Accurate analysis method for vibration response of moderately thick plate structures under random earthquake excitation

Abstract

The present invention belongs to the field of vibration analysis of medium-thick plate structures under random seismic excitation in civil engineering, and discloses an accurate analytical method for random vibration response. It includes: based on the accurate free vibration analysis of medium-thick rectangular plates with a set of simply supported boundaries on opposite sides, the corresponding accurate natural frequency and analytical vibration mode function are given; based on the virtual excitation method, combined with the accurate solution of free vibration, the analytically accurate power spectrum density and root mean square distribution of various vibration responses under random seismic excitation are obtained; through analytical integration followed by discretization in the spatial domain, fine integration in the time domain and discretization in the frequency domain, efficient response analysis of medium-thick plates under random seismic excitation is achieved without losing calculation accuracy. The analytical solution research on the vibration response of medium-thick plate structures under random seismic excitation proposed by the present invention has important engineering practical value and theoretical significance, and can provide a reference benchmark solution for the corresponding numerical analysis and experimental design.

Description

Technical Field

[0001] This invention belongs to the field of vibration analysis of medium-thick plates subjected to random seismic excitation in civil engineering. It involves a precise analytical method for the random vibration response of medium-thick plates under uniformly modulated non-stationary or completely time-frequency non-stationary random seismic excitation, taking into account the effects of first-order shear deformation. Background Art

[0002] Random seismic excitation has great destructive power on engineering structures. The study of the precise vibration response of engineering structures under random seismic excitation has important engineering application value. As a basic engineering structure, medium-thick plates that consider the influence of first-order shear deformation are widely used in engineering structures such as offshore platforms, ships, aircraft, bridges, and high-rise buildings. Researchers have done a lot of in-depth research on the dynamic response analysis of medium-thick plates. In order to obtain the natural frequency and vibration mode of medium-thick plates, the relevant free vibration analysis methods include numerical methods and analytical methods. Among them, approximate numerical methods are widely studied, while analytical closed-form solutions are relatively rare, and the boundary conditions are limited to six types of classical boundary conditions with a set of simple supports on opposite sides. In recent years, many scholars have conducted extensive research on the deterministic forced vibration of medium-thick plates, but there are few studies on the vibration response of medium-thick plates under random excitation, and the research results are limited to numerical approximate solutions.

[0003] Due to the current lack of analytical benchmark solutions for the vibration response of medium-thick plates under random seismic excitation, this paper incorporates the precise natural frequencies and analytical modes of six types of simply supported rectangular plates with analytical closed-form solutions into the decoupled structural vibration differential equations. Based on the virtual excitation method and mode superposition method, the precise power spectral density functions and root mean square distributions of the displacement, velocity, acceleration, and stress responses of medium-thick plates under non-stationary random seismic excitation are analytically obtained. In addition to benchmark solutions for the vibration response of medium-thick plates under random seismic excitation, this paper can also be used to explore key parameters for mitigating the vibration response of medium-thick plates under random seismic excitation. Summary of the Invention

[0004] The present invention solves the problem of lack of analytical benchmark solutions for the vibration response analysis of medium and thick plate structures under random seismic excitation. Based on the mode superposition method and the virtual excitation method, the present invention provides the analytical power spectrum density and root mean square of the vibration response of medium and thick plates under uniformly modulated non-stationary and time-frequency completely non-stationary seismic excitation, providing a reference benchmark solution for the corresponding numerical analysis and experimental design.

[0005] In order to achieve the above object, the technical solution of the present invention is:

[0006] Step 1: For the medium-thick plate structure based on the first-order shear deformation Mindlin theory, consider six types of simply supported boundary conditions, perform analytical derivation of undamped free vibration, use Newton iteration method to solve the transcendental equation, and obtain the accurate natural frequency and analytical mode shape of the medium-thick plate. It includes the following sub-steps:

[0007] Step 1-1: There are three unknown generalized displacements in the thick plate: the rotation angle of the line perpendicular to the mid-plane in the xoz and yoz planes and And the lateral displacement w. For the simple harmonic vibration of the thick plate, w(x,y,t)=W(x,y)e iωt , where W(x,y),ψ x (x,y) and ψ y (x,y) is the mode function, and ω is the natural vibration angular frequency of the structure. The free vibration differential equations of a moderately thick rectangular plate are given for three generalized displacements.

[0008] Step 1-2: Use the elimination method to obtain the values ​​containing only the vibration mode W and only the vibration mode ψ. x , including vibration mode ψ x and the partial differential equations of W.

[0009] Step 1-3: Assume that the two pairs of edges x=0 and x=a of the medium-thick rectangular plate are simply supported, and the two pairs of edges y=0 and y=a are freely combined with three types of boundaries: simply supported, clamped, and free. For the six types of classical boundary conditions with simply supported edges, assume that the mode function is Where m and n represent the half-wave numbers of the plate in the x and y directions, respectively.

[0010] Step 1-4: Substitute the mode shape functions from step 1-3 into the partial differential equations from step 1-2 to obtain the eigenvalue equations corresponding to the three desired generalized displacements. Solve the eigenvalue equations to obtain six eigenvalues ​​related to the desired natural frequencies, which in turn provide expressions for the undetermined coefficients of the three mode shape functions.

[0011] Step 1-5: Substitute the mode function expression containing the constant to be determined in step 1-4 into the six types of simply supported classical boundary conditions in step 1-3 to obtain the eigenvalue equations corresponding to each type of boundary combination, and rewrite them as the product of the eigenvalue matrix and the vector to be determined.

[0012] Step 1-6: To ensure that the constants in the mode function expressions in step 1-4 are not all zero, set the determinant of the eigenvalue matrix in step 1-5 to zero. Solving this transcendental equation using the Newton iteration method can yield the precise natural frequencies of the medium-thick plate.

[0013] Step 1-7: After obtaining the precise natural frequency, further obtain the ratio relationship between the constants to be determined in step 1-4, thereby obtaining the analytical vibration mode functions corresponding to the three generalized displacements to be determined.

[0014] Step 2: Introduce the mode superposition method to introduce the precise natural frequency and analytical mode shape into the random vibration control equation of the decoupled single-degree-of-freedom system; obtain the virtual excitation corresponding to the random ground motion based on the virtual excitation method, and use the time-domain Duhamel integral to obtain the analytical power spectrum density and root mean square formula of the displacement, velocity, acceleration, and stress response of the medium-thick plate. It includes the following sub-steps:

[0015] Step 2-1: Based on the modal superposition method, the generalized displacements of the thick plate in three directions are developed. Considering the orthogonality of the modal shapes, a series of decoupled single-degree-of-freedom systems are obtained for the thick plate continuum structure.

[0016] Step 2-2: Consider applying nonstationary random seismic excitation to a thick plate, including uniformly modulated nonstationary seismic excitation and time-frequency fully nonstationary random seismic excitation, and provide the power spectral density function of the excitation. Uniformly modulated nonstationary seismic excitation only considers temporal nonstationarity, described by the time modulation function; time-frequency fully nonstationary random seismic excitation considers both temporal and frequency nonstationarity, with frequency nonstationarity described by the time-varying filter parameter in the excitation power spectral density function.

[0017] Step 2-3: Based on the virtual excitation method, the random lateral seismic excitation with known power spectral density is rewritten into a virtual excitation form; the virtual excitation is substituted into the decoupled single-degree-of-freedom system in step 2-1 to convert the random vibration into deterministic transient vibration. The virtual generalized displacement response is obtained using the Duhamel integral. Based on the geometric relationship and material constitutive relationship of the medium-thick plate, the analytical power spectral density and root mean square of the generalized displacement, velocity, acceleration, and stress response are finally obtained.

[0018] Step 3: To further improve computational efficiency, the geometric space, virtual excitation, and generalized displacement field of the medium-thick plate are discretized by using analytical integration or differentiation followed by discretization in the spatial domain, frequency domain discretization, and time domain fine integration. This allows for the acquisition of discrete analytical solutions to the random vibration responses of medium-thick shell structures under random seismic excitation, and allows for the efficient batch acquisition of the distribution of random vibration responses of medium-thick cylindrical shells.

[0019] The analytical response analysis of the vibration response of medium-thick plates under random seismic excitation proposed in this invention can serve as a benchmark solution with extremely high calculation accuracy to compare and verify other numerical approximation methods and experimental parameter designs. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 This is a flow chart for implementing an accurate analysis method for the vibration response of medium-thick plate structures under random earthquake excitation.

[0021] Figure 2 is a schematic diagram of the geometric model of the medium and thick plate, the application of random seismic excitation and six types of simply supported boundary conditions provided by an embodiment of the present invention, wherein Figure 2(a) is the geometric model of the medium and thick plate under random seismic excitation, and Figure 2(b) is the six types of simply supported boundary conditions considered for the medium and thick plate.

[0022] FIG3 shows the root mean square displacement response at the center point of a medium-thick plate under two types of random seismic excitation provided by an embodiment of the present invention, where FIG3(a) considers uniformly modulated non-stationary excitation, and FIG3(b) considers time-frequency completely non-stationary excitation.

[0023] Figure 4 shows the power spectral density curves (t = 3s) of the displacement response at the center point of medium-thick plates with different relative thicknesses under two types of random seismic excitation provided by an embodiment of the present invention, where Figure 4(a) considers uniformly modulated non-stationary excitation, and Figure 4(b) considers time-frequency completely non-stationary excitation.

[0024] FIG5 shows the root mean square curves of the displacement responses at the center points of medium and thick plates of different relative thicknesses under two types of random seismic excitations provided by an embodiment of the present invention, where FIG5(a) considers uniformly modulated non-stationary excitation, and FIG5(b) considers time-frequency completely non-stationary excitation.

[0025] Figure 6 shows the power spectral density curves (t = 3s) of the displacement response at the center point of thick plates with different elastic moduli under two types of random seismic excitation provided by an embodiment of the present invention, where Figure 6(a) considers uniformly modulated non-stationary excitation, and Figure 6(b) considers time-frequency completely non-stationary excitation.

[0026] FIG7 shows the root mean square (RMS) curves of the displacement responses at the center point of thick plates with different elastic moduli under two types of random seismic excitations provided by an embodiment of the present invention, wherein FIG7(a) considers uniformly modulated non-stationary excitation, and FIG7(b) considers time-frequency completely non-stationary excitation.

[0027] Figure 8 shows the power spectral density curves (t = 3s) of the displacement response at the center point of a thick plate under two types of random seismic excitations under different boundary conditions provided by an embodiment of the present invention, where Figure 8(a) considers uniformly modulated non-stationary excitation, and Figure 8(b) considers time-frequency completely non-stationary excitation.

[0028] Figure 9 shows the root mean square curves of the displacement responses at the center point of a thick plate under different boundary conditions under two types of random seismic excitations provided by an embodiment of the present invention, where Figure 9(a) considers uniformly modulated non-stationary excitation, and Figure 9(b) considers time-frequency completely non-stationary excitation. DETAILED DESCRIPTION

[0029] Example 1:

[0030] Consider a medium-thick plate with four simply supported sides under non-stationary random earthquake excitation, where the plate length a = 10m, width b = 10m, Poisson's ratio υ = 0.2, and density ρ = 2600Kg / m 3 , shear correction coefficient κ = 0.8667, thickness h = 0.5m, elastic modulus E = 20GPa, structural damping ratio ζ = 0.05. Uniform modulation non-stationary and time-frequency completely non-stationary ground motion excitation are considered respectively, where the time modulation function To describe the temporal stationarity, the Clough-Penzien spectrum model is used for both types of non-stationary ground motion excitations, where the time-varying filter parameter θ(t) = [ω g (t),ξ g (t);ω f (t),ξ f (t)] describes the non-stationary nature of the frequency, S0 = 1m 2 / s 3 ,ζ g =ζ f =0.64,ω g =15.6rad / s (for uniformly modulated non-stationary ground motion excitation), T = 10s, ω0 = 39.4rad / s, ω T =4.86rad / s (for time-frequency completely non-stationary ground vibration excitation).

[0031] Based on the present invention and Monte Carlo simulations, Figure 3 shows the RMS displacement response curves for the center of a medium-thick plate under two types of nonstationary random seismic excitation. Furthermore, Table 1 provides the specific RMS response values ​​and CPU time consumption at t = 3 s. Comparison with Monte Carlo simulation results demonstrates the accuracy and efficiency of the proposed benchmark solution for the vibration response of medium-thick plates under random seismic excitation.

[0032] Table 1 RMS response of the center position of the medium-thick plate under non-stationary ground motion excitation (t = 3s)

[0033]

[0034] Figures 4 and 5 show the power spectral density and root mean square (RMS) of the displacement responses of medium-thick plates with different relative thicknesses under two types of random seismic excitation, respectively, with relative thicknesses (thickness h / plate length a) set to 0.05, 0.06, and 0.07. It can be seen that the peak position of the response power spectral density shifts later with increasing relative thickness, due to the increase in the fundamental frequency of the medium-thick plate with increasing relative thickness. Furthermore, under non-stationary seismic excitation, the amplitude of the response power spectral density decreases overall with increasing relative thickness, while the RMS of the response gradually decreases. This means that the variability of the response decreases, improving structural safety.

[0035] Figures 6 and 7 show the power spectral density and root mean square (RMS) of the displacement responses of medium-thick plates with different elastic moduli under two types of random seismic excitation, respectively, with the elastic modulus set to 20 GPa, 25 GPa, and 30 GPa. It can be seen that as the elastic modulus increases, the bending stiffness of the medium-thick plate increases, the peak of the power spectral density response shifts later as the fundamental frequency of the structure increases, and the RMS displacement response of the structure induced by the same random excitation decreases.

[0036] Figures 8 and 9 show the power spectrum density and root mean square (RMS) of the displacement response of medium-thick plates under two types of random seismic excitation, respectively, for three simply supported sides and one clamped side (SSSC), four simply supported sides (SSSS), and three simply supported sides and one free side (SSSF). It can be seen that as the boundary stiffness decreases, the resonance peak of the medium-thick plate response power spectrum shifts to the low-frequency range. At the same time, the RMS response increases with decreasing boundary stiffness. Considering Figures 4 to 9, increasing the relative thickness, elastic modulus, and boundary stiffness of the structure can achieve the goal of reducing vibration of medium-thick plates under random seismic excitation.

Claims

1. A precise analysis method for the vibration response of medium-thick plate structures under random seismic excitation. Based on the mode superposition method and the virtual excitation method, it provides the analytical power spectrum density and root mean square (RMS) of the vibration response of medium-thick plate structures under uniformly modulated non-stationary and time-frequency completely non-stationary seismic excitation, providing a reference benchmark solution for corresponding numerical analysis and design. Follow these steps: Step 1: For the medium-thick plate structure based on the first-order shear deformation Mindlin theory, consider six types of simply supported boundary conditions and perform analytical derivation of the undamped free vibration. Use the Newton iteration method to solve the transcendental equation and obtain the accurate natural frequency and analytical mode shape of the medium-thick plate. The following steps are involved: Step 1-1: There are three unknown generalized displacements in the thick plate: the rotation angle of the line perpendicular to the mid-plane in the xoz and yoz planes and And the lateral displacement w; for the simple harmonic vibration of the thick plate, w(x,y,t)=W(x,y)e iωt , where W(x,y),ψ x (x,y) and ψ y (x,y) is the mode function, ω is the natural vibration angular frequency of the structure; the free vibration differential equations of the moderately thick rectangular plate are given for three generalized displacements; Step 1-2: Use the elimination method to obtain the values ​​containing only the vibration mode W and only the vibration mode ψ. x , including vibration mode ψ x and the partial differential equations of W; Step 1-3: Assume that the two pairs of edges x = 0 and x = a of the medium-thick rectangular plate are simply supported, and the two pairs of edges y = 0 and y = a are freely combined with three types of boundaries: simply supported, clamped, and free. For the six types of classical boundary conditions with simply supported edges, assume that the mode function is Where m and n represent the half-wave numbers in the x and y directions of the medium and thick plate, respectively; Step 1-4: Substitute the mode shape functions in step 1-3 into the partial differential equations in step 1-2 to obtain the eigenvalue equations corresponding to the three generalized displacements to be determined; solve the eigenvalue equations to obtain six characteristic roots related to the natural frequencies to be determined, and further obtain the expressions for the undetermined coefficients of the three mode shape functions; Step 1-5: Substitute the mode function expression containing the constant to be determined in step 1-4 into the six types of simply supported classical boundary conditions in step 1-3 to obtain the eigenvalue equations corresponding to each type of boundary combination, and rewrite them as the product of the eigenvalue matrix and the vector to be determined; Step 1-6: To ensure that the constants to be solved in the mode function expression in step 1-4 are not all zero, set the determinant of the eigenvalue matrix in step 1-5 to zero; use the Newton iteration method to solve the transcendental equation to obtain the accurate natural frequency of the medium and thick plate; Step 1-7: After obtaining the accurate natural frequency, further obtain the ratio relationship between the constants to be determined in step 1-4, thereby obtaining the analytical vibration mode functions corresponding to the three generalized displacements to be determined; Step 2: Introduce the mode superposition method to introduce the precise natural frequency and analytical mode shape into the random vibration control equation of the decoupled single-degree-of-freedom system. Based on the virtual excitation method, obtain the virtual excitation corresponding to the random ground motion. Use the Duhamel integral in the time domain to obtain the analytical power spectrum density and root mean square formula of the displacement, velocity, acceleration, and stress response of the moderately thick plate. Step 2-1: Based on the mode superposition method, the generalized displacements of the thick plate in three directions are developed. Considering the orthogonality of the mode shapes, a series of decoupled single-degree-of-freedom systems for the thick plate continuum structure are obtained. Step 2-2: Consider applying non-stationary random seismic excitation to the medium-thick plate, including uniformly modulated non-stationary seismic excitation and time-frequency completely non-stationary random seismic excitation, and give the power spectrum density function of the excitation. Among them, the uniformly modulated non-stationary seismic excitation only considers the time non-stationarity, and the time non-stationarity is described by the time modulation function; the time-frequency completely non-stationary random seismic excitation considers both time and frequency non-stationarity, and the frequency non-stationarity is described by the time-varying filtering parameter in the excitation power spectrum density function. Step 2-3: Based on the virtual excitation method, rewrite the random lateral seismic excitation with known power spectral density into a virtual excitation form. Substitute the virtual excitation into the decoupled single-degree-of-freedom system in step 2-1 to convert the random vibration into deterministic transient vibration. Use the Duhamel integral to obtain the virtual generalized displacement response. Based on the geometric relationship and material constitutive relationship of the medium-thick plate, the analytical power spectral density and root mean square of the generalized displacement, velocity, acceleration, and stress response are finally obtained. Step 3: To further improve computational efficiency, the geometric space, virtual excitation, and generalized displacement field of the medium-thick plate are discretized by using analytical integration or differentiation followed by discretization in the spatial domain, frequency domain discretization, and time domain fine integration. This allows for the acquisition of discrete analytical solutions for the random vibration responses of medium-thick shell structures under random seismic motions, and allows for the efficient batch acquisition of the distribution of random vibration responses of medium-thick cylindrical shells.

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