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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

. In the above Eqs. 共4a兲 and 共4b兲. 共6e兲 and 共6f兲 is valid for i⫽ j and the In Eqs. .M t ... 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.org .sin . 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. The displacement zH and moment M zH at depth z can be obtained..0. d. Taking E p ⫽2⫻107 kPa. Young’s modulus and I p ⫽second moment of area of the pile sec- tion.⫺cos k hl dl 2 l 2l 2l M t ⫽lH t (7) 再 兵 B mH 其 T ⫽K r l 0. 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.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.. ˆH‰. k hl ⫽soil modulus of subgrade reaction at the pile toe.0. and pi. k hl . 兵 H 其 ⫽ 兵 H t . For the problem of a fixed- 1 z z kz head pile under a horizontal load. . This size is adopted in the spreadsheet calcula- (6b) tion procedure. 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 158.0.. 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. Both 关 h H 兴 and 关 h M 兴 are dimensionless and symmetrical ma- trices with their coefficients given as follows.关 k sM 兴 are the matrices for a moment loading. l⫽10 m..ascelibrary. . where E p ⫽pile 关 h M 兴 ⫺1 . respectively. 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.80.e. ␣. For convenience.5 m.0 其 T and 兵M其 second term for i⫽ j. 共6a兲–共6f兲. k hl dl l l l pile head can be derived based on the zero-rotation condition. Redistribution subject to ASCE license or copyright.. 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. 兵H其 and 兵M其 are the where the first term in Eqs..⫺cos . where the first term in Eqs. 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. ␣⫽0. and the rest of the terms for i⭓3 or j⭓3.. 共 i⫹ j⫺4 兲 2 共 i⫺ j 兲 2 2 termined. . 共5a兲 and 共5b兲. as l.5. . . i.⫺ . 共6a兲 and 共6b兲 is valid for i⬍3 or j ⬍3.⫺k 2 2 sin kz l .. d⫽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. ⫺ (6c) the cells with the values of these above stored parameters are 共 i⫹ j⫺4 兲 selected and named. and k hl ⫽50 000 kN/m3 . Visithttp://www. These quantities are described as follows. ... . 再 冎 which can be expressed as 1 z z 共 2k⫺1 兲 z 兵 B M 其 T ⫽ ⫺1. †hM‡.sin . K r . the moment generated at the 兵 B H 其 T ⫽ 1. respectively. 冎 The coefficient k⫹2 2 4 再 z 兵 B mM 其 T ⫽K r 0..124. K r is then calcu- lated in a cell 共see Table 1兲. . 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‡. cos . The coefficients in the rest of the rows pile head. 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..125. 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.⫺ 2 sin z l . Using the menu command insert/name/define..0. 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. ␣ is a coeffi- ⫽ 兵 0. the value of is 共 ⫺1 兲 i⫹ j⫺4 also typed in a cell.. respectively.

B mH .’’ ‘‘index. and 共6e兲. set as M t .2 兲 ⫺1 兲 * l * H t (8) where ‘‘sum. Spreadsheet Calculation of Pile Displacement and Bending Moment expressions. Eqs. These a free-head pile.e. and naming it z/l.2. and M. Similarly. the matrix 关 h M 兴 can be calculated in the involved in these equations with respect to row and column same way.2 共see Table 1兲. and 兵M其 are.. selected and named h H . Table 1. 关 h M 兴 . 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. 兵 BM其 . selected and named B H . The vector 兵M其. the second row of the vector 兵M其 just needs to be vectors are.80. based on Eq. 兵H其. B M . can be obtained by vectors 兵 B H 其 . h M . 共5a兲 and 共5b兲 in a way similar to built-in spreadsheet functions for matrix operations. Redistribution subject to ASCE license or copyright. 共7兲.0 兲 )/ 共 index共 minverse共 h M 兲 . The entire matrices 关 h H 兴 . 共6c兲.ascelibrary. Note that for the coefficient calculation of the matrices 关 h H 兴 and 关 h M 兴 . and 兵 BmM其 Calculation of Pile Displacement and Bending Moment Taking a value of z/l. The conditions applicable. 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. based on Eq.’’ and ‘‘minverse’’ are Microsoft Excel’s based on their definition in Eqs. 共5a兲. 兵 B mH 其 . It tained are shown in cells in Table 1.125. Visithttp://www.2. 兵 BmH其 . say 0.124. 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 . 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 .org . 兵 B M 其 . i. and B mM . The vector 兵H其 can be easily should be noted that in this way. Calculation of 兵 BH其 . 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. respectively. the The pile displacement z . respectively. H. 共6a兲.

The basic parameters for the pile are as follows. tical purpose. I M . Visithttp://www.mmult共 minverse共 h M 兲 . Comparison of displacement and bending moment profiles JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / AUGUST 2004 / 881 Downloaded 08 Jul 2010 to 158.1 m and a bending here兲. mmult共  H . rigidity E p I p ⫽320 kN m2 . 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. Comparison of displacement and rotations factors Fig. respectively. The computed displacement and bending moment distributions are plotted and compared with Comparison with Numerical Solutions the measured results in Fig. 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. 1. is invoked 共not described the pile is l⫽5. and I F . I M . respectively.H 兲兲 In the analysis.mmult共 minverse共 h H 兲 . I H . 共5b兲.ascelibrary. the soil modulus of subgrade reaction is assumed to increase with depth with the coefficient ␣ taken as ␣⫽0. Solutions from the present method are compared in Fig. ent values of k hl .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 . 2 and agreement is generally good. Using the established spreadsheet cal- It should be pointed out that once the above spreadsheet cal. match with the measured pile head displacement by trying differ- ting relevant pile and soil parameters.mmult共 minverse共 h H 兲 .124. the displace- ment factors due to horizontal load and moment for a free-head pile.75 mm at the above-mentioned load level. which is obtained as k hl ⫽187. 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. moment can also be easily determined if ‘‘Solver.org . I H and I M are. the displacement and bending moment profiles can be calculated by inputting relevant values of z/l. The results presented by Barber correspond to the case of ␣⫽0 in the present solutions. 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. The comparison in Fig. 1 with especially for the maximum bending moment. mmult共  mH . the rotation factors due to horizontal load and moment.M 兲兲 (9a) ing. and this is useful for prac.5 MN/m3 . The maximum bending and compared with the measured results. Good agreement can be observed between the present solutions and those by Barber 共1953兲. it can be easily used later on for evant parameters.’’ a built-in op. the bending moment M z can be 4. the value k hl can be back-analyzed based on a estimating the response of laterally loaded piles merely by input. culation procedure in Table 1 and typing in the cells in the rel- culation procedure is established.mmult共 minverse共 h M 兲 . The length of timization routine in Microsoft Excel.M 兲兲 (9b) their values at the working load level are determined through The obtained results are shown in Table 1. and I F is the displacement factor due to horizontal load for a fixed-head pile. and ⫹mmult共  mM . based on Eq. Thus. Redistribution subject to ASCE license or copyright.80. Application to Field Pile Test The spreadsheet calculation procedure is used to calculate the displacement and bending moment profiles of one instrumented Fig. The instrumented pile Pile IN1 at a working load level of Similarly.125. Displacement and back-analysis from the measured pile head displacement.90 kN is selected for analysis. 1 is presented in terms of a set of influence factors I H .25 m with a diameter d⫽0. where I H and I M are. 2.

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

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