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30.4 Long-Span Structures Subjected to Stationary Random Ground Excitations

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30.4.1 The Solution of Equations of Motion Using the Pseudoexcitation

Method

In Equation 30.93, the PSD matrix of u€ b; that is, SðivÞ in Equation 30.55, is an N-dimensional Hermitian

matrix, while R is an N-dimensional real symmetric matrix. Both matrices are usually positive definite or

semipositive definite. If the rank of R is rðr # NÞ; then by using Equation 30.111 it can be readily

decomposed into the product of an N £ r matrix Q and its transposition; that is

R ¼ QQT ð30:163Þ

Thus, Equation 30.55 can be written as

SðivÞ ¼ BpJQQTJB ¼ PpPT ð30:164Þ

in which

P ¼ BJQ ð30:165Þ

To solve Equation 30.93, the right-hand side u€ b can be replaced by the pseudoground acceleration

€~Ub ¼

expðivtÞ ð30:166Þ

Thus, Equation 30.93 becomes the following sinusoidal equations of motion:

€~

Yr þ Cs

Yr þ Ks

Y~ r ¼ MsK21

s KsbEmN P expðivtÞ ð30:167Þ

The stable solution of Equation 30.167 is the pseudorelative displacement vector Y~ r; whilst the

pseudostatic displacement vector Y~ s can be computed by

Y~ s ¼

v2 K21

s KsbEmN P expðivtÞ ð30:168Þ

Thus, the pseudoabsolute displacement vector is

X~ s ¼ Y~ r þ Y~ s ð30:169Þ

If necessary, any arbitrary pseudointernal force vector N~ s can be computed from X~ s by means of a quasistatic

analysis. Then, the corresponding PSD matrix is

SNs Ns ðvÞ ¼ N~ p

N~ T

s ð30:170Þ

Seismic Random Vibration of Long-Span Structures 30-27

If it is assumed that SX€ 1 ¼ SX€ 2 ¼ · · · ¼ SX€ N

; denoted as Sa; then Equation 30.165 becomes

P ¼

ffiffiffi

BQ ð30:171Þ

If only the wave passage effect is considered, that is, all lrijl ¼ 1 in Equation 30.58, then matrix Q will

reduce to a vector q0 with all its elements unity; that is

Q ¼ q0 ¼ {1; 1; …; 1}T ð30:172Þ

Thus, Equation 30.171 reduces to

P ¼

ffiffiffi

e0 ð30:173Þ

in which e0 is a complex vector

e0 ¼ {expð2ivt1Þ; expð2ivt2Þ; …; expð2ivtN Þ}T ð30:174Þ

Therefore, when only the wave passage effect is considered, Equation 30.167 reduces to

y€~r þ Cs

y_~r þ Ksy~r ¼ MsK21

s KsbEmN e0

ffiffiffi

expðivtÞ ð30:175Þ

Furthermore, if the structure is subjected to a uniform ground motion, then the vector e0 in Equation

30.174 should be replaced by q0; and so Equation 30.175 can be further reduced to

y€~r þ Cs

y_~r þ Ksy~r ¼ MsK21

s KsbEmN q0

ffiffiffi

expðivtÞ ð30:176Þ

30.4.2 Numerical Comparisons with Other Methods

30.4.2.1 Song-Hua-Jiang Suspension Bridge

The Song-Hua-Jiang suspension bridge (see

Figure 30.10) is located in Jilin Province of

China. Its overall length is 450 m, with a main

span of 240 m and a width of 28 m. The finite

element model had 2076 DoF, 445 nodes (including

12 supports) and 574 elements. The static

equilibrium position of the bridge included

the effects of the initial tensions of the cables. The

earthquake action was determined based on the

Chinese National Standard (Code for Seismic

Design of Buildings GB 50011-2001), which

directly gives the ground ARS curve for the bridge.

The PSD curve was obtained in terms of the Kaul

method from which the samples of the ground

acceleration time-history can be produced (Kaul, 1978). For the analyses associated with the SH and SV

waves, 100 modes were used for mode-superposition, whereas for the P waves, only 30 modes were used.

The apparent wave speeds used were 3 km/s for P waves and 2 km/s for SH or SV waves.

30.4.2.1.1 All Supports Move Uniformly

Figure 30.11(a) gives the axial force distribution of this bridge along the deck due to the seismic P waves,

which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move

uniformly. The following four computational models were used:

1. Response spectrum method

2. Pseudoexcitation method

3. Time-history method using three samples

4. Time-history method using ten samples

FIGURE 30.10 Song-Hua-Jiang suspension bridge.

30-28 Vibration and Shock Handbook

0.0E+0

1.0E+2

2.0E+2

3.0E+2

4.0E+2

0 50 100 150 200 250 300 350 400

RSM PEM THM (3 samples) THM (10 samples)

(a)

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)

RSM PEM THM (3 samples) THM (10 samples)

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)

FIGURE 30.11 Deck force distribution of Song-Hua-Jiang bridge due to uniform ground motion: (a) axial force

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

Seismic Random Vibration of Long-Span Structures 30-29

Figure 30.11(b) gives the transverse shear force distribution along the deck due to the seismic SH

waves traveling along the deck. All supports move uniformly. The above four computational models

were used.

Figure 30.11(c) gives the vertical shear force distribution along the deck due to the seismic SV waves

traveling along the deck. All supports move uniformly. The above four computational models were also

used here.

All computations were executed on a P3-750 personal computer. The computation times for different

methods are listed in Table 30.1.

Figure 30.11 and Table 30.1 show that, when ground motion is assumed uniform, that is, the

earthquake spatial effects are not taken into account, the RSM, PEM, and THM (using ten samples) give

very close results if the excitations are properly produced. The RSM is the most popular method, but the

newly developed PEM may be the more efficient one. The THM needs to be executed for a number of

ground acceleration samples and so was inefficient.

30.4.2.1.2 Wave Passage Effect Is Taken into Account

Figure 30.12(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves,

which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move

with certain time lags; that is, the wave passage effect is taken into account. The apparent P wave speed is

3 km/sec. The following four computational models were used:

1. RSM (uniform ground motion is assumed for comparison only)

2. PEM (wave passage effect is considered)

3. THM (wave passage effect is considered using three ground-acceleration samples)

4. THM (wave passage effect is considered using ten ground-acceleration samples)

Figure 30.12(b) gives the transverse shear force distribution along the deck due to the seismic SH waves

traveling along the deck and Figure 30.12(c) gives the corresponding vertical shear-force distribution.

The above four computational models were used.

Figure 30.12 and Table 30.1 show that when the seismic wave-passage effect is taken into account, that

is, the earthquake spatial effects are partly taken into account, the PEM and THM (using ten samples)

give very close results. The RSM, which does not consider the wave-passage effect, may give quite

different results; these may appear larger or smaller than, or very close to, the results by the more

reasonable PEM or THM analyses. Therefore, such computations are necessary for evaluating the seismic

spatial effects of long-span structures. The PEM gives the most reliable results with the least

computational effort and, therefore, this method is strongly recommended.

30.4.2.1.3 Wave-Passage Effect and Incoherence Effect Are Jointly Taken into Account

Figure 30.13(a) gives the axial force distribution of the bridge along the deck due to the seismic P

waves, which travel along the longitudinal direction of the deck. All supports of the bridge are assumed

to move with certain time lags; that is, the wave passage effect is taken into account. In addition, the

incoherence effects are also taken into account. Two coherence models, that is, the Loh – Yeh model

TABLE 30.1 Central Processing Unit (CPU) Times Required by Different Methods for Stationary Analysis

Method Used RSM (sec) PEM (sec) THM (for One Sample) (sec)

Uniform ground motion 80.6 15.1 29.9

Wave passage effect 24.2 36.7

Wave passage effect and incoherence effect 209.1

Note: the CPU time for mode extraction is not included; extracting 100 modes needs 180.9 sec.

30-30 Vibration and Shock Handbook

FIGURE 30.12 Deck force distribution of Song-Hua-Jiang bridge due to wave passage effect: (a) axial force

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

Seismic Random Vibration of Long-Span Structures 30-31

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)

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

RSM (uniform) PEM (uniform) PEM (v=3km/s)

PEM (Loh model) PEM (QWW model)

(a)

RSM (uniform) PEM (uniform) PEM (v=2km/s)

PEM (Loh model) PEM (QWW model)

RSM (uniform) PEM (uniform) PEM (v=2km/s)

PEM (Loh model) PEM (QWW model)

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)

FIGURE 30.13 Deck force distribution of Song-Hua-Jiang 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;

30-32 Vibration and Shock Handbook

and the QWW model were used for evaluating such effects. The following five computational models

were used:

1. RSM (uniform ground motion is assumed)

2. PEM (uniform ground motion is assumed)

3. PEM (wave passage effect is considered)

4. PEM (wave passage effect is considered also using the Loh – Yeh coherence model)

5. PEM (wave passage effect is considered also using the QWW coherence model)

Figure 30.13(b) gives the transverse shear force distribution along the deck due to the seismic SH

waves traveling along the deck, and Figure 30.13(c) gives the corresponding vertical shear

force distribution due to the seismic SV waves. For both analyses the above five computational models

were used.

The above computations as well were executed on a P3-750 personal computer, and the computation

times required by different methods are listed in Table 30.1.

From Figure 30.13 and Table 30.1, we can conclude that:

1. When ground motion is assumed to be uniform, the RSM and PEM give very close internal force

responses (i.e., demands), with the PEM being more efficient.

2. The wave passage effect is an important factor that affects the seismic responses of long-span

structures. To execute such seismic analyses, the PEM is not only theoretically quite reasonable,

but also very efficient.

3. The incoherence effect appears to diverge when using different coherence models. Herein, the

influence caused by the QWW model is more evident than that caused by the Loh – Yeh model.

However, compared with the wave passage effect, the influence of the incoherence effect is of less

importance.

30.4.2.2 San Joaquin Concrete Bridge

San Joaquin Bridge, located in California (see

Figure 30.14), is a reinforced concrete bridge

built in 2001. Its length is 36 þ 50 þ 50 þ

50 þ 36 ¼ 222 m, its width is 12 m, and the

height of all piers is 16.76 m. The finite element

model had 367 nodes (including 10 ground

nodes) and 366 elements. Its basic natural period

is 0.811s. Twenty modes were used in the modesuperposition

analysis with all damping ratios

being 0.05. The seismic analysis was carried out

using the RSM and PEM, respectively. The RSM

analysis was conducted according to the CALTRANS

Code (1999) with ARS ¼ 0.2 g, Type D

soil profile and magnitude Mw ¼ 7.0. The

equivalent ground-acceleration power spectral

density curve was produced by means of the

Kaul method. All seismic waves were assumed to

travel along the longitudinal direction of the

bridge. The apparent P and S wave speeds were

3000 and 2000 m/s, respectively. The internal forces in the deck (i.e., the axial forces due to P waves,

the transverse shear forces due to SH waves, and the vertical shear forces due to SV waves) were all

computed using the following computational models:

1. RSM (uniform ground motion is assumed)

2. PEM (uniform ground motion is assumed)

z y

36m 50 m 50m 50 m 36 m

FIGURE 30.14 Structural model of San Joaquin

bridge.

Seismic Random Vibration of Long-Span Structures 30-33

3. PEM (wave-passage effect is considered)

4. PEM (wave-passage effect is considered also using the Loh – Yeh coherence model)

5. PEM (wave-passage effect is considered also using the QWW coherence model)

The computational results are shown in Figure 30.15(a) – (c). This bridge is not very long. However,

similar phenomena to those found for the bridge of Example 30.1 are still found. Clearly, when the

ground motion is assumed to be uniform, the RSM and PEM still give very close results. If the wavepassage

effect is taken into account, then the internal force distribution with the PEM will change

considerably, particularly at the midpoint of the deck. It is known that, for symmetric bridges, the

antisymmetric modes will not participate in the symmetric motions under the assumption of uniform

ground motion. However, when the wave-passage effect is taken into account, this conclusion does not

hold. It is obvious that, even for this shorter bridge, the wave-passage effect seems to be quite

significant. The incoherence effect is comparatively not so important.

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