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 s4d uE I = -k Dudz

(1)

in which, Ep = Young’s modulus of the pile; Ip = moment of inertia of the pile cross section; ks = dynamic sub-grade 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 3u = e (C cospz + C sinpz) + e (C cospz + C sinpz) (2)

in which 4s p pp = k D 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 1u = 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: ( )-βz /Lφ(z) = e cos(βz/L) + ηsin(βz/L)⎡ ⎤⎣ ⎦ (5)

in which 44s p pβ = k DL 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)]: 22

p p 2h2

E I d u KdΠ = + u2 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)):

0x

0

8Gr hK = 1+(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:

( )

2L L2p p 2

20 0

E I d u 8GΠ = dz + u dz2 2 2 - νdz

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫ (8)

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

(z)dzφ(z)φν)(2

8G(z)dz(z)φφIEK j

L

0

i

L

0

jippij ∫∫ −+″″= (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:

dzφ(z)ν)(2

8Gdz(z)φIEK2L

0

L

0

2ppij ∫∫ −

+″= (10)

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

∫∫ ⎟⎠⎞

⎜⎝⎛ ++

−+⎟

⎠

⎞⎜⎝

⎛ −−=−− L

0

L2L

0

L2

4pp

4

pile dzL

2ηsinL

2cos2Y

2Xe

ν)(28Gdz

L2ηsin

L2

cos2Y

2Xe

L

IE4βK ξβξβξβ ξβξβ ξβ

(11)

where, 2X = 1+ η ; 2Y = 1- η 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

4 1 1p p -2βξ -2βξ

pile 30 0

4p p

1 23

4β E I X Y 8GL X YK = e - cos2βξ - ηsin2βξ dξ + × e + cos2βξ + ηsin2βξ dξL 2 2 (2 - ν) 2 2

4β E I 8GL = I + IL (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 2k = k DLI (13) Equating eqn. (13) to the second term of eqn. (12) we can establish that: sk = 8G/(2 - ν)D (14)

This gives 44p pβ = 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 12K = 8GL/(2 - ν) χ⎡ ⎤⎣ ⎦ (15)

in which, χ12=I1+I2. The values of χ12 for various values of β and α( )/LL1≈ 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 11 p p p p-2β ξ -2β ξ

pile 1 1 1 13 3 31-α 1-α

4β E I 12E IX Y 8GL X YK = e - cos2β ξ - ηsin2β ξ dξ + × e + cos2β ξ + ηsin2β ξ dξ +L 2 2 (2 - ν) 2 2 L (1- α)

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫ ∫ (16)

Where with respect to Figure 1, α=L1/L, and 10 ≤≤ α and 4 441 p pβ = 2Gα L /(2 - ν)E I while the stiffness becomes

p ppile 12 3 3

12E I8GLK = χ +(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)]:

x x i jM = m φ (z)φ (z)dz∫ (18)

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

L

p p 2x

0

γ AM = 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 2M = 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 dM = 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 0c = (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 0c = (r K /V ) 0.68 + 0.57 dz/r⎡ ⎤

⎣ ⎦ (23)

Now considering, y = m , where 0m = dz/r , 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: ( )elog y 1.5m - 0.92≅ (24)

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

s 0

r K dzC = 0.855V r

(27)

For systems having continuous response function, the damping may be expressed as (Paz(1987)): x i jC = 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 2x

s 1-α

K LC = 0.855 φ(ξ) dξ

V ∫ (29)

Here, 10 ≤≤ α ; when fully embedded α=1 and for partial embedment, α<1. The damping ratio of the pile is given by: x x cζ = C /C , here c pile pileC = 2 K × M (30)

Based on above one arrives at an expression: x f 2ζ = (0.43Lω /Vs)I (31)

In eqn. (31) ωf is the natural frequency of the pile @ pile pileK 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 ( mζ ) 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

( )

33pp12

pile

α)(1ν)]/L(2I[12E8GLχ

ν2M2πT

−−+

−= (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 bT = 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

( )2d

col 2b

4π P /gK =

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

{ }col pile col col pile col1 1 1 1g

2 2 2 2col col col col

C + C -C K + K -Km 0 u u u+ + = - M u

0 m u u u-C C -K K

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎡ ⎤⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎣ ⎦

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦

&& &&&

&& & (35)

In the above equation col col col dC = ζ × 2 K .P /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:

2d i F aS = 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

W 2 - ν SaS = κ C8GLχ g

⎛ ⎞⎜ ⎟⎝ ⎠

, where W=Mpilex g. (36a)

The displacement along the pile length can now be expressed as:

( ) ( ) ( )( )-βξi F

12

W 2 - ν Sau(z) = κ C e cos βξ + ηsin βξ8GLχ g

⎛ ⎞⎜ ⎟⎝ ⎠

(37)

For partial embedment case the maximum displacement (up) at pile head can be estimated as:

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

−−+−

=g

Sa]α)(1/[Lν2I12E8GLχ

ν2WCκ(z)u 33pp12

Fip (38)

The modal mass participation factor can be expressed as:

2i i i i iκ = m φ m φ∑ ∑ (39)

For the present problem this can be expressed as:

L L

2 2i 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 iκ 1→

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

p pM = -E I u′′ ( ) ( )

2p p -βξ

F a312

E I Wβ 2 - νM(z) = -2C S /g e sin(βξ) - ηcos(βξ)

8GL χ⎡ ⎤⎣ ⎦ (41)

51

p pV = -E I u′′′ ( ) ( )

3p p -βξ

F a412

E I Wβ 2 - νV(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

-β ξd(L-L1) a2

p p

P Lu = S /g e [(1/β +1- α)cosβ ξ - (1- α)sinβ ξ )]2E I β

′ ′ ′ ′ ′ ′ ′′

(43)

where 1ξ = z/L′ and 4 44p pβ = 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) zE I u = M′′ and

1p p (L-L ) zE I u = V′′′ when -β ξ

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

1

-β ξz(L-L ) d aV = 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:

( )

2F af s

f 2

32C S γ H πzu = cosπ 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 2F af s

f 2

32C S γ µ L πzu = cosπ π + 2 Gg 2µL

(47)

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

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=L2

πzcosG

IEg

S2πγ8C

M ppafsff µ

and ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=L2

πzsinG

IEg

SµL2πγ4π

V ppafsff 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

-50.00000.0000

50.0000100.0000

150.0000200.0000250.0000

0 0.2 0.4 0.6 0.8 1

z/L

Mom

ent(

kN.m

)

Moment consideringpile stif fness

Moment consideringfixed base

Comparison of shear

-200

-150

-100

-50

0

50

0 0.2 0.4 0.6 0.8 1

z/L

Shea

r for

ce(K

N)

Shear consideringpile stiffness

Shear consideringfixed base

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

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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

Ti me st e ps

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

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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.

Ishibashi I and Zhang K 1993 “Unified dynamic shear modulii and damping ratios of sand and clay” Soils 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.

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