A Practical Approach for Estimation of Lateral Load on Piles under

Earthquake

The 12th International Conference of

International Association for Computer Methods and Advances in Geomechanics (IACMAG)

1-6 October, 2008

Goa, India

A Practical Approach for Estimation of Lateral Load on Piles under

Earthquake

Indrajit Chowdhury

Petrofac International Limited, Sharjah, UAE.

Sambhu.P.Dasgupta

Dept.of Civil Engineering, Indian Institute of Technology (Kharagpur), Kharagpur, India.

Keywords: Dynamic response, Stiffness, Damping, Piles, Lateral load, Partial Embedment.

ABSTRACT: The problem of lateral load induced on piles under earthquake has been plaguing professional

engineers, geo-technical investigators and researchers alike for quite some time. The normal practice is to

ensure that the fixed base shear of column does not exceed the static shear load capacity of piles. The inertial

and stiffness effect of pile is usually ignored in dynamic earthquake analysis. The present paper proposes a

method where based on modal response or time history analysis pile load may be estimated under earthquake,

considering its stiffness, inertia and effect of material and geometric damping properties. The results are then

compared with the conventional methods.

The effect of partial embedment, a situation that may develop under soil liquefaction during earthquake has also

been derived.

The pile loads are estimated for two cases:

1) When the structure is a lumped mass having infinite stiffness: like a machine foundation or a heavy short

vessel supported directly on pile cap.

2) The superstructure has a finite stiffness and mass like a frame (building /pipe rack etc)

The paper also suggests a way of how a geo-technical investigator can estimate rationally the dynamic pile

loading with minimum information available at the outset of a project.

One of the major advantages of the method is that it does not warrant sophisticated software to be developed for

this analysis. A simple spread sheet is sufficient to produce an accurate result.

1 Introduction

Vibration of piles under lateral load is an important study for piles supporting machines and structures under

earthquake loading. In majority of the cases, of all the modes, the lateral vibration is the most critical and often

governs the design during an earthquake. Thus, a study of such motion is of paramount importance for piles

supporting important installations.

A number of researchers have proposed solution to the problem of pile dynamics, namely, Parmelee et al. (1964),

Tajimi (1966), Penzien (1970), Novak et al. (1974, 1983), Banerjee and Sen (1987), Dobry and Gazetas (1988)

only to name a few. However, most of these solutions are based on harmonic analysis and are valid for the

design of machine foundations, where the dynamic stiffness and damping of pile remain frequency dependent.

The application of these theories are though well established for design of machine foundations except for an

approximate method as proposed by Chandrashekaran (1974) and Prakash (1973), a comprehensive analytical

tool to predict the pile response under earthquake load still remains uncertain.

2 The Proposed Method

The present paper deals with a semi-analytic solution for predicting the lateral load on a pile under earthquake

forces. For obtaining the time period vis-a vis the stiffness and mass of the system, one may start with a pile

embedded in homogeneous elastic medium under plane strain condition as shown in Figure 1. To start with, the

pile is taken as long and slender. Under static condition, the equation of equilibrium in the x-direction is given by:

4

p p 4 s

E I d u =-k Du

dz

(1)

in which, Ep = Young’s modulus of the pile; Ip = moment of inertia of the pile cross section; ks = dynamic subgrade

modulus of the soil (kN/m3), u = displacement in the x-direction and D= diameter of the pile.

The general solution of eqn.(1) may be written as:

-pz pz

0 1 2 3 u = e (C cospz + C sinpz) + e (C cospz + C sinpz) (2)

in which 4

s p p p= kD 4E I . (3)

For a long pile subjected to load or moment at its head, it is reasonable to assume that at a significant distance

from the pile head (where the load is applied), the curvature along the pile axis vanishes. This condition can only

46

be satisfied when C2 and C3 in eqn. (2) are considered insignificant and the deflection equation is taken as:

-pz

0 1 u = e (C cospz + C sinpz) . (4)

Considering the pile head undergoing specified deflection and rotation as well as its head is fixed on to the pile

cap, one has the boundary condition at z=0, u=u0 and θ=θ0, where for a small value of θ, θ0 ≅ u0/L. With this

boundary condition it can be shown that, the generic shape function of the pile in dimensionless form for any

arbitrary loading can be written as:

φ(z) = e(-βz)/L⎡⎣cos(βz/L) + ηsin(βz/L)⎤⎦ (5)

in which 4 4

s pp β= kDL 4E I , η =1+1/β and L being the length of the pile.

Potential energy dΠ of an element of depth dz, shown in Figure 1, is then given by [Shames and Dym (1995)]:

2 2

p p h 2

2

dΠ=E I d u +K u

2 dz 2

⎡ ⎤

⎢ ⎥

⎣ ⎦

(6)

in which, Ep = Young’s modulus of pile; Ip = moment of inertia of pile; Kh = lateral dynamic stiffness of soil in

kN/m; u = displacement of the pile in the x direction and may be written as [φ(z) q(t)].

M

P

Soil undergone liquefaction

L H

L1

Bedrock Level.

Figure 1. Conceptual Model of Pile under Lateral Loads

For a rigid circular disc embedded in soil for a depth h the stiffness under earthquake force can be expressed as

(Newmark (1971) and Wolf (1988)):

0

x

0

K = 8Gr 1+ h

(2 - ν) r

⎛ ⎞

⎜⎜ ⎟⎟

⎝ ⎠

(7)

in which, Kx = static foundation stiffness in horizontal direction in kN/m; G = dynamic shear modulus of the soil ;

r0 = radius of the foundation; h= depth of embedment ν=Poisson’s ratio of soil.

Ignoring the first term within parenthesis in eqn. (7) which represents the contribution of base resistance, and

substituting the same in eqn. (6), for a cylindrical element of depth dz embedded in soil, the potential energy Π ,

of the pile of length L ,may be expressed as:

( )

L 2 2 L

p p 2

2

0 0

Π=E I d u dz + 8G u dz

2 dz 22 - ν

⎡ ⎤

⎢ ⎥

⎢⎣ ⎥⎦ ∫ ∫ (8)

Considering u(z, t) = φ(z) q(t), it can be proved (Hurty and Rubenstein(1967)) that:

φ (z)φ (z)dz

(2 ν)

K E I φ (z)φ (z)dz 8G j

L

0

i

L

0

ij p p ∫ i j ∫ −

= ″ ″ + (9)

in which the shape function of the problem is given by eqn. (5).

For the fundamental mode, stiffness of the pile is then given by:

φ(z) dz

(2 ν)

K E I φ (z) dz 8G

L 2

0

L

0

2

ij p p ∫ ∫ −

= ″ + (10)

Eqn. (10) on expansion and simplification finally gives:

∫ ∫ ⎟⎠

⎞

⎜⎝

⎛ + +

−

+ ⎟⎠

⎞

⎜⎝

= ⎛ − −

− L −

0

L

L 2

0

L

2

4

p p

4

pile dz

L

ηsin 2

L

cos 2

2

Y

2

e X

(2 ν)

dz 8G

L

ηsin 2

L

2

cos

2

Y

2

e X

L

4β E I

K

ξβ ξβ ξβ ξβ ξβ ξβ

(11)

where, X =1+η2 ; Y =1- η2 and η is as given in eqn. (5).

Now considering ξ=z/L, Ldξ=dz and as z → 0 ; ξ → 0 and as z → L ;ξ → 1, eqn. (11) can be expressed as:

dz

X

Z

47

41 1

p p -2βξ -2βξ

pile 3

0 0

4

p p

3 1 2

K =4β E I e X-Ycos2βξ - ηsin2βξ dξ+8GL×e X+Ycos2βξ + ηsin2βξ dξ

L 2 2 (2 - ν) 2 2

= 4β E I I + 8GL I

L (2-ν)

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ∫ ∫

(12)

Here I1 and I2 are integral functions as shown in eqn. (12) that need to be determined numerically. However prior

to that, relationship between the dynamic sub-grade modulus ks and Newmark’s parameter (as modified by Wolf

for embedment), as shown in eqn. (7) needs to be established.

Observing eqn. (12) it is seen that the first term represents the structural stiffness of pile and the second terms

expresses the contributing soil stiffness. Thus in term of ks dynamic sub-grade modulus the soil stiffness part in

eqn. (12) can be expressed as:

soil s 2 k =kDLI (13)

Equating eqn. (13) to the second term of eqn. (12) we can establish that:

s k =8G/(2-ν)D (14)

This gives 4 4

p p β= 2GL/(2-ν)E I (14a)

Based on β above and in terms of dynamic shear modulus G, eqn. (12) can now be expressed simply by:

pile 12 K =⎡⎣8GL/(2-ν)⎤⎦χ (15)

in which, χ12=I1+I2. The values of χ12 for various values of β and α( ≈ L1/L) are furnished in Table-1.

Table 1. Typical design values of χ12 for various embedment ratio of pile α and β

β χ12(for α=1) χ12(α=0.9) χ12 (α=0.8) χ12 (α=0.7) χ12 (α=0.6)

4.00 0.345 0.149 0.068 0.032 0.015

4.50 0.296 0.116 0.048 0.021 8.89E-03

5.00 0.259 0.092 0.035 0.014 5.22E-03

5.50 0.23 0.075 0.026 9.13E-03 3.08E-03

6.00 0.206 0.061 0.019 6.10E-03 1.83E-03

6.50 0.187 0.05 0.014 4.10E-03 1.09E-03

7.00 0.171 0.042 0.011 2.77E-03 6.53E-04

7.50 0.158 0.035 8.27E-03 1.88E-03 3.96E-04

8.00 0.147 0.029 6.30E-03 1.28E-03 2.43E-04

8.50 0.137 0.025 4.82E-03 8.70E-04 1.50E-04

9.00 0.128 0.021 3.70E-03 5.96E-04 9.42E-05

9.50 0.12 0.018 2.85E-03 4.10E-04 5.96E-05

10.00 0.114 0.015 2.20E-03 2.84E-04 3.80E-05

10.50 0.107 0.013 1.70E-03 1.97E-04 2.44E-05

11.00 0.102 0.011 1.31E-03 1.38E-04 1.57E-05

11.50 0.097 9.88E-03 1.02E-03 9.69E-05 1.01E-05

12.00 0.093 8.55E-03 7.92E-04 6.84E-05 6.53E-06

12.50 0.089 7.41E-03 6.16E-04 4.86E-05 4.20E-06

13.00 0.085 6.44E-03 4.80E-04 3.46E-05 2.69E-06

13.50 0.081 5.60E-03 3.75E-04 2.47E-05 1.72E-06

14.00 0.078 4.88E-03 2.94E-04 1.77E-05 1.10E-06

In the above formulation it will be observed that static spring effect of the soil is only considered. The dynamic

part, which is frequency-dependent, has been ignored.

This is justified in this case for it has been observed by Wolf et al. (2004), that for vertical and horizontal motion,

the spring constants are almost independent of the dimensionless frequency number a0 (ωr/vs). The same

conclusion has also been arrived at by Hall (1976) and Kramer (2002) wherein it is suggested that use of static

soil spring adequately serves the purpose of earthquake analysis.

For the pile to be partially embedded when some part near the surface soil looses its strength due to liquefaction

and needs to be ignored from the pile strength calculation (Figure 1), eqn. (12) is modified to:

1 1

4 1 1

1 p p -2β ξ -2β ξ p p

pile 3 1 1 1 1 3 3

1-α 1-α

K =4β E I e X-Ycos2β ξ - ηsin2β ξ dξ+8GL× e X+Ycos2β ξ + ηsin2β ξ dξ + 12E I

L 2 2 (2 - ν) 2 2 L (1- α)

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ∫ ∫ (16)

Where with respect to Figure 1, α=L1/L, and 0 ≤ α ≤ 1 and 4 4 4

1 pp β = 2Gα L /(2- ν)E I while the stiffness becomes

p p

pile 12 3 3

K = 8GL χ + 12E I

(2 - ν) L(1-α)

(17)

Here χ12=I1+I2 whose boundary conditions are as given in eqn. (16) and the last term is valid only when the pile is

partially embedded. For full embedment this term is to be ignored.

48

3 Computation of pile mass

The pile mass consists of two parts, a) the self weight and b) lumped mass at its head. The contribution of the

self weight of the pile can be expressed as [Meirovitch(1967)]:

Mx=mx∫φi(z)φj(z)dz (18)

For the present case, eqn. (19) can be expressed as:

L

p p 2

x

0

γ A

M = φ(z) dz

g ∫ (19)

where, γp = unit weight of the pile material; Ap = cross sectional area of the pile; g = acceleration due to gravity.

The above in natural co-ordinates can be simply expressed as

( ) x p p 2 M = γ A L/g I (20)

in which, I2 is the integral function as explained in eqn. (12) and given in Table 2..

Table 2. Typical design values of I2 for various embedment ratio of pile α and β.

β I2(for α=1) Ι2(α=0.9) Ι2 (α=0.8) I2(α=0.7) I2 (α=0.6)

4.00 0.221 0.122 0.051 0.016 0.00375

4.50 0.193 0.096 0.034 0.008 0.00146

5.00 0.171 0.077 0.022 0.004 0.00066

5.50 0.154 0.062 0.015 0.0020 0.00044

6.00 0.139 0.050 0.010 0.0010 0.00038

6.50 0.128 0.041 0.006 0.0005 0.00035

7.00 0.118 0.033 0.004 0.0004 0.00030

7.50 0.109 0.027 0.002 0.0003 0.00024

8.00 0.102 0.022 0.002 0.0003 0.00018

8.50 0.095 0.018 0.001 0.0003 0.00012

9.00 0.090 0.015 0.001 0.0002 0.00008

9.50 0.085 0.012 0.00041 0.0002 0.00005

10.00 0.080 0.010 0.00030 0.0002 0.00003

10.50 0.076 0.008 0.00024 0.0001 0.00001

11.00 0.072 0.0065 2.108X10-04 0.000108 7.11E-06

11.50 0.069 0.0052 1.966X10-04 8.05X10-05 3.24E-06

12.00 0.066 0.0042 1.878X10-04 5.77X10-05 1.40E-06

12.50 0.063 0.0034 1.797X10-04 3.99X10-05 6.33E-07

13.00 0.061 0.0027 1.703X10-04 2.65X10-05 3.76E-07

13.50 0.058 0.0022 1.587X10-04 1.7X10-05 3.14E-07

14.00 0.056 0.0017 1.452X10-04 1.05X10-05 3.00E-07

Now the question comes as to what will be the lumped mass at the top of the pile?

The most logical inference is that this must be equal to the static vertical design load of the pile. For this is what a

designer would always restrict his load on pile to.

Thus based on above argument, the total contributing mass of the pile will be

( ) ( ) pile p p 2 d M = γ A L/g I + P /g (21)

Here Pd is the allowable static vertical load on the pile. For partial embedment case I2 as given in the second

part of eqn. (16) is to be considered.

Damping of the pile embedded in soil medium will consist of two parts: material and radiation damping of the pile.

Material damping of soil is also a part of the vibration system, however, it has been found that for translational

vibration their effect is insignificant and may be neglected. As a first step for calculating the total damping, one

may ignore the material damping of the pile for the time being.

For a rigid circular disc embedded in soil for a depth h, Wolf (1988) has shown that radiation damping may be

expressed as:

x 0 pile s 0 c = (r K /V ) ⎡0.68 + 0.57 h/r ⎤

⎣ ⎦ (22)

where Vs = shear wave velocity of the soil.

Thus for an infinitesimally thin circular disc of depth dz of the pile, eqn (22) can be expressed as:

x 0 pile s 0 c = (r K /V ) ⎡0.68 + 0.57 dz/r ⎤

⎣ ⎦ (23)

Now considering, y = m , where m = dz/r0 , one can write taking logarithmic function on both sides and then

expanding logem, as a series of m where higher orders of m are ignored for being very small:

( ) e log y ≅ 1.5m - 0.92 (24)

Thus y = e(1.5m-0.92) (25)

Expanding the right hand side of eqn. (25) in power series and ignoring the higher orders of m being exceedingly

49

small since it contains higher order of dz, one can finally arrive at:

y ≅ 1.5m + 0.083 (26)

Substituting this value in eqn. (23) and ignoring the first term within the parenthesis which is due to the base

resistance one can have:

0 pile

x

s 0

C =0.855r K dz

V r

(27)

For systems having continuous response function, the damping may be expressed as (Paz(1987)):

x i j C =c(x)∫φ (z)φ (z)dz (28)

The above for a pile partially or fully embedded in soil can be generically expressed as

1

pile 2

x

s 1-α

K L

C = 0.855 φ(ξ) dξ

V ∫ (29)

Here, 0 ≤ α ≤ 1; when fully embedded α=1 and for partial embedment, α<1.
The damping ratio of the pile is given by:
x x c ζ = C /C , here c pile pile C =2 K ×M (30)
Based on above one arrives at an expression:
x f 2 ζ = (0.43Lω /Vs)I (31)
In eqn. (31) ωf is the natural frequency of the pile @ pile pile K M and I2 is the integral function as furnished in
second part of eqn. (16) for partial embedment and eqn. (12) for full embedment.
To eqn. (31), now a suitable material damping ratio of the pile ( m ζ ) depending on what constitute the pile
material (like concrete, steel etc.) may be added to arrive at the total damping ratio of the system.
4 Dynamic response of pile
Having established the stiffness, mass and damping ratio of the pile for the fundamental mode the time period of
the pile can be generically expressed as
( )
3 3
12 p p
pile
8GLχ [12E I (2 ν)]/L (1 α)
M 2 ν
T 2π
+ − −
−
= (32)
In the above formulation it is assumed that the structure has infinite stiffness (T → 0 ), like a machine sitting on a
pile cap or a heavy vessel supported by pile cap, whose own fixed base stiffness is far too high compared to the
pile stiffness and may be ignored from the calculation. For full embedment the second term in the denominator of
eqn. (32) is to be ignored.
For case where the super structure has finite stiffness, the problem can be tackled as explained hereafter. Let us
assume that for a project, building dimensions are known (Height Hb and width D) then the fundamental time
period of the building can be established as per UBC(97) as
b b T = 0.09H / D (33)
Based on the above equation it can be stated that in the fundamental mode the whole building mass (all parts) is
moving with time period Tb and the acceleration thus generated is a function of Tb. Thus for any arbitrary mass
which forms the part of the building will be subjected to an acceleration Sa which is a function of this time period
Tb. Thus for the mass (Pd/g), the static design load at the top of the pile should also move with an acceleration
that is a function of Tb. Now if one assumes a fictitious column above a pile supporting this mass, Pd/g, its
stiffness can be expressed as
2 ( )
d
col 2
b
4π P /g
K =
T
` (34)
Based on above we can now mathematically model the superstructure and the pile as a two mass lumped model
as shown in Figure 2. The equation of motion in terms of stiffness, mass and damping matrix can be expressed
u2 m2 =Pd/g
Kco l(Eqn(34))
u1 m1=γp.Ap.L.I2/g
Kpile (Eqn (17))
Figure 2. A Two-mass Lumped Model for Pile and Superstrucutre
50
as
1 1 col pile col 1 col pile col 1 { }
g
2 2 col col 2 col col 2
m 0 u C +C -C u K +K -K u
+ + = - M u
0 m u -C C u -K K u
⎡ ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫
⎢ ⎥⎨ ⎬ ⎢ ⎥⎨ ⎬ ⎢ ⎥⎨ ⎬ ⎡⎣ ⎤⎦
⎣ ⎦⎩ ⎭ ⎢⎣ ⎥⎦⎩ ⎭ ⎢⎣ ⎥⎦⎩ ⎭
&& &
&&
&& &
(35)
In the above equation Ccol=ζcol ×2 Kcol.Pd/g where ζcol is usually 0.02 for steel and 0.05 for RCC, ζpile is as
derived in eqn. (32) plus material damping of the pile.
It should be realized, that in this case modal solution is not possible as the damping matrix is non classical and a
time history analysis has to be performed from which the force induced on the pile can be established.
Above theory can now be extended to an interesting hypothesis. The fictitious stiffness of the column (which
would give same base shear as the fixed based building) was developed based on the fundamental time period
Tb. Now considering Pd/g is a constant for the pile, Tb can be any arbitrary value for which the force induced on
pile would be (Pd/g)XSa where Sa is a function of this arbitrary time period. In a particular industrial project there
could be umpteen structures (like pipe racks, compressor foundations, buildings etc) whose value may vary from
0.2 sec to 1.5 sec say.
Hence for each of these time periods it is possible to generate a column stiffness (so long as the mass on the pile
head remains constant) and arrive at the force on pile head for each of these mass and arbitrary stiffness.
Thus, in essence the geo-technical consultant need not know the stiffness of structure so long as he knows the
static design capacity of the pile. He can simply select a range of time period from 0.1 sec to 2 sec say at an
increment of 0.1 sec and arrive at a range of lateral capacity for the piles for each of these time periods. The
structural designer who will undertake the analysis of the structure can check on this table later (furnishing the
lateral load) against various time period of the system while doing the final design and arrive at the number of
piles to be selected based on this capacity. Intermediate values as usual can be linearly interpolated.
With respect to the modal analysis as per eqn. (33), the maximum amplitude at pile head can be expressed as:
2
d i F a S =κ C (S /ω ) (36)
Here κι is modal mass participation factor , CF is the code factor constituting of importance factor, zone factor and
response reduction factor, Sa is the acceleration corresponding to the time period of the pile and ω is the natural
frequency of the pile . Considering ω=2π / T, eqn. (37) can be simplified to:
( )( )
d i F
12
S =κ C W 2-ν Sa
8GLχ g
⎛ ⎞
⎜ ⎟
⎝ ⎠
, where W=Mpilex g. (36a)
The displacement along the pile length can now be expressed as:
( ) -βξ( ( ) ( ))
i F
12
u(z) = κCW 2-ν Sae cos βξ + ηsin βξ
8GLχ g
⎛ ⎞
⎜ ⎟
⎝ ⎠
(37)
For partial embedment case the maximum displacement (up) at pile head can be estimated as:
( )
( ) ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
+ − −
−
=
g
Sa
8GLχ 12E I 2 ν /[L (1 α) ]
u (z) κ C W 2 ν 3 3
12 p p
p i F (38)
The modal mass participation factor can be expressed as:
2
i i i i i κ =Σmφ Σm φ (39)
For the present problem this can be expressed as:
L L
2 2
i p p d p p d
0 0
κ = (γ A L/g)∫φ(z) + (P /g)φ(0) (γ A L/g∫φ(z) + (P /g)φ(0) (40)
Considering Pd/g >> γp.Ap.L/g i κ →1

The bending moment and shear force in pile fully embedded in soil along its depth can be thus expressed as:

p p M =-E I u′′

( )( )

2

p p -βξ

F 3 a

12

E I Wβ 2 - ν

M(z) = -2C S /g e sin(βξ) - ηcos(βξ)

8GL χ

⎡⎣ ⎤⎦ (41)

51

p p V = -E I u′′′

( )( )

3

p p -βξ

F 4 a

12

E I Wβ 2 - ν

V(z) = -2C S /g e (1+ η)cos(βξ) - (η -1)sin(βξ)

8GL χ

⎡⎣ ⎤⎦ (42)

Based on eqns. (32) and (33) for partially embedded pile one can obtain the value of the acceleration Sa. This will

induce a shear (Pd/g)Sa at the pile cap level and which in turn will induce a shear of same magnitude at the soil

line( z=L-L1) and an additional moment of (Pd/g).Sa.L(1-α) at this level. Now considering the level z=L-L1 as z=0,

displacement at the soil line level may be obtained from the expression:

( ) 3

d -β ξ

(L-L1) 2 a

p p

u = P L S /g e [(1/β +1- α)cosβ ξ - (1- α)sinβ ξ )]

2E I β

′ ′ ′ ′ ′ ′ ′

′

(43)

where ξ′ = z/L1 and 4 4 4

p p β′= 2Gα L /(2- ν)E I

The bending Moment and shear force profile can now be obtained as usual from the expression:

p p (L-L1) z E I u′′ =M and p p (L-L1) z E I u′′′ =V when

-β ξ

z(L-L1) d a M =P(S/g)L(1-α)e [(1/β +1- α)sinβ ξ - (1- α)cosβ ξ )] ′ ′ ′ ′ ′ ′ ′ (44)

1

-β ξ

z(L-L ) d a V =P(S/g)e [β (1/β + 2- 2α)sinβ ξ + cosβ ξ )] ′ ′ ′ ′ ′ ′ ′ ′ (45)

What has been discussed till now is the kinematical interaction between the superstructure and the pile. Other

than this, free field displacement of the site also influences the stress in pile.

For a site having depth H to the bedrock and shear wave velocity Vs the free field time period can be estimated as

Tf = 4H/Vs.

Considering a suitable material damping of soil based on, say Ishibashi and Zang(1993), one can estimate the

acceleration (Saf) induced on the ground due to this free field motion.

It has been shown by Chowdhury and Dasgupta(2007) that the shape function for such free field motion of ground

in fundamental mode(Figure 1) can be expressed as φ(z) = cos(πz/2H) in one dimension. The displacement of

the soil can be expressed as:

( )

2

F af s

f 2

u = 32C S γ H cos πz

π π + 2 Gg 2H

(46)

Here γs=Weight density of the soil. Now considering H=μL(Figure 1) , where 0<μ<1 the displacement at soil free

surface can be expressed in terms of pile length L as:

( )

2 2

F af s

f 2

u = 32C S γ μ L cos πz

π π + 2 Gg 2μL

(47)

The bending moment and shear force in pile is then expressed as:

( ) ⎟

⎟⎠

⎞

⎜ ⎜⎝

⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎟⎠

⎞

⎜ ⎜⎝

⎛

+

=

2 L

cos πz

G

E I

g

S

π 2

8C γ

M f s af p p

f μ

and ( ) ⎟

⎟⎠

⎞

⎜ ⎜⎝

⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎟⎠

⎞

⎜ ⎜⎝

⎛

+

=

2 L

sin πz

G

E I

g

S

π 2 μL

4π γ

V f s af p p

f μ

C

(48)

These have to be added to eqns. (41) and (42) to get the final dynamic response of piles under earthquake. For

very long and slender pile it is apparent that the ratio (Ep.Ip/G) is low when the free field effect is not profound and

may well be ignored.

5 Results and Discussions

Results are shown hereunder of a 30 m long RCC pile of diameter 1.0 m embedded in a soil of average shear

wave velocity of 130 m/sec having Poisson’s ratio of 0.33 with static design capacity of 1000 kN. It is supporting a

vessel of weight 10000 kN of operating weight.

The moments and shear are plotted for the pile considering with and without the dynamic effect in Figure 3. The

site as per Indian code is zone 4 lying on soft soil.

Comparison of Bending Moment

-50.0000

0.0000

50.0000

100.0000

150.0000

200.0000

250.0000

0

0.2

0.4

0.6

0.8

1

z/L

Moment(kN.m)

Moment considering

pile stiffness

Moment considering

fixed base

Comparison of shear

-200

-150

-100

-50

0

50

0

0.2

0.4

0.6

0.8

1

z/L

Shear force(KN)

Shear considering

pile stiffness

Shear considering

fixed base

Figure 3. Bending Moment & Shear force of Pile with and without dynamic effect

52

Force on pile from superstructure(kN)

-150

-100

-50

0

50

100

150

1 114 227 340 453 566 679 792 905 1018 1131 1244 1357 1470

Time st eps

Force on pile (kN)

Figure 4. Time history response of load on pile from an air cooler structure.

In the above problem 31% damping was considered for pile (vide eqn.33+5% material damping) and 5% for the

vessel. It is apparent that the moment and shear has undergone amplification in-spite of such high damping in

this case. Considering fixed base analysis base shear is 124 kN and considering dyamic pile stiffness it is

155kN.The corresponding moments are found to be 154 kN.m and 192 kN.m respectively

We finally show a time history response of the same pile vide eqn.(35) for a air-cooler supporting steel structure

having fundamental fixed base time period of 0.4sec.In this case damping considered is 2% for the structure and

31% for the piles- the response curve was considered as per IS-1893(2002).

It is seen maximum shear on pile estimated in this case is 115 kN. While fixed base estimate(ignoring the pile)

was found to be 211kN. The force on pile is given in Figure 4.

Based on the above method the design steps for the pile including the algorithm for development of a spreasheet

can be summarized as hereafter.

• Read values of Dynamic Shear Modulus (G) and Poissons ratio(ν) from soil report

• Read basic pile data like Ep, Ip, L, Pd, γp etc. from soil report.

• Determine β from Equation (14a).

• Determine χ12 and I2 for a given β and α from Table 1 and 2 respectively.

• Determine Mpile from Equation (21).

• Determine Time period T and damping ratio ζ from Equation (32) and (31) respectively.

• For the given T and ζ read off Sa/g from the code and select the paremters Z,I and R.

• Determine displacement(u) , bending moment( M) and shear(V) in pile from Equation (38),(41) and (42)

repsectively.

• Determine free field moment and shear in pile from Equation (48).

• Add to M and V to get the final Design moment and shear.

For two mass lumped system

• Determine Kpile and M1 and M2 from Equation (17) and Fig-2 respectively

• Determine Kcol and Kpile as shown in Fig-2 .

• Determine Cpile and Ccol as stated in the paper.

• Form Equation (35) to perform time history to determine the displacement(u) moment(M) and shear(V)

in pile.

6 Conclusion

It is evident from the above two cases that lateral load on pile is dependent on the soil- pile-structure stiffness and

damping property. And without doing a proper dynamic analysis it cannot be estimated as to what is the actual

load on pile. Recommendations furnished in some codes (like IS2911), of considering lateral load as 5% of the

axial load may seriously underrate the load at times.

Present method gives a rational and practical way for estimation of such forces on piles under earthquake force

including partial embedment. The formulas for time period, moment, shear etc are direct and can very well be

developed in a spread sheet for dynamic analysis of the pile based on steps as explained above.

7 References

Banerjee, P.K and Sen, R. 1987, Dynamic behavior of axially and laterally loaded piles and pile groups”,

Chapter 3, Developments in Soil Mechanics and Foundation Engineering, Vol. 3, Ed. Banerjee, P.K. and

Butterfield, R., Elsevier Applied Science, London

Chandrashekharan V. 1974 - Analysis of Pile Foundations under Static and Dynamic Loads- PhD thesis

University of Roorke.

Chowdhury I & Dasgupta S.P.2007 – Dynamic earth pressure on rigid unyielding walls under earthquake

forces, Indian Geotechnical Journal, 37(1), pp.-81-93.

Dobry, R. and Gazetas, G. 1988- ”Simple Method for dynamic stiffness and damping of floating piles

53

groups”, Geotechnique, 38, No 4, pp-557-574.

Hall J.R and Kissenpfennig J.F 1976- Special topics on Soil-Structure Interaction- Nuclear Engg Design 38.

Hurty, W.C. and Rubenstein, M.F. 1967- Dynamics of Structures, Prentice-Hall of India, New Delhi.

IS-1893-2002 Criteria for Earthquake resistant Design of Structures ISI India.

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and Foundations Vol. 33 No1 pp182-191

Kramer S 2002 Dynamic Stiffness of piles in Liquefiable Soil Technical report # WA-RD-514.1 University of

Washington.

Meirovitch, L. 1967, Analytical Methods in Vibration, Macmillan Publication.

Newmark N. and Rosenblueth E 1971- Fundamentals of Earthquake Engineering Prentice Hall New Jersey.

Novak, M. 1974, “Dynamic stiffness and damping of piles”, Can. Geotech. J., Vol.11, pp.574-598.

Novak, .M. and El Sharnouby. B. 1983, “Stiffness and damping constants for single piles”, J. Geotech.

Engng. Div., ASCE, 109, pp. 961 –974.

Parmelee,.R.A., Penzien J, Scheffey, C.F, Seed, H.B and Thiers, G. 1964, Seismic effects on structures

supported on piles extending through deep sensitive clays”, University of California, Berkeley, Report-

SESM 64-2.

Prakash Shamsher 1973 Pile Foundations under Lateral Dynamic Loads 8tth ICSMFE, Moscow, Vol-2.

Paz, Mario 1987, Structural Dynamics, CBS Publishers Ltd., New Delhi.

Penzien, J. 1970, Soil Pile Foundation Interaction in Earthquake engineering, Ed. R.L. Wiegel, Prentice Hall,

Englewood Cliff, New Jersey.

Shames, I.H. and Dym, C.L. 1995, Energy and Finite Element Method in Structural Mechanics, New Age,

International Publishers Ltd., New Delhi.

Tajimi H. (1966), Earthquake response of Foundation Structures (in Japanese) Report, Faculty of Science

and Engineering, Nihon University Tokyo 1.1-3.5

Uniform Building Code Part II -1997 Design of Building under Seismic Loading

Wolf J 1988 Dynamic Soil Structure Interaction in Time Domain, Prentice Hall, New York.

Wolf J and Deeks A-2004- Vibration of Foundations: A Strength of Material Approach, Elsevier,UK.

54

# A Practical Approach for Estimation of Lateral Load on Piles under Earthquake

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