Rod system structure system reliability analysis method without searching failure mode
By constructing a virtual stiffness matrix and using its minimum eigenvalue as the function, the problems of long calculation time and insufficient accuracy in the reliability analysis of high-order statically indeterminate bar structure systems are solved, and efficient and accurate reliability assessment is achieved.
Patent Information
- Application Number
- CN202510996245.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-18
- Publication Date
- 2025-10-31
AI Technical Summary
Existing technologies for analyzing the reliability of highly statically indeterminate bar structures require the identification of numerous failure modes, resulting in long calculation times and insufficient accuracy. Furthermore, approximate methods that rely on a few failure modes are prone to missing critical modes.
By constructing a virtual stiffness matrix and using its minimum eigenvalue as a function, failure mode identification is avoided. The reliability of the structural system is evaluated by combining the probability density function, and the probability density evolution method is used to calculate the probability of structural failure.
It enables efficient calculation of the system reliability of highly statically indeterminate bar structures, avoiding failure mode identification and correlation analysis, and improving calculation accuracy and speed.
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Figure CN120874447A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of engineering structure technology, specifically relating to a reliability analysis method for bar structure systems that does not require failure mode search. Background Technology
[0002] Truss structures, such as trusses, space frames, and frames, are widely used in engineering structures across various industrial sectors due to their excellent mechanical properties. To ensure the safety of these structures during service, their system reliability needs to be predicted and assessed. Currently, the analysis methods for the reliability of truss structure systems typically rely on the identification of structural failure modes. Based on failure modes, the structure is abstracted into a series, parallel, or hybrid system, and then methods such as upper and lower bound estimation techniques, probabilistic network evaluation techniques, and narrow bound estimation techniques are used to approximate the failure probability of this abstract system.
[0003] The number of failure modes increases exponentially with the degree of static indeterminacy of a structure. For a highly statically indeterminate engineering structure, the number of failure modes can easily reach millions. This makes identifying all failure modes virtually impossible. Although various methods have been developed to approximate the reliability of structural systems by identifying only a few failure modes, including the load increment method, branch and bound method, β-bound method, and deep learning-based methods, these methods are still prone to missing major failure modes, thus failing to fully guarantee the accuracy of the structural system reliability calculation. In addition, the large amount of repetitive structural analysis and correlation calculations between failure modes required for highly statically indeterminate structures directly affect the calculation time and accuracy of the structural system reliability. Therefore, proposing a reliability analysis method for truss structural systems that does not rely on failure mode search has significant theoretical and engineering practical value. Summary of the Invention
[0004] The purpose of this invention is to address the shortcomings of existing technologies by providing a reliability analysis method for bar structure systems that does not require failure mode searching. This method avoids determining the failure modes of the structure and their correlation analysis, thus solving a key problem in the reliability calculation of highly statically indeterminate bar structure systems.
[0005] The specific steps of this invention are as follows: Step (1) Derive the elastic stiffness matrix based on the finite element model of the structure; The elastic stiffness matrix of a hinged rod system with n degrees of freedom and b members is: ; It is a diagonal matrix with b rows and b columns, and the pivot element of the diagonal matrix is the axial stiffness of the element. , , , and These represent the elastic modulus, cross-sectional area, and length of element k, respectively. Let n be the n x b balance matrix of the structure, denoted as: The k-th column vector of the balance matrix , and Let i and j represent the node coordinates of node i and j in element k, respectively, and the superscript T indicates transpose.
[0006] Step (2) Based on the finite element model of the structure, output the internal forces of all elements; The load factor is taken as the generalized time. According to the structural finite element method, at any time step Internal, structural displacement increment satisfy , Indicates load, For all nodes Displacement at any moment This indicates that after considering the nonlinearity of the material, This is a diagonal matrix consisting of b rows and b columns. At each time step, Newton's method is used to solve for the nodal displacement increments. ; Unit internal force increment Calculate and output the internal forces of all elements at each generalized time point. .
[0007] Step (3) Subtract the contribution of the section of the member that needs to bear the internal force in the structural elastic stiffness matrix, and construct the virtual stiffness matrix of the bar structure; (3-1) Load Internal forces generated in element k As a reduction in its axial stiffness, the reduced element stiffness is: ; Let k be the bearing capacity corresponding to buckling or yielding, which is a known constant; when When this occurs, it indicates that unit k has failed. Degenerate to zero; (3-2) will diagonal elements Replace with ,but Convert to a virtual structural stiffness matrix considering element stiffness degradation ; It is determined by the reduction factor The resulting b-row, b-column diagonal matrix is represented as , It is the identity matrix. This is the element bearing capacity vector.
[0008] Step (4) performs eigenvalue decomposition on the virtual stiffness matrix of the structure and outputs its minimum eigenvalue; the minimum eigenvalue characterizes the degree of geometric variability of the distance of the rod structure under load, and the minimum eigenvalue serves as a functional function for judging the failure of the structural system; (4-1) Perform eigenvalue decomposition. ; Arranged in ascending order eigenvalues The diagonal matrix formed For the corresponding feature vector The orthogonal matrix formed If the failure of some elements causes geometrical changes in the structure, then the structural system fails. In this case, the virtual stiffness matrix... Singularity, its smallest eigenvalue Reduced to 0. If For positive definiteness, its smallest eigenvalue is... A value greater than 0 indicates that the structure remains geometrically unchanged.
[0009] (4-2) Structural design parameters include the cross-sectional area of the members. Elastic modulus Load Unit bearing capacity A function of node coordinates. Considering that some or all design parameters are random, the function function of the structural system is expressed as: ; The vector consisting of design parameters to be considered as random variables. Let represent the s-th random variable, and S represent the number of random variables. The function of a structural system is its minimum eigenvalue. This indicates that the structure remains geometrically invariant and the system is in a safe state; when When this happens, the structure becomes geometrically variable, meaning the system fails.
[0010] For a truss structure composed of beam elements, the members need to be checked for individual or combined action of axial force, bending moment, and torque, which are resisted by the axial stiffness, bending stiffness, and torsional stiffness of the beam, respectively. The portion of the beam stiffness used to resist the three types of internal forces is deducted. A virtual stiffness matrix of the structure is established according to steps (1)-(4), and its minimum eigenvalue is used as the function to calculate the reliability of the structural system.
[0011] Step (5) Solve for the probability density function of the function; the joint probability density function of the design parameters as random variables is: The probability density function of the function in step (4) is determined using the probability distribution evaluation method. .
[0012] Step (6) Calculate the system reliability and failure probability of the rod structure; System reliability of truss structures ; failure probability of bar structure .
[0013] The beneficial effects of this invention include: 1. This invention constructs a virtual stiffness matrix for a rod structure under load.
[0014] 2. This invention uses the minimum eigenvalue of the virtual stiffness matrix of the structure as the functional function for judging the failure of the structural system, thus avoiding the need to determine the failure mode of the structure and perform correlation analysis.
[0015] 3. The method proposed in this invention is applicable to the system reliability analysis of highly statically indeterminate bar structures.
[0016] 4. By subtracting the contribution of the component sections required to bear the internal forces generated by external loads from the structural stiffness matrix, a similar virtual stiffness matrix can be established for other types of structures. Based on the same idea, this structural system reliability analysis method can be extended to other structural types. Attached Figure Description
[0017] Figure 1 This is a schematic diagram of a regular square pyramidal space frame structure; Figure 2 This is a flowchart of the analysis method of the present invention; Figure 3 This is a schematic diagram of the probability density function of the minimum eigenvalue of a virtual stiffness matrix. Detailed Implementation
[0018] The invention will be further described below with reference to specific embodiments.
[0019] Figure 1 A regular square pyramidal space frame with a thickness of 1.5m and a length and width of 30m is shown. It is supported by 20 fully constrained top chord nodes around its perimeter. The finite element model of the space frame contains 800 rod elements with hinged ends and 201 free nodes. The Young's modulus of the rod elements is... The bar elements are divided into three groups corresponding to the top chord, bottom chord, and web members. The cross-sectional areas of each group of bar elements are as follows: , and It is assumed that the tensile and compressive strengths of the elements are the same and that both are ideal elastoplastic materials. Their load-bearing capacity These are considered to be independent random variables. Random vertical nodal loads are applied to all upper chord nodes of the space frame. It consists of independent dead loads and live load They are summed. All random variables follow a normal distribution, and their means and coefficients of variation are shown in Table 1.
[0020] Table 1 Random variables and their mean and coefficient of variation
[0021] A reliability analysis method for truss structure systems without the need for failure mode search, such as Figure 2 As shown: Step (1) Derive the elastic stiffness matrix based on the finite element model of the structure.
[0022] The elastic stiffness matrix of a hinged rod system with n degrees of freedom and b members is: ; It is a diagonal matrix with b rows and b columns, and the pivot element of the diagonal matrix is the axial stiffness of the element. , , , and These represent the elastic modulus, cross-sectional area, and length of element k, respectively. Let n be the n x b balance matrix of the structure, denoted as: The k-th column vector of the balance matrix , and Let i and j represent the node coordinates of node i and j in element k, respectively, and the superscript T indicates transpose.
[0023] Step (2) Output the internal forces of all elements based on the finite element model of the structure.
[0024] The load factor is taken as the generalized time. According to the structural finite element method, at any time step Internal, structural displacement increment satisfy , Indicates load, For all nodes Displacement at any moment This indicates that after considering the nonlinearity of the material, The diagonal matrix consists of b rows and b columns. In this embodiment, the materials of the members are assumed to be ideal elastic-plastic models, but nonlinear models for other materials are also applicable to the method of this invention. At each time step, the Newton method is used to solve for the nodal displacement increments. ; Unit internal force increment Calculate and output the internal forces of all elements at each generalized time point. .
[0025] Step (3) Subtract the contribution of the section of the member that needs to bear the internal force in the structural elastic stiffness matrix, and construct the virtual stiffness matrix of the bar system.
[0026] (3-1) Load Internal forces generated in element k As a reduction in its axial stiffness, the reduced element stiffness is: ; Let element k be the bearing capacity corresponding to buckling or yielding, which is usually a known constant; when When this occurs, it indicates that unit k has failed. Degenerate to zero; (3-2) will diagonal elements Replace with ,but Convert to a virtual structural stiffness matrix considering element stiffness degradation ; It is determined by the reduction factor The resulting b-row, b-column diagonal matrix is represented as , It is the identity matrix. This is the element bearing capacity vector.
[0027] Step (4) performs eigenvalue decomposition on the virtual stiffness matrix of the structure and outputs its minimum eigenvalue; the minimum eigenvalue characterizes the degree of geometric variability of the distance of the rod structure under load, and the minimum eigenvalue serves as a functional function for judging the failure of the structural system.
[0028] (4-1) Perform eigenvalue decomposition. ; Arranged in ascending order eigenvalues The diagonal matrix formed For the corresponding feature vector The orthogonal matrix formed If the failure of some elements causes geometrical changes in the structure, then the structural system fails; in this case, the virtual stiffness matrix... Singularity, its smallest eigenvalue Reduced to 0; if For positive definiteness, its smallest eigenvalue is... A value greater than 0 indicates that the structure remains geometrically unchanged. In this embodiment, when all random variables take the mean, .
[0029] (4-2) Structural design parameters include the cross-sectional area of the members. Elastic modulus Load Unit bearing capacity The function of node coordinates; in this embodiment, there are a total of 5 independent random variables. and Therefore, the structural system function of the space frame is: The function of a structural system, i.e., its minimum eigenvalue. This indicates that the structure remains geometrically invariant and the system is in a safe state; when When this happens, the structure becomes geometrically variable, meaning the system fails.
[0030] Step (5) Solve for the probability density function of the function.
[0031] In this embodiment, the five independent random variables all follow a normal distribution, and their means and coefficients of variation are shown in Table 1. The probability density function of the function in step (4-2) is determined using a probability distribution evaluation method. This embodiment employs a probability density evolution method, the specific steps of which include: (5-1) Using point set renormalization techniques in random variables Distribution area Internal selection Representative points. In this embodiment... Each representative point corresponds to a representative unit. ; Let represent the distribution domain corresponding to the q-th representative unit. The assigned probability within this representative unit is... ;in, , Describes the joint probability density function of random variables; (5-2) Based on the output of step (2), the internal forces of all elements at each generalized time point Repeat steps (3) and (4) to obtain the minimum eigenvalue of the structural virtual stiffness matrix at each generalized time point. According to the central difference method, the rate of change of the smallest eigenvalue is... ; (5-3) According to the principle of probability conservation, the generalized probability density evolution equation of the minimum eigenvalue is: (1); express time and The joint probability density function. For each representative point, equation (1) is discretized as: (2); .
[0032] The corresponding initial conditions are , ; Absorption boundary conditions are ; (5-4) Solving the discrete generalized probability density evolution equation (2) using the finite difference method with a one-sided difference scheme yields the result. probability density function .
[0033] Figure 3 The probability density function of the minimum eigenvalue of the virtual stiffness matrix of the space frame is shown.
[0034] Step (6) Calculate the system reliability and failure probability of the rod structure.
[0035] System reliability of truss structures Correspondingly, the failure probability of the rod structure .
[0036] The failure probabilities of the space frame obtained using the method of this invention are listed in Table 2. For comparison, a sample size of [missing information] was used. Monte Carlo simulations were used to estimate the failure probability of the space frame, and the results are also listed in Table 2. Simultaneously, Monte Carlo simulations were used to estimate the probability density function of the minimum eigenvalue of the virtual stiffness matrix of the structure, such as... Figure 3 As shown in Table 2, the computation time of the three methods on a workstation configured with an Intel(R) Core(TM) i7-10700 CPU @ 2.90GHz and 16.0 GB of RAM is also presented. It can be seen that, compared with the results of Monte Carlo simulations, the computation time obtained by the method of this invention is significantly faster. The relative error is 7.62%, while the calculation time is 1 / 59.7 of that.
[0037] Table 2 Calculation results of different methods
[0038] The static indeterminate number of this space frame is 197, and the maximum number of failure modes of the structure is [missing information]. Its main failure modes number in the hundreds of millions. Therefore, determining all failure modes of the structure using methods such as branch and bound is an impossible task. It is clear that failure mode-based structural system reliability analysis methods are not suitable for this structure.
[0039] The specific embodiments described in this invention are preferred embodiments and are not intended to limit the scope of this invention. That is, all equivalent changes and modifications made within the scope of the claims of this invention should be considered within the technical scope of this invention.
Claims
1. A reliability analysis method for bar structure systems without failure mode search, characterized in that, The specific steps are as follows: Step (1) Derive the elastic stiffness matrix based on the finite element model of the structure; Step (2) Based on the finite element model of the structure, output the internal forces of all elements; Step (3) Subtract the contribution of the section of the member that needs to bear the internal force in the structural elastic stiffness matrix, and construct the virtual stiffness matrix of the bar structure; Step (4) Perform eigenvalue decomposition on the virtual stiffness matrix of the structure and output its minimum eigenvalue; The minimum eigenvalue characterizes the degree of geometric variability of the distance of the rod structure under load, and the minimum eigenvalue serves as a functional function for judging the failure of the structural system. Step (5) Solve for the probability density function of the function; the joint probability density function of the design parameters as random variables is: The probability density function of the function in step (4) is determined using the probability distribution evaluation method. ; Step (6) Calculate the system reliability and failure probability of the rod structure; System reliability of truss structures Failure probability of the rod structure .
2. The reliability analysis method for a rod structure system without failure mode search as described in claim 1, characterized in that, Step (1) specifically involves: The elastic stiffness matrix of a hinged rod system with n degrees of freedom and b members is: ; It is a diagonal matrix with b rows and b columns, and the pivot element of the diagonal matrix is the axial stiffness of the element. , , , and These represent the elastic modulus, cross-sectional area, and length of element k, respectively. Let n be the n x b balance matrix of the structure, denoted as: The k-th column vector of the balance matrix , and Let i and j represent the node coordinates of node i and j in element k, respectively, and the superscript T indicates transpose.
3. The reliability analysis method for a rod structure system without failure mode search as described in claim 1, characterized in that, Step (2) specifically involves: The load factor is taken as the generalized time. According to the structural finite element method, at any time step Internal, structural displacement increment satisfy , Indicates load, For all nodes Displacement at any moment This indicates that after considering the nonlinearity of the material, it is... The matrix is composed of b rows and b columns of diagonal matrix; at each time step, Newton's method is used to solve for the nodal displacement increments. ; Unit internal force increment Calculate and output the internal forces of all elements at each generalized time point. .
4. The reliability analysis method for a rod structure system without failure mode search as described in claim 1, characterized in that, Step (3) specifically involves: (3-1) Load Internal forces generated in element k As a reduction in its axial stiffness, the reduced element stiffness is: ; Let k be the bearing capacity corresponding to buckling or yielding, which is a known constant; when When this occurs, it indicates that unit k has failed. Degenerate to zero; (3-2) will diagonal elements Replace with ,but Convert to a virtual structural stiffness matrix considering element stiffness degradation ; It is determined by the reduction factor The resulting b-row, b-column diagonal matrix is represented as , It is the identity matrix. This is the element bearing capacity vector.
5. The reliability analysis method for a rod structure system without failure mode search as described in claim 1, characterized in that, Step (4) specifically involves: (4-1) Perform eigenvalue decomposition. ; Arranged in ascending order eigenvalues The diagonal matrix formed For the corresponding feature vector The orthogonal matrix formed If the failure of some elements causes geometrical changes in the structure, then the structural system fails; in this case, the virtual stiffness matrix... Singularity, its smallest eigenvalue Reduced to 0; if For positive definiteness, its smallest eigenvalue is... A value greater than 0 indicates that the structure remains geometrically unchanged; (4-2) Structural design parameters include the cross-sectional area of the members. Elastic modulus Load Unit bearing capacity The function of node coordinates; the functional expression of the structural system is as follows: ; The vector consisting of design parameters to be considered as random variables. Let S represent the s-th random variable, and S represent the number of random variables. The function of a structural system, i.e., the minimum eigenvalue. This indicates that the structure remains geometrically invariant and the system is in a safe state; when When this happens, the structure becomes geometrically variable, meaning the system fails.
6. The reliability analysis method for a rod structure system without failure mode search as described in claim 5, characterized in that, In step (4), for the rod structure composed of beam elements, the members need to be checked for axial force, bending moment and torque individually or in combination, and resisted by the axial stiffness, bending stiffness and torsional stiffness of the beam respectively. The three types of stiffness of the beam are deducted from the parts used to resist the three types of internal forces respectively; according to steps (1)-(4), the virtual stiffness matrix of the structure is established, and its minimum eigenvalue is used as the function to calculate the reliability of the structural system.