. Geotech. Div. Houston.org . 共2002兲. 共1980兲. 26. ‘‘Analysis of laterally loaded pile ing with depth have been presented. H..124. 677– 684.. 52共3兲. the response of laterally loaded piles. W. L.. M. E. ‘‘Nonlinear behaviour of single vertical piles under lateral loads. ‘‘Discussion to paper by S. 96 –99. ‘‘Laterally loaded piles: Program documentation. H. H. can be easily loaded piles. 201–208. 3rd Annual Offshore Techni- cal Conf. G. and Teh.. Conclusions Poulos. the use of the established spreadsheet calculation procedure. The example also shows that with Barber. 154. C. S. C.’’ Comput.. E. Div. 共1953兲.’’ ASTM Spec. 103共4兲. bending moment profiles of laterally loaded piles can be evalu.125. of laterally loaded piles at working load levels may be predicted fairly well provided that the soil modulus of subgrade reaction is determined properly through back-analysis. Eng. ‘‘Generalised solutions for laterally moment profile. Publ. ‘‘Analysis of capped pile groups ated just through spreadsheet calculation using Microsoft Excel. and Shrivastava.. P. Visithttp://www. Civ. for analyzing laterally loaded piles in soils with stiffness increas. H. A variational solution and its spreadsheet calculation procedure Geotech. H. It potential to be used in engineering practice for the analysis of appears that the displacement and bending moment profiles laterally loaded piles. 86共5兲. Wiley. J. Redistribution subject to ASCE license or copyright. Reese. Eng. L. Vol. 91–97. especially the bending Matlock.’’ J... Soc. I. estimated. 共1960兲. Am. 2. 共2000兲. Shen. Soil Mech. As the displacement and groups using a variational approach.ascelibrary. 287–305.is also observed in the analysis of other laterally loaded piles. which is critical in design. the present method has the 1–21.

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