ABSTRACT
In the design of piles, we are frequently bound to study both the pile head deflection and the maximum bending moment. By using the subgrade reaction theory with a simplified p-y soil response, an analytical method is developed for the solution of the pile bending equations for a pile subjected to flexion and horizontal load. This paper describes this analytical method applied to a single pile installed in clay. Predicted responses by this method are compared with experimental results given in literature. A comparison of computed and measured lateral load-deflection response of single pile in clay shows a good agreement. In order to make useful this analytical method graphical non-dimensional relationships are proposed. Lateral deflection at ground level, rotation at ground level and maximum bending moment in pile are given versus the pile lateral load.
KEYWORDS: Pile, Analytical, Lateral load, soil, Elastic-plastic, reaction, clay, deflection, bending moment, graphical.
See Discussion
INTRODUCTION
Single pile and pile groups are often subject to significant lateral loads in addition to axial compressive and uplift loads. The lateral loads on piles are derived from earth pressures, wind pressures, water forces and impact loads from vessels. Structures are often constructed in coastal areas and in the flood plains of rivers.
Over the years, several methods have been proposed to predict the response of laterally loaded piles. The most frequently used is the well-known p-y analysis, which has been included into several codes of practice (Fascicule 62 (M.E.L.T.) 1993, P.H.R.I. 1980, A.P.I. 1993). In this approach, the soil around a pile is simulated by a series of independent non-linear springs, each spring representing the behaviour of a soil layer of unit height. This procedure requires to input series of p-y curves. To derive the load-deformation characteristics of piles we usually need to use numerical methods except in some cases where analytical solutions may be found. Subgrade reaction approach treats a laterally loaded pile as a beam on elastic foundation. It is assumed that the beam is supported by a Winkler soil model according to which the are represented by a series of independent and elastic springs.
This paper presents an example of analytical solution developed to predict the response of laterally loaded piles in clay. In this method the subgrade reaction approach is used with the simplification of linear elastic-perfectly-plastic p-y soil response. In order to verify the validity of this analytical method the results obtained for pile deflection and bending moment are compared with experimental results reported by Poulos (1980). The principal results are presented by graphical non-dimensional relationships which allow an easy evaluation of pile head deflection and maximum bending moment in the pile.
LATERAL SOIL REACTION
It is well known that the soil behavior in the case of piles under lateral load in non-linear. The non-linear behavior of soils in the analysis of laterally loaded piles is generally incorporated using p-y curve approach. Then the relationship between lateral force P on the soil and pile deflection y at any point along a pile is non-linear. A simplest approach to account for this nonlinearity is to consider the elastoplastic model. This model, used in the present method, is defined by a limit lateral force P_{u} below which the soil reaction is linear and then the soil pressure is expressed as P(z) = E_{s}y(z), where E_{s} is the modulus of lateral subgrade reaction of the soil and y is the lateral displacement of the pile. This model needs to define the lateral limit load and the modulus of lateral subgrade reaction of the soil.
It is usually assumed that the lateral limit load for cohesive soils increases from 2c_{u}d to 3c_{u}d at ground level up to 8c_{u}d to 12c_{u}d at a depth of about 2d to 3d and beyond this depth it remains constant (Matlock, 1970; Reese and al., 1975), c_{u} being the undrained shear strength of the soil and d the diameter or width of pile. According to Broms (1964a) the proposed distribution of P_{u} is shown in Figure 1. The soil resistance in the upper zone is neglected up to a depth equal to 1.5d. Below this depth ultimate lateral load P_{u} is taken equal to 9c_{u}d as in the tip capacity of vertically loaded piles. For the mathematical solution this is represented by taking the oringin, z = 0, at a distance 1.5d below the soil surface, and restricing the soil reaction development with P_{u} (Figure 1).
The conventional modulus of subgrade reaction in cohesive soils is expressed as a function of the undrained shear strength. Banerjee and Davies (1978) backfigured values of E_{s} between 100c_{u} and 180c_{u}. In practice, a possible way to derive E_{s} values is the use of the results of pressuremeter tests (Fascicule 62 (M.E.L.T.) 1993).
MATHEMATICAL FORMULATION
When the pile is subject to lateral load it moves in this direction until the counteracting pressure is distributed in such a way that the equilibrium conditions are satisfied. The resulting distribution of soil reaction against the pile may be decomposed into two zones. A plastic zone which extends to a depth z_{p} which is a function of the horizontal force level and its point of action, and an elastic zone below this depth. The governing equation for the pile deflection at any depth z is given by:
(1) | |
The integration of this equation leads to the following expressions giving the load per unit of length P, the shear force V, the bending moment M, the rotation of pile section w and the lateral deflection y in each zone.
For the plastic zone ():
(2) | |
(3) | |
(4) | |
(5) | |
(6) | |
For the elastic zone ():
(7) | |
Assuming that the pile is long and the soil modulus is independent from depth, the pile deflection is
(8) | |
where b is the reciprocal of the characteristic length l_{0} of the pile:
The expressions that give the rotation w, the bending moment M, the shear force V and the load per unit of length P are found by successive differentiations. The unknown parameters C_{i} and z = z_{p} are obtained by imposing the equilibrium for bending moment and shear force at z = 0 and z = z_{p}, and the compatibility for displacement and slope at z = z_{p}. The last boundary condition is obtained from the fact that at depth z = z_{p}, the rate of shear variation is equal to the ultimate soil resistance/unit length.
The following solution is obtained for pile subject to a lateral force H_{0} and bending moment M_{0} at z = 0.
For load level is such that soil is elastic. Thus z = z_{p} = 0 and deflection and bending moment are given by
(9) | |
(10) | |
The position and value of maximum bending moment are
(11) | |
(12) | |
For z_{p} > 0 a plastic zone develops. The integration parameters are
C_{1} = H_{0} and C_{2} = M_{0} |
(13) |
The depth of the plastic zone in this case is given by
(14) | |
Furthermore,
(15) | |
(16) | |
(17) | |
(18) | |
where
(19) | |
and
(20) | |
The two values C_{3} and C_{4} divided by the bending rigidity EI of pile give respectively rotation and horizontal deflection of pile at z = 0.
According to the two cases the position and value of maximum bending moment are
For the maximum bending moment occurs in elastic zone at depth:
(21) | |
where
(22) | |
For , the maximum bending moment occurs in plastic zone at depth:
(23) | |
and its value is
(24) | |
The lateral deflection of pile at a level L (above z = 0) where the load is applied is given by the following equation
(25) | |
in which y(0) and w(0) are respectively lateral deflection and rotation of pile at z = 0. When the modulus of lateral subgrade reaction is linearly increasing with depth (z) the resolution of equation (1) needs to use numerical methods. This inconvenience is overcome by assuming an average value of Es taken in a significant part of the pile and giving the appropriate pile head displacement in elastic conditions. Based on the above considerations an implicit equation that gives Esm was developed and graphical dimensionless solution is shown in Figure 2.
Figure 1. Pile movement and lateral soil pressure distribution
Figure 2. Curve giving average value of lateral soil modulus
RESULTS AND DISCUSSION
In order to check the suggested formulation, predicted results are compared with some of the available published data obtained from tests. The test results presented in this paper were reported by Poulos (1980). The piles were installed in clay and subject to lateral load. The lateral deflection at ground level was measured. The details of pile and soil materials are given in Table 1. The undrained shear strength c_{u} and the lateral subgrade reaction modulus Es were obtained from plate-bearing tests.
Table 1. Pile and soil properties in Poulos (1980) tests
Test no | Pile diameter (m) | Pile length (m) | Pile bending rigidity EI (kN/m) | Soil shear strength c_{u} (kPa) |
Lateral reaction modulus Es (kPa) |
1 | 0.01 | 3.00 | 622000 | 14.4 | 3500 |
2 | 0.038 | 5.25 | 31600 | 14.4 | 3500 |
3 | .0076 | 5.25 | 246000 | 14.4 | 3500 |
4 | 0.001 | 5.25 | 320000 | 14.4 | 3500 |
Figure 3 shows typical experimental and computed load-deflection curves using the analytical method presented here. The test results and predicted values of deflection were matched at different load levels. The results of this comparison (Figure 3 below) show that the agreement between measured and computed deflection seems to be better for high load level. Nevertheless the predicted responses by this method are reasonably in good agreement with the experimental values.
Figure 3. Comparison of measured and calculated load-deflection curves
------Measured , ---o--- predicted
GRAPHS OF DIMENSIONLESS SOLUTIONS
In the design of piles we are usually interested in the pile head deflection and the maximum bending moment. By using the above solutions, it is possible to define respectively the dimensionless lateral deflection (y_{0} E_{s}/c_{u} d), the pile rotation (w E_{s})/c_{u} at level z = 0 and the corresponding maximum bending moment M_{0} in the pile section. These dimensionless parameters are expressed as a function of the dimensionless applied load and the dimensionless length bL as shown in Figure 4. These practical graphics can be used by engineers to predict pile head deflection and maximum-bending moment in a pile structure project, or to verify the lateral load resistance of existing piles.
In the design of piles we are usually interested in the pile head deflection and the maximum bending moment. By using the above solutions, it is possible to define respectively the dimensionless lateral deflection
y_{0} E_{s} / (c_{u} d),
the pile rotation
w_{0}E_{s} / c_{u}
at level z = 0, and the corresponding maximum bending moment
M_{max} b^{2} / (c_{u} d)
in the pile section. These dimensionless parameters are expressed as a function of the dimensionless applied load
H_{0}b / (c_{u} d)
and the dimensionless length bL as shown in Figure 4. These practical graphics can be used by engineers to predict pile head deflection and maximum-bending moment in a pile structure project, or to verify the lateral load resistance of existing piles.
Figure 4(a) Ground line deflection versus applied load
Figure 4(b) Ground line rotation versus applied load
Figure 4(c) maximum bending moment versus applied load
CONCLUSIONS
A simple analytical formulation has been proposed to predict the response of laterally loaded piles in clay. This method is based on a simplified subgrade modulus approach. The comparison of calculated solutions by this method with measured test data show a good agreement for load-displacement curves. Nevertheless this simplified analytical approach presents more validity for high lateral load levels. Dimensionless relationships to estimate easily pile head deflection and maximum bending moment in pile are presented as curves.
REFERENCES
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