30.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations

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A typical strong motion earthquake record consists of three stages. In the first stage, the intensity of the

ground motion increases, which mainly reflects the motion of P waves. The intensity of the ground

motion remains the strongest in the second stage, which mainly reflects the motion of S waves. The

ground motion will die down in the last stage. Such a complete seismic motion is usually regarded as a

nonstationary random process. If the nonstationary property is assumed to takes place only for the

intensity of the motion, then this random process is regarded as a uniformly modulated evolutionary

random process. However, if the shape of the ground motion PSD curve also varies with time (in other

words, the intensity and the distribution with frequency of the ground motion energy both depend on

time), then the ground motion is regarded as a nonuniformly modulated evolutionary random process.

It is usually accepted that when the intensity of the seismic motion in the second stage appears quite

stationary while the time interval of this stage is much longer (e.g., three times or over) than the basic

period of the structure under consideration, a simplified, stationary-based random analysis may be

acceptable as a substitute of the nonstationary analysis. In fact, the basic periods of many long-span

bridges range from 10 to 20 sec, and the stationary portion of a typical strong earthquake is usually less

than 1 min, being only 20 to 30 sec in most cases. Therefore, nonstationary analyses are appropriate for

such problems. Previously, such nonstationary random analyses have been considered very difficult.

However, by using the recently developed PEM, combined with the precise integration method, such

analyses have become relatively easy.

30.5.1 Modulation Functions

Some popular uniform modulation functions are listed below:

gðtÞ ¼

I0ðt=t1Þ2 0 # t # t1

I0 t1 # t # t2

I0 exp{cðt 2 t2Þ} t $ t2

8>><

>>:

ð30:177Þ

gðtÞ ¼

1 t $ 0

0 t , 0

(

ð30:178Þ

gðtÞ ¼ a½expð2a1tÞ 2 expð2a2tÞ􀀉; 0 # a1 , a2 ð30:179Þ

gðtÞ ¼ sin bt ð30:180Þ

30-34 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

3

2

2

FIGURE 30.15 Deck force distribution of San Joaquin bridge due to incoherence effect: (a) axial force distribution

along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear

force distribution along the deck under SV waves.

Seismic Random Vibration of Long-Span Structures 30-35

© 2005 by Taylor & Francis Group, LLC

Nonuniform modulation models have rarely been investigated. Lin et al. (1997a, 1997b) suggested the

following nonuniform modulation model:

Aðv; tÞ ¼ bðv; tÞgðtÞ ¼ exp 2h0

vt

va ta

􀀏 􀀐

gðtÞ ð30:181Þ

in which gðtÞ is an amplitude modulation function; bðv; tÞ is a frequency modulation function; and va

and ta are the reference frequency and time, which are introduced to transform v and t into

dimensionless parameters. In principle, va and ta can be arbitrarily selected. Once they have been

selected, the factor h0ðh0 . 0Þ can be adjusted accordingly to make the high-frequency components of

the nonstationary random process decay more quickly than the low-frequency components and, thus,

simulate the seismic motion more accurately. When h0 ¼ 0; that is, bðv; tÞ ¼ 1; Aðv; tÞ reduces to the

uniform modulation function gðtÞ:

30.5.2 The Formulas for Nonstationary Multiexcitation Analysis

For nonuniformly modulated multiexcitation problems, the pseudoexcitation for the corresponding

stationary problems, that is, Equation 30.166, is extended to

€~

Ubðv; tÞ ¼ Aðv; tÞPexpðivtÞ ð30:182Þ

in which the kth diagonal element of the N £ N diagonal matrix Aðv; tÞ is the modulation function

Akðv; tÞ of the excitation which is applied to the kth support of the structure. In the case of uniformly

modulated excitations, it is only necessary to replace all the nonuniform modulation functions

Akðv; tÞ by the uniform modulation functions gkðtÞ: Other formulae remain entirely unchanged. The

N £ r matrix P can be generated by means of Equation 30.165 to Equation 30.174. Each column of

€~

Ubðv; tÞ can be regarded as a deterministic acceleration excitation vector. By substituting it into the

right-hand side of Equation 30.93 and solving the equations of motion, a column of the matrix

Y~ rðv; tÞ can be produced.

Because Ajðv; tÞ is a time-dependent and slowly varying function, the pseudoground displacement

matrix can be computed approximately from

U~ bðv; tÞ ¼ 2

1

v2

€~

Ubðv; tÞ ð30:183Þ

The pseudoquasi-static displacement matrix Y~ sðv; tÞ can then be computed from Equation 30.92.

Then, the PSD matrix of the absolute displacement vector Xsðv; tÞ is

SXs Xs ðv; tÞ ¼ ðY~ rðv; tÞ þ Y~ sðv; tÞÞpðY~ rðv; tÞ þ Y~ sðv; tÞÞT ð30:184Þ

If a group of pseudointernal forces, denoted as N~ e; has been computed, then the PSD matrix of the

corresponding internal forces Ne can be computed from

SNe Ne ðv; tÞ ¼ N~ p

e

N~ Te

ð30:185Þ

When the ground acceleration PSD matrix is known, the corresponding pseudoacceleration vector €~

ub

is easy to generate according to Equation 30.163 to Equation 30.166. If instead, the ground displacement

PSD matrix or velocity PSD matrix is known, then the acceleration PSD matrix can be obtained by

multiplying the displacement or velocity PSD matrices by v4 or v2; respectively.

30-36 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

30.5.3 Expected Extreme Values of Nonstationary Random Processes

The evaluation of the peak amplitude responses of structures subjected to nonstationary seismic

excitations has also received much attention (Shrikhande and Gupta, 1997; Zhao and Liu, 2001).

Previously, only very simple structures could be computed. However, by using the PEM, complicated

structures can be analyzed, as is briefly described below.

To evaluate the expected extreme value responses of a structure subjected to nonstationary Gaussian

excitations, the duration of which the intensity of the excitation peaks exceeds 50% of the maximum peak

intensity denoted by ½t0; t0 þ t􀀉 is taken as the equivalent stationary duration in order to use Equation

30.40 to Equation 30.49 to evaluate the desired expected extreme values. Provided that the timedependent

PSD of any arbitrary response yðtÞ; that is Syy ðv; tÞ, has been computed over that equivalent

duration using the PEM, then the equivalent stationary PSD over that duration is

S0yy ðvÞ ¼

1

t

ðt0 þt

t0

Syy ðv; tÞdt ð30:186Þ

To compute the extreme value responses based on Equation 30.177, the parameters t0 and t are

chosen as

t0 ¼ t1

􀀋 ffiffi

2 p ; t ¼ t2 þ ln 2=c 2 t1=

ffiffi

2 p ð30:187Þ

Thus, the equivalent stationary random responses are obtained and the subsequent processing can still

use Equation 30.40 to Equation 30.49.

30.5.4 Numerical Comparisons with the Corresponding Stationary Analysis

The example of the Song-Hua-Jiang suspension bridge of the last section is used here for the seismic

nonstationary random vibration analysis. The results are compared with those from the corresponding

stationary random-vibration analyses with the ground assumed to move uniformly (i.e., at an apparent

wave speed vapp ¼ 1), or to move at a limited apparent wave speed vapp (with the wave-passage effect is

taken into account), which is 3 km/sec for P waves and 2 km/sec for S waves.

The nonstationary random excitation model zðtÞ ¼ gðtÞxðtÞ was used in which the auto-PSD of xðtÞ is

assumed to be identical to that used for the stationary excitation in the preceding section. The frequencydomain

parameters also remained the same. The modulation function had the form of Equation 30.177

with t1 ¼ 8:0; t2 ¼ 20:0; and c ¼ 0:20: The duration of the earthquake was t [ ½0; 25􀀉; and the time stepsize

was Dt ¼ 0:5:

The nonstationary analysis results are shown in Figure 30.16(a) to (c), and are compared with the

results of the corresponding stationary random vibration analyses. Clearly, for such a long-span bridge,

the wave passage effect is quite significant in its seismic analysis, as seen in Figure 30.11 to Figure 30.13. In

addition, whether for uniform ground motion or for differential ground motion (i.e., the wave-passage

effect is considered), the nonstationary responses are always smaller than the corresponding stationary

responses. The maximum difference between their corresponding peak values may reach up to 23.1% for

the present problem, as shown in Table 30.2. For very slender bridges, this nonstationary property will be

even stronger.

By means of the PEM combined with the precise integration method (its HPD-E form for the

modulation function used in this example), such modification can be fulfilled quickly and

conveniently. The computational effort required by the nonstationary analysis is only about 25 min

(see Table 30.3).

Seismic Random Vibration of Long-Span Structures 30-37

© 2005 by Taylor & Francis Group, LLC

0.0E+0

1.0E+2

2.0E+2

3.0E+2

4.0E+2

5.0E+2

0 50 100 150 200 250 300 350 400

Uniform-Nonstationary Uniform-Stationary

v=3km/s-Nonstatinary v=3km/s-Stationary

(a)

m

kN

0.0E+0

1.0E+3

2.0E+3

3.0E+3

4.0E+3

0 50 100 150 200 250 300 350 400

(b)

m

Uniform-Nonstationary Uniform-Stationary

v=2km/s-Nonstatinary v=2km/s-Stationary

kN

Uniform-Nonstationary Uniform-Stationary

v=2km/s-Nonstatinary v=2km/s-Stationary

kN

0.0E+0

1.0E+2

2.0E+2

3.0E+2

4.0E+2

5.0E+2

0 50 100 150 200 250 300 350 400

(c)

m

FIGURE 30.16 Deck-force distribution of Song-Hua-Jiang bridge due to uniform and differential, and stationary

and nonstationary random ground motion: (a) axial force distribution along the deck under P waves; (b) transverse

shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under

SV waves.

30-38 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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