A truss structure reliability analysis method based on error analysis
By optimizing the adaptive Kriging model using a learning function based on error analysis and a new stopping criterion, the problem of low computational efficiency in the reliability analysis of high-reliability structures is solved, and efficient and accurate failure probability estimation is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIHANG UNIV
- Filing Date
- 2026-04-15
- Publication Date
- 2026-06-19
AI Technical Summary
Existing adaptive kriging models are computationally inefficient in the reliability analysis of structures with high reliability and low failure probability, and suffer from severe oversampling, making it difficult to meet the real-time and economic requirements of engineering analysis.
By employing a learning function based on error analysis, combining spatial correlation between samples and probability distribution of input variables, the optimal sample points are selected through the new learning function to update the Kriging model, and a new stopping criterion is introduced to accurately estimate the failure probability.
It significantly improves computational efficiency, reduces computational costs, ensures the accuracy of failure probability estimation and the robustness of the model, and is suitable for structural reliability assessment of complex multivariate coupled problems.
Smart Images
Figure CN122241374A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural reliability analysis and evaluation technology, and specifically to a method for reliability analysis of truss structures based on error analysis. Background Technology
[0002] As modern engineering systems evolve towards higher performance and reliability, structural reliability analysis has become a crucial research area in various engineering applications. Its aim is to efficiently quantify the failure probability of a product structure under the influence of uncertainties such as material properties, geometric dimensions, and load conditions during service. However, traditional reliability methods often face challenges such as difficulty in solving problems and low computational efficiency when dealing with situations involving low failure probabilities, implicit or highly nonlinear true function properties. Against this backdrop, reliability analysis methods based on surrogate models have emerged. These methods construct approximate surrogate models, such as response surface models, Kriging models, radial basis functions, and neural networks, to replace time-consuming real experiments or finite element analysis, thereby significantly reducing computational costs and improving analysis efficiency.
[0003] To further improve computational efficiency and avoid wasting computational resources, an adaptive kriging model was proposed by introducing an active learning strategy based on the kriging model. Due to its accurate interpolation characteristics and ability to quantify uncertainty, it has become a research focus in the field of structural reliability analysis and assessment. AK-MCS is the cornerstone of this research. Its basic process involves first establishing an initial model based on a small number of initial samples, then adding sample points through a learning function to gradually update the surrogate model, and finally using the final model to estimate the failure probability after meeting certain conditions. The learning function is a crucial component of the adaptive kriging model; it selects the optimal sample points based on the distribution information of the predicted sample response. Currently, mainstream learning functions include U, EFF, LIF, and REIF.
[0004] In modern engineering, products requiring increasing precision, complexity, and high reliability with long lifespans, such as critical structures in aerospace and aviation, often exhibit low failure probabilities. Furthermore, their function parameters are frequently computationally expensive "black box" models, making it extremely difficult to obtain accurate responses. Existing adaptive kriging learning functions applied to active learning strategies are often limited to constructing globally accurate surrogate models. While this strategy ensures computational accuracy, it inevitably leads to oversampling in non-critical regions far from the failure domain. Moreover, existing learning functions do not adequately mine information such as the probability distribution of input variables and the correlation between sample spaces, ultimately resulting in low model iteration efficiency. When solving real-world engineering problems, issues such as inefficient sample utilization, insufficient model convergence speed, and the need for large sample sizes for low failure probability estimation arise, leading to high computational costs and making it difficult to meet the real-time and economic requirements of engineering analysis. Summary of the Invention
[0005] In view of the above problems, the present invention provides a reliability analysis method for truss structures based on error analysis, which solves the technical problem of insufficient accuracy in failure probability calculation in the prior art.
[0006] This invention provides a reliability analysis method for truss structures based on error analysis, comprising the following steps: Step S1: Obtain the structural reliability function model, which is used to calculate the reliability function value based on the values of multiple structural parameters; obtain the probability distribution of each structural parameter; Step S2: Randomly generate multiple candidate samples according to the probability distribution of each structural parameter to form a candidate sample set; perform Latin hypercube sampling based on the candidate sample set to obtain multiple training samples to form a training sample set; establish an initial Kriging model based on the training sample set and the structural reliability functional model. Step S3: Input each sample in the candidate sample set into the current Kriging model, obtain the corresponding reliability function value statistics, obtain new training samples based on the statistics, and update the training sample set based on the new training samples; Step S4: Input each sample in the current training sample set into the structural reliability function model to obtain the corresponding reliability function value; update the Kriging model based on the training sample set and the corresponding reliability function value. Step S5: Determine whether the stopping criterion is met. If not, return to step S3. If met, obtain the current Kriging model. Step S6: Input each sample in the candidate sample set into the current Kriging model to obtain the failure probability.
[0007] Preferably, in step S1, the expression for the reliability functional model is:
[0008] Indicates the allowable displacement of the roof truss. Indicates the maximum vertical displacement. This indicates the vertical load on the roof truss. Indicates the length of the lower chord. This represents the cross-sectional area of a reinforced concrete member. This represents the elastic modulus of a reinforced concrete member. This represents the cross-sectional area of the steel component. This indicates the elastic modulus of a steel component. The structural parameters include: roof truss vertical load. Length of the lower chord Cross-sectional area of reinforced concrete members The elastic modulus of reinforced concrete members Cross-sectional area of steel components The elastic modulus of steel components The reliability function value is the roof truss displacement.
[0009] Preferably, the step of obtaining the probability distribution of each structural parameter specifically includes: Obtain the probability distribution of each structural parameter, and use the RF method to transform the random variable into the standard normal space. Obtain the mean and variance of each structural parameter in the standard normal space, thereby obtaining the probability distribution of each structural parameter.
[0010] Preferably, step S2 specifically includes: Randomly generated in standard normal space A candidate sample set consists of 10 candidate samples. ,in, They represent the 1st, 2nd, and 3rd respectively. One candidate sample; The Latin hypercube method was used to perform initial sampling within the standard normal space [-5,...,5], and the samples were selected. These samples constitute the initial training sample set. ,in, They represent the 1st, 2nd, and 3rd respectively. One training sample; Input each sample from the initial training sample set. The initial response set is obtained. ,in, They represent the 1st, 2nd, and 3rd respectively. One response value; Based on the initial training sample set and its corresponding initial response set The initial Kriging model was constructed using the DACE toolbox.
[0011] Preferably, step S3 specifically includes: Step S3-1: Input each sample in the candidate sample set C into the current Kriging model in sequence to obtain the predicted mean and variance of the reliability function value corresponding to each candidate sample; Step S3-2: Obtain new training samples based on the predicted mean and variance of the reliability function values of each candidate sample; update the training sample set using the new training samples.
[0012] Preferably, in step S3-2, the expression for obtaining new training samples based on the predicted mean and variance of the reliability function values of each candidate sample is:
[0013] in, This represents the new training samples selected by the learning function. This indicates that selection is made from the candidate sample set. Indicates the maximum value. Represents the new learning function. Indicates the radius of the region affected by the sample. Represents the weight parameters. , Indicates the distance between samples. This represents the mean of the predicted values from the Kriging model. This represents the variance of the predicted values from the Kriging model. The cumulative distribution function represents the standard normal distribution. In This represents a sample in the sample space.
[0014] Preferably, the expression for the stopping criterion in step S5 is:
[0015] Where i is the index of a sample in the sample space. Represents variance. and These represent the indicator function and the prediction indicator function, respectively. The cumulative distribution function represents the standard normal distribution. This indicates the calculation of absolute value. This indicates the threshold for judgment.
[0016] Preferably, step S6 specifically includes: All candidate samples in C Each candidate sample is input into the final Kriging model to obtain the mean predicted function response for each candidate sample. The failure probability is calculated based on the predicted mean of the function response for each candidate sample, expressed as follows:
[0017]
[0018]
[0019] in, This represents the probability of failure. Let i be the index of a sample in the sample space, representing the predicted failure probability calculated based on the current Kriging model. Represents the mathematical expectation. For indicator functions, For prediction indicator function, This indicates that a prediction indicator function is applied to sample i. This represents the mean of the predicted values of the function response corresponding to the candidate sample.
[0020] Compared with the prior art, the present invention has at least the following beneficial effects: (1) The learning function proposed in this invention utilizes the predicted distribution of the Kriging model, the spatial correlation between samples, and the probability distribution of the input variables, so that each new sample accurately points to the region that contributes most to the failure probability estimation. Compared with the traditional sampling criteria that only rely on the predicted mean or variance, this method can significantly reduce the number of calls to the true reliability function and converge to a stable failure probability estimate more quickly, thereby greatly improving the overall computational efficiency and saving computational resources.
[0021] (2) By co-designing the new stopping criterion with the learning function, this invention can quantitatively determine when enough information has been obtained to terminate sampling during the adaptive kriging modeling process. This avoids the generation of too many redundant samples and unnecessary high-cost function evaluations, ensuring the accuracy of failure probability estimation while controlling computational costs, thus enabling high-quality reliability analysis results to be obtained within a limited budget.
[0022] (3) This invention conducts qualitative and quantitative analysis on the sources of error that may be introduced by the adaptive Kriging method in structural reliability analysis, improves the robustness of the model to complex, multivariate coupled problems, and enables more accurate and stable structural reliability assessment, which is beneficial to engineering decision-making and risk management. Attached Figure Description
[0023] The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of the invention.
[0024] Figure 1 This is a schematic diagram illustrating the probability of a sample being misclassified based on Kriging model response prediction information provided by the present invention.
[0025] Figure 2 The flowchart shows the learning function based on error analysis and its structural reliability analysis method provided by this invention.
[0026] Figure 3 The convergence curves of the learning function based on error analysis and other learning functions used in this invention to calculate the failure probability are shown.
[0027] Figure 4 This is a schematic diagram illustrating the process of adding samples with zero iterations, as provided by the present invention.
[0028] Figure 5 This is a schematic diagram illustrating the process of adding samples with 9 iterations as provided by the present invention.
[0029] Figure 6This is a schematic diagram illustrating the process of adding samples with 18 iterations as provided in this invention.
[0030] Figure 7 This is a schematic diagram showing that the number of iterations during the sample addition process provided by the present invention is 27.
[0031] Figure 8 This is a schematic diagram illustrating the process of adding samples with 35 iterations as provided in this invention.
[0032] Figure 9 This is a schematic diagram of the roof truss structure provided by the present invention.
[0033] Figure 10 The convergence curves of the learning function based on error analysis and other learning functions provided by this invention for calculating the failure probability of the roof truss structure. Detailed Implementation
[0034] To better understand the above-described objectives, features, and advantages of the present invention, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other. Furthermore, the present invention can be implemented in other ways different from those described herein; therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.
[0035] This invention provides a novel learning function that utilizes the predicted distribution of sample responses and combines spatial correlation between samples with probability distribution information of input variables to identify samples that can significantly improve the accuracy of failure probability calculation, thereby achieving efficient sampling. Furthermore, considering the impact of the deviation between the Kriging model and the true function on the estimated failure probability, a new structural reliability analysis method is proposed, achieving accurate estimation of failure probability.
[0036] First, the steps for obtaining the failure probability using the adaptive Kriging model method in the field of structural reliability analysis are introduced in detail below.
[0037] In the process of structural reliability analysis, the expression for solving the failure probability is: ; in, This represents the probability of failure. For the i-th sample, The total number of samples, express The corresponding indicator function, the larger N is The closer it is to the actual failure probability.
[0038] in For indicator functions, the expression is: .
[0039] in, This refers to the actual functional function, which, in the context of this invention, will be described later as a reliability functional model.
[0040] The adaptive kriging model method obtains the true function by constructing and calling a kriging model. The predicted values, and the predicted values follow a normal distribution. Therefore, the expression for solving the failure probability using the adaptive Kriging method is: ; in Let be the indicator function for the predicted values based on the Kriging model, expressed as follows: .
[0041] in, This is the predicted value of the failure probability. express The corresponding indicator function for prediction. This represents the mean of the predicted values of the function response corresponding to the candidate sample.
[0042] Due to the discrepancy between the Kriging model and the true function, the sign of the mean predicted by the Kriging model for the same sample may differ from the sign of the true function, leading to misclassification of the sample and affecting the calculation of the failure probability. However, if the number of misclassified samples can be accurately estimated, a highly accurate estimate of the failure probability can still be obtained even if there are still differences between the current surrogate model and the true function.
[0043] For a given sample, there are two possible misclassification scenarios: one where the true value is positive and the predicted value is negative, and the other where the true value is negative and the predicted value is positive. Both scenarios will lead to incorrect indicator function values, thus causing bias in the calculation of the failure probability. Based on the distribution information of the predicted values from the Kriging model, the probability of a sample being misclassified can be quantified, as shown in the following expression: , .
[0044] in, This represents the mean of the predicted values from the Kriging model. This represents the variance of the predicted values from the Kriging model. This represents the probability that the difference between the indicator function of the true value and the indicator function of the mean of the predicted values is -1. The cumulative distribution function represents the standard normal distribution. This indicates the calculation of absolute value.
[0045] Figure 1 The illustration shows a diagram of the probability of a sample being misclassified based on the response prediction information of the Kriging model.
[0046] Therefore, the mean and variance information of the impact of sample classification errors on the failure probability can be obtained. ; .
[0047] in, Represents the mathematical expectation. Represents variance. Represents a symbolic function. This represents the probability of misclassification.
[0048] Based on this strategy, the inventors proposed a new learning function and improved the adaptive Kriging model method, achieving accurate estimation of failure probability.
[0049] like Figure 2 As shown, this invention discloses a reliability analysis method for truss structures based on error analysis, and the specific implementation steps are as follows: Step S1: Obtain the structural reliability function model, which is used to calculate the reliability function value based on the values of multiple structural parameters; The probability distribution of the structural parameters is converted into a normal distribution, and the probability distribution of each structural parameter is obtained. In this step, a structural reliability functional model is first constructed by analyzing information such as the product structure's working principle, geometric dimensions, and working environment. This structural reliability functional model refers to the functional function... , ,in This represents a vector consisting of multiple structure parameters, each of which is a random variable.
[0050] This invention obtains the probability distribution of each structural parameter through analysis or statistics, and uses the Random Range (RF) method to transform the random variables into a normal distribution. The mean and variance of each structural parameter after transformation to a normal distribution are then obtained, thus yielding the probability distribution of each structural parameter.
[0051] In one specific embodiment, the present invention can analyze information such as the working principle, geometry, and working environment of a roof truss structure. The roof truss structure consists of a top chord, a bottom chord, and intermediate compression and tension members. The top chord and compression members are made of reinforced concrete, while the bottom chord and other tension members are made of steel.
[0052] Based on the mechanical properties of roof truss structures, a structural reliability functional model is established using roof truss displacement as an evaluation index.
[0053] The expression for the reliability functional model is:
[0054] This is the allowable displacement of the roof truss. It is the maximum vertical displacement. In the model, This represents a vector of random variables affecting structural reliability. The specific meanings of each structural parameter are as follows: Indicates the vertical load of the roof truss. Indicates the length of the lower chord. This represents the cross-sectional area of a reinforced concrete member. This represents the elastic modulus of a reinforced concrete member. This represents the cross-sectional area of the steel component. This indicates the elastic modulus of a steel component.
[0055] In one specific embodiment, the Rosenblatt-Fessler (RF) transformation method is used to uniformly transform these random variables into a standard normal space, and the mean and variance of each structural parameter in the standard normal space are obtained, thereby obtaining the probability distribution of each structural parameter, providing data for subsequent sample generation and reliability analysis.
[0056] Step S2: Randomly generate multiple candidate samples according to the probability distribution of each structural parameter to form a candidate sample set; perform Latin hypercube sampling based on the candidate sample set to obtain multiple training samples to form a training sample set; each sample includes the values of multiple structural parameters. An initial Kriging model is established based on the training sample set and the structural reliability functional model; In this step, the structural parameters are first randomly generated in the standard normal space based on their probability distributions. A candidate sample set consists of 10 candidate samples. ,in, They represent the 1st, 2nd, and 3rd respectively. There are 10 candidate samples. Each candidate sample contains multiple values for structural parameters. These candidate samples constitute the sample pool for subsequent adaptive learning and failure probability calculation.
[0057] In some embodiments, to ensure sufficient coverage of the sample space, the following settings can be configured: .
[0058] Next, the Latin hypercube method is used to perform initial sampling within the standard normal space [-5,...,5], and the samples are selected. These samples are used as the initial training sample set. The initial training sample set is represented as: ,in, They represent the 1st, 2nd, and 3rd respectively. There are 10 training samples. Each training sample also contains multiple values for structural parameters.
[0059] In this step, the invention also performs the step of establishing an initial Kriging model. Specifically, each sample from the initial training sample set is substituted into the structural reliability functional model established in step S1. Calculate the corresponding true reliability function value to obtain the initial response set. ,in, They represent the 1st, 2nd, and 3rd respectively. Each response value.
[0060] Based on the initial training sample set and its corresponding initial response set To construct the initial Kriging model.
[0061] The construction of Kriging models is a prior art in this field. In some embodiments, the DACE toolbox in MATLAB can be used to prepare the initial training sample set. and its corresponding initial response set As input, the DACE toolbox automatically optimizes the relevant parameters of the kriging model to obtain the initial kriging model. Other software modules can also be used to construct the kriging model; this invention does not limit the specific method of constructing the kriging model.
[0062] Step S3: Input each sample in the candidate sample set into the current Kriging model, obtain the corresponding reliability function value statistics, obtain new training samples based on the statistics, and update the training sample set based on the new training samples; A key characteristic of the Kriging model is that its predicted values follow a normal distribution. For input candidate samples, the Kriging model can output the predicted mean and variance of the reliability function value.
[0063] In this step, each sample in the candidate sample set C is sequentially input into the current Kriging model to obtain the predicted mean and variance of the reliability function value corresponding to each candidate sample.
[0064] New training samples are obtained based on the predicted mean and variance of the reliability function values of each candidate sample, as detailed below.
[0065] To select the optimal new training sample, this invention provides a new learning function that updates the current Kriging model by selecting the sample with the maximum learning function value as the new training sample.
[0066] The expression for the step of obtaining new training samples in this invention is:
[0067] in, This represents the new training samples selected by the learning function. This indicates that selection is made from the candidate sample set. Indicates the maximum value. Represents the new learning function. Indicates the radius of the region affected by the sample. Represents the weight parameters. , Indicates the distance between samples. This represents the mean of the predicted values from the Kriging model. This represents the variance of the predicted values from the Kriging model. The cumulative distribution function represents the standard normal distribution. In This represents a sample in the sample space.
[0068] In some embodiments, The value can be set to 0.05 times the size of the candidate sample space.
[0069] After acquiring new training samples, add the new training samples to the training sample set. .
[0070] The adaptive sampling strategy of the novel learning function of this invention can efficiently identify the sample points that can best improve the accuracy of failure probability estimation, avoiding unnecessary sampling in unimportant areas, thereby significantly improving computational efficiency.
[0071] Step S4: Input each sample in the current training sample set into the structural reliability function model to obtain the corresponding reliability function value; update the Kriging model based on the training sample set and the corresponding reliability function value. In this step, the present invention calculates the true response of the new training sample function and updates the current Kriging model.
[0072] Based on the updated training sample set and its corresponding reliability function values, the Kriging model is re-acquired. The Kriging model update process includes re-estimating the parameters of the correlation function and recalculating the regression coefficients, so that the updated Kriging model can more accurately fit the response behavior of the true function.
[0073] Step S5: Determine whether the stopping criterion is met. If not, return to step S3. If met, obtain the current Kriging model. In this step, we calculate whether the stopping criterion is met to determine whether adaptive learning needs to continue.
[0074] The expression for the stopping criterion is:
[0075] in, This indicates the threshold for judgment, which can be set to... .
[0076] The stopping criterion states that when the average misclassification effect of all samples in the candidate sample set is sufficiently small, it indicates that the impact of the deviation between the current Kriging model and the true function on the failure probability estimate can be accurately quantified, and the model update process can be stopped at this point.
[0077] If the stopping criterion is not met, it means that the current Kriging model's prediction accuracy is insufficient to support accurate failure probability estimation, and adaptive learning needs to continue to improve the model. In this case, return to step S3.
[0078] If the stopping criterion is met, it indicates that the current kriging model has reached the required accuracy, and the impact of the deviation between the kriging model and the true function on the estimated failure probability can be precisely quantified. At this point, the current kriging model is adopted as the final surrogate model, and the iterative update process is stopped.
[0079] Step S6: Input each sample in the candidate sample set into the current Kriging model to obtain the failure probability.
[0080] Once the iteration process meets the stopping criterion, all candidates in the sample set C will be... Each candidate sample is input into the final Kriging model, and the predicted mean of the function response for each candidate sample is obtained. The number of samples with a mean less than or equal to zero is counted, i.e., the number of failure samples. The ratio of the number of failure samples to the total number of samples is calculated. Based on this ratio and considering the impact of the deviation between the Kriging model and the true function response on the estimated failure probability, the failure probability is calculated, expressed as:
[0081]
[0082]
[0083] in, This represents the mean of the predicted values of the function response corresponding to the candidate sample.
[0084] To illustrate the effectiveness of the method proposed in this invention, the above technical solution of this invention will be described in detail below through specific embodiments.
[0085] Example 1 In this embodiment, the present invention performs a reliability analysis on a four-branch series system. The four-branch series system consists of four interconnected branches, which are logically connected in a "series" manner.
[0086] In this embodiment, the limit state function of a four-branch series system is used as the structural reliability functional model of the present invention. The expression of the limit state function of the four-branch series system is:
[0087] in, It is a normally distributed random variable. , Let represent the first and second structural parameters, respectively, and their specific probability distributions are shown in Table 1.
[0088] Table 1
[0089] Monte Carlo simulation, due to its high accuracy and robustness, is often used as a benchmark method. The computational accuracy of other reliability methods can be evaluated by the error between their values and this reference value. The adaptive kriging model method based on the U-learning function is foundational in this direction; therefore, Monte Carlo simulation, the adaptive kriging model method based on the U-learning function, and the reliability analysis method based on the MDEF learning function proposed in this invention are used to evaluate the reliability of four-branch cascaded systems. The computational efficiency of each method can be evaluated by the number of calls to the actual function.
[0090] Table 2 shows the computational accuracy and efficiency of different methods. Table 2
[0091] As shown in Table 2, the Monte Carlo simulation method first generates... A set of random samples is generated, and the corresponding true response values are calculated. Then, the number of failed samples is counted, and the ratio of the number of failed samples to the total number of samples is calculated to obtain the corresponding failure probability. This failure probability is considered the accurate failure probability. The total number of function calls for this method is... The adaptive kriging model method based on the U learning function, as an efficient existing method, first generates... The initial training samples are set up and the corresponding true responses are calculated to construct the initial Kriging model. Then, the best sample is selected based on the U learning function, and its true response is calculated to update the Kriging model until a preset stopping criterion is met. Each model update increases the number of function calls. The final number of function calls for the adaptive Kriging model method based on the U learning function is: The calculated failure probability is Unlike the two methods mentioned above, the reliability analysis method provided in this invention selects the optimal sample based on a novel learning function MDEF and considers the impact of the deviation between the Kriging model and the true function on the estimated failure probability. Similar to the adaptive Kriging model method based on the U learning function, it first generates... The initial training samples are set up and the corresponding true responses are calculated to construct the initial Kriging model. Furthermore, each model update increases the number of function calls. Ultimately, the number of function calls in the method proposed in this invention is [number missing]. The calculated failure probability is . Figure 3 The convergence curve for calculating the failure probability of a four-branch series system is shown.
[0092] The structural reliability analysis method proposed in this invention achieves similar accuracy in calculating failure probability to other methods, with an error of only 0.403% compared to the standard failure probability calculated using Monte Carlo simulation. Compared to the adaptive Kriging model method based on the U-learning function, the calculated failure probability is more accurate, and the number of function calls is significantly reduced to only 51% of the former. In conclusion, the method proposed in this invention can perform reliability analysis accurately and efficiently.
[0093] Figure 4 , Figure 5 , Figure 6 , Figure 7 , Figure 8 This diagram illustrates the process of adding new training samples to the learning function proposed in this invention. In the figure, the black dashed line represents the failure boundary obtained from the true function, the red solid line represents the failure boundary predicted by the current Kriging model, the black hollow dots are the initial training samples, the black solid dots are the added training samples, and the red pentagram dots are the optimal samples under the current Kriging model. It can be seen that the learning function proposed in this invention identifies samples near the failure boundary more quickly, and the corresponding stopping criterion avoids unnecessary resource waste, making the algorithm more efficient. Although the failure boundary predicted by the current Kriging model differs slightly from the true failure boundary after the stopping criterion is met, this does not affect the calculation of the failure probability.
[0094] Example 2 In this embodiment, the present invention performs a reliability analysis on the roof truss structure. For example... Figure 9 As shown, the top chord and compression members of the roof truss structure are made of concrete, while the bottom chord and tension members are made of steel. The structure bears a uniformly distributed vertical load. This load can be converted into three nodal loads, each with a load size of [value missing]. .
[0095] In this embodiment, the roof truss displacement is used as the structural reliability functional model of the present invention.
[0096] The expression for the roof truss displacement is:
[0097] It is a normally distributed random variable, and the specific probability distribution is shown in Table 3.
[0098] Table 3
[0099] Monte Carlo simulation, adaptive Kriging model based on the U learning function, and the reliability analysis method based on the MDEF learning function proposed in this invention were used to evaluate the reliability of roof truss structures. Table 4 shows the computational accuracy and efficiency of different methods.
[0100] Table 4
[0101] As can be seen, the structural reliability analysis method proposed in this invention is still effective for events with a low probability of failure. The accuracy of the calculation results is similar to that of other methods, and the error between the method and the standard value of failure probability calculated by Monte Carlo simulation is less than one percent. Figure 10 The convergence curve for calculating the failure probability of a four-branch series system is shown. Compared with the adaptive Kriging model method based on the U learning function, the method provided in this embodiment significantly reduces the number of function calls, to only 58% of the former. In summary, the method proposed in this invention can accurately and efficiently perform reliability analysis on roof truss structures.
[0102] While the specific embodiments of the present invention depict actions or steps in a particular order, this should be understood as requiring such actions or steps to be performed in the specific order shown or in sequential order, or requiring all illustrated actions or steps to be performed to achieve the desired result. In certain environments, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are included in the above discussion, these should not be construed as limiting the scope of this disclosure. Certain features described in the context of individual embodiments may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple implementations.
[0103] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.
Claims
1. A reliability analysis method for truss structures based on error analysis, characterized in that, Includes the following steps: Step S1: Obtain the structural reliability function model, which is used to calculate the reliability function value based on the values of multiple structural parameters; obtain the probability distribution of each structural parameter; Step S2: Randomly generate multiple candidate samples according to the probability distribution of each structural parameter to form a candidate sample set; perform Latin hypercube sampling based on the candidate sample set to obtain multiple training samples to form a training sample set. An initial Kriging model is established based on the training sample set and the structural reliability functional model; Step S3: Input each sample in the candidate sample set into the current Kriging model, obtain the statistics of the corresponding reliability function value, and obtain new training samples based on the statistics; The training sample set is updated based on the new training samples; Step S4: Input each sample in the current training sample set into the structural reliability function model to obtain the corresponding reliability function value; update the Kriging model based on the training sample set and the corresponding reliability function value. Step S5: Determine whether the stopping criterion is met. If not, return to step S3. If met, obtain the current Kriging model. Step S6: Input each sample in the candidate sample set into the current Kriging model to obtain the failure probability.
2. The reliability analysis method for truss structures based on error analysis according to claim 1, characterized in that, In step S1, the expression for the reliability functional model is: Indicates the allowable displacement of the roof truss. Indicates the maximum vertical displacement. Indicates the vertical load of the roof truss. Indicates the length of the lower chord. This represents the cross-sectional area of a reinforced concrete member. This represents the elastic modulus of a reinforced concrete member. This represents the cross-sectional area of the steel component. This indicates the elastic modulus of a steel component. The structural parameters include: roof truss vertical load. Length of the lower chord Cross-sectional area of reinforced concrete members The elastic modulus of reinforced concrete members Cross-sectional area of steel components The elastic modulus of steel components The reliability function value is the roof truss displacement.
3. The reliability analysis method for truss structures based on error analysis according to claim 2, characterized in that, In step S1, the step of obtaining the probability distribution of each structural parameter specifically includes: Obtain the probability distribution of each structural parameter, and use the RF method to transform the random variable into the standard normal space. Obtain the mean and variance of each structural parameter in the standard normal space, thereby obtaining the probability distribution of each structural parameter.
4. The reliability analysis method for truss structures based on error analysis according to claim 3, characterized in that, Step S2 specifically includes: Randomly generated in standard normal space A candidate sample set consists of 10 candidate samples. ,in, They represent the 1st, 2nd, and 3rd respectively. One candidate sample; The Latin hypercube method was used to perform initial sampling within the standard normal space [-5,...,5], and the samples were selected. These samples constitute the initial training sample set. ,in, They represent the 1st, 2nd, and 3rd respectively. One training sample; Input each sample from the initial training sample set. The initial response set is obtained. ,in, They represent the 1st, 2nd, and 3rd respectively. One response value; Based on the initial training sample set and its corresponding initial response set The initial Kriging model was constructed using the DACE toolbox.
5. The reliability analysis method for truss structures based on error analysis according to claim 4, characterized in that, Step S3 specifically includes: Step S3-1: Input each sample in the candidate sample set C into the current Kriging model in sequence to obtain the predicted mean and variance of the reliability function value corresponding to each candidate sample; Step S3-2: Obtain new training samples based on the predicted mean and variance of the reliability function values of each candidate sample; update the training sample set using the new training samples.
6. The reliability analysis method for truss structures based on error analysis according to claim 5, characterized in that, In step S3-2, the expression for obtaining new training samples based on the predicted mean and variance of the reliability function values of each candidate sample is as follows: in, This represents the new training samples selected by the learning function. This indicates that selection is made from the candidate sample set. Indicates the maximum value. Represents the new learning function. Indicates the radius of the region affected by the sample. Represents the weight parameters. , Indicates the distance between samples. This represents the mean of the predicted values from the Kriging model. This represents the variance of the predicted values from the Kriging model. The cumulative distribution function represents the standard normal distribution. In This represents a sample in the sample space.
7. The reliability analysis method for truss structures based on error analysis according to claim 6, characterized in that, The expression for the stopping criterion mentioned in step S5 is: Where i is the index of a sample in the sample space. Represents variance. and These represent the indicator function and the prediction indicator function, respectively. The cumulative distribution function represents the standard normal distribution. This indicates the calculation of absolute value. This indicates the threshold for judgment.
8. The reliability analysis method for truss structures based on error analysis according to claim 7, characterized in that, Step S6 specifically includes: All candidate samples in C Each candidate sample is input into the final Kriging model to obtain the mean predicted function response for each candidate sample. The failure probability is calculated based on the predicted mean of the function response for each candidate sample, expressed as follows: in, This represents the probability of failure. Let i be the index of a sample in the sample space, representing the predicted failure probability calculated based on the current Kriging model. Represents the mathematical expectation. For indicator functions, For prediction indicator function, This indicates that a prediction indicator function is applied to sample i. This represents the mean of the predicted values of the function response corresponding to the candidate sample.