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Discrete Layer Analysis of Laterally Loaded Piles
C. Y. Lee Department of Civil Engineering, College of Engineering, Email: cylee@uniten.edu.my
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ABSTRACT
A discrete layer analysis of laterally loaded piles embedded in homogeneous and non-homogeneous soils is presented. The soil mass is divided into uncoupled discrete soil layers. Basically, the formulation of the analysis is similar to the subgrade reaction theory but the response of the discrete soil layers is evaluated by a semi-analytical solution, which is related to the actual soil properties and the pile geometry instead of the conventional subgrade reaction modulus. The non-linear response of the soil around the pile is represented by a simple hyperbolic soil model. The linear elastic solutions computed by the discrete layer and the more rigorous continuum approaches are compared and are found to be in satisfactory agreement. The computed non-linear response of two piles embedded in stiff clay compares favorably with field measurements.
KEYWORDS: laterally loaded piles, discrete layer analysis, linear and non-linear response.
INTRODUCTION
The subgrade reaction theory, which idealizes a pile as an elastic transversely loaded beam supported by a series of discrete linear springs representing the soil, is the most common approach used in the analysis of laterally loaded pile (Hetenyi 1946). The p-y analysis is also widely used in analyzing the non-linear load-deflection behavior of the piles (Matlock and Reese 1961, Matlock 1970, Reese et al 1975 and Reese and Welch 1975). The main attractions of these two approaches are their simplicity and the relatively straightforward computations, but their main disadvantage lies in the difficulty of choosing an appropriate subgrade reaction modulus or p-y relationship for a given combination of pile size and soil type. These are usually estimated by empirical correlations that may lead in some cases to uncertainties and inaccurate solutions.
The pile behavior may also be analyzed by some more rigorous continuum approaches, the most versatile and powerful being the finite element approach (Randolph 1981, Chow 1987 and Hull 1987) which is relatively more expensive and cumbersome in data preparation. The modified boundary element approach (Poulos and Davis 1980) is a more practical approach which employs the analytical point load solution of Mindlin (1936) in an elastic homogeneous half-space and the effect of soil inhomogeneity is approximated by using some averaging process to obtain the soil moduli. The boundary integral approach (Banerjee and Davies 1978) utilizes the fundamental solution for point loads acting at the interface of two-layer elastic half-space, while Lee and Small (1991) have suggested an efficient and economical finite layer approach for piles embedded in layered isotropic and cross-anisotropic soils. All the above continuum approaches consider the interaction between soil elements along a pile, and this results in fully populated soil influence factor matrices. Hence the computer storage and computing time requirements are substantially more than the subgrade reaction approach. These continuum approaches are mainly limited to linear elastic behavior although allowance can be made for some soil non-linearity in the boundary element approach by considering the yielding of soil elements.
This paper presents a practical and efficient discrete layer approach for the analysis of laterally loaded piles. The soil mass may be divided into discrete and uncoupled layers and the formulation of the approach is basically similar to the conventional subgrade reaction theory, except that the response of these discrete soil layers is evaluated explicitly by a semi-analytical solution which is related to the actual soil properties and the pile geometry instead of the subgrade reaction modulus. An important feature of the proposed method is that the non-linear soil response is represented by a simple hyperbolic soil model. The validity of the present approach is verified by comparing the results with solutions computed by more rigorous continuum approaches. Finally, the present approach is used to compute the behavior of two field piles embedded in stiff clay.
METHOD OF ANALYSIS
The conventional subgrade reaction theory for the deflection of a laterally loaded pile embedded in a soil mass is governed by the equation:
![]() | (1) |
where Ep = pile Young’s modulus, Ip = second moment of area of pile section, y = pile deflection, z = depth in the soil, d = width or diameter of pile and kh= subgrade reaction modulus.
In the past few decades, there have been many empirical correlations being suggested to relate kh to the soil modulus Es (e.g. Skempton 1951, Terzaghi 1955, Rowe 1956, McClelland and Focht 1958, Broms 1964, Davisson 1970, Poulos 1971a, Baguelin et al 1977, Pyke and Beikae 1982 and Habibagahi and Langer 1982). These correlations, which were usually derived from specific tests, depend on the pile and soil type. Douglas and Davis (1964) have derived analytical expressions for the deflections at the corners of a thin vertical rectangular element embedded in a semi-infinite mass subjected to a uniform horizontal pressure. These expressions may also be used to evaluate the central deflection of the rectangular element by a superposition technique. The central deflection yc of the rectangular element subjected to a uniform horizontal pressure p may be expressed as follows:
![]() | (2) |
where Es = soil modulus, d = width of the rectangular element and F = dimensionless factor which may be evaluate explicitly by using the expressions derived by Douglas and Davis (1964).
If equation (2) is used to represent the deflection of a pile element subjected to the same horizontal uniform pressure, and it is expected to be underestimated since the additional deflections caused by the interaction of the remaining elements in the pile are not considered. Indeed, it is found empirically from the study of the theoretical solutions of laterally loaded piles obtained by finite element (Randolph 1981, Chow 1987 and Hull 1987) and finite layer (Lee and Small 1991) approaches that the value of F from equation (2) needs to be doubled to achieve adequate approximation in the pile analysis. Hence equation (2) may be re-expressed as follows:
![]() | (3) |
where x=1/(2F).
If the soil stiffness per unit length (khd) is assumed to be equal to xEs then equation (1) may be re-expressed as follow:
![]() | (4) |
Equation (4) forms the basis of the analysis of a laterally loaded pile in this paper. In the present analysis, the pile may be considered to be supported by a series of uncoupled discrete soil layers as shown schematically in Figure 1. These discrete soil layers may be represented by linear or non-linear soil models, which will be described below. This analysis avoids the main difficulty in choosing an appropriate subgrade reaction modulus in the conventional spring model. Since it retains the similar subgrade reaction theory formulation (i.e. the discrete soil layers result in only a diagonal ‘soil matrix’ in equation (4)), it represents a versatile and practical approach for routine analysis.
Figure 1. Discrete Layer Analysis of Laterally Loaded Pile
The pile is represented by discrete two-noded elastic cylindrical solid beam elements. The soil pressures are assumed acting on the pile segments and lumped at the pile nodes. The pile load-deflection relationship may be expressed as:
![]() | (5) |
where [Kp]=pile stiffness matrix, {yp}=pile nodal deflection and rotation vector, {Hp}=pile internal force vector and {H}=external applied nodal load and moment vector.
The load-deflection relationship for the discrete soil layers is expressed as:
![]() | (6) |
where {Hs}=internal soil load vector, [Ks]=diagonal soil stiffness matrix and {ys}=soil deflection vector.
The equilibrium of the internal pile-soil forces and compatibility of the pile-soil displacement yield the following equations respectively:
![]() | (7) |
![]() | (8) |
Using equations (5), (6), (7) and (8), the load-deflection relationship of the pile may be expressed as:
![]() | (9) |
The linear elastic soil stiffness Ksi for the nodes at the pile is given as:
![]() | (10) |
where Esi = soil modulus at node i and Isi = soil influence factor at node i derived from equation (3).
Non-Linear Soil-Pile Response
The non-linear soil-pile response is represented in this model by a simple hyperbolic soil model and the resultant soil tangent stiffness Ksti at the nodes is given as:
![]() | (11) |
where Esti = initial tangent soil modulus at node i, pi = soil pressure at node i, pfi = limiting pressure at node i and Rf = hyperbolic soil parameter.
COMPARISON WITH MORE RIGOROUS CONTINUUM SOLUTIONS
The validation of the present approach is assessed by comparison with the linear elastic solutions from the more rigorous modified boundary element BE (Poulos and Davis 1980), boundary integral BI (Banerjee and Davies 1978), infinite layer IL (Guo et al 1987), finite element FE (Randolph 1981, Chow 1987 and Hull 1987) and finite layer FL (Lee and Small 1991) approaches. In the present approach, the pile was discretized into 30 elements and all of the analyses were performed on a personal computer.
The effect of pile slenderness (L/d where L=embedded pile length and d=pile diameter) on the normalized head deflection yo due to a horizontal load H on a free-head pile embedded in homogeneous and non-homogeneous soils is shown in Figure 2. The present solutions are consistent with those by the boundary element approach and both indicate that the normalized head deflections increase with increasing slenderness ratio.
Figure 2. Effect of L/d on Free-Head Pile Response
The influence on normalized deflection of the pile flexibility factor for homogeneous (KR = EpIp/EsL4) and non-homogeneous soils (‘Gibson soil’, KN = EpIp/EsLL4 where EsL=soil modulus at pile base) is depicted in Figure 3. In homogeneous soils, the present solutions are close to those from the finite element and finite layer approaches except for very flexible piles where the present solutions are closer to those by the boundary integral approach. The boundary element solutions are much higher than the other solutions in the non-homogeneous soils and the present solutions agree reasonably well with those from the finite element, boundary integral and finite layer approaches.
Figure 3. Normalized Head Deflection Response for Free-Head Piles
Figure 4 illustrates the variation of the deflection normalized with respect to moment (which is also equivalent to head rotation qo normalized with respect to load) in both homogeneous and non-homogeneous soils. The present solutions are comparable with those from the finite element and finite layer approaches except for very flexible piles where the present approach computes higher deflections in homogeneous soils. The boundary element solutions are generally higher than the other solutions in non-homogeneous soils whereas the finite element, finite layer and present solutions are in good agreement except for very flexible piles where the present solutions are closer to those by the boundary element approach.
Figure 4. Normalized Head Deflection (or Rotation) Response for Free-Head Piles
The effect on the normalized head rotation qo with respect to a moment M of the pile flexibility factor for a free-head pile is plotted in Figure 5. In both homogeneous and non-homogeneous soils, the present solutions agree reasonably well with those from the more rigorous finite element and finite layer approaches.
Figure 5. Normalized Head Rotation Response for Free-Head Piles
Figure 6 shows the influence on the normalized head deflection of Ep/Es and Ep/Esr (where Esr is the soil modulus at the depth of one pile radius, r) in homogeneous and non-homogeneous soils respectively for a fixed-head pile. The normalized head deflections computed by the finite element (Chow, 1987) and present approach in homogeneous soils agree well whereas the other approaches give higher values. In non-homogeneous soils, the present solutions lie between those from the boundary element approach and the other approaches.
Figure 6. Normalized Head Deflection Response for Fixed-Head Piles
COMPARISON WITH FIELD MEASUREMENTS
Reese et al (1975) performed a series of field tests on laterally loaded steel-pipe piles that were 15m long driven into stiff clay at a site near Austin in Texas, USA. The instrumented test piles were 625mm and 170mm in diameter. The wall thickness of the top 7m of the 625mm diameter pile was increased from the original 9.5mm to 25.4mm. The wall thickness of the top 8.5m of the 170mm diameter pile was 18.2mm. The undrained shear strength profile of the test site was investigated extensively by triaxial, unconfined compression and penetrometer tests. The undrained shear strength used in the analysis is taken to be cu = 0.05z MPa where z is the soil depth. For the present analysis, the initial tangent soil modulus and pile-soil limiting pressure are assumed to be Est = 800cu and pf = 9cu respectively. The undrained soil Poisson’s ratio and Rf are taken to be 0.5 and 0.9 respectively.
The measured load-deflection behavior of the test piles is shown in Figure 7. The present solutions are in good agreement with the measured results and it is noticed that they are even better than those computed by the p-y approach (Reese et al 1975). Figure 8 shows that the present approach computes slightly higher maximum bending moments Mm than those measured at higher load levels for the 625mm diameter pile, however the present solutions and measured results are in very good agreement for the 170mm diameter pile.
CONCLUSIONS
A practical and efficient discrete layer approach for the analysis of laterally loaded piles embedded in homogeneous and non-homogeneous soils is presented. Although the formulation of the present approach is basically similar to the conventional subgrade reaction theory, the uncoupled discrete soil layers are represented by a semi-analytical soil model, which is related to the actual soil properties and the geometry instead of the conventional subgrade reaction modulus. The non-linear pile response may be represented by a simple hyperbolic soil model. The present solutions agree satisfactorily with those obtained from the more rigorous continuum approaches. The computed response from the present approach for the two field test piles embedded in stiff clay is in good agreement with the actual measured behavior.
Figure 7. Load-Deflection Response for 625mm and 170mm Piles
Figure 8. Load-Maximum Moment Response for 625mm and 170mm Piles
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