WO2021082959A1

WO2021082959A1 - Nonlinear dynamic time history analysis method for complex building structure

Info

Publication number: WO2021082959A1
Application number: PCT/CN2020/121713
Authority: WO
Inventors: 徐俊杰; 黄羽立
Original assignee: 李鲁
Priority date: 2019-10-31
Filing date: 2020-10-17
Publication date: 2021-05-06
Also published as: CN110795790B; CN110795790A

Abstract

A nonlinear dynamic time history analysis method for a complex building structure. The method comprises the following steps: step one, performing spatial finite element discretization on a complex building structure, and establishing a motion equation set of a discrete system; step two, selecting parameters and determining a global invariant; and step three, performing calculation by time steps, and calculating the displacement, speed and acceleration at an end moment of each time step. In the method, two key improvements are made to the nonlinear iteration process of a Newmark method, such that the calculation workload is greatly reduced and the calculation efficiency is improved on the premise of ensuring the precision of calculations.

Description

A nonlinear dynamic time-history analysis method for complex building structures

Technical field

The invention belongs to the technical field of building structure design, and specifically relates to a fast analysis method for nonlinear dynamic time history analysis of complex building structures under dynamic external loads.

Background technique

In recent years, earthquakes have occurred frequently, causing structural damage and even collapse of buildings, seriously threatening the safety of people’s lives and property. Therefore, the domestic building structure design code requires that, in the design stage, a nonlinear dynamic time history analysis of complex building structures is required to better grasp the seismic capacity of the structure, and thereby ensure the safety of the structure under the action of an earthquake.

Structural nonlinear dynamic time history analysis is a spatial discrete model of building structure. In the entire seismic time history, the differential equations of the motion of each particle are solved to obtain the displacement, velocity, and acceleration response of each particle in the model, and then the model is obtained The peak values of axial force, shear force and bending moment of each component are used to guide the design of the component section.

At present, the existing technology mainly uses two types of methods for dynamic time history analysis of complex building structures. One type is an explicit method, such as the central difference method. However, the explicit method is stable and the time step of the analysis is limited by the highest frequency of the model. For large-scale structural models, the highest frequency is very large, and when the model has zero mass degrees of freedom or rigid connections, the frequency is infinite, and the explicit algorithm fails. The other type of methods are implicit methods, such as Newmark method, HHT method, generalized α method, etc. These methods are all unconditionally stable methods, which are suitable for dynamic analysis of complex structures. However, when these methods are used for nonlinear dynamic time history analysis When, non-linear iteration cannot be avoided. Because of the large number of complex building structure components, its refined finite element model has a large number of degrees of freedom, and nonlinear iterations, resulting in a huge time-consuming analysis process. For general complex building structures, a dynamic time history analysis often lasts more than a dozen Hours or even days, seriously hindering the research and design of complex building structures.

Summary of the invention

In view of the shortcomings of the existing dynamic time history analysis methods of complex building structures in terms of calculation amount, solution efficiency, and result accuracy, the purpose of the present invention is to make two key improvements to the nonlinear iterative process of the Newmark method and provide a A fast analysis method for nonlinear dynamic time history analysis of complex building structures under dynamic external loads.

The technical scheme of the present invention is as follows:

A method for nonlinear dynamic time history analysis of complex building structures. The method is a fixed iteration number analysis method for nonlinear dynamic time history analysis of complex building structures under earthquake action. The steps are as follows:

The first step is to perform spatial finite element discretization of the complex building structure, and establish a finite element model discrete system for the building structure. The beams and columns adopt fiber beam models, and the shear walls and floor slabs adopt layered shell elements and Rayleigh damping. The element damping matrix integrates the overall stiffness matrix, the overall mass matrix and the overall damping matrix by the element stiffness matrix, the element mass matrix and the element damping matrix, and the motion equations of the discrete system are derived from the Hamilton principle, and the motion equations of the discrete system are established:

Where u,

And ü are the displacement, velocity and acceleration vectors of each particle of the finite element model; M is the mass matrix, C is the damping matrix, F _S is the nonlinear restoring force, which is the nonlinear function of the displacement vector; P is the external dynamic load, earthquake When applied, P=-Mü _g , ü _g is the ground motion acceleration input from the base of the building structure;

The second step is to select parameters and determine global invariants:

1) Select γ, β, σ;

2) Select the time step Δt;

3) According to the known initial displacement u ₀ and initial velocity

Determine the initial acceleration ü ₀ ;

4) Calculate the equivalent stiffness matrix

Among them, K ₀ is the Jacobian matrix of the nonlinear restoring force F _S (u) at u ₀ , that is, the initial stiffness matrix;

Step-by-step calculation time, the end of each time step is calculated time displacements, velocity and acceleration for the i-th time step, known _at time _i t displacement velocity V _i and u _i, is calculated by the i Displacement u _i+1 and velocity _{at t i+1} on two time steps

And acceleration ü _i+1 ;

1) Select the initial solution of the iteration:

2) A total of n non-linear iterations are performed, n≥2, and the kth iteration is as follows:

3) Take the result of the nth iteration as the target solution

Preferably, the value range of γ is γ≥0.5.

Preferably, γ is taken as 0.5.

Preferably, the value range of β is β≥0.25.

Preferably, β is taken as 0.25.

Preferably, the value range of σ is σ≥1.

Preferably, σ is taken as 1.

Preferably, the time step Δt is taken as _{N times the time interval of the base input acceleration record ü g} , and N is a positive integer, preferably 1.

Preferably, the initial displacement and initial velocity are both 0, that is, u ₀ =0 and

The initial acceleration ü ₀ =-ü _g (0).

Preferably, n takes 2 in the n non-linear iterations.

Compared with the prior art, the present invention has the advantages of:

1) The fast dynamic time history analysis method of the present invention only uses the initial stiffness matrix K ₀ and does not need the tangent stiffness matrix K _T. Therefore, it is not necessary to update the equivalent stiffness matrix. It is only necessary to calculate the equivalent stiffness matrix before the time step calculation.

Triangular decomposition can be performed once, which avoids the traditional method of recalculating and decomposing a large number of calculations of the equivalent stiffness matrix at each time step.

2) The present invention adopts the initial stiffness matrix amplification factor σ. For structures containing special hardened materials and considering geometrically nonlinear structures, the value of σ≥1 can be selected to ensure the unconditional stability of the algorithm.

3) The fast dynamic time history analysis method of the present invention only needs to perform a fixed number of nonlinear iterations at each time step, generally two times, which avoids the huge amount of calculation consumed by the traditional method to iterate to meet the tolerance.

4) The fast dynamic time history analysis method of the present invention has a second-order accuracy, which is equivalent to the accuracy of the traditional Newmark method.

5) The dynamic time history analysis method of the present invention has simple steps, only needs to modify the equivalent stiffness update mechanism of the traditional Newmark method, and restricts the number of iterations, even without the need to prepare a new program, and is easy to popularize and apply.

Description of the drawings

Figure 1 Schematic diagram of complex high-rise structure;

Figure 2 Schematic diagram of El-Centro wave;

Fig. 3 A comparison schematic diagram of the horizontal x-direction displacement of the roof calculated by the Newmark method and the dynamic time history analysis method of the present invention;

Fig. 4 A comparison schematic diagram of the horizontal x-direction velocity on the roof of a building calculated by the Newmark method and the dynamic time history analysis method of the present invention;

Fig. 5 A comparison schematic diagram of the vertical x-direction acceleration on the roof of a building calculated by the Newmark method and the dynamic time history analysis method of the present invention;

Detailed ways

The present invention will be further described below in conjunction with specific examples, but the present invention is not limited to these specific embodiments. Those skilled in the art should realize that the present invention covers all alternatives, improvements and equivalents that may be included in the scope of the claims.

The structural principle and working principle of the present invention will be described in detail below in conjunction with the accompanying drawings:

Taking a complex high-rise structure as an example, the nonlinear fast dynamic time history analysis method of the present invention is described in detail. The complex high-rise building is shown in Figure 1. The first three periods of the model are T ₁ =1.815s and T ₂ =1.579. s and T ₃ =0.890s. The dynamic time history analysis method includes the following steps:

The first step is to carry out spatial finite element discretization of the high-rise building to establish a finite element model discrete system of the building structure; the high-rise building includes 23945 nodes, 9744 fiber beam elements defined by 8244 reinforced concrete members and 4704 layered shell elements defined by 177 shear wall components; fiber beam models are used for beams and columns, layered shell elements are used for shear walls and floor slabs, and Rayleigh damping is used to establish element damping matrix, which is composed of element stiffness matrix and element mass matrix The overall stiffness matrix, overall mass matrix and overall damping matrix are integrated with the element damping matrix, and the motion equations of the discrete system are derived from the Hamilton principle, and the motion equations of the discrete system are established:

Where u,

And ü are the displacement, velocity and acceleration vectors of each particle of the finite element model. M is the mass matrix, C is the damping matrix, and F _S is the nonlinear restoring force, which is the nonlinear function of the displacement vector. P is the external dynamic load. If it is an earthquake, P=-Mü _g , ü _g is the ground motion acceleration input from the base of the building structure. In this example, ü _g uses the El Centro seismic wave shown in Figure 2. The acceleration recording interval is 0.01s, and the analysis step length is Δt=0.01s. The discrete process of this step space finite element method is a widely used routine operation, and the details will not be repeated here. When calculating the damping matrix C, a damping ratio of 5% is used, and the first and ninth modes are selected to calculate the Rayleigh damping coefficient, including mass damping and initial stiffness proportional damping.

The second step is to select parameters and determine global invariants:

1) Select γ, β, σ

The value range of γ is γ≥0.5, generally 0.5;

The value range of β is β≥0.25, generally 0.25;

The value range of σ is σ≥1, and it is generally 1;

In this embodiment, the parameters are selected as γ=0.5, β=0.25, and σ=1;

2) Select the time step Δt, which is generally _{N times the time interval of the base input acceleration record ü g} , and N is generally equal to 1. In this embodiment, the El Centro seismic wave shown in Figure 2 is input into the base. The acceleration recording interval is 0.01s, and the analysis step length is Δt=0.01s;

3) According to the known initial displacement u ₀ and initial velocity

Determine the initial acceleration ü ₀ ;

In this embodiment, the initial displacement and initial velocity are both 0, that is, u ₀ =0 and

So the initial acceleration ü ₀ =-ü _g (0);

4) Calculate the equivalent stiffness matrix

Among them, K ₀ is the Jacobian matrix of the nonlinear restoring force F _S (u) at u ₀ , that is, the initial stiffness matrix.

And acceleration ü _i+1 ;

1) Select the initial solution of the iteration:

2) A total of n non-linear iterations are performed, n≥2, generally 2, where the kth iteration is as follows:

3) Take the result of the nth iteration as the target solution

In order to demonstrate the efficiency and accuracy of the dynamic time history analysis method of the present invention, the comparative example adopts the most commonly used second-order precision Newmark method for analysis. For the convenience of comparison, the parameters of the Newark method are also selected as γ=0.5, β=0.25, The step size is Δt=0.01s, and the nonlinear iterative convergence tolerance is selected as Tol=0.001. The analysis result is shown in Figure 3-5. The calculation time is 300 hours. With the dynamic time history analysis method of the present invention, the result is very close to the result using the Newmark method (the two response curves almost overlap), and the specific results are shown in Figures 3-5. However, the calculation time using the method of the present invention is only 5 hours. Under the premise of ensuring the calculation efficiency, the dynamic time history analysis method of the present invention greatly reduces the calculation workload and increases the calculation efficiency by 60 times.

It can be seen from the embodiments that the fast power time history analysis method of the present invention has the following advantages:

It should be understood that the steps of the method described in the present invention are only exemplary descriptions, and there is no special requirement on the time sequence of their sequential execution, unless they have a certain sequence relationship.

As shown above, although the present invention has been described with reference to limited embodiments and drawings, anyone with ordinary knowledge in the field to which the present invention pertains can make various modifications and variations from this description. Therefore, other embodiments, claims, and equivalents fall within the protection scope of the claims.

Claims

A nonlinear dynamic time history analysis method for complex building structures. The method is a fixed iteration number analysis method for nonlinear dynamic time history analysis of complex building structures under external dynamic loads. The steps are as follows:

The first step is to perform spatial finite element discretization of the complex building structure, and establish a finite element model discrete system for the building structure. The beams and columns adopt fiber beam models, and the shear walls and floor slabs adopt layered shell elements and Rayleigh damping. The element damping matrix integrates the overall stiffness matrix, the overall mass matrix and the overall damping matrix by the element stiffness matrix, the element mass matrix and the element damping matrix, and the motion equations of the discrete system are derived from the Hamilton principle, and the motion equations of the discrete system are established:

Where u,
with
They are the displacement, velocity, and acceleration vectors of each particle of the finite element model; M is the mass matrix, C is the damping matrix, and F S is the nonlinear restoring force, which is a nonlinear function of the displacement vector; P is the external dynamic load, during earthquake action ,

Ground motion acceleration input for the base of the building structure;

The second step is to select parameters and determine global invariants:

1) Select γ, β, σ;

2) Select the time step Δt;

3) According to the known initial displacement u 0 and initial velocity
Determine the initial acceleration

4) Calculate the equivalent stiffness matrix

Among them, K 0 is the Jacobian matrix of the nonlinear restoring force F S (u) at u 0 , that is, the initial stiffness matrix;

Step-by-step calculation time, the end of each time step is calculated time displacements, velocity and acceleration for the i-th time step, known at time i t displacement velocity V i and u i, is calculated by the i Displacement u i+1 and velocity at t i+1 on two time steps
And acceleration

1) Select the initial solution of the iteration:

2) A total of n non-linear iterations are performed, n≥2, and the kth iteration is as follows:

3) Take the result of the nth iteration as the target solution
The nonlinear dynamic time history analysis method of a complex building structure according to claim 1, wherein the value range of γ is γ≥0.5.
The nonlinear dynamic time history analysis method of a complex building structure according to claim 2, wherein γ is taken as 0.5.
The nonlinear dynamic time history analysis method of a complex building structure according to claim 1, wherein the value range of β is β≥0.25.
The nonlinear dynamic time history analysis method of a complex building structure according to claim 4, wherein β is taken as 0.25.
The nonlinear dynamic time history analysis method of a complex building structure according to claim 1, wherein the value range of σ is σ≥1.
The nonlinear dynamic time history analysis method of a complex building structure according to claim 6, wherein σ is taken as 1.
The nonlinear dynamic time history analysis method of a complex building structure according to any one of claims 1-7, wherein the time step Δt is taken as the base input acceleration record
The time interval is N times, and N is a positive integer.
The nonlinear dynamic time history analysis method for complex building structures according to claims 1-8, wherein the initial displacement and initial velocity are both 0, that is, u 0 =0 and
Initial acceleration
The nonlinear dynamic time history analysis method of a complex building structure according to any one of claims 1-9, wherein n is 2 in the n-th nonlinear iteration.