A seismic optimization design method for frame building structures considering multiple randomness

Abstract

A seismic optimization design method for frame building structures considering multiple randomness includes: based on the direct probability integral method, the sensitivity formulas of the structural dynamic reliability to the distribution parameters of random variables and deterministic variables are derived respectively, and an efficient and accurate analysis method for the dynamic reliability sensitivity is proposed; the construction cost of the building structure is used as the objective function, and the dynamic reliability is used as the constraint function to construct a frame building structure parameter optimization design framework, and the optimization problem is solved based on the dynamic reliability sensitivity analysis method and the moving asymptote method proposed in the present invention. Compared with the traditional building structure design scheme, the present invention takes into account the random excitation of near-fault earthquake motion that the frame building structure may suffer and the randomness of the structure's own parameters, and provides a design scheme for the frame building structure that takes into account both safety and economy.

Description

Technical Field

[0001] The invention relates to an earthquake-resistant optimization design method for a frame building structure considering multiple randomness, and is applicable to the field of earthquake-resistant design of building structures. Background Art

[0002] Random uncertainties are common in building structures, including randomness of structural parameters caused by building material properties, geometric dimensions, manufacturing errors and material degradation, as well as random excitations caused by earthquakes and wind loads suffered by the structure during service. These random uncertainties have an important impact on the safety, applicability and durability of building structures, requiring engineers to consider the structural reliability under uncertain factors during the structural design process. Structural optimization design uses systematic methods and theories to achieve the best performance of the structure with the least materials and the lowest cost, which is in line with the current national development plan and current economic conditions. Structural optimization design combines optimization theory with engineering design, so that people can find the optimal structure from a large number of design schemes while carrying out engineering design, greatly improving the efficiency and quality of engineering design. At present, it has been widely used in many fields such as national defense, civil engineering, and industry. Traditional deterministic structural design can no longer meet the safety and economic requirements of the structure. Therefore, the present invention, under the premise of considering random uncertainties, performs frame building structure optimization design based on dynamic reliability to provide a theoretical reference for building structure design.

[0003] Based on the principle of probability conservation, the direct probability integration method realizes the decoupled calculation of physical evolution and probability density evolution, and can efficiently and accurately calculate the dynamic reliability of linear and nonlinear structures. In addition, the direct probability integration method provides a clear expression of dynamic reliability, which brings great convenience and higher calculation efficiency to reliability sensitivity analysis. In view of this, the present invention uses the direct probability integration method to develop an optimization design method for frame building structures based on dynamic reliability, and comprehensively optimizes the design of frame building structures with random parameters under the random excitation of near-fault ground motion. Summary of the invention

[0004] The present invention proposes an optimization design method for a shear frame building structure with random parameters under random excitation of near-fault seismic motion, which is suitable for the optimization design of frame building structures under random seismic motion excitation. It mainly includes: establishing an optimization formula for a frame building structure with random structural parameters under random near-fault seismic motion excitation; using the direct probability integration method to accurately and efficiently calculate the dynamic reliability of the structure; calculating the sensitivity of the dynamic reliability of the structure to the design variables based on the direct probability integration method; using the moving asymptote method to update the design variables, and iteratively obtaining the optimal structural parameters that meet the reliability requirements of the structure.

[0005] The technical solution of the present invention is:

[0006] (1) Establish an optimization formula for frame building structures with random structural parameters under random near-fault ground motion excitation with construction cost as the objective function and dynamic reliability as the constraint function;

[0007] (2) Explicitly calculate the objective function and the sensitivity of the objective function to the design variables;

[0008] (3) Using direct probability integration method to solve the structural dynamic reliability, that is, solving the constraint function;

[0009] (4) Based on step (3), the sensitivity of the structural dynamic reliability to two types of design variables, namely, random variable distribution parameters and deterministic variables, is calculated respectively;

[0010] (5) Update the design variables using the moving asymptote method based on the objective function, constraint function and their sensitivity calculation results;

[0011] (6) Calculate the difference between the design variables of this cycle and the previous cycle to determine whether convergence has occurred. If not, repeat the process from step (2) until the structural design with the minimum construction cost that satisfies the dynamic reliability constraint is obtained.

[0012] Specifically, step (3) includes:

[0013] The point selection method based on generalized F-deviation is used to divide the probability space to obtain representative points and assigned probabilities.

[0014] A near-fault ground motion model based on Gabor wavelet transform is used to generate random near-fault ground motions.

[0015] Solve the structural dynamics equations at different representative points;

[0016] Solve the probability density integral equation to obtain the structural dynamic reliability;

[0017] Specifically, step (4) includes:

[0018] The relevant parameters are substituted into the explicit integral sensitivity of the structural dynamic reliability to the random variable distribution parameters and the efficient sensitivity analysis formula to the deterministic design variables given by the present invention, and the sensitivity of the structural dynamic reliability to the random variable distribution parameters and the deterministic variables are calculated respectively.

[0019] The beneficial effects of the present invention are:

[0020] The direct probability integral method is used to solve the structural dynamic reliability, and based on this method, efficient and accurate calculation methods for the sensitivity of dynamic reliability to the distribution parameters of random variables and deterministic variables are developed respectively;

[0021] This method is applicable to the optimal design of frame building structures with random parameters under random earthquake excitation;

[0022] This method constructs a building structure optimization design framework with minimum construction cost as the objective function and structural dynamic reliability as the constraint function, realizes the optimal design of frame building structural parameters considering randomness, significantly improves the seismic safety of the structure, and reduces the construction cost of the building.

[0023] Description of Figures / Tables

[0024] Figure 1 The present invention is a flow chart for implementing a frame building structure optimization design method based on dynamic reliability.

[0025] Figure 2 A schematic diagram of a 10-story shear frame building structure with a tuned mass damper provided in an embodiment of the present invention.

[0026] Figure 3(a) shows the standard deviation of 2000 random near-fault ground acceleration time histories.

[0027] Figure 3(b) is a representative ground acceleration time history curve.

[0028] Figure 4 This is an optimized iterative history diagram of the mean stiffness of each floor of a building structure provided by an embodiment of the present invention.

[0029] Figure 5 An optimization iteration history diagram of the stiffness of a tuned mass damper provided in an embodiment of the present invention.

[0030] Figure 6 An optimization iteration history diagram of the objective function provided by an embodiment of the present invention.

[0031] Figure 7 An optimization iteration history graph of a constraint function provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0032] Figure 1 This is a flow chart for implementing a frame building structure optimization design method based on dynamic reliability. It includes the following steps:

[0033] (1) Establish an optimization formula for frame building structures with random structural parameters under random near-fault ground motion excitation with construction cost as the objective function and dynamic reliability as the constraint function;

[0034] (2) Explicitly calculate the objective function and the sensitivity of the objective function to the design variables;

[0035] (3) Using direct probability integration method to solve the structural dynamic reliability, that is, solving the constraint function;

[0036] (4) Based on step (3), the sensitivity of the structural dynamic reliability to two types of design variables, namely, random variable distribution parameters and deterministic variables, is calculated respectively;

[0037] (5) Update the design variables using the moving asymptote method based on the objective function, constraint function and their sensitivity calculation results;

[0038] (6) Calculate the difference between the design variables of this cycle and the previous cycle to determine whether convergence has occurred. If not, repeat the process from step (2) until the structural design with the minimum construction cost that satisfies the dynamic reliability constraint is obtained.

[0039] Step (1): Establish an optimization formula for a frame building structure with random structural parameters under random near-fault seismic excitation with construction cost as the objective function and dynamic reliability as the constraint function.

[0040] The seismic optimization design formula of frame building structure based on dynamic reliability is as follows:

[0041]

[0042] Where d represents the design variable vector, which can be a random variable distribution parameter, a deterministic variable, or both; Θ is the random variable vector related to the structural parameters and random excitations; f(d) is the objective function, which is a function related to the construction cost; h(d) is the constraint function; R 0 is the reliability threshold; R is the structural dynamic reliability, g(d,Θ,t) represents the functional function, which is related to the structural dynamic response. When g(d,Θ,t)>0, the structure is safe; d L and d U represent the lower and upper bounds of the design variables respectively.

[0043] Step (2): Explicitly calculate the objective function and the sensitivity of the objective function to the design variables.

[0044] The objective function f(d) is a function related to the construction cost and is usually in an explicit cumulative form. Therefore, the objective function and its sensitivity analysis to the design variables can be simply solved explicitly.

[0045] Step (3): Use the direct probability integration method to solve the structural dynamic reliability, that is, to solve the constraint function.

[0046] The generation of random near-fault ground motion is divided into two steps: 1) low-frequency-long period components, using Gabor wavelets to establish random pulse function models at different fault distances; 2) high-frequency components, using spectral expressions based on random functions to generate residual acceleration. Through simulation, a series of near-fault random ground motion samples can be generated for structural analysis.

[0047] The motion equation of the frame building structure under random earthquake excitation is:

[0048]

[0049] Where X is the displacement vector, M, C, K represent the mass matrix, inherent damping matrix and stiffness matrix of the model respectively, and a g is the input seismic acceleration. Construct the equivalent extreme value mapping of the functional function in (0, t] time

[0050]

[0051] x represents the dynamic response of attention, x 0 is the response threshold. When the structural dynamic response exceeds the threshold, the structure fails. t represents a given duration, which is the duration of the earthquake in this invention. Based on the direct probability integration method, the structural dynamic reliability can be expressed as

[0052]

[0053] Where H[·] is the Heaviside function, Ω Θ is the probability space composed of random variables, p Θ (θ) is the joint probability density function of the random variables. Based on the generalized F deviation point selection method, a series of representative points θ are generated. j ,j=1,2,…,N, and its assigned probability p θj ,j=1,2,…,N, then formula (4) can be expressed as

[0054]

[0055] The structural dynamic reliability is obtained by solving equation (5).

[0056] Step (4): Based on step (3), calculate the sensitivity of the structural dynamic reliability to two types of design variables: random variable distribution parameters and deterministic variables.

[0057] Dynamic reliability sensitivity is the partial derivative of reliability with respect to design variables, and is an important parameter required by the optimization solver. Then the sensitivity of structural dynamic reliability to random variable distribution parameters (such as mean, standard deviation, etc.) is:

[0058]

[0059] In the formula represents the distribution parameter of the ith random variable. Since the distribution parameter of a random variable only determines the probability distribution characteristics of the random variable, the performance function is determined by structural analysis and has nothing to do with the distribution parameter of the random variable. In other words, the performance function is related to the coordinates of the random variable, but has nothing to do with the probability density value of the random variable at the coordinates. Therefore, the partial derivative of the performance function with respect to the distribution parameter of the random variable is 0, that is, the first part of formula (6) is equal to 0, so

[0060]

[0061] The solution of formula (7) also refers to the process of formula (4) in the direct probability integration method, that is,

[0062]

[0063] It is worth pointing out that formula (8) only has one more thing than formula (5): One term, and this term can be solved explicitly, so the reliability sensitivity analysis is carried out simultaneously with the reliability analysis without any additional work. The reliability sensitivity analysis is carried out using the explicit integral form shown in formula (8), which is not only accurate but also efficient.

[0064] The sensitivity of structural dynamic reliability to deterministic variables is:

[0065]

[0066] In the formula, x i represents the ith deterministic variable. Obviously, the joint probability density function of random variables is independent of the deterministic variables, so the second term in equation (9) is equal to 0. Since the derivative of the Heaviside function is the Dirac delta function, using the chain rule of derivation, equation (9) can be expressed as

[0067]

[0068] To numerically calculate equation (10), the Gaussian function is used to approximate the Dirac delta function. Similarly, the solution of equation (10) in the probability space after segmentation is

[0069]

[0070] Where σ is the standard deviation of the Gaussian function, determined based on kernel density estimation

[0071]

[0072] In the formula, std[·] represents the standard deviation, iqr[·] represents the quartile, N represents the number of representative points, and the smoothing factor a∈(0,1) is recommended to be between 0.6 and 0.9. Due to the particularity of the Dirac delta function, the standard deviation of the Gaussian function used to replace it is very small. The Gaussian function has the 3σ principle, that is, when |g(θ j ,X)|>>3σ, Using this feature, the solution of equation (11) only needs to consider representative points where the performance function value is close to 0. Just in case, we can consider representative points where the response is within the range of 5σ. Then, equation (11) is simplified to

[0073]

[0074] Formula (13) expresses the reliability sensitivity as the sensitivity form of the performance function, and the sensitivity of the performance function can be solved by the finite difference method. Since the performance function value of each representative point needs to be solved in the reliability solution process, in the sensitivity solution in Formula (13), it is only necessary to screen out the representative points whose performance function values ​​are within the range of 5σ and repeatedly calculate the performance function sensitivity, and then accumulate them to obtain the reliability sensitivity, which greatly reduces the reliability sensitivity calculation cost and improves the calculation efficiency.

[0075] In this step, by numerically solving equations (8) and (13), the sensitivity of the structural dynamic reliability to the random variable distribution parameter and the deterministic variable can be obtained respectively. This is a key step in this invention.

[0076] Step (5): Based on the objective function, constraint function and their sensitivity calculation results, the design variables are updated using the moving asymptote method;

[0077] The moving asymptote method is a high-performance optimization solver that requires the objective function value, objective function sensitivity, constraint function value, constraint function sensitivity, and some other information in each iteration. The moving asymptote method can update the design variable values ​​toward the optimal solution.

[0078] Step (6): Calculate the difference between the design variables of this cycle and the previous cycle to determine whether convergence has occurred. If not, repeat the process from step (2) until the minimum construction cost structural design that meets the dynamic reliability constraint is obtained.

[0079] The iterative convergence criterion in this method is to judge whether the change of the design variable meets the requirements. If the change of the design variable meets the requirements, the iteration converges and the final design variable value is the optimal solution. Otherwise, the iteration continues and the structural analysis and design variable update are repeated.

[0080] Table 1 shows the final optimization results of the objective function, the mean stiffness of each layer of the structure and the stiffness of the tuned mass damper when the reliability threshold is 0.95 provided in the embodiment of the present invention.

[0081] Table 1

[0082]

[0083] Reference example: Optimal design of a 10-story shear frame structure with tuned mass damping subjected to random near-fault ground motion.

[0084] Based on the dynamic reliability optimization design method provided by the present invention, a 10-story shear frame structure with tuned mass damping under random near-fault ground motion (schematic diagram is shown in FIG. Figure 2 ) as an example. Assume that the mass of each floor of the 10-story building structure is the same, m = 8.75 × 10 4 kg, the mass of the tuned mass damper is m 0 =7.16×10 4 kg. The modal damping ratio of the system is the same, that is, ζ = 0.05. The interlayer stiffness of every two layers of the 10-story building structure is the same, expressed as k from bottom to top. 1 ,k 2 ,k 3 ,k 4 and k 5 ; The stiffness of the tuned mass damper is k 0 . Assume that the building structure stiffness is a normally distributed random variable with a coefficient of variation of 0.2, and its mean is the design variable; the stiffness of the tuned mass damper is a deterministic design variable. This problem involves 14 random variables, of which 5 normally distributed variables are structural stiffness, and the other 9 random variables are used to generate random near-fault ground motions. The maximum peak acceleration of the generated random near-fault ground motions is adjusted to 0.5g. The specific calculation steps are as follows:

[0085] 1. The seismic optimization design of a 10-story shear frame building with a tuned mass damper under random excitation is listed as follows:

[0086]

[0087] Among them, x drift represents the inter-layer displacement. When the maximum inter-layer displacement exceeds 0.02m, the structure fails. In this problem, the reliability threshold R 0 =0.95.

[0088] 2. Accumulate the structural stiffness and tuned mass damper stiffness, and calculate the objective function and its sensitivity value.

[0089] 3. The probability space composed of random variables is divided by the point selection method based on generalized F deviation to generate representative points and assigned probabilities. The generated representative points are used to generate a corresponding number of near-fault random ground motions using the near-fault ground motion generation method based on Gabor wavelet transform. The generated acceleration sample standard deviation and an acceleration sample time history are shown in Figure 3. The dynamic response of the frame building structure under the action of near-fault ground motion is solved. The response solution result is substituted into formula (5) to solve the dynamic reliability of the structure.

[0090] 4. Calculate the structural dynamic reliability for the mean stiffness k of each layer using formula (8): i , i = 1, 2, 3, 4, 5, the sensitivity of the structural dynamic reliability to the stiffness k of the tuned mass damper is calculated using formula (13) 0 sensitivity.

[0091] 5. Based on the calculated objective function, constraint function and its sensitivity information, the moving asymptote method is used to update the values ​​of the design variables.

[0092] 6. Convergence criterion: Determine whether the change of design variables in two adjacent iteration steps satisfies ||d m –d m-1 ||≤10 -2 , if satisfied, the iteration converges, d m This is the optimal design value of the structure; if it is not satisfied, repeat the above steps from step 2 until the preset structural reliability requirements are met, and the optimal design values ​​of the stiffness of each layer of the structure and the tuned mass damper are obtained.

[0093] This application provides a method for optimizing the design of frame building structures based on dynamic reliability, and proposes an optimization design scheme for frame building structures with the goal of minimizing construction costs and the constraints of structural dynamic reliability. It uses the direct probability integral method to efficiently and accurately solve the structural dynamic reliability and its sensitivity, and uses the moving asymptote method to update the design variables and optimize the design parameters of the building structure. This invention realizes the optimization design of frame building structure parameters based on dynamic reliability, fully considers the seismic motion that the structure may suffer and the randomness of the structure's own parameters, significantly improves the seismic safety of the structure, and reduces the construction cost of the structure.

Claims

1. A seismic optimization design method for frame building structures considering multiple randomness, characterized in that The following steps, Step (1): Establish an optimization formula for a frame building structure with random structural parameters under random near-fault ground motion excitation with construction cost as the objective function and dynamic reliability as the constraint function; The seismic optimization design formula of frame building structure based on dynamic reliability is as follows: Where d represents the design variable vector, which is a random variable distribution parameter, a deterministic variable, or both; Θ is a random variable vector related to structural parameters and random excitations; f(d) is the objective function, which is a function related to construction cost; h(d) is the constraint function; R0 is the reliability threshold; R is the structural dynamic reliability; g(d,Θ,t) represents the functional function, which is related to the structural dynamic response. When g(d,Θ,t)>0, the structure is safe; d L and d U denote the lower and upper bounds of the design variables respectively; Step (2): Explicitly calculate the objective function and the sensitivity of the objective function to the design variables; The objective function f(d) is a function related to the construction cost; Step (3): Use direct probability integration method to solve the structural dynamic reliability, that is, solve the constraint function; The generation of random near-fault ground motions is divided into two steps: 1) For the low-frequency and long-period components, Gabor wavelet is used to establish a random pulse function model at different fault distances; 2) For high-frequency components, a spectrum expression based on random functions is used to generate residual acceleration; a series of near-fault random ground motion samples are generated through simulation for structural analysis; The motion equation of the frame building structure under random earthquake excitation is: Where X is the displacement vector, M, C, K represent the mass matrix, inherent damping matrix and stiffness matrix of the model respectively, and a g is the input seismic acceleration; construct the equivalent extreme value mapping of the functional function in (0, t] time x represents the dynamic response of interest, x0 is the response threshold, when the structural dynamic response exceeds the threshold, the structure fails, t represents the given duration, which is the earthquake duration in this invention; based on the direct probability integration method, the structural dynamic reliability can be expressed as Where H[·] is the Heaviside function, Ω Θ is the probability space composed of random variables, p Θ (θ) is the joint probability density function of the random variables. Based on the generalized F-deviation point selection method, a series of representative points θ are generated. j ,j= 1 ,2,…, N , and its assigned probability Then formula (4) can be expressed as The structural dynamic reliability is obtained by solving equation (5); Step (4): Based on step (3), the sensitivity of the structural dynamic reliability to two types of design variables, namely, random variable distribution parameters and deterministic variables, is calculated respectively; The dynamic reliability sensitivity is the partial derivative of the reliability with respect to the design variable; the sensitivity of the structural dynamic reliability with respect to the distribution parameter of the random variable is In the formula represents the distribution parameter of the i-th random variable; the partial derivative of the performance function with respect to the distribution parameter of the random variable is 0, that is, the first part of formula (6) is equal to 0, so The solution of formula (7) also refers to the process of formula (4) in the direct probability integration method, that is, Formula (8) only has one more thing than formula (5): One item; The sensitivity of structural dynamic reliability to deterministic variables is: In the formula, x i represents the ith deterministic variable. Obviously, the joint probability density function of random variables is independent of the deterministic variables, so the second term in equation (9) is equal to 0. Since the derivative of the Heaviside function is the Dirac delta function, using the chain derivation rule, equation (9) is expressed as To numerically calculate equation (10), the Gaussian function is used to approximate the Dirac delta function. Similarly, the solution of equation (10) in the probability space after segmentation is Where σ is the standard deviation of the Gaussian function, determined based on kernel density estimation Where std[·] represents the standard deviation, iqr[·] represents the quartile, N represents the number of representative points, and the smoothing factor a∈(0,1) is recommended to be between 0.6 and 0.

9. Due to the particularity of the Dirac delta function, the standard deviation of the Gaussian function used to replace it is very small. The Gaussian function has the 3σ principle, that is, when |g(θ j ,X)|>>3σ, Using this feature, the solution of equation (11) only needs to consider representative points whose performance function values ​​are close to 0 and representative points whose responses are within the range of 5σ. Then equation (11) is simplified to Formula (13) expresses the reliability sensitivity as the sensitivity form of the performance function. The sensitivity of the performance function is solved by the finite difference method. The representative points whose performance function values ​​are within the range of 5σ are selected to repeatedly calculate the performance function sensitivity, and then the reliability sensitivity is accumulated. In this step, by numerically solving equations (8) and (13), the sensitivity of the structural dynamic reliability to the random variable distribution parameters and the deterministic variables are obtained respectively; Step (5): Based on the objective function, constraint function and their sensitivity calculation results, the design variables are updated using the moving asymptote method; Step (6): Calculate the difference between the design variables of this cycle and the previous cycle to determine whether convergence has occurred. If not, repeat the process from step (2) until the minimum construction cost structural design that meets the dynamic reliability constraint is obtained. The iterative convergence criterion in this method is to judge whether the change of the design variable meets the requirements. If the change of the design variable meets the requirements, the iteration converges and the final design variable value is the optimal solution. Otherwise, the iteration continues and the structural analysis and design variable update are repeated.

2. The method for seismic optimization design of frame building structures considering multiple randomness according to claim 1, characterized in that: The step (3) comprises: The point selection method based on generalized F-deviation is used to divide the probability space to obtain representative points and assigned probabilities. A near-fault ground motion model based on Gabor wavelet transform is used to generate random near-fault ground motions. Solve the structural dynamics equations at different representative points; Solve the probability density integral equation to obtain the structural dynamic reliability.

3. A method for seismic optimization design of frame building structures considering multiple randomness according to claim 1 or 2, characterized in that: The step (4) comprises: substituting the relevant parameters into the given explicit integral form sensitivity of the structural dynamic reliability to the random variable distribution parameters and the efficient sensitivity analysis formula to the deterministic design variables, and calculating the sensitivity of the structural dynamic reliability to the random variable distribution parameters and the deterministic variables respectively.

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