Seismic response and design method of single large span structures to spatially random ground motions

Abstract

This article aims at studying the seismic response rules and design method of single large span structures to spatially varying ground motions. A corrected earthquake power spectral density model is first proposed and applied to the following solution process. The practical solution form of the multi-support pseudo excitation method is put forward and used to analyze the seismic response. The total response is divided into the pseudo-static and relative dynamic response. A simplified single large span structural model is used as the research object to study the response law when both the wave-passage effect and incoherence effect are considered. The result shows that the wave-passage effect and incoherence effect can both increase the pseudo-static response. The relative dynamic response is influenced by the form of trigonometric function, while wave-passage frequency influences the period and coherence function results in amplitude attenuation. When the ratio of natural frequency to wave-passage frequency equals to 0.5 n (n = 0, 1, 2,…), the structural seismic response reaches extreme value. Based on the derived response law, a method of quickly determining the most unfavorable conditions of the multi-support excitation effect is proposed for seismic design of single large span structures.

Keywords

incoherence effect random seismic response seismic design method single large span building structures spatially varying ground motions wave-passage effect

Introduction

The spatial effects, such as the wave-passage effect and incoherence effect, are usually ignored in conventional seismic response analysis. In other words, the earthquake ground motions between different supports are considered to be uniform. This assumption does not produce a large error when the span is small. However, for large span building structures, the seismic response of a structure to multi-support excitations may have a large difference from a structure to uniform excitation. Current researches indicate that the seismic response of a large span building structure may become larger when the spatial effects are considered (Allam and Datta, 1999; Sevket et al., 2005; Soyluk and Dumanoglu, 2004; Thomas and Marc, 1998). Ignoring the spatial effects may lead to a dangerous situation. Thus, studying the seismic response and design method of large span structures to multi-support ground motions has great significance.

Specific engineering structures are usually used as research objects to study the seismic response to multi-support excitations (Bai et al., 2010; Ernesto and Adrian, 2003; Hao, 1998; Li et al., 2006; Liang, 2004; Liu et al., 2004; Luo and Wang, 2005; Su et al., 2007; Sun et al., 2007; Yang et al., 2008; Ye et al., 2008). However, it is difficult to study the influence mechanism of seismic response to multi-support excitations based on the above researches. Thus, the simplified model of a single large span structure is built to study the influence mechanism in this article. The analytical formulae for the seismic response power spectral density (simplified as power spectrum density (PSD) below) of this simplified model to multi-support excitations are derived using the pseudo excitation method (Lin, 1992; Lin et al., 2004). The influence mechanism of multi-support excitation effects, including the wave-passage effect and incoherence effect, are studied by analyzing the composition and numerical characteristic of the analytic formulae.

The multi-support earthquake PSD used in the analysis can be divided into two parts, the single-support earthquake PSD and the coherence function. The former reflects the earthquake characteristic of a single point and the latter reflects the spatial correlation. Existing studies often pay more attention to the construction of the spatial correlation of the multi-support earthquake PSD model (Harichandran and Vanmarcke, 1986; Luco and Wong, 1986) and the influence on the structural response (Allam and Datta, 1999; Sevket et al., 2005; Soyluk and Dumanoglu, 2004; Thomas and Marc, 1998). However, the reasonability of the single-support earthquake PSD, which also has a significant effect on the structural seismic response to multi-support excitations, is often ignored. Thus, the PSD value at zero-frequency and the maximum ground displacement, which are used as judging standards for the reasonability of the power spectrum, are studied in this article. The corrected earthquake PSD model is also proposed for the following analysis.

There are few structural seismic design methods considering the spatial effects. Usually, many multi-support input conditions need to be chosen for trial calculations. However, there are few guiding principles for how to choose the unfavorable conditions. Arbitrary selection will lead to heavy calculations or inaccurate results. Based on the response law derived from the simplified model, the method of quickly determining the most unfavorable conditions of the multi-support excitation effect is proposed in this article. This method can both reduce the amount of calculations and consider enough unfavorable conditions to guarantee accuracy of the result.

Corrected earthquake PSD model

The PSD function is considered as a reasonable form to describe the ground motion. Currently, the widely used PSD models are the filtered white noise models (Kanai, 1957; Tajimi, 1960) and their improved forms (Hong et al., 1994; Ou and Niu, 1990). These models are expressed by analytical expressions. The parameters can usually be determined by either the practical PSD model or fitting the response spectrum curves of different seismic codes (Hong et al., 1994; Xue et al., 2003). The derived PSD still remains different from the code’s response spectrum. In order to further combine with the seismic code, the PSD model can be directly transferred by the response spectrum of the code (Kaul, 1978; Wei et al., 2000; Zhang et al., 2007). However, the response spectra between different seismic codes have a large disparity, especially in the long period segment. Thus, the corresponding low-frequency range of the earthquake PSD, transformed from the long period segment of the response spectrum, may carry a large difference. The reasonability of the PSD at low-frequency range can be reflected by the maximum ground displacement. It is also stated that an earthquake PSD should satisfy energy finiteness in the reference (Li et al., 2008). Thus, these two judging principles are used to evaluate the reasonability of the PSD transformed from the response spectrum of the Chinese seismic code (GB 50011, 2010) in this section. Then, the corrected response spectrum of the Chinese code and the corresponding earthquake PSD model are put forward.

Synthesis method of earthquake PSD

Maximum value estimation of random process

In the random vibration theory, the input and output quantities of the PSD function participate in the calculation directly, while only the input and output maximum values are considered in seismic design. Therefore, the expectation of extreme value of stationary random response is selected as the equivalent quantity to the structural response. The expectation of extreme value of a zero mean stationary random process can be denoted by (Davenport, 1964)

y_{m} = f σ_{y}

(1)

f = \frac{\sqrt{2 \ln (ν t_{d})}}{\sqrt{2 \ln (ν t_{d})}} + γ

(2)

where y_m is the expectation of extreme value, f is the peak factor, σ_y is the mean square deviation, ν is the zero cross rate, t_d is the duration of ground motion, and γ = 0.5772 is the Euler constant

σ_{y} = \sqrt{λ_{0}}

(3)

ν = \frac{\sqrt{λ_{2} / λ_{0}}}{2 π}

(4)

λ_{i} = 2 \int_{0}^{\infty} ω^{i} S_{y} (ω) d ω

(5)

in which S_y(ω) is the PSD of the random process at frequency ω.

Iterative synthesis method of earthquake PSD

The response PSD $S_{\overset{\cdot\cdot}{x}} (ω)$ of a single freedom system can be derived as follows

S_{\overset{\cdot\cdot}{x}} (ω) = \frac{1 + 4 ζ {(ω / ω_{0})}^{2}}{{[1 - {(ω / ω_{0})}^{2}]}^{2} + 4 ζ {(ω / ω_{0})}^{2}} S_{a} (ω)

(6)

where ω₀ is the natural frequency, ζ is the damping ratio, and S_a(ω) is the earthquake acceleration PSD.

The response spectrum value R_a(ω₀) at frequency ω₀ (i.e. the expectation of extreme value of the response PSD $S_{\overset{\cdot\cdot}{x}} (ω)$ ) can be obtained by substituting the result of equation (6) into equations (1) to (5). Hereby, Zhang et al. (2007) gives the accurate PSD synthesis method iterated by the target response spectrum:

The initial value is estimated according to the approximate relationship between response spectrum and power spectrum. The approximate formula put forward by Kaul (1978) is used to find the PSD initial value $S_{a}^{(0)} (ω_{i})$ , i = 1, 2, 3,…, n

S_{a}^{(0)} (ω_{i}) = \frac{\frac{2 ζ}{π ω_{i}} R_{a}^{2} (ω_{i})}{{- 2 \ln (- \frac{π}{ω_{i} t_{d}} \ln P)}}

(7)

where R_a(ω_i) is the target response spectrum, ζ is the damping ratio, t_d is the duration of ground motion, and P is the probability that the response value does not exceed response spectrum value. P = 0.85 is adopted here (Kaul, 1978).

The response PSD $S_{\overset{\cdot\cdot}{x}}^{(k)} (ω_{i})$ is calculated by substituting the PSD $S_{a}^{(k)} (ω_{i})$ into equation (6), where k denotes the kth (k = 0, 1, 2,…) time of iteration. The response spectrum $R_{a}^{(k)} (ω_{i})$ can be determined by equations (1) to (5).

Comparing $R_{a}^{(k)} (ω_{i})$ with R_a(ω_i), the errors at each frequency value ω_i are calculated by

E (ω_{i}) = \frac{| R_{a}^{(k)} (ω_{i}) - R_{a} (ω_{i}) |}{R_{a} (ω_{i})} \times 100 %

(8)

If the error exceeds the limit value, the PSD $S_{a}^{(k)} (ω_{i})$ can be adjusted as follows

S_{a}^{(k + 1)} (ω) = {[\frac{R_{a} (ω)}{R_{a}^{(k)} (ω_{i})}]}^{2} S_{a}^{(k)} (ω)

(9)

Procedures (2) to (4) should be circularly executed until satisfying the error demand. The corresponding displacement PSD S_u(ω) can be calculated by acceleration PSD S_a(ω) as follows

S_{u} (ω) = \frac{S_{a} (ω)}{ω^{4}}

(10)

Analysis on reasonability of practical earthquake PSD

Comparison between different response spectra

The reasonability of the synthetic PSD is decided by the target response spectrum. As mentioned before, the earthquake displacement characteristic is significantly influenced by the PSD value at low-frequency range, corresponding to the long period segment of the response spectrum. However, there is limited research on the characteristic of long period earthquake ground motion; large differences exist between the seismic codes of different countries. Table 1 lists the expressions of the acceleration response spectrum for distinct codes. The Chinese code for seismic design of highway bridges (JTG/T B02-01, 2008), American code (FEMA 450, 2004), and Europe code (EN 1998-1, 2005) adopt the acceleration response spectrum directly, while the Chinese code for seismic design of buildings (GB 50011, 2010) uses the earthquake affecting coefficient instead. All the response spectrum curves include an ascending segment, horizontal segment, and descending segment. However, the descent rate of each long period descending segment contains large differences.

The GB 50011 response spectrum adopts a linear descending segment after T^−0.9 as its descent rate, which is small.

The JTG/T B02-01 response spectrum only adopts T⁻¹ as its descent rate.

The EN 1998-1 and FEMA 450 response spectrums both adopt T⁻¹ and T⁻² as their descent rates, which are larger than the above two.

Table 1.

Response spectra in the codes of different countries.

	GB 50011		JTG/T B02-01
Ascending segment	$α_{max} (5.5 T + 0.45)$	0 ≤ T ≤ 0.1	$S_{max} (5.5 T + 0.45)$	0 ≤ T ≤ 0.1
Horizontal segment	$α_{max}$	0 < T ≤ T_g	$S_{max}$	0 < T ≤ T_g
T ⁻¹ descending segment	$α_{max} (T_{g} / T)^{0.9}$	T _g < T ≤ 5 T_g	$S_{max} (T_{g} / T)$	T _g < T ≤ 10
Linear descending segment	$α_{max} [{0.2}^{0.9} - 0.02 (T - 5 T_{g})]$	5 T_g < T ≤ 6
	EN 1998-1		FEMA 450
Ascending segment	$a_{g} S (1.5 T / T_{B} + 1)$	0 ≤ T ≤ T_B	$S_{DS} (0.6 T / T_{0} + 0.4)$	0 ≤ T ≤ T₀
Horizontal segment	$a_{g} S 2.5$	T _B ≤ T ≤ T_C	$S_{DS}$	T ₀ ≤ T ≤ T_S
T ⁻¹ descending segment	$a_{g} S 2.5 (T_{C} / T)$	T _B ≤ T ≤ T_C	$S_{D 1} (1 / T)$	T _S ≤ T ≤ T_L
T ⁻² descending segment	$a_{g} S 2.5 (T_{C} T_{D} / T^{2})$	T _D ≤ T ≤ 4	$S_{D 1} (T_{L} / T^{2})$	T _L ≤ T

Damping ratio of each expression is 0.05. Definition and value of each parameter can refer to the corresponding codes.

Analysis on reasonability of PSD

There is little difference between the response spectrum of EN 1998-1 and FEMA 450 in the long period segment. Since EN 1998-1 gives the estimation formula of maximum displacement, which allows for easy analysis of the displacement reasonability of PSD, the result of EN 1998-1 is retained in the following examples.

PSD value at zero-frequency

The ground motion parameters are selected according to GB 50011 and JTG/T B02-01: degree of earthquake intensity of 8, peak acceleration a_g = 0.7 m/s², site class II, the second design group, eigen-period T_g = 0.4 s, and a damping ratio of 0.05. Corresponding to the parameters above, the response spectrum curve with A class site of EN 1998-1 is selected, and peak acceleration a_g = 0.7 m/s². The acceleration and displacement PSD are shown in Figures 1 and 2. Figure 2(a) and (b) only gives the displacement PSD value with frequencies larger than 0.2 and 0.1π rad/s, respectively, as the value is very large near zero-frequency.

Figure 1.

Acceleration PSD of ground motion: (a) GB 50011, (b) JTG/T B02-01, and (c) EN 1998-1.

Figure 2.

Displacement PSD of ground motion: (a) GB 50011, (b) JTG/T B02-01, and (c) EN 1998-1.

As shown in Figures 1 and 2, there is a large difference in PSD values at zero-frequency between different response spectra (see Table 2). Obviously, the acceleration and displacement PSD synthesized by the EN 1998-1 response spectrum with a T⁻² descending segment can both satisfy finite values at zero-frequency. By contrast, the other two are not reasonable.

Table 2.

Characteristic of PSD at zero-frequency.

Code	GB 50011	JTG/T B02-01	EN 1998-1
Rate of descent	Linear	T ⁻¹	T ⁻²
Acceleration PSD	∞	0	0
Displacement PSD	∞	∞	0

PSD: power spectrum density.

Maximum displacement of ground

Substituting the synthesized acceleration and displacement PSDs into equations (1) to (5), the calculated values of peak acceleration a_max and peak displacement u_max can be obtained. Parameter ρ is used to reflect the peak displacement using the following expression

ρ = \frac{u_{max}}{S_{a max}}

(11)

where S_a_max is the maximum of the acceleration response spectrum.

The reasonability of displacement and acceleration characteristics can be judged by comparing the errors between the calculated and target values. The target peak acceleration is equal to the parameter of ground motion a_g. The target values of ρ refer to the formula in EN 1998-1 as follows

ρ = 0.01 T_{C} T_{D}

(12)

where T_C and T_D are the beginning periods of the T⁻¹ and T⁻² descending segment of the response spectrum in EN 1998-1, respectively.

This article adopts the value of ρ from 0.007 to 0.021 as the reference value, which is put forward according to the SMART-1 array seismic records (Wang and Fan, 1998). The earthquake parameters of this section are selected as follows: degree of earthquake intensity of 8, peak acceleration a_g = 0.7 m/s², a damping ratio of 0.05, the second design group with I–IV class sites of the Chinese code, and A–D class sites of EN 1998-1. Tables 3 and 4 show the comparison of peak acceleration and peak displacement between the calculated and target values.

Table 3.

Comparison of calculated value and target value of peak acceleration.

Site class	GB 50011			JTG/T B02-01			EN 1998-1
Site class	Target value	Calculated value	Error (%)	Target value	Calculated value	Error (%)	Target value	Calculated value	Error (%)
A/I	0.70	0.674	−3.7	0.63	0.595	−5.6	0.70	0.683	−2.4
B/II	0.70	0.703	0.4	0.70	0.693	−1.0	0.84	0.854	1.7
C/III	0.70	0.733	4.7	0.84	0.870	3.6	0.81	0.807	0.2
D/IV	0.70	0.761	8.7	0.91	0.981	7.8	0.95	0.992	5.0

Target peak accelerations of JTG/T B02-01 and EN 1998-1 are related to class of site.

Table 4.

Comparison of calculated value and target value of peak displacement.

Site class	GB 50011		JTG/T B02-01		EN 1998-1
Site class	u _max	Calculated value of ρ	u _max	Calculated value of ρ	u _max	Calculated value of ρ	Target value of ρ	Error of ρ (%)
A/I	0.0874	0.0557	0.0220	0.0155	0.0144	0.0082	0.0080	2.9
B/II	0.0944	0.0602	0.0327	0.0208	0.0218	0.0104	0.0100	3.8
C/III	0.1041	0.0664	0.0538	0.0285	0.0252	0.0125	0.0120	4.3
D/IV	0.1168	0.0745	0.0789	0.0385	0.0397	0.0168	0.0160	5.0

Error of ρ refers to EN 1998-1.

Table 3 shows that there is a large difference in the long period segment between different response spectra, but all the calculated values of peak acceleration have little difference with target values, and the result of EN 1998-1 has higher precision. This indicates that the peak acceleration cannot estimate the reasonability of the long period segment of the response spectrum.

In Table 4, the calculated values of ρ of GB 50011 are considerably larger than the reference value from 0.007 to 0.021 given by Wang and Fan (1998). The calculated value of ρ of JTG/T B02-01 is smaller than the former, but still larger than the reference value. The calculated value of ρ of EN 1998-1, which is within the range of the reference value and similar to the estimated value given by the EN 1998-1 code, has the least error.

Preliminary conclusions can be obtained based on the analysis of the results: the PSD synthesized by the response spectrum with a T⁻² long period descending segment can satisfy the zero value at zero-frequency; correspondingly, the peak displacement can also satisfy the precision demand. However, the linear and T⁻¹ descending segments, which cannot satisfy all the conditions, are not reasonable.

Corrected response spectrum of the Chinese code

The analysis above shows that the low-frequency segment of PSD, which is synthesized by the long period of the response spectrum of the Chinese code (especially GB 50011), does not satisfy the reasonability demand. So the T⁻² descending segment is involved in the long period of the response spectrum of GB 50011, which can both remain the original characteristics and improve the reasonability of the long period segment.

The corrected response spectrum of GB 50011 is shown in Figure 3. The parameter values refer to the code (GB 50011, 2010). The value T_L of the beginning point of T⁻² descending segment needs to be determined by trial calculation. The original 5 T_g and 2 s referring to EN 1998-1 are used to conduct this trial calculation. The earthquake parameters are the same as those mentioned above in section “Analysis on reasonability of PSD.” The calculation results of peak displacement are shown in Table 5.

Figure 3.

Corrected acceleration response spectrum in GB 50011.

Table 5.

Peak displacement obtained from corrected response spectrum in GB 50011.

Site class	T _L = 5 T_g		T _L = 2 s
Site class	u _max	Calculated value of ρ	u _max	Calculated value of ρ
A/I	0.0086	0.0055	0.0115	0.0073
B/II	0.0151	0.0096	0.0151	0.0096
C/III	0.0279	0.0178	0.0240	0.0153
D/IV	0.0501	0.0320	0.0271	0.0173

Table 5 shows that if the value of T_L is 5 T_g, the value of ρ exceeds the reference range (0.007–0.021) (Wang and Fan, 1998). T_L of 2 s can stay within the range and will be chosen as the beginning point of the T⁻² descending segment. In order to ensure the reasonability of the ground displacement characteristics, the long period segment of the response spectrum is corrected: the linear descending segment of the response spectrum is replaced by the form of the T⁻² descending segment with the beginning period of 2 s.

Calculation method of random seismic response to multi-support excitations

Dynamic equation of multi-support excitations

To facilitate formula derivation later, this section first provides a brief introduction on the solving method of the dynamic equation for structures subject to multi-support excitations (Clough and Penzien, 1993). The general form of the dynamic equation for the response of structures to multi-support excitations is shown as follows

\begin{matrix} (\begin{matrix} M_{ss} & M_{sb} \\ M_{bs} & M_{bb} \end{matrix}) (\begin{matrix} {\overset{\cdot\cdot}{X}}_{s} \\ {\overset{\cdot\cdot}{X}}_{b} \end{matrix}) + (\begin{matrix} C_{ss} & C_{sb} \\ C_{bs} & C_{bb} \end{matrix}) (\begin{matrix} {\overset{\cdot}{X}}_{s} \\ {\overset{\cdot}{X}}_{b} \end{matrix}) \\ + (\begin{matrix} K_{ss} & K_{sb} \\ K_{bs} & K_{bb} \end{matrix}) (\begin{matrix} X_{s} \\ X_{b} \end{matrix}) = (\begin{matrix} 0 \\ P_{b} \end{matrix}) \end{matrix}

(13)

where the indices s and b represent structure and foundation, respectively; X is the displacement vector; M , C , and K are the mass, damping, and stiffness matrix, respectively; and P _b is the applied force of bearing by foundation. The absolute displacement is generally divided into pseudo-static displacement and relative dynamic displacement (with superscript s and d)

(\begin{matrix} X_{s} \\ X_{b} \end{matrix}) = (\begin{matrix} X_{s}^{s} \\ X_{b} \end{matrix}) + (\begin{matrix} X_{s}^{d} \\ 0 \end{matrix})

(14)

Ignoring the inertial force and damping force items in equation (13) gives the following equation for the response of pseudo-static displacement by static calculation

X_{s}^{s} = - {K_{ss}}^{- 1} K_{sb} X_{b} = R X_{b}, R = - {K_{ss}}^{- 1} K_{sb}

(15)

Substituting equation (15) into equation (13) and ignoring the damping items (Clough and Penzien, 1993) give the dynamic equation of relative dynamic displacement response as follows

M_{ss} {\overset{\cdot\cdot}{X}}_{s}^{d} + C_{ss} {\overset{\cdot}{X}}_{s}^{d} + K_{ss} X_{s}^{d} = - (M_{ss} R + M_{sb}) {\overset{\cdot\cdot}{X}}_{b}

(16)

The pseudo-static displacement and relative dynamic displacement can be found in equations (15) and (16), and the total displacement can be obtained by equation (14).

Multi-support pseudo excitation method

In the pseudo excitation method, the virtual ground acceleration and displacement excitations are first determined according to the spatial effects of ground motion (Lin et al., 2004)

\begin{matrix} {\tilde{\overset{\cdot\cdot}{U}}}_{b} = P \exp (i ω t), {\tilde{U}}_{b} = - \frac{1}{ω^{2}}, \\ {\tilde{\overset{\cdot\cdot}{U}}}_{b} = - \frac{1}{ω^{2}} P \exp (i ω t) \end{matrix}

(17)

where the superscript ∼ represents the virtual item, and matrix P can be obtained by the Cholesky decomposition of earthquake PSD matrix S (iω) as follows (Lin et al., 2004)

S (i ω) = P^{*} P^{T}

(18)

Transforming the virtual ground motion into virtual acceleration and displacement of the structure degrees of freedom (DOFs) leads to the following equation

\begin{matrix} {\tilde{\overset{\cdot\cdot}{X}}}_{b} = E {\tilde{\overset{\cdot\cdot}{U}}}_{b} = EP \exp (i ω t), \\ {\tilde{X}}_{b} = E {\tilde{U}}_{b} = - \frac{1}{ω^{2}} EP \exp (i ω t) \end{matrix}

(19)

where E , which reflects the relationship between the propagating direction and the structural DOFs, is the transformation matrix between ground motion and structure DOFs.

Substituting equation (19) into equations (15) and (16) gives the following

{\tilde{X}}_{s}^{s} = - \frac{1}{ω^{2}} REP \exp (i ω t)

(20)

\begin{matrix} M_{ss} {\tilde{\overset{\cdot\cdot}{X}}}_{s}^{d} + C_{ss} {\tilde{\overset{\cdot}{X}}}_{s}^{d} + K_{ss} {\tilde{X}}_{s}^{d} \\ = - (M_{ss} R + M_{sb}) EP \exp (i ω t) \end{matrix}

(21)

Then, substituting equations (20) and (21) into equation (14) gives the virtual displacement response as follows

{\tilde{X}}_{s} = {\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d}

(22)

According to the principle of the pseudo excitation method, the total displacement response spectrum can be written thusly

S_{X_{s} X_{s}} (i ω) = {\tilde{X}}_{s}^{*} {\tilde{X}}_{s}^{T} = ({\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d})^{*} ({\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d})^{T}

(23)

Equation (23) can be expanded as follows

\begin{matrix} S_{X_{s} X_{s}} (i ω) & = ({\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d})^{*} ({\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d})^{T} \\ = {\tilde{X}}_{s}^{s *} {\tilde{X}}_{s}^{sT} + {\tilde{X}}_{s}^{d *} {\tilde{X}}_{s}^{dT} + 2 real ({\tilde{X}}_{s}^{s *} {\tilde{X}}_{s}^{dT}) \\ = S_{X_{s} X_{s}}^{s} (i ω) + S_{X_{s} X_{s}}^{d} (i ω) + S_{X_{s} X_{s}}^{sd} (i ω) \end{matrix}

(24)

where $S_{X_{s} X_{s}}^{s} (i ω)$ is the pseudo-static displacement PSD, $S_{X_{s} X_{s}}^{d} (i ω)$ is the relative dynamic displacement PSD, and $S_{X_{s} X_{s}}^{sd} (i ω)$ is the cross PSD of the coupled item.

The structural response can be finally obtained by calculating the expectation of extreme value of the corresponding PSD (Davenport, 1964).

Practical solution form of multi-support pseudo excitation method

In the theory of the pseudo excitation method, exp(iωt) is a very important component. In the derivation process of section “Analysis on reasonability of practical earthquake PSD,” the time factor exp(iωt) always exists as a relative independent item and does not take part in the calculation directly. Therefore, this section will focus on the solving method of equations (20) and (21) with the aim of simplifying the solving method of the structural pseudo response and the pseudo excitation form of bearing by taking the time factor exp(iωt) out of the solution process. A more practical solution form of the pseudo excitation method for structures subject to multi-support excitations is put forward.

Equation (20) is a static calculation process, so exp(iωt) can be ignored. Then equation (20) can be expressed as follows

{\underset{~}{\tilde{X}}}_{s}^{s} = {\underset{~}{\tilde{X}}}_{s}^{s} \exp (i ω t)

(25)

{\underset{~}{\tilde{X}}}_{s}^{s} = - \frac{1}{ω^{2}} REP

(26)

where the subscript ∼ denotes the corresponding virtual item without the time factor exp(iωt).

The right side of equation (21) is the typical harmonic excitation load. If considering the stationary response only, the structural response of equation (21) can be written as follows

{\tilde{X}}_{s}^{s} = {\underset{~}{\tilde{X}}}_{s}^{s} \exp (i ω t)

(27)

(- ω^{2} M_{ss} + i ω C_{ss} + K_{ss}) {\underset{~}{\tilde{X}}}_{s}^{s} = - (M_{ss} R + M_{sb}) EP

(28)

Accordingly, the total response is shown as follows

{\tilde{X}}_{s} = {\tilde{X}}_{s}^{s} + {\tilde{X}}_{s}^{d} = ({\underset{~}{\tilde{X}}}_{s}^{s} + {\underset{~}{\tilde{X}}}_{s}^{d}) \exp (i ω t) = \underset{\tilde{s}}{\underset{~}{X}} \exp (i ω t)

(29)

{\underset{~}{\tilde{X}}}_{s} = {\underset{~}{\tilde{X}}}_{s}^{s} + {\underset{~}{\tilde{X}}}_{s}^{d}

(30)

Substituting equation (29) back to equation (23) to solve the response PSD gives the following

S_{X_{s} X_{s}} (i ω) = {\tilde{X}}_{s}^{*} {\tilde{X}}_{s}^{T} = {\underset{~}{\tilde{X}}}_{s}^{*} {\underset{~}{\tilde{X}}}_{s}^{T}

(31)

We can see from the above equations, ${\tilde{X}}_{s}$ is equivalent to ${\underset{~}{\tilde{X}}}_{s}^{s}$ when solving for the response PSD. Therefore, the original virtual response solving process of equations (20), (21), and (29) are equivalent to the solving processes of equations (26), (28), and (30). The latter does not contain the time factor exp(iωt). In order to correspond with the change in solving process, the form of pseudo excitation is also transformed without the time factor exp(iωt)

{\underset{~}{\tilde{\overset{\cdot\cdot}{X}}}}_{b} = EP, {\underset{~}{\tilde{X}}}_{b} = - \frac{1}{ω^{2}} EP

(32)

Thus, equations (26) and (28) can be indicated by converting the pseudo excitation item as follows

{\underset{~}{\tilde{X}}}_{s}^{s} = R {\underset{~}{\tilde{X}}}_{b}

(33)

(- ω^{2} M_{ss} + i ω C_{ss} + K_{ss}) {\underset{~}{\tilde{X}}}_{s}^{d} = - (M_{ss} R + M_{sb}) {\underset{~}{\tilde{\overset{\cdot\cdot}{X}}}}_{b}

(34)

Based on the derivation above, the practical solving process of the multi-support pseudo excitation method has been put forward:

Constructing the pseudo excitation of bearing without time factor exp(iωt) according to equation (32);

Solving equations (33) and (34) to find the pseudo-static response and relative dynamic response without the time factor, adding these two to obtain the total pseudo response according to equation (30);

Substituting the structural pseudo response without the time factor into equation (31) to get the response PSD.

This solving process is a practical adjustment based on the theory of the multi-support pseudo excitation method. The whole process takes the time factor exp(iωt) out, so the form is more concise and easier.

Random seismic responses for single large span structures to multi-support excitations

Simplified structural model

A simplified single large span structural model with two supports is shown in Figure 4. The concentrated mass is m, the axial stiffness is k₁, the lateral stiffness is k₂, and the span is L.

Figure 4.

Two-mass and two-supporting model.

This structure has two modes: the first mode is homodromous and the second one is reverse. The natural frequency is expressed thusly

ω_{1} = \sqrt{\frac{k_{2}}{m}}, ω_{2} = \sqrt{\frac{2 k_{1} + k_{2}}{m}}

(35)

The corresponding mode vector is shown as follows

φ_{1} = (\begin{matrix} 1 & 1 \end{matrix})^{T}, φ_{2} = (\begin{matrix} 1 & - 1 \end{matrix})^{T}

(36)

The parameter values of the model refer to an actual arch truss. The parameter values are listed as follows: m = 2.0 × 10⁴ kg, k₁ = 1000 kN/m, and k₂ = 400 kN/m. As the incoherence effect is significantly related to the span value, L = 50 m and L = 200 m are considered for the incoherence effect.

Parameters of multi-support ground motions

Assuming that the seismic wave propagates along the longitudinal direction, two spatial effects, the wave-passage effect and the incoherence effect, are considered. Two conditions are selected as follows:

Condition 1: only the wave-passage effect is considered

In this condition, the wave-passage effect is reflected by apparent wave velocity v_app. For easy analysis, the wave-passage frequency ω_app = 2πv_app/d (Ding et al., 2008) is introduced as the analysis parameter. In this condition, the seismic ground motions of two supports are assumed completely coherent, that is, the coherence function of the two supports ρ(ω) = 1.

Condition 2: both the wave-passage effect and the incoherence effect are considered

The influence of the incoherence effect can be further studied by comparing the calculation results between Condition 1 and Condition 2. The parameters of the wave-passage effect are the same as those in Condition 1, and the incoherence effect can be reflected by coherence function ρ(ω). The commonly used Luco–Wong (simplified as L–W) (Luco and Wong, 1986) and Harichandran–Vanmarcke (simplified as H–V) (Harichandran and Vanmarcke, 1986) coherence function models, as shown in equations (37) and (38), are adopted. Considering two spans of 50 and 200 m, all the coherence functions are shown in Figure 5

ρ_{L - W} (ω) = \exp (- α^{2} ω^{2} L^{2}), α = 2.5 \times 1 0^{4}

(37)

\begin{matrix} ρ_{H - V} (ω) = A \exp [- \frac{2 L}{α θ (ω)} (1 - A + α A)] \\ + (1 - A) \exp [- \frac{2 L}{θ (ω)} (1 - A + α A)] \\ θ (ω) = K {[1 + {(\frac{ω}{2 π f_{0}})}^{b}]}^{- 1 / 2}; \\ A = 0.736, α = 0.147, K = 5210 m, \\ f_{0} = 1.09 Hz, b = 2.78 \end{matrix}

(38)

Figure 5.

Coherence function.

The earthquake acceleration PSD function of each support is S_a(ω). The practical PSD model, which is iteratively synthesized by the corrected response spectrum in Chinese code GB 50011 mentioned above, is adopted in this section. Figure 6 shows the synthetic ground motion acceleration PSD S_a(ω) and displacement PSD S_u(ω) assuming that there is a frequent earthquake with a degree of intensity of 7, III class site, and the first design earthquake group.

Figure 6.

PSD curves of earthquake ground motion: (a) acceleration and (b) displacement.

Response variables to be studied

The changing rules of the pseudo-static response, relative dynamic response, and total response are studied in this section. The characteristics of relative dynamic response can be reflected by the excited degree of the mode. Therefore, the following response variables are selected to study the influence mechanism of spatial effects:

Relative dynamic modal coordinate

The relative dynamic modal coordinate q_k directly reflects the excited degree of the kth stage mode to relative dynamic response

X^{d} = \sum_{k = 1}^{2} φ_{k} q_{k}

(39)

where X^d denotes the structural relative dynamic response and φ_k denotes the kth stage mode vector.

Structural internal force

The structural internal forces include beam axial force F₁ and column shear force V₁. Each response includes a corresponding pseudo-static response, relative dynamic response, and total response.

Analytic expression of response PSD to simplified model

The total response PSD is expressed as follows

S (ω) = S^{s} (ω) + S^{d} (ω) + S^{sd} (ω)

(40)

S ^s(ω) and S^d(ω) are the pseudo-static and relative dynamic response PSD, respectively. S^sd(ω) is the coupling item of two response PSDs above. Since S^sd(ω) is much smaller than S^s(ω) and S^d(ω) (Kiureghian and Neuenhofer, 1992), this article ignores the coupling item and only studies the pseudo-static and relative dynamic response, respectively. The pseudo-static and relative dynamic response PSD analytic expressions of the simplified model are derived by the pseudo excitation method in this section.

Pseudo-static response

The pseudo-static responses of beam axial force F₁ and column shear force V₁ are equal. The pseudo-static response PSD is

S_{V_{1}}^{s} (ω) = {(\frac{2 k_{1} k_{2}}{2 k_{1} + k_{2}})}^{2} S_{u} (ω) \cdot \frac{1}{2} [1 - ρ (ω) \cos (2 π \frac{ω}{ω_{app}})]

(41)

in which S_u(ω) = S_a(ω)/ω⁴ is the earthquake displacement PSD and ω_app = 2πv_app/L is the wave-passage frequency.

Relative dynamic response

The characteristics of relative dynamic response can be reflected by the excited degree of the mode. The excited degree can be described by the modal coordinates of the relative dynamic response. The PSDs of the two modal coordinates q₁ and q₂ are shown as follows

S_{q_{1}}^{d} (ω) = {| H_{1} (ω) |}^{2} S_{a} (ω) \cdot \frac{1}{2} [1 + ρ (ω) \cos (2 π \frac{ω}{ω_{app}})]

(42)

\begin{matrix} S_{q_{2}}^{d} (ω) = & \frac{{k_{2}}^{2}}{{(2 k_{1} + k_{2})}^{2}} {| H_{2} (ω) |}^{2} S_{a} (ω) \\ \cdot \frac{1}{2} [1 - ρ (ω) \cos (2 π \frac{ω}{ω_{app}})] \end{matrix}

(43)

H_i(ω) is the transfer function of the ith stage mode

H_{i} (ω) = \frac{1}{ω_{i}^{2} - ω^{2} + i 2 ζ ω_{i} ω}, i = 1, 2

(44)

where ζ = 0.05 is the structural damping ratio and $i = \sqrt{- 1}$

Accordingly, the relative dynamic response PSDs of beam axial force F₁ and column shear force V₁ are shown as follows

S_{F_{1}}^{d} (ω) = k_{1}^{2} S_{q_{1}}^{d} (ω)

(45)

S_{V_{1}}^{d} (ω) = k_{2}^{2} (S_{q_{1}}^{d} (ω) + S_{q_{2}}^{d} (ω))

(46)

The modal coupling item of the relative dynamic response is ignored in equation (46). It can be seen from equations (41) to (43) that the influence of the spatial effect can be reflected by the trigonometric function items as follows

E_{V_{1}}^{s} (ω) = \frac{1}{2} [1 - ρ (ω) \cos (2 π \frac{ω}{ω_{app}})]

(47)

E_{q_{1}}^{d} (ω) = \frac{1}{2} [1 + ρ (ω) \cos (2 π \frac{ω}{ω_{app}})]

(48)

E_{q_{2}}^{d} (ω) = \frac{1}{2} [1 - ρ (ω) \cos (2 π \frac{ω}{ω_{app}})]

(49)

$E_{V_{1}}^{s} (ω)$ is the multi-support effect item of pseudo-static response. $E_{q_{1}}^{d} (ω)$ and $E_{q_{2}}^{d} (ω)$ are the relative dynamic multi-support effect items of the first- and second-stage modal coordinates, respectively.

The structural response can finally be obtained by calculating the expectation of extreme value of corresponding PSD (Davenport, 1964).

Influence mechanism of the multi-support effect

Wave-passage effect

The influence mechanism of the wave-passage effect under Condition 1, which considers the wave-passage effect only, is studied in this section. It can be seen from equations (41) to (49) that the wave-passage frequency ω_app = 2πv_app/d is the major parameter for reflecting the wave-passage effect. Thus, the wave-passage frequency is studied below as the variable parameter.

Figures 7 and 8 show the relationship of different multi-support effect items in equations (41) to (43) when ω_app = 4π. Figures 9 to 12 show the relationship between wave-passage frequency and the expectation of extreme value of pseudo-static internal force, relative dynamic modal coordinate, relative dynamic internal force, and total internal force, respectively.

Figure 7.

Pseudo-static multi-excitation effect item and displacement PSD function curve.

Figure 8.

Relative dynamic multi-excitation effect item and transfer function: (a) first-stage mode and (b) second-stage mode.

Figure 9.

Expectation of extreme value of pseudo-static internal force.

Figure 10.

Expectation of extreme value of relative dynamic modal coordinate: (a) first-stage mode and (b) second-stage mode.

Figure 11.

Expectation of extreme value of relative dynamic internal force: (a) beam axial force and (b) column shear force.

Figure 12.

Expectation of extreme value of total internal force: (a) beam axial force and (b) column shear force.

The influence of the wave-passage effect on the pseudo-static, relative dynamic, and total responses can be drawn as follows:

The influence of the wave-passage effect on the pseudo-static response

As can be seen from equation (41), the pseudo-static response PSD is mainly influenced by the displacement PSD and the pseudo-static multi-support effect item. The frequency distribution of these two is shown in Figure 7. Since the displacement PSD is mainly distributed in a low-frequency band, only the value of the pseudo-static multi-support effect item in a low-frequency band has a large effect on the pseudo-static response.

Figure 9 shows that the pseudo-static response increases and tends to remain at a constant value, that is, the wave-passage effect enhances with the increasing variable 1/ω_app. As the wave-passage frequency ω_app decreases, the time delay and the relative displacement between different supports increase and then the pseudo-static response also increases. However, the maximum ground displacement is limited, so the pseudo-static response tends to remain constant.

The influence of the wave-passage effect on the relative dynamic response

As can be seen from equations (42) and (43), the relative dynamic modal coordinate PSD is influenced by the transfer function, earthquake acceleration PSD, and the multi-support effect item. Comparing Figures 6(a) and 8, the distribution regularity of those three items mentioned above is different from each other. The transfer function has a narrow band characteristic with value only concentrating near the natural frequency band. The earthquake acceleration PSD has a large distribution. The multi-support effect item has a trigonometric distribution. Therefore, the transfer function with narrow band characters mainly influences the result. This is because only the acceleration PSD and the multi-support effect item distributed near the natural frequency have a large effect.

Figure 10 shows that the wave-passage effect leads to a trigonometric fluctuation distribution on the expectation of extreme value of the relative dynamic modal coordinate. This is also coincident with the wave-passage effect on the relative dynamic response PSD. Figure 10 also directly shows how the relative dynamic response is influenced by the variable ω_k/ω_app. When the ratio of natural frequency to wave-passage frequency ω_k/ω_app = 0.5 n (n = 0, 1, 2,…), the expectation of extreme value of the relative dynamic modal coordinate reaches the peak point. The characteristic is shown in Table 6.

The dynamic response of component internal force has a direct correlation with the modal excited degree (refer to equations (45) and (46)). Thus, as shown in Figure 11, the expectation of extreme value of the relative dynamic modal coordinate has the same peak point distribution as Figure 10 and Table 6.

The influence of the wave-passage effect on total response.

Table 6.

Distribution of expectation of the extreme modal coordinate.

ω_k/ω_app	First-stage mode	Second-stage mode
n	Maximum	Minimum
(2n + 1)/2	Minimum	Maximum

n = 0, 1, 2,….

Figure 12 shows that the expectation of extreme value of the total internal force also presents a trigonometric fluctuation distribution, similar to the relative dynamic response. The mid-value of the fluctuation curve tends to increase, similar to the pseudo-static response.

Different response variables have different response regularities. Figure 12(a) shows that the column shear force V₁ obtains the maximum value in the case of uniform excitation. The response value decreases when considering the multi-support excitation effect. Figure 12(b) shows that the beam axial force F₁ is zero in the case of uniform excitation. The multi-support excitation effect produces this additional force. This kind of response variable should be given more attention in the seismic design method of multi-support excitation.

Incoherence effect

Condition 2 is compared with Condition 1 to study the influence mechanism of the incoherence effect in this section. Taking the pseudo-static multi-excitation effect item as an example, Figure 13 shows how it is influenced by the coherence function when ω_app = 4π. Figures 14 and 15, respectively, show how the expectations of extreme value of the pseudo-static internal force and relative dynamic modal coordinate are influenced by the incoherence effect. Figure 16 shows the relationship between the wave-passage frequency and the expectation of extreme value of pseudo-static, relative dynamic, and total internal force when considering both the wave-passage effect and the incoherence effect (H–V = 200 m).

Figure 13.

Pseudo-static multi-excitation effect.

Figure 14.

Expectations of extreme value of pseudo-static internal force.

Figure 15.

Expectations of extreme value of relative dynamic modal coordinate: (a) first-stage mode and (b) second-stage mode.

Figure 16.

Expectation of extreme value of total internal force: (a) beam axial force and (b) column shear force.

It can be seen from Figures 13 to 16:

Because the coherence function is less than 1, the amplitude of the pseudo-static multi-support effect function attenuates when the incoherence effect is considered as shown in Figure 13. The attenuation rate depends on the span and the selected coherence function as shown in Figure 5: the faster the coherence function decreases, the faster the curve amplitude will attenuate.

Figure 14 shows that the incoherence effect enhances the pseudo-static response and different coherence functions affect it to distinct degree. The influence of the H–V model is larger than the L–W model, as the earthquake PSD is mainly distributed in the low-frequency band and the value of the H–V model is less than that of the L–W model at this frequency segment, as shown in Figure 5.

Figure 15 shows that the incoherence effect would not change the peak point distribution of the expectations of extreme value of the relative dynamic response curve but attenuate the amplitude. The lower the correlation is, the larger the amplitude attenuation will be.

It can be seen from Figures 14 and 15(b) that the additional response caused by the multi-support excitation effect, such as the pseudo-static response and the second-stage modal coordinate, will be affected greatly when the wave-passage effect is low. However, when the wave-passage effect is great, the incoherence effect has a small effect on the response, which can be ignored.

Comparing Figure 16 with Figure 12, the pseudo-static response does not increase when considering the incoherence effect, but the amplitudes of the relative and total responses reduce. This can also indicate that considering the wave-passage effect only is more conservative than considering these two spatial effects simultaneously in the structural seismic design.

Determination method of the most unfavorable conditions of structures subject to multi-support excitations

The multi-support response regularity of the simplified single span model has been derived above. The wave-passage effect and incoherence effect should be considered in the seismic design of structures subject to multi-support excitations. The wave-passage effect can be reflected by the apparent wave velocity when the span is determined. Therefore, the determination method of the most unfavorable conditions of the multi-support excitation effect is studied through establishing how to select the most unfavorable apparent wave velocity values and how to consider the incoherence effect. The final conditions can be named “the most unfavorable multi-support input conditions” (Jiang, 2010).

Most unfavorable apparent wave velocity

Since the characteristics of earthquake ground motions are unpredictable and the site condition is complicated, the most unfavorable apparent wave velocity is often difficult to obtain exactly. Usually, a variety of apparent wave velocity values are adopted to ensure the accuracy of the results. As the apparent wave velocity distributes continuously, the issue of how to choose the reasonable ones is worth studying.

It can be seen from Figures 10 and 11 that the wave-passage effect leads to a trigonometric fluctuation distribution on the relative dynamic response. The distribution of peak points is obvious. The ratio of each modal natural frequency ω_k (k = 1, 2) to the wave-passage frequency ω_app = 2πv_app/L may have influence on the distribution of peak points. If ω_k/ω_app = 0.5 n (n = 0, 1, 2,…), the kth stage modal coordinate may reach extreme value. Each apparent wave velocity v_app, which meets the above formula, can be determined as an unfavorable apparent wave velocity. Because the peak point distribution of total response is similar to the relative dynamic response (see Figure 12), these unfavorable apparent wave velocities are also applicable to the total response. The flowchart of the determination method of the most unfavorable multi-support input conditions is shown in Figure 17; the meaning of each parameter is shown in Table 7.

Figure 17.

Determination method of the most unfavorable multi-support input conditions.

Table 7.

Meaning of the parameters in the flowchart below.

Parameters	Meaning	Parameters	Meaning
I	Number of the most unfavorable apparent wave velocity	I _max	Quantity of the most unfavorable apparent wave velocity
k	Number of the mode	k _max	Quantity of mode (k_max = 2)
f_k	Natural frequency of the kth stage mode, f_k = ω_k/2π	D	Bearing span
n	Number of extreme points on the same curve	v _app,min	Lower limit of the apparent wave velocity
v _app,0	Extreme point apparent wave velocity	v _app,I	Unfavorable apparent wave velocity
v _app,0,min v _app,0,max	Effective range of the extreme point apparent wave velocity	v _app,I,min v _app,I,max	Effective range of the unfavorable apparent wave velocity

Combining the flowchart as shown in Figure 17, the procedure to determine the most unfavorable multi-support input conditions is introduced as follows:

1. Determine the lower limit v_app,min and quantity I_max of the apparent wave velocity

This step is the preparation work before the process. The selection of I_max is very important. A small value may lead to inadequate consideration, while a large value may lead to heavy calculation. The apparent wave velocity quantity I_max should be selected considering the span, apparent wave velocity range, and the natural frequency of primary modes.

2. Calculating the apparent wave velocity corresponding to each extreme point

With the ratio of each natural frequency ω_k to wave-passage frequency ω_app, we can obtain many apparent wave velocities, and it is difficult to judge which is the most unfavorable. Therefore, the principle of selection is introduced in this section.

The natural frequency f_k (equation (50)) corresponding to the larger modal coordinate is priority.

The smaller value n with the same natural frequency is priority.

Because it is difficult to judge whether the extreme value is the maximum or minimum, the type of extreme value need not be distinguished. The apparent velocity can be directly obtained by the equation ω_k /ω_app = 0.5 n (n = 0, 1, 2,…)

v_{app, 0} = \frac{2 f_{k} d_{J}}{n}, n = 0, 1, 2, \dots

(50)

Except for the unfavorable apparent wave velocities calculated above, the lower limit of apparent wave velocity v_app,min should also be an unfavorable one.

3. Combination of the close apparent wave velocities

The unfavorable apparent wave velocity values obtained by different natural frequencies may be close. Combination of those close values is necessary to cut down the amount of calculation.

In this section, the range of 0.1 times the period on both sides of the peak point is supported as the effective range, as shown in Figure 18. For the trigonometric function, the lower limit value of the effective range is 0.9 times the extreme value. In other words, the largest error may be 10% due to the velocity combination. The later obtained unfavorable wave velocity, which falls within the effective range of an existing unfavorable one, can be combined. Adopting the most unfavorable apparent wave velocities and comparing the corresponding responses, the maximum response will be used for structural seismic design.

Figure 18.

Schematic of the unfavorable apparent wave velocity combination.

Consideration of the incoherence effect

The influence regularity of the wave-passage effect and incoherence effect has been summarized above: the wave-passage effect leads to a trigonometric fluctuation distribution on the structural response, while the incoherence effect leads to amplitude attenuation as shown in Figure 15. As stated above, if the wave-passage effect is great, the incoherence effect has a small effect on the response, which can be ignored. Therefore, if adequate apparent wave velocities are considered, the response given by considering the wave-passage effect only is larger than considering the wave-passage effect and incoherence effect simultaneously. In this situation, ignoring the incoherence effect seems to be safer.

Conclusion

In this article, the practical earthquake PSD model based on the response spectrum of the Chinese seismic code regulation is first introduced. The long period segment of the response spectrum in the Chinese seismic code is corrected for the displacement reasonability of the PSD. The practical solution form of the multi-support pseudo excitation method is put forward. The PSD analytic formulae of a simplified single span structural model are also derived for the study on the influence mechanism of structural seismic response to multi-support excitations. Based on the response law derived from a simplified model, the method of quickly determining the most unfavorable conditions of the multi-support excitation effect is finally proposed. The main conclusions are summarized as follows:

Numerical analysis implies that the PSD displacement characteristic is determined by the long period segment of the target response spectrum. It is more reasonable to adopt the T⁻² form as the descending segment of the response spectrum curve. So the T⁻² descending segment is involved in the response spectrum of GB 50011 with the beginning period of 2 s. The displacement reasonability of PSD synthesized by the corrected response spectrum is improved and used in the following study.

The practical solution form of the multi-support pseudo excitation method is proposed. This solving process is a practical adjustment based on the theory of the multi-support pseudo excitation method. The whole process takes the time factor exp(iωt) out, so the form is more concise and easier.

By theoretical analysis and numerical calculation, the influence mechanism of a simplified single span structural model subject to multi-support excitations is studied. The wave-passage effect and incoherence effect are considered. Both the spatial effects increase the pseudo-static response. The influence of relative dynamic response by the wave-passage effect and incoherence effect is reflected in the form of a trigonometric function, while the wave-passage frequency influences the period and coherence function results in the amplitude attenuation. When the ratio of natural frequency to wave-passage frequency ω_k/ω_app = 0.5 n (n = 0, 1, 2,…), the structural seismic response reaches extreme value.

Based on the response law derived from the simplified single span model, the method of quickly determining the most unfavorable conditions of the multi-support excitation effect is proposed. This quick determination method can not only reduce the calculation amount but also consider enough conditions to attain greater accuracy of the result. The influence of the incoherence effect need not be considered separately if adequate apparent wave velocities are considered using the determination method. The complex calculation for considering the incoherence effect can be avoided.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China No. 51038006 and Specializes Research Fund for the Doctoral Program of Higher Education No. 20090002110045. The authors gratefully acknowledge financial support from the China Scholarship Council.

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