The equation for the beam-column must be solved for implementation of the p-y method, and a brief derivation is shown in the following section. An abbreviated version of the equation can be solved by a closed-form method for some purposes, but a general solution can be made only by a numerical procedure. Both of these kinds of solution are presented in this chapter.
2-2-1 Derivation of the Differential Equation
In most instances, the axial load on a laterally loaded pile is of such magnitude that it has a small influence on bending moment. However, there are occasions when it is desirable to include the axial loading in the analytical process. The derivation of the differential equation for a beam-column foundation was presented by Hetenyi (1946) and is shown in the following paragraphs.
The assumption is made that a bar on an elastic foundation is subjected not only to the vertical loading, but also to the pair of horizontal compressive forces Q acting in the center of gravity of the end cross-sections of the bar.
If an infinitely small unloaded element, bounded by two verticals a distance dx apart, is cut out of this bar (see Figure 2-13), the equilibrium of moments (ignoring second-order terms) leads to the equation
(M + dM) – M + Qdy – Vv dx = 0 ...(2-1) or
Vv M
Px
Vv+dVv M+dM x
dx
y
y+dy Px
Vv Vn
S y
x
Vv M
Px
Vv+dVv M+dM x
dx
y
y+dy Px
Vv Vn
S y
x
Figure 2-13 Element of Beam-Column (after Hetenyi, 1946)
=0
−
+ Vv
dx Qdy dx
dM ...(2-2)
Differentiating Equation 2-2 with respect to x, the following equation is obtained
2 0
2 2
2 + − =
dx dV dx
y Qd dx
M
d v
...(2-3)
The following definitions are noted:
4 4 2
2
dx y EI d dx
M
d =
dx p dVv
=
p = –Esy
where Es is equal to the secant modulus of the soil-response curve.
And making the indicated substitutions, Equation 2-3 becomes
2 0
2 4
4 + +E y=
dx y Qd dx
y
EI d s ...(2-4)
The direction of the shearing force Vv is shown in Figure 2-13. The shearing force in the plane normal to the deflection line can be obtained as
Vn = Vv cos S – Q sin S ...(2-5)
Because S is usually small, we may assume the small angle relationships: cos S = 1 and sin S = tan S = dy/dx. Thus, Equation 2-6 is obtained.
dy
Vn will mostly be used in computations, but Vv can be computed from Equation 2-6 where dy/dx is equal to the rotation S.
The ability to allow a distributed force W per unit of length along the upper portion of a pile is convenient in the solution of a number of practical problems. The differential equation then becomes as shown below.
2 0
2 4
4 + −p+W =
dx y Qd dx
y
EId ...(2-7)
where:
Q = axial thrust load in the pile,
y = lateral deflection of the pile at a point x along the length of the pile, p = soil reaction per unit length,
EI = flexural rigidity, and
W = distributed load along the length of the pile.
Other beam formulas that are needed in analyzing piles under lateral loads are:
dx Qdy dx
y EId
Vv = 33 + ...(2-8)
2 2
dx y EId
M = ...(2-9)
and,
dx
S = dy ...(2-10)
where
V = shear in the pile,
M = bending moment in the pile, and
S = slope of the elastic curve defined by the axis of the pile.
Except for the axial load Q, the sign conventions that are used in the differential equation and in subsequent development are the same as those usually employed in the mechanics for
beams, with the axes for the pile rotated 90degrees clockwise from the axes for the beam. The axial load Q does not normally appear in the equations for beams. The sign conventions are presented graphically in Figure 2-14. A solution of the differential equation yields a set of curves such as shown in Figure 2-15. The mathematical relationships for the various curves that give the response of the pile are shown in the figure for the case where no axial load is applied.
The assumptions that are made in deriving the differential equation are:
1. The pile is straight and has a uniform cross section,
2. The pile has a longitudinal plane of symmetry; loads and reactions lie in that plane, 3. The pile material is homogeneous,
4. The proportional limit of the pile material is not exceeded,
5. The modulus of elasticity of the pile material is the same in tension and compression, 6. Transverse deflections of the pile are small,
7. The pile is not subjected to dynamic loading, and 8. Deflections due to shearing stresses are small.
Assumption 8 can be addressed by including more terms in the differential equation, but errors associated with omission of these terms are usually small. The numerical method presented later can deal with the behavior of a pile made of materials with nonlinear stress-strain properties.
y
x
y
x
y
x
y
x
y
x
y
x
y(+) S (+) M (+)
V (+)
p (+)
Q (+)
Deflection (L) Slope (L/L) Moment (F*L)
Shear (F) Soil Resistance (F/L) Axial Force (F)
y
x
y
x
y
x
y
x
y
x
y
x
y(+) S (+) M (+)
V (+)
p (+)
Q (+)
Deflection (L) Slope (L/L) Moment (F*L)
Shear (F) Soil Resistance (F/L) Axial Force (F)
Figure 2-14 Sign Conventions
y S M V p
y S M V p
Figure 2-15 Form of Results Obtained for a Complete Solution
2-2-2 Solution of Reduced Form of Differential Equation
A simpler form of the differential equation results from Equation 2-4, if the assumptions are made that no axial load is applied, that the bending stiffness EI is constant with depth, and that the soil modulus Es is constant with depth and equal to α. The first two assumptions can be satisfied in many practical cases; however, the last of the three assumptions is seldom or ever satisfied in practice.
The solution shown in this section is presented for two important reasons: (1) the resulting equations demonstrate several factors that are common to any solution; thus, the nature of the problem is revealed; and (2) the closed-form solution allows for a check of the accuracy of the numerical solutions that are given later in this chapter.
If the assumptions shown above are employed and if the identity shown in Equation 2-11 is used, the reduced form of the differential equation is shown as Equation 2-12.
EI E EI
s
4 4
4 = α =
β ...(2-11)
0 4 4
4
4 + y =
dx y
d β ...(2-12)
The solution to Equation 2-12 may be directly written as:
)
The coefficients C1,C2,C3, and C4 must be evaluated for the various boundary conditions that are desired. A pile of any length is considered later but, if one considers a long pile, a simple set of equations can be derived. An examination of Equation 2-13 shows that C1 andC2 must approach zero because the term eβx will increase without limit.
The boundary conditions for the top of the pile that are employed for the solution of the reduced form of the differential equation are shown by the simple sketches in Figure 2-16. A more complete discussion of boundary conditions for a pile is presented in the next section. The boundary conditions at the top of the pile selected for the first case are illustrated in Figure 2-16(a) and in equation form are:
at x = 0,
The differentiations of Equation 2-13 are made and the substitutions indicated by Equation 2-14 yield the following.
4 2
2EIβ
C −Mt
= ...(2-16)
The substitutions indicated by Equation 2-15 yield the following.
4 3
3 2EIβ
C P
C + = t ...(2-17)
y y y
Spring (takes no shear, but restrains pile head rotation)
Mt
Pt Pt Pt
Free-head Fixed-Head Partially Restrained
y y y
Spring (takes no shear, but restrains pile head rotation)
Mt
Pt Pt Pt
Free-head Fixed-Head Partially Restrained
(a) (b) (c) Figure 2-16 Boundary Conditions at Top of Pile
Equations 2-16 and 2-17 are used and expressions for deflection y, slope S, bending moment M, shear V, and soil resistance p can be written as shown in Equations 2-18 through 2-22.
⎥⎦⎤
It is convenient to define some functions that make it easier to write the above equations.
These are:
A1 = e–βx ( cosβx + sinβx) ...(2-23) B1 = e–βx ( cosβx – sinβx) ...(2-24) C1 = e−βx cosβx ...(2-25) D1 = e−βx sinβx...(2-26)
Using these functions, Equations 2-18 through 2-22 become:
2 1
1 2
2 B
EI C M
y Pt t
β α
β +
= ...(2-27)
1 1
2 2
EI C A M
S Pt t
β α
β −
= − ...(2-28)
1
1 M A
P D
M = t + t
β ...(2-29) V = PtB1 – 2MtβD1 ...(2-30) p = –2PtβC1 – 2Mtβ2B1 ...(2-31)
Values for A1, B1, C1, and D1, are shown in Figure 2-17 as a function of the nondimensional distance βx along the pile.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
βx
A1, B1, C1, D1
A1 B1 C1 D1
Figure 2-17 Values of A1, B1, C1, D1
For a pile whose head is fixed against rotation, as shown in Figure 2-16(b), the solution may be obtained by employing the boundary conditions as given in Equations 2-32 and 2-33.
At x = 0, =0
dx
dy ...(2-32)
Pt
dx y
EI d33 = ...(2-33)
Using the procedures as for the case where the boundary conditions were as shown in Figure 2-4(a), the results are as follows.
4 3
3 4EIβ
C P
C = = t ...(2-34)
The solution for long piles is given in Equations 2-35 through 2-39.
A1
y Pt α
= β ...(2-35)
2 1
2 D
EI S Pt
− β
= ...(2-36)
2P B1
M t
− β
= ...(2-37)
V = Pt C1...(2-38)
p = –PtβA1...(2-39)
It is sometimes convenient to have a solution for a third set of boundary conditions describing the rotational restraint of the pile head, as shown in Figure 2-16(c). For this boundary condition, the rotational spring does not take any shear, but does restrain the rotation of the pile head. These boundary conditions are given in Equations 2-40 and 2-41. At the pile head, where x
= 0, the rotational restrain is controlled by
t t
S M dx
dydx y EId
2 =
2
...(2-40)
and the pile-head shear force is controlled by
Employing these boundary conditions, the coefficients C3 and C4 can be evaluated, and the results are shown in Equations 2-42 and 2-43. For convenience in writing, the rotational restraint Mt /St is given the symbol kθ.
These expressions can be substituted into Equation 2-13, differentiation performed as appropriate, and substitution of Equations 2-23 through 2-26 will yield a set of expressions for the long pile similar to those in Equations 2-27 through 2-31 and 2-35 through 2-39.
Timoshenko (1941), S. P. “ stated that the solution for the “long” pile is satisfactory where βL is greater than 4; however, there are occasions when the solution of the reduced differential equation is desired for piles that have a nondimensional length less than 4. The solution can be obtained by using the following boundary conditions at the tip of the pile. At x = L,
When the above boundary conditions are used, along with a set for the top of the pile, the four coefficients C1, C2, C3, and C4 can be evaluated. The solutions are not shown here, but new values of the parameters A1, B1 ,C1, and D1 can be computed as a function of βL. Such computations, if carried out, will show readily the influence of the length of the pile.
The reduced form of the differential equation will not normally be used for the solution of problems encountered in design; however, the influence of pile length and other parameters can be illustrated with clarity. Furthermore, the closed-form solution can be used to check the accuracy of the numerical solution shown in the next section.
2-2-3 Solution by Finite Difference Equations
The solution of Equation 2-7 is necessary for dealing with numerous problems that are encountered in practice. The formulation of the differential equation in finite difference form and a solution by iteration mandates a computer program. In addition, the following improvements in the solutions shown in the previous section are then possible.
• The effect of the axial load on deflection and bending moment can be considered and problems of pile buckling can be solved.
• The bending stiffness EI of the pile can be varied along the length of the pile.
• And perhaps of more importance, the soil modulus Es can vary with pile deflection and with distance along the pile.
• Soil displacements around the pile due to slope movements or seepage forces can be taken into account.
In the finite difference formulations, the derivative terms are replaced by algebraic expressions. The following central difference expressions have errors proportional to the square of the increment length h.
h
If the pile is subdivided in increments of length h, as shown in Figure 2-18, the governing differential equation, Equation 2-7, in difference form with collected terms for y is as follows:
ym+2
Figure 2-18 Representation of deflected pile
0
The assumption is implicit in Equation 2-46 that the magnitude of Q is constant with depth. Of course, that assumption is not strictly true. However, experience has shown that the maximum bending moment usually occurs a relatively short distance below the ground line at a point where the value of Q is undiminished. This fact plus the fact that Q, except in cases of buckling, has little influence on the magnitudes of deflection and bending moment, leads to the conclusion that the assumption of a constant Q is generally valid. For the reasons given, it is thought to be unnecessary to vary Q in Equation 2-46; thus, a table of values of Q as a function of x is not required.
If the pile is divided into n increments, n+1 equations of the sort as Equation 2-46 can be written. There will be n+5 unknowns because two imaginary points will be introduced above the top of the pile and two will be introduced below the bottom of the pile. If two equations giving boundary conditions are written at the bottom and two at the top, there will be n+5 equations to solve simultaneously for the n+5 unknowns. The set of algebraic equations can be solved by matrix methods in any convenient way.
The two boundary conditions that are employed at the bottom of the pile involve the moment and the shear. If the possible existence of an eccentric axial load that could produce a moment at the bottom of the pile is discounted, the moment at the bottom of the pile is zero. The assumption of a zero moment is believed to produce no error in all cases except for short rigid piles that carry their loads in end bearing, and when the end bearing is applied eccentrically. (The case where the moment at the bottom of a pile is not equal to zero is unusual and is not treated by the procedure presented herein.) Thus, the boundary equation for zero moment at the bottom of the pile requires
0 2 0 1
1− + =
− y y
y ...(2-47)
where y0 denotes the lateral deflection at the bottom of the pile. Equation 2-47 is expressing the condition that EI(d2y/dx2) = 0 at x = L (The numbering of the increments along the pile starts with zero at the bottom for convenience).
The second boundary condition involves the shear force at the bottom of the pile. The assumption is made that soil resistance due to shearing stress can develop at the bottom of a short pile as deflection occurs. It is further assumed that information can be developed that will allow V0, the shear at the bottom of the pile, to be known as a function of y0 Thus, the second equation for the zero-shear boundary condition at the bottom of the pile is
(
2 1 1 2) (
1 1)
03 0
2 2
2 2 y y V
h y Q y y h y
R − − − + − + − − = ...(2-48)
Equation 2-48 is expressing the condition that there is some shear at the bottom of the pile or that EI(d3y/dx3) + Q(dy/dx) = V0 at x = L. The assumption is made in these equations that the pile carries its axial load in end-bearing only, an assumption that is probably satisfactory for short piles for which V0 would be important. The value of V0 should be set equal to zero for long piles (2 or more points of zero deflection along the length of the pile).
As noted earlier, two boundary equations are needed at the top of the pile. Four sets of boundary conditions, each with two equations, have been programmed. The engineer can select the set that fits the physical problem.
Case 1 of the boundary conditions at the top of the pile is illustrated graphically in Fig 2-19. (The axial load Q is not shown in the sketches, but Q is assumed to be acting at the top of the pile for each of the four cases of boundary conditions.). For the condition where the shear at the top of the pile is equal to Pt, the following difference equation is employed.
+Mt
Figure 2-19 Case 1 of Boundary Conditions
(
2 1 1 2) (
1 1)
For the condition where the moment at the top of the pile is equal to Mt, the following difference equation is employed.
(
1 1)
Case 2 of the boundary conditions at the top of the pile is illustrated graphically in Figure 2-20. The pile is assumed to be embedded in a concrete foundation for which the rotation is known. In many cases, the rotation can be assumed to be zero, at least for the initial solutions.
Equation 2-49 is the first of the two equations that are needed. The second of the two needed equations reflects the condition that the slope St at the top of the pile is known.
yt
Figure 2-20 Case 2 of Boundary Conditions
h
Case 3 of the boundary conditions at the top of the pile is illustrated in Figure 2-21. It is assumed that the pile continues into the superstructure and becomes a member in a frame. The solution for the problem can proceed by cutting a free body at the bottom joint of the frame. A moment is applied to the frame at that joint, and the rotation of the frame is computed (or estimated for the initial solution). The moment divided by the rotation, Mt/St, is the rotational restraint provided by the superstructure and becomes one of the boundary conditions. The boundary condition has proved to be useful in some cases.
yt
Pile extends above ground surface and in effect becomes a column in the superstructure
Pile extends above ground surface and in effect becomes a column in the superstructure
Figure 2-21 Case 3 of Boundary Conditions
To implement the boundary conditions in Case 3, it may be necessary to perform an initial solution for the pile, with an estimate of Mt/St, to obtain a preliminary value of the moment at the bottom joint of the superstructure. The superstructure can then be analyzed for a more accurate value of Mt/St, and then the pile can be re-analyzed. One or two iterations of this sort should be sufficient in most instances.
Equation 2-49 is the first of the two equations that are needed for Case 3. The second equation expresses the condition that the rotational restraint Mt/St is known.
( )
Case 4 of the boundary conditions at the top of the pile is illustrated in Figure 2-22. It is assumed, for example, that a pile is embedded in a bridge abutment that moves laterally a given
bending moment is known. If the embedment amount is small, the bending moment is frequently assumed to be zero. The first of the two equations expresses the condition that the moment Mt at the pile head is known, and Equation 2-50 can be employed. The second equation merely expresses the fact that the pile-head deflection is known.
yt = Yt...(2-53)
yt yt-1 yt-2 yt+1 yt+2 Foundation
moves laterally
Mt
h Pile-head moment is
known, may be zero
yt yt-1 yt-2 yt+1 yt+2 Foundation
moves laterally
Mt
h Pile-head moment is
known, may be zero
Figure 2-22 Case 4 of Boundary Conditions
Case 5 of the boundary conditions at the top of the pile is illustrated in Figure 2-23. Both the deflection yt the rotation St at the top of the pile are assumed to be known. This case is related to the analysis of a superstructure because advanced models for structural analyses have been recently developed to achieve compatibility between the superstructure and the foundation. The boundary conditions in Case 5 can be conveniently used for computing the forces at the pile head in the model for the superstructure. Equation 2-53 can be used with a known value of yt and Equation 2-51 can be used with a known value of St.
The five sets of boundary conditions at the top of a pile should be adequate for virtually any situation but other cases can arise. However, the boundary conditions that are available in LPile, with a small amount of effort, can produce the required solutions. For example, it can be assumed that Pt and yt are known at the top of a pile and constitute the required boundary condi-tions (not one of the four cases). The Case 4 equacondi-tions can be employed with a few values of Mt
being selected, along with the given value of yt. The computer output will yield values of Pt. A simple plot will yield the required value of Mt that will produce the given boundary condition, Pt. LPile solves the difference equations for the response of a pile to lateral loading.
Solutions of some example problems are presented in the User’s Manual. Also, case studies are included in which the results from computer solutions are compared with experimental results.
Because of the obvious approximations that are inherent in the difference-equation method, a discussion is provided of techniques for the verification of the accuracy of a solution that is essential to the proper use of the numerical method. The discussion will deal with the number of
Because of the obvious approximations that are inherent in the difference-equation method, a discussion is provided of techniques for the verification of the accuracy of a solution that is essential to the proper use of the numerical method. The discussion will deal with the number of