Estimating Laterally Loaded Pile Response
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1 Estimating Laterally Loaded Pile Resonse C. Y. Lee, PhD Deartment of Civil Engineering College of Engineering Universiti Tenaga Nasional ABSTRACT A simlified and ractical aroach for estimating the laterally loaded iles in an elastic homogeneous soil is resented. The formulation of the aroach is similar to the conventional subgrade reaction theory but the resonse of the soil mass is evaluated by a semi-analytical solution that is related to the actual soil roerties and the ile geometry instead of the conventional subgrade reaction modulus. Modified closed-form solutions for a free-head ile embedded in an elastic homogeneous soil are resented. The solutions comuted by using the roosed simlified aroach and the more rigorous boundary element aroach for free-head iles embedded in homogeneous soils are in reasonably good agreement. Keywords: Piles, Lateral Loading, Homogeneous Soil, Elastic Solutions
2 INTRODUCTION In designing ile foundations to resist lateral loads, the critical factor in some cases is the maximum ile deflection instead of the ultimate of the lateral caacity esecially for long and relatively flexible iles. The common methods of analysing the behaviour of laterally loaded iles are the subgrade reaction theory and the more rigorous continuum aroaches. The subgrade reaction theory, which idealizes a ile as an elastic transversely loaded beam suorted by a series of discrete linear srings reresenting the soil, is the most common aroach used in the analysis of laterally loaded iles (Hetenyi, 1946). The main attraction of the subgrade reaction theory is its simlicity and the relatively straightforward comutations, but its main disadvantage lies in the difficulty of choosing an aroriate subgrade reaction modulus for a given combination of ile size and soil tye. It is usually estimated by emirical correlation that may lead in some cases to uncertainties and inaccurate solutions. The behaviour of laterally loaded iles may also be analysed by the more rigorous continuum aroaches. The most versatile and owerful continuum aroach is the finite element aroach (Randolh 1981, Chow 1987 and Hull 1987), which is relatively more exensive and cumbersome in data rearation. The modified boundary element aroach (Poulos & Davis 1980) is a more ractical aroach which emloys the analytical oint load solution of Mindlin (1936) in an elastic homogeneous half-sace. The boundary integral aroach (Banerjee & Davies 1978) utilizes the fundamental solution for oint loads acting at the interface of two-layer elastic half-sace, while Lee and Small (1991) have suggested an efficient and economical finite layer aroach for iles embedded in layered isotroic and cross-anisotroic soils. This aer resents a simlified and ractical aroach for estimating the laterally loaded ile resonse. The aroach is basically similar to the conventional subgrade reaction theory, excet that the resonse of the ile is evaluated exlicitly by a semi-analytical solution, which is related to the actual soil roerties and the ile geometry instead of the subgrade reaction modulus. Modified closed-form solutions for a free-head ile embedded in an elastic homogeneous soil are resented. The validity of the resent aroach is verified by comaring the results with solutions comuted by more rigorous boundary element aroach.
3 METHOD OF ANALYSIS The governing equation for the deflection of a laterally loaded ile using the subgrade reaction theory is exressed as: 4 u E I + k d u = 0 4 h (1) z where E = ile Young s modulus, I = second moment of area of ile section, u= ile deflection, z= deth in the soil, d= diameter of ile and k h = subgrade reaction modulus. Equation (1) may be solved either analytically or numerically. Hetenyi (1946) has obtained closed form solutions for equation (1). There have been many emirical correlations being suggested to relate k h to the soil modulus E s (e.g. Skemton 1951, Terzaghi 1955, Rowe 1956, McClelland & Focht 1958, Broms 1964, Davisson 1970, Baguelin et al 1977, Pyke & Beikae 198 and Habibagahi & Langer 1983). These correlations, which were usually derived from secific tests, deend on the ile and soil tye. The average deflection u of a ile element subjected to a uniform lateral ressure may be exressed as follows: d = ξ E s u or k d = ξ () h E s where 8π (1 ν s ) ξ = (Glick, 1948) L 1.13(1 + ν s )(3 4ν s ) ln d E s = soil modulus, ν s = soil Poisson s ratio and L= embedded ile length. Substituting equation () into (1), 4 u E I + ξ E u = 0 4 s (3) z
4 The solutions for equation (1) derived by Hetenyi (1946) may be modified by relacing (k h d) with (ξe s ) for equation (3) and the solutions for a freehead ile resonse shown in Figure 1 are exressed as follows: (a) Alied Head Load H Hβ sinh β Lcos β z cosh β ( L z) sin β Lcosh β z cos β ( L z) (i) u = ξe s sinh β L sin β L (4) sinh β L(sin β z cosh β ( L z) Hβ 1 (ii) + cos β z sinh β ( L z)) θ = (5) ξe sinh sin + sin L(sinh z cos ( L z) s β L β L β β β + cosh β z sin β ( L z)) (iii) M H sinh β Lsin β z sinh β = β sinh ( L z) sin β Lsinh β z sin β ( L z) βl sin βl (6) (b) Alied Head Moment M o M oβ (i) u = ξ Es sinh 1 βl sin sinh β L(cosh β ( L z)sin β z sinh β ( L z)cos β z) βl + sin βl(sinh β z cos β ( L z) cosh β z sin β ( z L)) (7) (ii) 3 4M oβ sinhβlcoshβ( L z)cosβ z + sin βlcoshβz cosβ( L z) = (8) ξ Es sinh βl sin βl θ
5 M sinhβl o (iii) M = sinh βl sin βl sinβl ( sinhβ( L z)cosβ z+ coshβ( L z)sinβz) ( ) sinhβ zcosβ( L z) + coshβz sinβ( L ) z (9) where 4 ξes β = E I 1 4, u=ile deflection, θ= ile rotation and M=ile moment. Figure 1 : Definition of Laterally Loaded Pile Problem
6 COMPARISON WITH MORE RIGOROUS BOUNDARY ELEMENT SOLUTIONS The validation of the resent aroach is assessed by comarison with the linear elastic solutions from the more rigorous boundary element aroach BE (Poulos & Davis 1980) for free-head iles embedded in an elastic homogeneous soil. Figure shows the effect of ile flexibility factor (K R =E I /E s L 4 ) and ile slenderness ratio (L/d) on the normalized ile head deflection (u o E s L/H). The resent solutions are comarable with those comuted by the boundary element aroach excet for very flexible iles in which the resent aroach comuted higher ile head deflections. Figure : Normalized Head Deflection Resonse for Free-Head Pile
7 The influence on normalized head rotation with resect to moment (which is also equivalent to head rotation θ o normalized with resect to load) of the ile flexibility factor and ile slenderness ratio is shown in Figure 3. The solutions comuted by both aroaches are in close agreement excet for very flexible iles. Figure 3 : Normalized Head Deflection (or Rotation) Resonse for Free-Head Pile Figure 4 illustrates the effect on the normalized head rotation with resect to a moment M o of the ile flexibility factor and ile slenderness ratio. The solutions comuted by both aroaches are similar. Figure 5 shows the effect of ile slenderness ratio on tyical ile deflection rofile along a free-head ile. The resent aroach comuted slightly lower ile deflections than those comuted by the boundary element aroach. The variation of moment distribution along a ile with ile flexibility factor and slenderness ratio is deicted in Figure 6. For horizontal head loading only, the maximum moment tyically occurs at a deth of between 0.1L and 0.4L below the ground surface, the greater deth occurring for stiffer iles. For moment loading only, the maximum moment always occur at the ground surface and equals to the alied moment. The moment distributions along a ile comuted by the resent aroach and boundary element aroach are in good agreement.
8 Figure 4 : Normalized Head Rotation Resonse for Free-Head Pile Figure 5 : Deflection Profile Along Free-Head Pile
9 Figure 6 : Moment Profile Along Free-Head Pile The deflection and rotation for very flexible or slender iles comuted by the boundary element aroach are generally lower than those comuted by the resent aroach. Because of the limited number of elements used in the boundary element aroach, the solutions may be somewhat inaccurate for iles that are very slender or flexible, and may lead to underestimates of deflection and rotation (Poulos &Davis 1980). CONCLUSIONS A simlified and ractical aroach for estimating the resonse of laterally loaded iles embedded in an elastic homogeneous soil is resented. Although the formulation of the resent aroach is basically similar to the conventional subgrade reaction theory the resonse of the soil is evaluated by a semi-analytical solution, which is related to the actual soil roerties and the ile geometry instead of the conventional subgrade reaction modulus. Modified closed-form solutions for a free-head ile embedded in an elastic homogeneous soil are resented. Desite some differences in magnitude between the resent aroach and the more rigorous boundary element aroach for very slender or flexible iles, the overall results are in reasonably good agreement.
10 REFERENCES Baguelin, F., Frank, R. and Said, Y.M., 1977 Theoretical Study of Lateral Reaction Mechanism of Piles. Geotechnique, Vol. 7, No. 3, ,. Banerjee, P.K. and Davies, T.G., The Behavior of Axially and Laterally Loaded Single Piles embedded in Non-Homogeneous Soils. Geotechnique, Vol. 1, No. 3, Broms, B.B., 1964 The Lateral Resistance of Piles in Cohesive Soils. J. Soil Mech. Fdns. Div., ASCE, 90, SM, Part I, Chow, Y.K., 1987 Axial and Lateral Resonse of Pile Grous Embedded in Non- Homogeneous Soils. Int. J. for Numerical and Analytical Methods in Geomechanics, Vol. 11, Davisson, M.T., 1970 Lateral Load Caacity of Piles. National Academy of Sciences, Highway Research Record, No. 333, Washington, D.C., Glick, G.W., 1948 Influence of Soft Ground on the Design of Long Piles. Proc. Int. Conference On Soil Mechanics and Foundation Engineering, Vol. IV, Habibagahi, K. and Langer, J.A., 1983 Horizontal Subgrade Modulus of Granular Soils. Symosium on Laterally Loaded Dee Foundations: Analysis and Performance, ASTM, Hetenyi, M., 1946 Beams on Elastic Foundations. University of Michigan Press, Ann Arbor, Michigan. Hull, T.S., 1987 The Behavior of Laterally Loaded Piles. Ph.D. Thesis, University of Sydney, Australia. Lee, C.Y. and Small, J.C., Finite Layer Analysis of Laterally Loaded Piles in Cross-Anisotroic Soils. Int. J. for Numerical and Analytical Methods in Geomechanics, Vol. 15, Matlock, H., 1970 Correlations for Design of Laterally Loaded Piles in Soft Clay. Proc. Offshore Technology Conf., Paer 104, McClelland, B. and Focht, J.A., 1958 Soil Modulus of Laterally Loaded Piles. Transaction, ASCE, Vol. 13, Mindlin, R.D., 1936 Force at a Point in the Interior of a Semi-Infinite Solid. Physics, Vol. 7, Poulos, H.G. and Davis, E., 1980 Pile Foundation Analysis and Designs. Wiley, New York. Pyke, R. and Beikae, M., 198 A new Solution for the Resistance of Single Piles to Lateral Loading. Symosium on Laterally Loaded Dee Foundations: Analysis and Performance, ASTM, Randolh, M.F., 1981 The Resonse of Flexible Piles to Lateral Loading. Geotechnique, Vol. 31, No., Rowe, P.W., 1956 The Single Pile Subjected to Horizontal Force. Geotechnique, 6,
11 Skemton, A.W., 1951 The Bearing Caacity of Clays. Proc. Building Research Congress, Division I, London, Part III, Terzaghi, K., 1955 Evaluation of Coefficients of Subgrade Reaction. Geotechnique, Vol. 5, No. 4,
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