Multiple failure modes tend to be identified in the reliability analysis of a redundant truss structure. This identification process involves updating the model for identifying the next potential failure members. Herein we intend to update the finite element model automatically in the identification process of failure modes and further perform the system reliability analysis efficiently. This study presents a framework that is implemented through the joint simulation of MATLAB and APDL and consists of three parts: reliability index of a single member, identification of dominant failure modes, and system-level reliability analysis for system reliability analysis of truss structures. Firstly, RSM (response surface method) combines with a constrained optimization model to calculate the reliability indices of members. Then the
For a complex structure, multiple failure modes can result in its failure. It is essential to analyze the system-level reliability of a structure to achieve an accurate reliability estimation. Consequently, many research efforts to estimate the failure probability at the system level had been put forward for the past few decades. The identification of dominant failure modes and the joint failure probability of dominant failure modes are the major challenges of the system reliability analysis for a structure.
Innumerable failure modes can lead to the failure of a redundant structural system, and estimating the system reliability by means of dominant failure modes is sufficiently accurate, which was demonstrated in the past research. Many methods have been proposed to identify dominant failure modes such as the incremental loading method [
Through the dominant failure modes obtained, the joint failure probability or system-level reliability index can be computed, which is another challenge. A number of methods have been proposed to overcome this difficulty, such as the wide bound method [
Besides, the performance function of the structure is usually implicit in practical engineering, which facilitated the rapid development of surrogate models such as RSM (response surface method), Kriging model, and neural network etc. [
Herein, we propose a reliability evaluation framework that can calculate the system reliability index and system failure probability to analyze the system reliability for a truss structure with the implicit performance function. This framework is realized through the joint simulation of MATLAB and APDL. Its program is written in MATLAB, which can drive APDL to build a model, output responses, etc. for further calculating the system reliability index of the truss automatically. The performance function of each member is firstly constructed by RSM to avoid the implicit performance function problem, and the reliability index of each member is computed by a constrained optimization model that is built according to the geometric meaning of the reliability index. Then the dominant failure modes are identified by the
A practical truss structure consisting of numerous members is usually redundant, and it is usually a complex system incorporating multiple failure modes. Therefore, the reliability analysis of truss structures should be considered at the system level to perform an accurate reliability evaluation.
In structural reliability analysis, one challenge is building the performance function due to the nonlinearity of the structure that may result in the implicit performance function. It is assumed that the performance function is
The RSM [
where the coefficients
Determine the initial iteration point
Calculate the performance function values of
Solve the equation set of response surface functions. The equation set is given as:
where the subscripts
where the vector
The
The unknown coefficient matrix Perform the calculation of the reliability index If accuracy meets the requirement, output the reliability index
During this process of RSM, the reliability index plays an important role in performing response surface fitting on account of its relation to the final convergence. According to the geometric meaning of the reliability index, a constrained optimization model [
Accordingly, the reliability index can be obtained by solving the constrained optimization model. It is essential to search for the optimal solution in the variable space to find the accurate reliability index efficiently. Many algorithms were proposed to solve this problem such as differential evolution algorithm, genetic algorithm, particle swarm optimization, etc. The proposed framework is based on MATLAB programming, hence, considering the convenience of programming and computational efficiency, the fmincon function of MATLAB is applied to perform a search for the reliability index. The fmincon function is a nonlinear programming solver that can find the minimum of a constrained nonlinear multivariable function. When we use this function, an initial point is required so that the search for the optimal solution starts at the initial point. Moreover, the MCS and RSM combined with the fmincon function are adopted respectively to calculate the reliability indices of case 1 and case 2 to verify the accuracy of RSM combined with the fmincon function. For the two cases, the Latin Hypercube Sampling (LHS) method is used for generating 106 samples uniformly according to the distribution of variables to obtain accurate reliability indices. The results of RSM combined with the fmincon function for case 1 and case 2 converged after 5 and 4 iterations respectively shown in
Case 1.
Case 2.
For a redundant truss structure, the failure of one component does not destroy it immediately, instead, the failure of a series of components results in structural failure. The
As
Through the mentioned operation above, the updates of the finite element model and load distribution can be realized automatically.
There may be a great number of dominant failure modes identified by the
The response surface function of each member:
When the stress of the member of the truss is positive, the minus ‘’-” should be used, rather use the plus “+” if the stress is negative.
Perform Taylor expansion:
In the n-dimension standard normal space, the
The linear performance function of the next failure member is defined as:
The vector
The equivalent performance function can be denoted as:
According to the differential equivalence recursion algorithm introduced above, the performance function and the reliability index of each failure mode can be obtained. Then utilize the PNET to divide all dominant failure modes into several groups in accordance with their coefficients and choose the failure mode with the greatest failure probability to be the representative failure mode to compute the system reliability index of a truss structure. The steps of PNET are as follows:
Calculate the failure probability of each failure mode and put the failure probability in order from the largest to the smallest. Determine the relative coefficient Calculate the relative coefficients Calculate the system failure probability or system reliability probability according to
Thus, the system reliability probability is expressed as:
Based on these theories about the system reliability introduced above, a system reliability analysis framework that aims to assess the system-level reliability of a truss structure is proposed as shown in
The 2-dimensional truss bridge example is considered in
Members | Cross section areas (m2) |
---|---|
1–6 | 15 × 10−4 |
7–12 | 14 × 10−4 |
13–17 | 12 × 10−4 |
18–25 | 13 × 10−4 |
Random variables | Distribution | Mean | c.o.v |
---|---|---|---|
P1/(kN) | Lognormal | 160 | 0.1 |
P2/(kN) | Lognormal | 160 | 0.1 |
Normal | 276 | 0.05 |
The loads P1 and P2 are considered as the random variables to fit the response surface of each member, and 5 groups of sample points are required to obtain the response surface function of each member. Thus the performance function of each member can be built as
By the proposed framework, forty-one failure modes were identified, and the failure of this truss structure can be considered as a series system of forty-one failure modes which are shown in
Dominant failure modes | Reliability index | ||
---|---|---|---|
Proposed method | Reference [ |
Relative error (%) | |
−3 → 9 | 2.7104 | 2.592 | 4.57 |
9 →−3 | 2.9320 | 2.9542 | 0.75 |
−3 → −2 → 9 | 3.0069 | 3.2330 | 6.99 |
−2 → −3 → 9 | 3.0488 | 3.3170 | 8.06 |
−2 → 9 → −3 | 3.0442 | 3.5581 | 14.44 |
9 → −2→ −3 | 2.9999 | 3.5609 | 15.75 |
−1 | 3.6657 | 3.6334 | 0.88 |
−3 → −4 → 9 | 3.5798 | 3.6715 | 2.50 |
−3 → −1 | 3.6657 | 3.7461 | 2.15 |
−4 → −3 → 9 | 3.6518 | 3.7965 | 3.81 |
System | 2.5839 | 2.5478 | 1.42 |
A space truss with 25 bars [
Type | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
No. of | 1 | 2 | 3 | 6 | 7 | 10 | 12 | 14 | 15 | 18 | 19 | 22 | 23 |
members | 5 | 4 | 9 | 8 | 11 | 13 | 17 | 16 | 21 | 20 | 25 | 24 | |
Area/cm2 | 4.36 | 4.56 | 7.47 | 2.39 | 7.52 | 1.51 | 1.77 | 4.88 | 1.89 | 1.78 | 2.63 | 4.89 | 7.66 |
Random variables | Distribution | Mean | c.o.v |
---|---|---|---|
F1/(kN) | Normal | 88.9 | 0.2 |
F2/(kN) | Normal | 22.6 | 0.2 |
Normal | 276 | 0.05 |
As described in
The failure probabilities of identified dominant failure modes are calculated by the differential equivalence recursion algorithm, and the system reliability index of the space truss is obtained by PNET. The failure probability of the example is 1.43 × 10−6 which can be transformed into the reliability index, and after calculating, the system reliability index of this structure in reference [
System | Reliability index | ||
---|---|---|---|
Proposed method | Reference [ |
Relative error (%) | |
4.8437 | 4.6806 | 3.48 |
A computational framework that can update the finite element model automatically for identifying the dominant failure modes and performing system reliability analysis of the truss structure is proposed in this study. The proposed framework can be divided into three parts: (1) reliability index of a single member, which includes constructing the performance function of each member by RSM to solve the problem that the performance function is implicit and utilizing the fmincon function of MATLAB to solve the constrained optimization model for the reliability index; (2) identification of dominant failure modes, which consists of the
The advantages of the proposed framework are drawn as follows: (1) The RSM can get the explicit performance function and make the framework possible to analyze the large and complex truss structures whose performance functions are usually implicit. (2) The failure of members in the finite element model can be updated automatically with the utilization of system function in MATLAB and EKILL command in APDL. This helps improve computing efficiency. (3) The application of the differential equivalence recursion algorithm that considers the correlation of each failure member can obtain the performance function of each failure mode efficiently and accurately. And the PNET considering the correlation of each failure mode gives high accuracy of system reliability analysis.
Eventually, a 25-bar plane truss and a space truss with 25 members are presented to demonstrate the effectiveness of the proposed framework. The relative errors between two numerical examples and corresponding references are 1.42% and 3.48%, respectively, which verify the accuracy in the system reliability analysis of the truss. As a result, these examples indicate that the proposed framework gives high precision and efficiency in the system reliability analysis of truss structures.
The joint simulation of MATLAB and APDL is extremely important in the proposed framework that involves updating the finite element model automatically. Herein, the system function in MATLAB and EKILL command in APDL which are the key to achieving this automation process are introduced briefly.
A snippet of code, shown below, is provided to use the system function in MATLAB for reference.
system (‘SET KMP_STACKSIZE = 2048k & “D:\ANSYS Inc\v192\ansys\bin\winx64\ANSYS192.exe”-b-ane3fl-i “C:\Users\Desktop\file.txt”-o “C:\Users\Desktop\file.out"’).
In the command above, “D:\ANSYS Inc\v192\ansys\bin\winx64\ANSYS192.exe” represents the installation path of ANSYS software, “b” indicates specifying to run ANSYS in batch mode, “I” is specifying the input file (APDL command flow file) and “o” specifies the output file.
Meanwhile, another important command EKILL in APDL can be written as:
*do,i1,1,len_node(1) $ time,i1 $ nlgeom,on $ nropt,full $ ekill,kill_node(i1, 1)
estif,0 $ esel,s,live $ eplot $ *enddo
which is realized by a loop structure to remove all failure members. In this command, “len_node(1)” defined by yourself represents the number of failure members, and “kill_node(i1, 1)” which also can be defined by yourself represents the member to be killed. Similarly, the forces calculated by MATLAB are applied to the corresponding nodes of failure members also through a loop structure in APDL.