dynamic response of substructures under earthquake force ppr12.145elr

7/30/2019 Dynamic Response of Substructures Under Earthquake Force Ppr12.145elr

1/21

- 1741 -

Dynamic Response of Substructures

under Earthquake Force

Indrajit Chowdhury

Petrofac International Limited;

Sharjah, UAE.

[email protected]

Shambhu P Dasgupta

Department of Civil Engineering,

Indian Institute of Technology Kharagpur, India

[email protected]

ABSTRACTWith increase in infrastructure development in metro-cities like construction of underground

railways, pedestrian subways etc., there exists a significant debate among designers as to -

what is the effect of earthquake on these substructures? While there exists widely diverging

opinion on the topic like i) ignoring its effect completely below ground to ii) reducing the

spectral acceleration to half its value up to 30 meter depth below ground or iii) simply using

Mononobe and Okabes (M-O) method to estimate the seismic pressure on a substructure, a

number of such substructures have been observed to have undergone damage under recentearthquakes like Kobe (1995), Bhuj (2001), Fukushima (2011) to name some of the major

few. Recent research in USA sponsored by US Nuclear Regulatory Corporation (USNRC)

shows that substructures below ground can be significantly affected by seismic force and

procedures in vogue as mentioned above are not at all realistic, to the extent that USNRC has

now stopped using any of the M-O based method for design of their substructures in their

Nuclear power plants. However this research is valid for simplified case like shear modulus

constant with depth and requires extensive use of software like SAASI/SHAKE. A simplified

formula based on this study has now been accepted by Federation of Emergency Management

Agency (FEMA/NEHRP) for estimating dynamic pressure on such substructures. Present

paper proposes an analytical method based on Galerkins weighted residual technique to

arrive at the dynamic pressure on such substructures with soil having generalized constitutive

model- (where G could vary with depth having different powers). The method does notrequire any special software to be used thus engineers not having access to such special

purpose software can yet handle this problem with confidence. The results obtained by the

proposed method matches well within limits of civil engineering design with the established

method as proposed now by FEMA/NEHRP and can be used for any general constitutive

model of soil.

KEYWORDS: Wave propagation, Energy principle, Plane strain, Galerkin WeightedResidual technique, Dynamic pressure.
http://www.ejge.com/Index.htm


2/21

Vol. 17 [2012], Bund. L 1742

INTRODUCTIONWith a spurt of recent underground infrastructure construction, namely metro rail tunnels,

pedestrian subways etc. being built in a number of metropolitan cities, the primary debate that has

come to the fore is

What is the effect of earthquake on these types of structures built below ground?

The debate has gained significant importance as the arguments put forward for the design of

such structures under seismic force are highly diverging.

While some recommend (most popular) to totally ignore the effect of any earthquake below

ground arguing, only structures above ground are affected by seismic forces, others prescribe to

reduce the Sa/g value linearly up to a depth of 30m below the ground and consider half the value of

(Sa/g) thereafter. Again, there are others who in the absence of any specific guidelines prefer to use

Mononobes earth pressure coefficients (Mononobe & Matsuo1929) to determine the seismic

pressure on the wall of such structures, that are usually considered rigid.

It was only in the recent past, research carried out in USA (Ostadan & White 1997), (Ostadan2004)based on extensive field observation and then back checking the same by finite element

method (FEM) through software like SHAKE(Schnabel et al 1972) and SASSI (Lysmer et al 2000)

came up with a pressure equation that is best fit over a number of soil data. This formula is now

adapted by NEHRP [2000] and also by US Nuclear Regulatory Corporation for all theirsubstructures for Nuclear Power plants. However, one of the major limitations of this method is that

the expression is valid only for G constant with depth and would still require software like SHAKE,

SASSI to arrive at an acceptable result.

Chowdhury & Dasgupta (2007, 2008) presented an analytical solution for determination of such

dynamic pressure, having G constant with depth and a few selected cases like sandy or gravelly

soils. The solution was restricted to the case where the underground structure was directly resting on

the bedrock. In the present paper an analysis is outlined which takes care of underground structures

that may be floating or resting on bedrock. The soil with a generalized constitutive model can be

adapted to estimate the dynamic pressure.

PROPOSED METHODThe problem statement along with free field characteristics of the site is shown in Fig. 1.

Figure 1: An underground structure built below the ground


3/21

Vol. 17 [2012], Bund. K 1743

The propagation of SV waves through the medium in two dimensions can be expressed as

2

2

22

2

2

2),,(1),,(),,(

t

tzxu

Vsz

tzxu

x

tzxu

=

+

(1)

where Vs = shear wave velocity of the soil medium; u(x, z, t) = the displacement function and can be

considered as u = H(x).Q (z).P (t) [H, P, Q are the three independent functions of x, z and t

respectively].

Eqn. (1) can be broken up into three ordinary differential equations of second order, given by

022

2

=+ Pdt

Pd (2)

where = iVs where i is a constant

0)()( 2

2

2

=+ xHkdx

xHd (3)

where k is another constant

0)()( 2

2

2

=+ zQpdz

zQd (4)

where p, i, and k are related through p2 = i2 - k2 .

The solution of the eqns. (3) and (4) is given by

kxBkxAxH sincos)( += (5)

pzDpzCzQ sincos)( += (6)

the boundary conditions

At x = 0, u = 0 H (x) = 0, which implies A = 0.

At x = a (where a may be very large), u = 0

H (a) = 0 which implies H (a) = 0sin =kaB

a

mk = (7)

and hence

a

xmxHm

sin)( = (8)


4/21

Vol. 17 [2012], Bund. L 1744

At the free surface i.e. where z = 0 (Fig. 1), the boundary conditions are

At z = 0, shear strain, 0=

z

uor 0

)(=

dz

zdQ

which implies D = 0.

At z = H, displacement, 0=u , i.e. 0)( =HQ . It implies that

H

np

2

)12( = (9)

and hence

H

znzQ

2

)12(cos)(

= (10)

The eigenvectors of the problem can be established as

H

zn

a

xmzQxHzx

2

)12(cossin)()(),(

== (11)

where, m, n = 1,2,3

Again, from the description of eqns. (2) and (3)

22 kpVs +=

Substituting the value of p and k from eqns. (9) and (7), one can have

2

2

2

2

4

)12(

H

n

a

mVs

+= .

For the fundamental mode considering m, n = 1 and lim a , the value of reduces to

24

10

HVs +== and

H

Vs

2

= (12)

The period, T can be derived from eqn. (12) as T = 4H/Vs which is basically the free field time

period for the site in one dimension.

For lim a , the first term of eigen-function (in x direction) can be dropped in eqn. (11)which gives the eigen-function in one dimension as

( )

H

znz

2

12cos)(

= (13)


5/21

Vol. 17 [2012], Bund. K 1745

In this case the eigen-value vis--vis the free field time period and eigen-vectors are derived by

direct solution of the differential equation of motion of the wave propagation in two dimensions for

constant G.

However the same can be derived from the energy equation also and are furnished hereafter.The strain energy equation of a soil body, in general, is given by (Timoshenko et al 1983)

( ) ( )2

2 2 2 2 2 2

2 2x y z xy yz xz

e GU G

= + + + + + + (14)

where, U = strain energy density of the soil body; = )21/(2 G ; G = dynamic shear modulus

of the soil medium and its Poisson ratio; e = x+y+z; x,y,z = strain in the x, y and z direction andxy,yz,zx = shear strains in the xy, yz and zx planes respectively.

With reference to Fig. 1 and assuming the condition of plane strain, eqn. (14) can be rewritten as

( ) ( ) ( )2222221

xzzxzx

GG

GU

++++

= (15)

For impulsive seismic response, z = 0 which reduces eqn. (15) further to

( ) 22221

1xyx

GGU

+

= (16)

Considering u(x,z) = (x,z), q(t) one can have

+

=

z

u

qz

uG

x

u

qx

uG

q

U

rrr

21

)1(2

That is

( )ri

ri

ri

ri

r

qqzz

Gqqxx

G

q

U

+

=

21

12

(17)

where

=),( zx generalized shape function with respect to x and z co-ordinate,

q(t) = displacement function with respect to time in generalized co-ordinate.

The stiffness and mass matrix can be written as (Hurty and Rubenstein 1967)

dzdxzz

Gxx

GK

H a

riri

ir .21

)21(2

0 0

+

=

(18)


6/21

Vol. 17 [2012], Bund. L 1746

and

=H a

ri

s

ir dzdxg

M0 0

.

(19)

where, K = stiffness matrix of the soil medium; M = mass matrix of the soil medium;

i and r are different modes 1,2,3..

K and M for the fundamental mode are given by

( )dzdx

zG

x

GK

H a

.21

12

0 0

22

11

+

=

(20)

( ) =H a

s dzdxg

M

0 0

2

11 .

(21)

It was shown earlier that when lim a , the first term can be dropped and eqns. (20) and (21)reduce to

dzz

GKH

.0

2

11

=

(22)

( )=

H

s dzg

M0

2

11

(23)

Considering the shape function given in eqn. (13) as z = cos

(i.e. for n = 1) and

substituting it in eqns. (22) and (23) for a constant G value and by integrating, one can have

H

GK

8

2

11

=

(24)

and,

g

HM s

211

=

(25)

Considering

= 2/substituting eqns. (24) and (25) one can arrive at the same

expression as as T = 4H/Vs derived earlier. This shows the correctness of the energy principle as

adapted herein.

For multi-degree freedom, the stiffness and mass matrices can be expressed as

[ ] dzzz

GKH

ri .0

=

(26)


7/21

Vol. 17 [2012], Bund. K 1747

[ ] ( )( )=H

ri

s dzg

M0

(27)

For the first four modes, expanding eqns. (26) and (27), it gives

[ ]

2

2

2

2

/ 8 0 0 0

0 9 / 8 0 0

0 0 25 / 8 0

0 0 0 49 / 8

GK

H

=

(28)

[ ]

1/ 2 0 0 0

0 1/ 2 0 0

0 0 1/ 2 0

0 0 0 1/ 2

sHMg

=

(29)

Solving for the eigenvalue from [ ] [ ] 0= MK and knowing = 2 where T==2/,, wefinally have

{ } { }4 1.333 0.8 0.571T

s

HT

V=

(30)

The corresponding eigenvectors [] are obtained as

[ ]

=

1000

0100

0010

0001

(31)

Considering modal analysis, the amplitude of displacement can be expressed as (Clough and

Penzien 1983)

2a

dSS =

(32)

Based on Code definition, the amplitude of vibration may be expressed as

H

zn

Su annn

2)12cos(

2

=

(33)


8/21

Vol. 17 [2012], Bund. L 1748

in which , is a Code factor expressed as = ZI/2R where Z = zone factor, I = importance factor,R = response reduction factor; is modal mass participation factor and is expressed as

= =

=n

i

n

i

iiiin mm1 1

2

For the first four modes this is given in Table 1.

Table 1: Values of modal mass participation factor.

Mode 1 2 3 4

8/(+2) -8(3-2) 8/(5+2) -8/(7-2)

The strain within the soil body is expressed as

0=xx andz

uzz

= which gives

( )H

zn

TS

H

n nannnzz

2)12sin(

42

122

2

= (34)

Eqn. (34) on simplification for the first four modes, it finally gives

( ) H

z

G

H

g

Saszz

2

sin

2

16 11

+

=

(35)

( ) Hz

G

H

g

Saszz

2

3sin

233

16 22

=

(36)

( ) Hz

G

H

g

Saszz

2

5sin

255

16 23

+=

(37)

( ) Hz

G

H

g

Saszz

2

7sin

277

16 44

=

(38)

=

xz

zz

xx

xz

zz

xxG

2

2100

01

01

)21(

2

(39)


9/21

Vol. 17 [2012], Bund. K 1749

Under plane strain condition

( )zzxxxx

GG

21

2

21

12

+

=

(40)

This gives

Now considering 0=xx one can have

zzxx

G

21

2

=

(41)

Considering, xx as the dynamic pressure on the wall we have for the first four modes

H

zH

g

Sp

s

a

2sin

21)2(

32 11

+

=

(42)

H

zH

g

Sp s

a

2

3sin

21)23(3

32 22

=

(43)

H

zH

g

Sp s

a

2

5sin

21)25(5

32 33

+

=

(44)

H

zH

g

Sp s

a

2

7sin

21)27(7

32 44

=

(45)

Eqns. (42) to (45) can also be generically expressed as

Hg

Scoeffp s

a

=

(46)

in which,

=

21

where, coeff is a coefficient and is a SRSS value of the above pressures for different modes areshown in Fig. 2.


10/21

Vol. 17 [2012], Bund. L 1750

Based on the above following are to be noted.

For G remaining constant with depth, cos[(2n-1) z/2H] is an exact solution of the differential

equation of motion and gives identical solution to the problem based on Rayleigh-Ritz basis (energy

principle) for obtaining the stiffness and mass matrices of the system.

The response for fundamental mode is most critical. The higher order modes have significantly

reduced response (Fig. 2).

The SRSS values of pressure for the first three modes vis--vis fundamental mode- there is

negligible difference (Fig. 2).

Thus, if one works with fundamental mode only, for practical engineering problem, it is

adequate.

SOLUTION FOR GIBSON TYPE SOILAs soil is heterogeneous in nature, it is only in the case of normally consolidated clay, G remains

constant with depth. For cohesion less soil, G is often found to vary with depth as G = G0.(z/H),

where G varies from 0 at z = 0 to G0 at depth H. This is often called Gibson soil.

Incorporating this soil constitutive model, the properties of the partial differential equation

furnished in eqn. (1) changes, giving rise to solutions with Bessel function of the first kind of order

0, where J0( nz) are the eigen vectors (Verruijt 2010) of the problem. Though this is easily

solvable; for other constitutive models like, G = G0.(z/H)2, G = G0(1+z/H), G = G0(1+z/H)2, and G

= G0(z/H)0.5 etc., it is extremely difficult to solve them analytically as they become inordinatelycomplex to handle or even not solvable.

Thus to circumvent this problem and yet arrive at a realistic result, the approach of the problem

is elaborated hereafter.

Considering eqn. (13) as the assumed shape function, it will generically satisfy the static

equation of a cantilever shear beam under loading q expressed as

Pressure coeeficient for first three modes including srss

value.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

z/H

pressure

coeff p1

p2

p3

psrss

Figure 2: Dynamic pressure coefficient on wall for G constant with depth


11/21

Vol. 17 [2012], Bund. K 1751

qdz

udA

H

zG =

2

2

0

(47)

q

dz

udGA =

2

2(48)

where G = G0.(z/H) , A = area of cross section of the shear beam and q = the externally applied load.

If ideally by eqn. (13)0.cos[(2 1) / 2 ]u u n z H = is the exact solution of eqn. (48) will yield

02

2

=

q

dz

udGA

(49)

However since it is not an exact solution, will have a residual error, Re expressed as

qdz

udGA

=

2

2

Re

(50)

Now as per Galerkin basis of weighted residual method, we minimize this error Re over the

domain that will give [Chowdhury & Dasgupta 2008)

0)()(0

2

2

0

=

dzzqdzz

dz

udGA

H

jj

H

(51)

where )(.0 zuu i= and )(zi is as defined in eqn. (13).

Integrating eqn. (51) by parts, we have

0)()(00

=

dzzqdzzdz

du

dz

dGA

H

jj

H

=

H

jj

H

j

H

dzzqdzzdz

duGA

dz

dz

dz

duGA

000

)()()(

(52)

The first term in eqn. (52) depicts the shear force Vb in beam which gives

b

H

jj

H

Vdzzqdzzdz

duGA

dz

d=

00

)()(

(53)

Now considering )(.0 zuu i= one can have


12/21

Vol. 17 [2012], Bund. L 1752

b

H

jj

i

H

Vdzzqudzzdz

zdGA

dz

d=

0

0

0

)(.)()(

(54)

From above it is apparent that based on Galerkin basis of weighted residual method, the stiffness

of the shear beam can be expressed as

[ ] dzzdz

zdGA

dz

dK j

i

H

)()(

0

=

(55)

[ ] dzzzGAKH

ji

=

0

)()(

(56)

It may be observed that stiffness matrix derived herein is different from eqn. (26) which is based

on Rayleigh Ritz basis.

For the mass matrix based on Galerkin basis, the matrix remains same as expressed by eqn. (27).

For soil type G = G0.(z/H) using = cos(2 1)2

zn

H

as the assumed shape function, for n =

1,2,3.., also satisfies the boundary condition of the equation (actual solution is = J0(nz/H), asdiscussed earlier) given by.

2

2

2

2

0t

u

dz

ud

H

zG

=

(57)

Now, applying Galerkin basis to derive the stiffness matrix [vide eqn. (56)], we have

[ ]

+

+

+

+

=

4

1

16

49

4

35

100

21

36

74

35

4

1

16

25

4

15

36

5100

21

4

15

4

1

16

9

4

336

7

36

5

4

3

4

1

16

2

2

2

2

H

GK (58)

The mass matrix remains same as eqn. (29).

Solving for the eigenvalue, from [ ] [ ] 0= MK and knowing =2 where T=2/we finallyhave

{ } { }5.19 2.133 1.228 0.755T

s

HT

V= (59)

The scaling factors for the eigenvectors are expressed as


13/21

Vol. 17 [2012], Bund. K 1753

{ }

=

733.0432.0199.0046.0668.0562.039.0086.0

125.0701.0723.0217.0

021.0078.0534.0971.0

(60)

Considering that eqn. (59) was derived based on assumed shape functionH

zn

2)12cos(

= in

lieu of =J0(nz/H) (which is exact), it would be enlightening to compare the values of time period

coefficient ( )sT VHCT= by both the method as shown in Table 2.

Table 2: Comparison of Time period coefficient (CT) exact versus proposed

Mode 1 2 3 4

=J0(nz/H). 5.19 2.14 1.232 0.769=cos(2n-1)z/2H. 5.19 2.133 1.228 0.755Error (%) 0 0.32% 0.32% 1.8%

The results are found to be in excellent agreement having error less than 0.5% for first three modes.

Considering fundamental mode to be the most critical, one can surely argue that this solution is

acceptable for practical design work. Advantage with approaching this problem based on Galerkin

weighted residual technique is that for other type of soil (G =G0(1+z/H), G = G0(z/H)0.5, G =

G0(z/H)2 etc) for which no exact solution exists one can come to a realistic solution.

As elaborated in eqn. (34) the amplitude of displacement in this case can be expressed as

[ ] Hz

n

TS

uT

n

nan

nn2)12cos(4 2

2

= (61)

Here n does not change and remains same as values derived earlier for G constant with depth, [n] isthe scale factor of the eigen vectors as derived in eqn. (60).

Eqn. (61) can be further expressed for the fundamental mode as

( )

+=

H

z

H

z

H

z

G

H

g

Su as

2

5cos.086.0

2

3cos.217.0

2cos.971.0

4

19.5

2

82

22

11

(62)

=

H

z

H

z

H

z

G

H

g

Su as

2

5cos.086.0

2

3cos.217.0

2cos.971.00616.1

21

1

Considering 0=xx andz

uzz

= , we have

++

=

H

z

H

z

H

z

G

H

g

Sazz

2

5sin675.0

2

3sin022.1

2sin525.10616.1 11

(63)


14/21

Vol. 17 [2012], Bund. L 1754

Considering zzxxG

p

21

2

==

++

= H

z

H

z

H

z

g

SHp as 2

5sin675.02

3sin022.12sin525.121123.2

11

(64)

Higher modes can be ignored as it was shown earlier (Fig. 2) that their effects are insignificant.

Proceeding in identical fashion and considering, )/( VsHCT T= , values of CT for different soil

type are presented in Table 3.

Table 3: Values of coefficient CT for various soils

Mode 1 2 3 4G=G0(z/H) 5.19 2.133 1.228 0.755G=G0(z/H)

0.5 4.486 1.668 1.005 0.679

G=G0(z/H)2

7.826 3.301 1.66 0.867G=G0(1+z/H) 3.094 1.095 0.66 0.462G=G0(1+z/H)

2 2.421 0.906 0.545 0.361

Now following the same steps as shown from eqns. (61) to (64) the dynamic pressure for soils with

different characteristics are shown hereafter:

i) For soil of type G=G0(z/H)0.5 the dynamic pressure is expressed as( )zzz

a

s CBAg

SHp 104.0372.0566.1

21586.1 ++

=

(65)

ii) For soil of type G=G0.(z/H)2( )zzz

a

s CBAg

SHp 26.1822.1424.1

21826.4 ++

=

(66)

iii) For soil of type G=G0.(1+z/H)( )zzz

as CBA

g

SHp 013.0240.0569.1

21754.0 ++

=

(67)

iv) For soil of type G=G0.(1+z/H)2( )zzz

a

s CBAg

SHp 075.051.0561.1

21462.0 ++

=

(68)

where Az= sin(z/2H), Bz= sin(3z/2H) and Cz= sin(5z/2H).

Eqns. (65) to (68) can be generically expressed as


15/21

Vol. 17 [2012], Bund. K 1755

Hg

Scoeffp s

a

= (69)

The variation of pressure profile for different type of soil over the depth of the soil to bed rock isshown in Figs. 3 and 4.

Figure 3: Variation of dynamic pressure over the soil depth

Figure 4: Variation of dynamic pressure over the soil depth

For quick computation, the pressure coefficients for different type of soil are as shown in Table 4.

Table 4: Coefficient of pressure along soil depth for various soil type

=z/H Coeff (G0)Coeff

(G0(z/H))

Coeff

(G0(z/H)

0.5

)

Coeff

(G0(z/H)

2

)

Coeff

(G0(1+z/H))

Coeff

(G0(1+z/H)

2

)0 0.000 0.00 0.000 0.000 0.000 0.000

0.05 0.155 1.30 0.396 4.919 0.139 0.125

0.1 0.310 2.50 0.773 9.367 0.274 0.2440.15 0.462 3.48 1.115 12.933 0.403 0.353

0.2 0.612 4.18 1.410 15.318 0.522 0.448

0.25 0.758 4.56 1.648 16.371 0.629 0.526

0.3 0.899 4.62 1.827 16.104 0.723 0.585

Table 4 continues on the next page.

Coefficient of pressure profile for different soil

0.000

5.000

10.000

15.000

20.000

0

0.15

0.3

0.45

0.6

0.75

0.9

z/H

Coefficient G0

G0(z/H)

G0(z/H)^0.5

G0(z/H)^2

Coefficient of presur e profile for differnt soil

0.000

0.500

1.000

1.500

2.000

2.500

00.

10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

z/H

Coeeficentof

pressure

G0(1+z/H)

G0(1+z/H)^2

G0


16/21

Vol. 17 [2012], Bund. L 1756

Table 4 continues from the previous page.

0.35 1.035 4.40 1.949 14.684 0.802 0.625

0.4 1.164 3.96 2.021 12.402 0.867 0.648

0.45 1.287 3.40 2.053 9.633 0.919 0.656

0.5 1.401 2.81 2.057 6.777 0.958 0.6520.55 1.506 2.27 2.044 4.202 0.985 0.639

0.6 1.603 1.85 2.027 2.196 1.003 0.622

0.65 1.689 1.60 2.012 0.932 1.014 0.601

0.7 1.765 1.53 2.004 0.448 1.019 0.581

0.75 1.830 1.61 2.006 0.657 1.020 0.563

0.8 1.884 1.80 2.015 1.367 1.019 0.547

0.85 1.926 2.04 2.030 2.323 1.016 0.535

0.9 1.957 2.27 2.044 3.253 1.014 0.527

0.95 1.975 2.44 2.055 3.919 1.012 0.522

1 1.981 2.50 2.059 4.160 1.012 0.520

Based on above table the steps for estimation of dynamic pressure on the wall of the substructure canbe summarized as follows.

1. Determine the depth of bedrock from where waves can emanate (in absence of bedrock thiscan be taken as depth where SPT value N 50).

2. Select the variation profile of dynamic shear modulus of soil body to the depth of bedrockwhich closely resembles expressions like G0(z/H), G0(1+z/H) etc

3. Estimate Poissons ratio vis--vis the value .4. Compute free field time period of the site from expression T = CT(H/Vs), where the

coefficient CT is as given in Table 3

5. Estimate the response reduction factor of soil, consider R = 3.0 for soft soil and R=2.0 forstiff soil.6. Estimate the code factor = ZI/2R.7. From the value of T as computed in step 4, determine the spectral acceleration Sa/g from

code.

8. To be on conservative side consider 15% damping for soft soil and 10% damping for stiffsoil while estimating the value Sa/g.

9. Now referring to Table-4 extract the pressure profile from eqn. (69).10. It is to be noted that the dynamic pressure profile will vary with location of the sub-structure

below ground, that is, whether it is near the surface or deeply embedded inside ground- for

this refer to the worked out example cited below.

RESULTS AND DISCUSSIONTo check validity of the proposed method the formulation has been compared with (Ostadan

2004) having G constant with depth having a recommended value of=1/3 as furnished in NEHRPand expressed as


17/21

Vol. 17 [2012], Bund. K 1757

2 3 4 5( ) 0.0015 5.05 15.84 28.25 24.59 8.14p z z z z z z= + + +

Figure 5: Comparison of pressure coefficient for= 1/3

It is observed in Fig. 5 that the values are well matching within the acceptable limit of civil

engineering design.

To further substantiate the matter a 30 ft deep basement wall having V s =1000ft/sec (constant

with depth), unit weight of soil 125 pcf and Poisson ratio=1/3 (Ostadan 2004) is compared by using

simplified methodof NEHRP and closed form solution as proposed herein in Fig. 6. The results are

again found to be closely matching.

Figure 6: Comparison of dynamic pressure for the 30ft basement wall with Vs=1000ft/sec.

While the method proposed by (Odstadan et al 1997, 2004) requires an estimation of natural

frequency based on SHAKE/SASSI and then compute the pressure, the proposed method does not

require any software to be used either to estimate the free field time period or the pressure for any

type of soil that fits in the profiles as mentioned above.

Comparison of pressure coefficient

-2.5

-2

-1.5

-1

-0.5

0

0.5

00.

10.

20.

30.

40.

50.

60.

70.

80.

9 1

z/H

Coefficient

Proposed Method

Ostadan

-500

0

500

1000

1500

2000

0 6 12 18 24 30

Height in feet

DynamicPressure(psf) Closed form

SimplifiedMethod


18/21

Vol. 17 [2012], Bund. L 1758

Computation of pressure on a metro rail tunnel at surface, and deeply

entrenched in the ground.

To elaborate how the system works for a metro tunnel 35m wide 20m deep is located at a site as

shown in Fig.7. It is, i) At the surface; ii) 16m below the ground.We need to determine the pressure on the wall when Vs= 200 m/s (varying linearly with depth

z/H) having = 0.3 and unit weight of 20 kN/m3. The site is seismic Zone III as per IS code. Thebedrock level is at 40m below the ground surface.

Since Vs varies as z/H, the time period of site is expressed as

5.19 / (5.19 40) / 200 1.038sT H V= = = s

Considering the soil as soft, damping ratio chosen is 15%.

For T=1.038s, Sa/g = 1.67/T =1.608.For damping ratio of 15% scaling factor = 0.7.

Thus the design Sa/g = 0.7x1.608 = 1.126.

For Zone III

Z = 0.16; I = 1.5, R = 3

/ 2 (0.16 1.5) / 6 0.04ZI R = = =

75.06.01

3.0

21=

=

=

For case- (i) 0.01 = and 2 20 / 40 0.5 = = the pressure distribution is shown below in Fig. 8.

Figure 7: A 35m x 20 m metro railway tunnel below ground

Case-(ii)

16m Case

-(i)

Bed Rock

20m

40m


19/21

Vol. 17 [2012], Bund. K 1759

For case-(ii)1 16 / 40 0.4 = = and 2 36 / 40 0.9 = = the pressure distribution is shown in Fig. 9.

Figure 9: Pressure on tunnel wall 16 m below ground surface (Case-2i).

The pressures are computed from the expression ( ) HgScoeffp sa = where thecoefficients are extracted for the corresponding values from Table 5.

Table 5: Pressure magnitude on tunnel wall at surface and 16.0m below ground level

=z/H

Pressure

Coefficient as per

Table-4

Dynamic

pressureCase-1

(kN/m2)

=z/H

PressureCoefficient as per

Table-4

Dynamicpressure

Case-2 (kN/m2)

0 0 0.00 0.4 3.967 107.200.05 1.309 35.37 0.45 3.404 91.990.1 2.505 67.70 0.5 2.81 75.940.15 3.489 94.29 0.55 2.272 61.40

Table 5 continues on the next page.

Pressure on wall caste at surface(Case-1)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

00.

05 0.1

0.15 0.

20.

25 0.3

0.35 0.

40.

45 0.5

z/H

Presrue(kN/m2

)

Pressure

Pressure on wall 16 m below ground (Case-2)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

z/H

Pressure(kN/m2)

Pressure

Figure 8: Pressure on tunnel wall at ground surface (Case -1)


20/21

Vol. 17 [2012], Bund. L 1760

Table 5 continues from the previous page.

0.2 4.189 113.20 0.6 1.857 50.180.25 4.567 123.42 0.65 1.607 43.430.3 4.626 125.01 0.7 1.532 41.400.35 4.403 118.99 0.75 1.612 43.560.4 3.967 107.20 0.8 1.804 48.750.45 3.404 91.99 0.85 2.047 55.320.5 2.81 75.94 0.9 2.278 61.56

Figs. 8 and 9 reflect that ignoring the seismic effect altogether below the ground is not a prudent

decision. Structures built below the ground are significantly affected by seismic forces. This is

logical, as because the waves propagating through the soil, interact with the rigid substructure and

the frequency content of the wave excites the structure generating pressure on the wall. The depth of

bedrock level with respect to the position of the substructure has a significant effect on the

magnitude of dynamic pressure. Shallower the bedrock level more intense will be the pressure on the

wall.

CONCLUSIONA comprehensive analytical model is proposed that can estimate dynamic pressure on the walls

of a substructure built below the ground. The method proposed is analytical in nature and generic

and can take care of any type of soil perceived in nature. It does not require any sophisticated FEM

software (like SASSI/PLAXIS) etc. and can be computed simply by a spread sheet. The results

match well with the established formulation given in NEHRP for constant G and Poissons ratio, but

can be extended to other type of soil for which no solution exists.

REFERENCES1. Chowdhury I. and Dasgupta S.P. (2008) Dynamics of Structures and Foundations - a

unified approach Volume 1 and 2 CRC Press, Leiden, Holland.

2. Chowdhury I. and Dasgupta S.P. (2007) Dynamic Earth pressure on rigid unyieldingwalls under earthquake forceIndian Geotechnical Journal Vol. 37(2). .

3. Clough R.W. and Penzien J. (1983)Dynamics of Structures; McGraw-Hill KogakushaLtd., Tokyo

4. Hurty W. C. and Rubenstein M.F. Dynamics of Structures (1967) Prentice HallPublication, New Delhi India.

5. Lysmer J., Ostadan F., and Chen C. C. SASSI 2000- A System for analysis of soilstructure interaction, Dept of Civil engineering, University of California, Berkeley

2000.

6. Mononobe N. and Matsuo H., (1929) On the determination of Earth Pressure duringEarthquakes, Proceeding of World Engineering Congress Tokyo Vol-9, Paper 388..

7. NEHRP (2000), Recommended provisions for seismic regulation for New Building andother Structures, FEMA 369 March 2001.


21/21

Vol. 17 [2012], Bund. K 1761

8. Ostadan F., (2004) Seismic Soil Pressure on Building Walls-An Updated approach,11th International Conference on Soil Dynamics and Earthquake Engineering.

University of California, Berkeley, January.

9.

Ostadan F. and White. W.H. (1997) Lateral Seismic Soil pressure, an updated approachBechtel Technical Grant Report, LA USA.

10.Schnabel B., Lysmer J. and Seed H.B., SHAKE, (1972) A Computer program forEarthquake response of Horizontally Layered Site, University of California, Berkeley

EERC 72-12 December.

11.Timoshenko S.P. and Goodier G., (1983) Theory of Elasticity, McGraw-Hill KogakushaLtd., Tokyo.

12.Verruijt, A. (2010)An Introduction to Soil Dynamics Springer Verlag Publication, NY.USA.

2012 ejge
http://www.ejge.com/copynote.htmhttp://www.ejge.com/copynote.htmhttp://www.ejge.com/Index.htm

dynamic response of substructures under earthquake force ppr12.145elr

Documents