RANDOM EXCITATION

Housner and Jennings(17

l

have shown that the important properties of recorded strong-motion earthquake accelerograms can be modelled with sections of a stationary, Gaussian, random process having a power spectral density found from the averaged undamped velocity spectrum. (More recently Jennings, Housner and Tsai(22

) have refined this model by using an envelope function to introduce the changing intensity at the beginning and end of an accelerogram.) The earthquake power spectral density given by Housner and Jennings<17

l

is plotted in Fig. 3.27 and expressed by

G(w)

0.01238(1+1~7.8)

2

=

2

2

2

( 1 - ;42)

+ 14~.

⁸

ft/ secZ )²

rad/sec) ^{(3. 46)}

Their relationship between the relative velocity response spectra and the power spectral density is expressed by

in which

J ^{~G(w) (i _}

e-21;.wT/2.44) Sy= 1 796 • 2tw

Sy = relative velocity response spectrum G{w} = power spectral density of earthquake

w = angular frequency i;.

=

damping ratio

T

=

accelerogram duration

(3. 4 7)

u

lJJ

~

c:i 1·5

<t

~

a:

Cl>

IO l·O 0

x ³

I ^I

(.'.) ^I

0·5

0 5

Figure 3. 27

I I I I ~

10 15 20 25

w RADIANS/SEC.

Power spectral density of strong-motion earthquake acceleration,

30

-

^N co

I

If T = 18,6 and {;. = O, then SV is approximately equal to 1.0 for a 3.0 sec period oscillator. This approximate relationship may be used to scale G{w) to the correct value for different velocity

spectra.

Statistical estimates of the earthquake wall forces and mo- ments can be computed from the earthquake power spectral density and the steady-state solutions for harmonic forcing by using some of the basic results of random vibration theory. The mean-square- response

in which

of a linear system to random excitation is given by

(3. 48)

H(w) = system transfer function

G (w) =power spectral density of input (w = 0 to oo) x

If the input is normally distributed with zero mean the output will also be normally distributed with zero mean. Assuming zero mean for the earthquake input and expressing (3.48) in terms of the wall problem parameters gives

in which

s~

=variance of wall force

s~

= variance of wall moment

(3. 49)

G(w) =power spectral density of earthquake

F' (w) and

M'

(w) are defined in dimensionless form by

r r

expressions {3.42). Expressions (3.49) can be numerically evalu- ated by using expressions (3.42) and the analytical expression for the power spectral density; alternatively, an approximate evaluation can be made by using the simple graphical interpretation of expres- sions (3. 49) and the plots given for

IF'

_r (w)

l/F ,

_sr

IM'(wll/M

_{r sr}

and G(w). Statistical estimates of the maximum forces and moments can be found from the mean-square responses by using the properties of the normal distribution curve. An example of the application of this method is given in Chapter 5.

Although not widely used the random vibration approach has general applicability to many earthquake engineering problems. De- tails of the application of this method to the estimation of responses of buildings to earthquakes are given by Tajimi(54

l.

The method has particular advantage if the system transfer function can be readily derived and if classical normal modes do not exist or are difficult to evaluate. This is frequently the case for many soil-structure interaction problems and in particular the random vibration approach was found to be a convenient method for computing the

response of the deformable-wall structures analyzed in Chapter 7.

3. 9. FINITE ELEMENT SOLUTIONS

The finite element method was used to compute normal mode solutions for a number of rigid-wall problems, The purpose of this study was to check the accuracy of the finite element method for this type of problem by a comparison with the analytical solution for the smooth wall, and to investigate the effect of the wall- soil contact assumption on the dynamic behavior.

The normal modes were computed using the second-order quadrilateral element and a consistent mass matrix (see Archer(1)).

Details of this element and the finite element formulation used are given in Appendix II. Solutions were computed for H equal to 1. 0 ^L and 3, 0 with a Poisson's ratio of O. 3 for each case. Two different meshs were used for each case and these are shown in Fig. 3, 28.

Both the smooth and bonded contact boundary conditions were analyzed. For the smooth wall problem the boundary conditions were taken to be the same as used previously for the analytical solution (Fig. 3.1). The bonded wall problems were identical in detail to their smooth wall counterparts except that the boundary condition on the end walls was taken as u z 0 and v

=

^0.

Solutions for the natural frequencies of the lower anti- symmetric modes are compared in Tables 3.1, 3.2, 3.3 and 3,4 below. The solutions for the static-one-g modal wall pressure distributions that contribute significantly to the static wall forces are shown in Figs. 3. 29 and 3. 30.

RIGID WALL

u•O Txy • 0 or v = 0

t

^{Y,V Uv =O}

~ u•O xy • 0 v• 0

--

x,u

-11= 0 v•O

L/H • l·O COARSE MESH I

RIGID WALL

u•O Txy • 0 or v = 0

r·v

-

(

CTv•OI

u•O v• 0

Antisymmetric

v•O

O")( = 0

-

x,u L/H • l·O FINE MESH

RIGID WALL

u•O Txy• 0 or v • 0

RIGID WALL u• 0 Txy: 0 or v= 0

I

^Y,v

r·v

0 ,

v•O

L/H • 3·0 COARSE MESH

0

'--" = 0 v•O

L/H • 3·0 FINE MESH

Antisymmetric

v•O ux=O

--

x,u

Antisymmetric V• 0

u._ = 0

-

x,u

Figure 3. 28 Finite element meshes for normal mode analyses.

~

w

N I

1·0

0·8

0·6 y/H

0-4

0·2

0 0

l·O

O·B

0·6 y/H

0-4

0·2

0 0

L/H 1·0 SMOOTH CONTACT

MODE 1,2 MOOE 1,3 MOOE 1,4

+o

ANALYTICAL + COARSE MESH

0 FINE MESH

O·I 0·2 0·3 0·4 0·5 0 O·I 0 0·1

P1,2/yH P1,3/yH P1,4/yH

L/H 1·0 BONDED CONTACT

MOOE ^MOOE 3 MODE 4

+ + +

+

+ + +

+ + COARSE MESH

-0----0- FINE MESH

+ + +

0·1 0·2 0·3 0-4 0·5 0 0·1 0 O·I

P1/rH p3/yH p4/yH

Figure 3. 29 Static-one-g modal pressure distributions.

Smooth and bonded rigid wall. L/H = 1. 0.

1·0

0·8

0·6 y/H

DA

0·2

0

l·O

0·8

0·6 y/H

0·4

0·2

0

0·2

0·2

L/H 3·0 SMOOTH CONTACT

MOOE 1,1 MODE 1,3

•

^•

ANALYTICAL x COARSE MESH o FINE MESH

0·8

L/H 3·0 BONDED CONT ACT

MODE

0·4 0·6

P1/yH

x COARSE MESH -o----o- FINE MESH

0·8 1·0

MOOE 4

MOOE 3,2

MODE 5

0 0·2

Ps/YH

Figure 3. 30 Static-one-g modal pressure distributions.

Smooth and bonded rigid wall. L/H = 3. O.

TABLE 3.1

Natural Frequencies, Smooth Contact

~

⁼ 1. 0

Mode Dimensionless Natural Frequency

n

_n,m

Type, Order Coarse Fine Analytical

1 • 1 1,2

*

1,3

*

1, 4

*

3,1 3,2

Mesh Mesh

2.23 2.19

3.51 3.45

4. 81 4.71

5.33 5.09

6.65 5.85

7.62 6.94

TABLE 3.2

Natural Frequencies, Bonded Contact H L = 1. 0

2.18 3.44 4.67 5.01 5.57 6.72

Mode Dim. Natural Fre. 52 n,m

Type, Order Coarse Fine

Mesh Mesh

1 * 3.70 3.63

2 3.98 3.79

3

*

^{5.07 4.89}

4 * 5.82 5. 50

5 7. 3 7 6.96

6 8.51 7.62

*These modes contribute significantly to the static wall force.

TABLE 3.3

Natural Frequencies, Smooth Contact H L

=

3.0

Mode Dimensionless Natural Frequency

n

_n,m

Type, Order Coarse Fine

Analytical

*

Mesh Mesh

1 ' 1

*

^{1. 521 1. 518}

1,2 1.852 1.844

3,1 2.24 2.21

5' 1 3.45 3.33

1,3

*

^{3.46 3.36}

3,2

*

^{3.55 3.49}

3,3 4.89 4.77

5,2 5.12 4.80

TABLE 3.4

Natural Frequencies, Bonded Contact H L = 3. 0

1. 510 1. 836 2.18 3. 1 7 3.28 3.44 4.67 4.74

Mode Dim. Natural Fre.

n

_n,m

Type, Order Coarse Fine

Mesh Mesh

1

*

^{1. 537 1.532}

2 2.03 2.01

3 2. 85 2.78

4

*

^{3.45 3.36}

5

*

^{3.61 3.56}

6 4.27 4.06

7 4.92 4.80

8 5.24 5.02

These modes contribute significantly to the static wall force.

The analytical and the fine mesh finite element results, for both the frequencies and the pressure distributions, show agreement to within 5%. (Agreement for the lowest mode frequencies is better than 1

%. )

Agreement between the fine and coarse mesh results is generally to about 10%. Thus the coarse mesh would be satisfactory for most applications. From these solutions it would appear that satisfactory results can be obtained for modes that have at least four elements within the modal wave length. It is of interest to note that when the

~ =

3. 0 coarse and fine meshes were used in a static finite element analysis for horizontal body force they gave pressure distributions within 10% and 7% respectively of the static analytical results.

Quite good agreement can be seen between the equivalent frequencies and modal pressure distributions of the smooth and bonded contact cases. Thus it is unlikely that the wall-soil interface condition will have a significant influence on the earthquake-induced pressures.