3.1.1 Central Point Method

For the limit state function of a structure

(1) When Z is a linear function

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

The linearization point of the checking point method is located at the failure boundary to overcome the problems with the central point method. It is also the location of the design checking point X* (where the distance from the origin of the coordinates to the limit state surface is the shortest) corresponding to the maximum possible failure probability of the structure. This kind of first-order second-moment method is known as the checking point method [3-3] or improved first-order second-moment method, and is the basis for structural reliability index calculations. The checking point method could be used in X-space or U-space.

(1) X-space

A linearized limit state equation can be established by selecting the design checking point .

(3.20)

Because X* is situated on the failure boundary, there is

So

Because , there are

The sensitivity coefficient, , represents the relative influence of the i^th variable on the entire standard deviation. And .

So

(3.21)

This can also be expressed as

It has

(3.22)

Let , so E(u_i) = 0. We have the limit state function from Equation (3.20)

(3.23)

The proposition of β_min may be calculated, or it can be expressed in optimal form in U space.

Condition

(3.24)

Where f(U) represents the probability density function of U.

Thus, Equation (3.24) can be expressed as , so f(u) → max.

Therefore, we can construct a function, Z(u) = f(u) + λg(u), where λ is a Lagrange coefficient

where

So

When g(u) = g(u₀) + g_u(u₀)(u−u₀), we have

So, the iteration for the design checking point is

(3.25)

(3.26)

(3.27)

(3.28)

The Breitung method [3-4] [3-5] could treat with random variables by mapping transformation and analyze the problem in standard space. Let Y be an independent standard normal random variable, expand the function Z into Taylor’s series at the checking point, and take the first and second terms to obtain:

(3.29)

(3.30)

Let the unit vector be

(3.31)

Because , and , Equation (3.30) can be written as

(3.32)

So

(3.33)

α_X is used to construct an orthogonal matrix H, H^T · H = I, α_X is a certain column of H, set to n for orthogonal transformation of Y space to U space.

(3.34)

Equation (3.34) is substituted into (3.32). It is noted that , α_X = (0, 0, ⋯, 0, 1)^T, so Equation (3.32) can be rewritten as

(3.35)

where (H^TQH)_n−1 is an n-1-order matrix generated from H^TQH with the n^th row and n^th column ruled out, .

The joint probability density function of Y is

(3.36)

By substituting Equation (3.34) into (3.36), we get

(3.37)

Structural failure probability

(3.38)

For Equation (3.35), according to Equations (3.37) and (3.38), the failure probability of the second-order second-moment method is

(3.39)

Let , and expand InΦ(t – β) by Taylor’s series at t = 0 and take the first term to obtain

(3.40)

(3.41)

By substituting Equation (3.41) into (3.39), we get

(3.42)

The integrand in Equation (3.42) is compared with the normal joint probability density function with a mean value of 0 and a covariance matrix of . Equation (3.42) can be simplified as

(3.43)

If the reliability index of the first-order second-moment method is obtained, then the failure probability of the second-order second-moment method can be obtained from Equation (3.43).

(3.44)

1. Use the first-order second-moment method to calculate the reliability index.
   
2. Calculate the unit vector in Equation (3.31)
   
3. Determine the orthogonal matrix H in Equation (3.34)
   
4. Calculate Q in Equation (3.33)
   
5. Calculate failure probability in Equation (3.42)

3.2.2 Laplace Asymptotic Method

The Laplace method is a type of second-order second-moment method, in which the second-order partial derivative of nonlinear functions is adopted. When Y obeys independent standard normal distribution, the structural failure probability is

(3.45)

(3.46)

The properties of integral Equation (3.46) are fully determined by the properties in the neighborhood of the integrand maximum. If functions h(x) and g(x) are twice continuously differentiable, while p(x) is continuous and h(x) only reaches its maximum at x^*, a point on the boundary of the integral domain, Equation (3.46) can be approximately expressed as

(3.47)

where

(3.48)

(3.49)

(3.50)

The Jacobi determinant of the transformation is det J_YV = λⁿ.

By substituting Equation (3.50) into Equation (3.47), we get

(3.51)

Equation (3.51) is also a Laplace integral shown in (3.46), and , .

If the function is twice differentiable, then the asymptotic integral value of Equation (3.51) is

(3.52)

(3.53)

Matrix B₁(v^*) is an adjoint matrix of matrix C₁(v^*)

(3.54)

By substituting Equation (3.54) into (3.53), we get

(3.55)

(3.56)

is an adjoint matrix of

(3.57)

Considering that β is generally a relatively large positive value, φ(β) ≈ βΦ(−β), Equation (3.54) can be written as

(3.58)

The following steps can be applied in the above derivation process:

1. Calculate β, y* and x*
   
2. Calculate in Equation (3.30)
   
3. Calculate in Equation (3.57)
   
4. Calculate det C
   
5. Calculate J in Equation (3.56)
   
6. Calculate P_fQ, Equation (3.58)

3.2.3 Maximum Entropy Method

In 1948, Shannon introduced the concept of thermodynamics into the information theory. If a random variable has n possible results, the probability of each result is p_i. To measure the uncertainty of this event, the following function is introduced:

(3.59)

(3.60)

The Shannon entropy is used to measure the uncertainty of an event before it occurs; after the occurrence of the event, the Shannon entropy is used to measure the information obtained from it. This is used to measure the uncertainty of an event or the amount of information it contains.

Under given conditions, there exists a distribution among all possible probability distributions that enables information entropy to reach a maximum. This is called Jaynes maximum entropy principle. The information entropy is maximized under known additional information constraints, and the obtained probability distribution is minimally biased, thus finding a way to construct an “optimal” probability distribution.

We consider taking the first m–order origin moment of a random variable x as constraints, i.e., the information entropy is maximized by meeting the conditions in the equation below

(3.61)

Using the Lagrange multiplier method, as well as Equations (3.60) and (3.61), a modified function is introduced

(3.62)

At the stable point there is , i.e., . Let , and the maximum entropy probability density function can be obtained as follows,

(3.63)

Equation (3.62) is equivalent to the central moment of a given x

(3.64)

(3.65)

where C_sx is the skewness coefficient and C_kx is the kurtosis coefficient.

In Pearson’s system, it is considered that the probability density function f_X(x) of a random variable x is determined by the ordinary differential equation below:

(3.66)

By integrating Equation (3.66), we can obtain a family of curves.

Equation (3.66) is a general form of Pearson’s family of curves, in which the parameters can be expressed by the first four-order central moment of X, as follows

(3.67)

The following recursive relation exists between all-order central moments of the family of curves:

(3.68)

(3.69)

According to Equation (3.69) and (3.70), it is noted that μ_y = 0, σ_y = 1, v_yi = μ_yi. The first four moments of y are as follows

(3.70)

For the standard random variable y, and with determined Pearson system parameters, Equation (3.67) can be written as

(3.71)

Equation (3.71) can be used to derive higher-order moments if x in Equation (3.71) is replaced by y.

Let the structural function be Z = g(X), where the statistical parameters of X_i in are , while the first four-order central moments are .

By expanding Z into Taylor’s series at the checking point and taking the quadratic term, we can get

(3.72)

According to Equation (3.72), the first four moments of Z_Q can be calculated as follows:

(3.73)

(3.74a)

(3.74b)

(3.74c)

Z is standardized as . The maximum entropy probability density function of random variable Y meeting the constraint condition remains in the form of Equation (3.63). By substituting Equations (3.73) and (3.74) into Equation (3.61), we get a system of integral equations

(3.75)

The coefficient in f(y) can be calculated.

The structural failure probability is

(3.76)

The following calculation steps can be applied in the above derivation process:

1. Calculate μ_Z in Equation (3.73)
   
2. Calculate μ_Zi in Equation (3.74)
   
3. Calculate C_sZ and C_kZ in Equation (3.65)
   
4. Calculate v_yi in Equation (3.70)
   
5. Calculate higher-order v_yi in Equation (3.70)
   
6. Solve a_i in Equation (3.75)
   
7. Calculate P_f in Equation (3.76)

3.2.4 Optimal Quadratic Approximation Method

If the moments of two random variables correspond to each other, then they also have the same probability distribution and eigenvalue. With the moments as constraints in a given inner product space, the undetermined coefficients of probability density function polynomials can be calculated, thus making it possible to determine the probability distribution form of Z and calculate the structural failure probability.

Let the function f(x) be continuous in [a, b], and p_i(x)(i = 0, 1, ⋯, m) be m+1 linearly independent continuous function in [a, b]. Its linear combination is utilized to approximate f(x) so that the, integral where λ_i(i = 0, 1, ⋯, m) is the coefficient and ρ(x) is the weight function in [a, b]. This is the least square approximation problem.

According to , the essential condition for the extreme value of the multivariate function, linear equations with determined coefficients can be obtained as follows

(3.77)

Or rewritten as

(3.78)

where the elements of matrix A and the components of vector b are as follows

(3.79)

(3.80)

Matrix A is an m+1-order nonsingular matrix, meaning Equation (3.78) has a unique solution.

Let the probability density function of random variable x be f(x). If p_i(x) = xⁱ, ρ(x) = 1, then the following can be derived from Equations (3.78), (3.79) and (3.80)

(3.81)

(3.82)

If the every-order origin moment of x is known, λ_i(i = 0, 1, ⋯, m) can be worked out by solving Equation (3.77), with the polynomial of least square approximation, p(x), obtained, leading to

(3.83)

(3.84)

The following steps can be applied in the above derivation process:

1. Calculate μ_Z in Equation (3.73)
   
2. Calculate μ_Zi in Equation (3.74)
   
3. Calculate C_sZ in Equation (3.65)
   
4. Calculate in Equation (3.70)
   
5. Calculate higher-order v_yi in Equation (3.70)
   
6. Calculate the matrix A and the vector b in Equation (3.78)
   
7. Calculate the coefficient A_ij and b_i in Equations (3.81) and (3.82)
   
8. Solve λ in Equation (3.83)
   
9. Calculate P_f in Equation (3.84)

3.3.1 R-F Method

The R-F (Rackwitz & Fiessler) method ^[3-6] is designed to solve the problem of arbitrary distribution of random variables. The following conditions are proposed in this method: (1) At the design checking point, F(x^*)=F_normal(x^*), and the probability distribution functions before and after transformation are equal; (2) at the design checking point, f(x^*)=f_normal(x^*), and the probability density functions before and after transformation are equal; under this condition, β and P_f can be calculated by the checking point method for normal variables.

So, from condition (1), the following can be obtained

From condition (2), the following can be obtained

So

(3.85)

The Rosenblatt transformation [3-7] is designed to solve the problem of random variables with arbitrary distribution and correlation. According to the principle of conditional probability, the non-normal random variables are transformed into independent standard normal random variables.

Conditions:

1. Dependent independent X
   
2. Arbitrary distribution om variables with distribution
   
3. Calculate β and P_f by means of the checking point method

According to condition (1), if r_i is an independent standard normal variable, then R = TX, where T refers to the Rosenblatt transformation

(3.86)

where .

Conditional probability density function

where

Therefore, the following can be derived by the inverting method

(3.87)

The above inverting method usually needs to be applied in combination with the numerical method, making it rather difficult to use; here, , there are i possible combination.

For example, n = 2

Apparently, this free combination leads to a difference in the method of solving for X.

But fortunately, the conditional probability density function or distribution function is not always known in engineering practice. Often, some estimates or correlations can be utilized. Particularly, when x_i is independent, is linear.

The Rosenblatt transformation can transform one type of variable distribution into another type, so

(3.88)

(3.89)

(3.90)

(3.91)

(3.92)

(3.93)

The P-H (Paloheimo-Hannus) method [3-8] lies between the central point method and the checking point method. Its basic starting point is as follows: (1) arbitrarily distributed random variables; (2) the limit state function (LSF) of multiple random variables.

For LSF is g(X), if there is

where is the probability function for the random variable x₁. So, the quantile can be determined by P_f or 1 − P_f.

For

For

β is the safety index, which can be worked out according to LSF g(X) = 0.

Under multivariable conditions, it is considered that is generally not at the quantile point. α_i can be used as a weighting coefficient to adjust the influence of each variable, so,

(3.94)

For the modified P-H method [3-9], which is based on the P-H method, is replaced by the equivalent normal random variable x_i, and

(3.95)

(3.99)

(3.100)

(3.101)

(3.102)

The symmetric functions[3-13] defining the Mercer conditions for kernel functions:

(3.103)

According to Equation (3.103), the optimization issue is transformed into linear equations:

(3.104)

Where, , , , , K_ji = K(x_j, x_i) and j, i = 1, 2, ⋯, l. Solving Equation (3.104) with the least squares method, and obtaining α and b, enables the predicted output to be obtained.

(3.105)

Different support vector machines can be constructed by using different kernel functions. Common kernel forms include:

1. Linear kernel: ;
   
2. Polynomial kernel of order d: ;
   
3. Radial basis kernel: ;
   
4. Two-layer perceptron neural network kernel: , where σ, κ and θ are adjustable constants.

Compared with the standard SVM, the least squares support vector machine (LS-SVM) replaces inequality constraints with equality constraints. With its fast solution speed, this algorithm can be transformed for solving linear equations.

2) Response Surface Methodology Combined with LS-SVM

A response surface analysis method for structural reliability based on LS-SVM is hereby proposed to solve the problem of weak function approximation of the response surface method. As a coupled form of the LS-SVM, FEA and Monte-Carlo numerical simulation methods, this method creates a new concept for structural reliability analysis. The LS-SVM program and Monte-Carlo simulation are compiled using MATLAB; the response (such as stress or displacement) of the structure is calculated using the finite element method, with all three integrated into a structural reliability analysis system. See Figure 3.3 for the program block diagram.

First, the structure is analyzed to determine the main failure modes; then the input vectors of the learning samples are obtained by an orthogonal experimental design method based on the probability distribution parameters of basic random variables. The structural response values of the input vectors of the learning samples are then obtained using standard finite element programs, such as ANSYS, SAP, or similar, to finally determine the learning samples. Next, the hyper-parameters and kernel parameters are determined using the cross validation method, so that the estimated LS-SVM value for the learning samples approaches the calculation results using the finite element method. The test samples are generated by means of sampling; a check is then required of whether the detection requirements of LS-SVM have been met or not. If not, the values of the hyper-parameters and kernel function parameters need to be adjusted until the requirements above are met, thus establishing a nonlinear mapping relationship between structural action and response. Finally, the failure probability of the structure is determined by the Monte-Carlo method and the LS-SVM nonlinear estimation function.

3) Generation and initialization of learning samples

To establish the nonlinear mapping relationship between structural action (input) and response (output) using LS-SVM, the learning samples of the LS-SVM must be selected or designed based on the specific purpose. If the complete combination method is used, there will be an excessive number of training samples, which will in turn lead to excessive workload. In this test, learning samples were selected using an orthogonal experimental design. Each random variable in the design sample is evenly selected at n levels within the range [m_i − 3σ_i, m_i + 3σ_i], with the value of n depending on the number of random variables. To be more specific, m_i and σ_i are the mean and standard deviations of each random variable. For a normally distributed random variable, the probability that its value deviates from the mean by more than 3 standard deviations is not greater than 0.13%. According to this statistical theory, the above learning samples are highly representative.

To eliminate the influence of various factors caused by differences in dimensions and units, the input and output of the sample are normalized by the following equation:

(3.106)

The linear limit state function is , where the basic variable x_i follows the rule of standard normal distribution. The exact solution of the reliability index for the limit state function is β=3. When n=2, 5, 7 and 10, SVM with a linear kernel function is used to reconstruct the response surface, with the reliability indexes analyzed by the SVM response surface method and the improved SVM response surface method converging to form accurate solutions. See Figure 3.4 to see how the number of FEM calculations varies with the number of variables n.

According to Figure 3.4, as the number of variables n increases, the effect of reducing the number of valid element calculations using the improved method becomes more pronounced, indicating that the actual effect of the improved method is related to the number of variables.

2) Example 2: Nonlinear limit state equation

If we choose the following three nonlinear limit state equations and calculate the corresponding reliability using different kernel functions. The results are shown in Table 3.2.

Example 2-1: , where x₁ ∼ N(10, 22), x₂ ∼ N(2.5, 0.3752).

Example 2-2: g(X) = 1 + x₁x₂ − x₂, where x₁ ∼ LN(2, 0.42) and x₂ ∼ N(4, 0.82).

Example 2-3: , where x₁ ∼ N(0.6, 0.07862), x₂ ∼ N(2.18, 0.06542) and x₃ ∼ LN(32.8, 0.9842)

3) Example 3 ^[3-14] : Portal plane frame

In the portal plane frame shown in Figure 3.5, the elastic modulus of each unit is E = 2.0×10⁶ kN/m², and the relationship between inertial moment and cross-sectional area is . The random variables are the sectional areas A₁ and A₂ of the elements and the external load P. See Table 3.3 for the random characteristics. If the horizontal displacement u₃ (unit: cm) of Node 3 is taken as the maximum deformation of the structure to be controlled, calculate its failure probability.

The limit state equation can be expressed as

(3.107)

Where, [u]=1cm is the maximum allowable horizontal displacement, and the relationship between u₃ and each random variable cannot be clearly expressed.

Firstly, learning samples need to be designed, and random variables selected from 5 levels {m_i −3σ_i, m_i −1.5σ_i, m_i, m_i + 1.5σ_i, m_i + 3σ_i} to create a 3-factor, 5-level orthogonal experimental design. 25 groups of random variables are selected, which are also taken as the input of the finite element program for calculating the horizontal displacements of Node 3, so as to obtain a learning sample for the LS-SVM.

The 25 learning samples are normalized according to Equation (3.105), and then input into the prediction model. LS-SVM is used for sample learning. In consideration of the favorable statistical performance of the radial basis function ^[3-15], this is selected for LS-SVM learning as . Kernel parameters σ and hyper-parameters γ exert a significant influence on the generalization performance of LS-SVM. After considering the fast solution speed of LS-SVM, the cross-validation method is used to select parameters γ and σ. The parameter set is determined for γ and σ, from which parameters are selected for combination. The LS-SVM is trained to select the best parameter combination of the model. Their generalization ability is at its best when hyper-parameter γ = 4 × 10⁸ and kernel parameter σ = 5. The estimated mean error ε_RMS of the learning sample and the test sample is calculated as follows:

(3.108)

Where, the ε_RMS of the learning sample is 2.027×10^-5 and the ε_RMS of the test sample is 3.9×10^-3. See Table 3.4 for the comparison between the FEM calculation results of OFEM and the estimated LS-SVM function values for the learning samples and the 10 test samples, which were all sampled according to the probability distribution of each random variable. To be specific, ε is the relative error. Due to space limitations, Table 3.4 only lists the training results of certain learning samples and test samples with relatively large errors. As shown in Table 3.4, the estimated values of the LS-SVM function of the learning samples and test samples are quite close to the results of the FEM calculation, indicating that LS-SVM can establish a correct nonlinear mapping relationship between the functions.

After learning and detecting, A1, A2 and P will generate random sampling points according to their probability distributions and the Monte-Carlo principle. Here, N= 100,000 sampling points are taken, and A1, A2 and P form the input vectors of the LS-SVM. By inputting these vectors into the learned estimation function relational expression, 100,000 displacement values of Node 3 can be obtained. If this is substituted into the function Z shown in Equation (3.107), then the number of samples nf for Z < 0 can be obtained, and the failure probability is . A failure probability P_f = 2.25×10⁻³, obtained by the traditional response surface method is found in the literature ^[3-14], while a failure probability P_f = 2.322×10⁻³ is obtained when using 2,000 iterations of the importance sampling method.

4) Example 4 ^[3-14]: Plane frame structure

In the calculation diagram of a plane frame structure for a 3-span 12-storey building (as shown in Figure 3.6), the elastic modulus of each unit is E = 2.0×10⁷kN/m², and the relationship between the unit section inertial moment and the section area is . See Table 3.5 for the sectional characteristics of each unit. The random variables are the sectional area A₁ of the unit and the external load P. The statistical parameters are shown in Table 3.5.

Assuming normal conditions and a maximum allowable deformation [u]=9.6cm, the following limit state equation can be established according to the code:

(3.109)

To study the influence of the number of learning samples l on the calculation results for failure probability, 5 and 7 levels of each random variable were taken, i.e. {m_i − 3σ_i, m_i − 1.5σ_i, m_i, m_i + 1.5σ_i, m_i + 3σ_i} and {m_i-3σ_i, m_i-2σ_i, m_i–σ_i, m_i, m_i + σ_i, m_i + 2σ_i, m_i + 3σ_i}; an orthogonal experimental design with 6 factors and 5 levels, and one with 6 factors and 7 levels were carried out. 25 and 49 learning samples were selected, respectively, and the failure probability of the structure was calculated using the response surface method based on LS-SVM. See Table 3.6 for the results. To be specific, Pf was obtained following N=100 000 iterations, and the average error ε_RMS was calculated using Equation (3.108).

As learnt from Table 3.6, a correct mapping relationship is established by LS-SVM between 6 random variables and the structural response u_A. The response surface method based on LS-SVM is insensitive to changes in sample number. It can thus be seen that this method has strong learning ability for small samples and can greatly reduce the workload of FEA. The failure probability P_f = 7.309×10⁻² obtained by the traditional response surface method is used in Literature ^[3-14], while the failure probability obtained after 2,000 iterations is simulated using the importance sampling method.