A multi-point excitation response spectrum calculation method considering fluid-solid coupling

Abstract

The present invention discloses a multi-point excitation response spectrum calculation method considering fluid-solid coupling, which is characterized by comprising the following steps: Step 1: Establish a structural finite element model based on general finite element software, and output the mass matrix M and stiffness matrix K of the structure through the software; Step 2: Calculate the dynamic water pressure P generated by the water-pier dynamic interaction based on the radiation wave theory; Step 3: Substitute the determined dynamic water pressure P into the vibration control equation of the structure, and simplify the control equation to obtain the vibration control equation considering the water-pier interaction; Step 4: Calculate the structural formation and natural frequency after considering the water-pier interaction according to the vibration control equation determined in the above step 3. The present invention can not only calculate the seismic response of the bridge under multi-point excitation, but also consider the dynamic interaction between the water body and the pier.

Claims

1. A multi-point excitation response spectrum calculation method considering fluid-solid coupling, Features: The specific steps include: Step 1: Establish a structural finite element model based on general finite element software, and output the mass matrix M and stiffness matrix K of the structure through the software; Step 2: Based on the radiation wave theory, calculate the dynamic water pressure P generated by the dynamic interaction between water and bridge piers; Step 3: Determine the dynamic water pressure P, substitute it into the vibration control equation of the structure, and simplify the control equation to obtain the vibration control equation considering the water-pier interaction; Step 4: Based on the vibration control equation determined in step 3 above, calculate the structural formation and natural frequency after considering the water-pier interaction; Step 5: According to the vibration control equation determined in step 3 above and the formation and natural frequency determined in step 4 above, based on the random vibration theory, the structural response expression is derived, and a calculation program is compiled to solve the average peak value of the structural seismic response.

2. According to the multi-point excitation response spectrum calculation method considering fluid-solid coupling effect as described in claim 1, Features: The method for determining the dynamic water pressure P in step 2 is: Dynamic water pressure P of the pier and the acceleration x″ at the pier bottom under seismic action g , relative acceleration x″ of the pier: P=-(M 1 x″ g +M 2 x″) (1) Where M 1 、M 2 are the rigid hydrodynamic added mass and elastic hydrodynamic added mass matrices respectively, The matrices M are 1 、M 2 elements; W is the cross-sectional length of the bridge pier perpendicular to the earthquake direction, ρ is the water density, h is the water depth of the bridge pier, k 0 , k m is the wave number, k i =(2i-1)π / (2h);σ=ω 2 / g, ω is the frequency of ground motion, g is the acceleration of gravity; H 1 (·) is the first-order Hankel function of the second kind, K 1 (·) is the modified second-kind first-order Bessel function; z j is the height of the bridge pier node j, Γ i is the integration range, z is the height of the dynamic water pressure calculation point.

3. According to claim 2, a multi-point excitation response spectrum calculation method considering fluid-solid coupling effect, Features: The method for determining the vibration control equation of the structure described in step 3 is: Under multi-point excitation, the structural vibration equation is: In the formula, x=[x 1 ,…,x n ] T The n-dimensional displacement column vector representing the unconstrained degrees of freedom of the superstructure, u = [u 1 ,…,u m ] T represents the m-dimensional displacement column vector of the constrained degrees of freedom at the pier support; M, C and K are the n×n-dimensional mass, damping and stiffness matrices of the unconstrained degrees of freedom of the structure, respectively. g , C g and K g are the m×m-dimensional mass, damping and stiffness matrices of the constrained degrees of freedom at the pier support, M c , C c and K c They represent the n×m-dimensional coupled mass, damping and stiffness matrices between the above two groups of degrees of freedom, respectively, and F is the m-dimensional reaction force column vector at the constrained degrees of freedom at the pier bottom; Substitute the hydrodynamic pressure P determined in step 2 into the load vector on the right side of the equal sign in equation (4): Decompose the displacement x of the unconstrained degree of freedom of the structure into the pseudo-static displacement x s and dynamic displacement x d , substitute into equation (5) and expand the first equation. When solving the quasi-static displacement, x related to the dynamic displacement d , x′, u′, x″, u″ are all 0, so the quasi-static displacement can be calculated by the following formula: x s =-K -1 K c u=Ru(6) Where R is the influence matrix; Substitute equation (6) into equation (5) and expand the second line. After simplification, we can get: (M+M 2 )x′ d ′+Cx′ d +Kx d =-(M+M 1 )Ru″(7) The above is the vibration control equation of the bridge structure considering the water-pier interaction.

4. According to claim 3, a multi-point excitation response spectrum calculation method considering fluid-solid coupling effect, Features: The method for determining the structural formation and natural frequency described in step 4 is: After considering the water-pier interaction, the structural system formation Φ=[φ 1 φ 2 …φ n ]、Self-oscillation frequency ω=(ω 1 ω 2 …ω n ), mass matrix M+M 2 , and the stiffness matrix K satisfy the following relationship: |K-ω i 2 (M+M 2 )|=0 (8) 5. According to the multi-point excitation response spectrum calculation method considering fluid-solid coupling effect as described in claim 4, Features: The method for deriving the structural response expression described in step 5 is: The structural dynamic displacement and structural formation satisfy the following relationship: x d =Φy (10) In the formula, y=[y 1 y 2 …y n ] is the coordinate vector; Substituting equation (10) into equation (7), and using the orthogonality of the formation, we can obtain: In the formula, r k is the kth column of the influence matrix R; For the natural frequency ω i , damping ratio ζ i A single degree of freedom system, under the excitation u′ k Response s under the action of ki satisfy: From formula (11) and formula (13), we can get: For any structural response RS, the bending moment, shear force, and displacement can all be expressed linearly by the structure, that is: RS=q T x=q T (x s +x d )(15) Where q is the transformation vector; Substituting equations (6), (10), and (14) into equation (13), we can obtain: a k =q T r k (17) According to random vibration theory, the power spectrum G of the response RS zz for: In the formula, a l Calculated by formula (17), subscript k is replaced by l; b lj Calculated by formula (18), subscript ki is replaced by lj, where k and l represent the numbers of the pier support, and i and j represent the structural formation numbers; H i (iω) is the frequency response function, G xy is the cross power spectrum of random processes x and y; By integrating both sides of the above equation in the frequency domain, we can get the mean square value σ of the response RS. Z for: Where σ ul and σ slj Calculate by equations (22) and (23) respectively, subscript u k and ki Replace with u l and lj ,u k 、u l is the excitation displacement time history at the pier support k and l, s ki 、s lj Calculated by formula (13); Usually, the maximum value of a random process and its root mean square satisfy the following equation: E(max|z|)=p z s z (27) In the formula, p z 、p uk 、p ski is the peak factor, D k (ω i ,ζ i ) is the incentive u k The response spectrum of i , i The value at Substituting equations (27)-(29) into (21), we can obtain: Calculations show that is approximately equal to 1, so:

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