## 2 The Principle of Structural Reliability Analysis
### Degree of Reliability for Structure  
In the current unified standards for structural reliability design, structural reliability is defined as the ability of structural components to fulfill predetermined functions within a specified time frame and under specified conditions. Furthermore, structural reliability is quantitatively described as the probability of successfully completing these functions within the given time and conditions. Thus, structural reliability can be regarded as a probabilistic measure of a structure's performance. Under the influence of various random factors, the capability of a structural system to perform its specified functions can be more accurately assessed when it is expressed in terms of probability. Let \(P_{s}\) denote the probability that the structure successfully completes its predetermined function within the specified time and conditions. This probability \(P_{s}\) represents the reliability of the structure. Conversely, the probability that the structure fails to complete the predetermined function is referred to as the failure probability, denoted as \(P_{f}\). Since reliability and failure are two mutually exclusive events, the reliability and failure probabilities are complementary, and this relationship can be expressed by Equation (3). \[P_{f}+P_{s}=1 \tag{3}\]  
According to the definition of structural reliability and the basic principles of probability theory, if the random vector composed of the basic random variables of the structure is denoted as \(\mathbf{X}=(X_{1},X_{2},\ldots,X_{n})\), the corresponding joint probability density function as \(f_{X}(\mathbf{x})\), and the limit state function represented by this random vector as \(Y=g(\mathbf{X})\), then the failure probability of the structure can be expressed as shown in Equation (4). \[P_{f}=P(Y<0)=\int_{F}f_{X}(\mathbf{x})\mathrm{d}\mathbf{x} \tag{4}\]  
where \(F=\{\mathbf{x}|g(\mathbf{x})\leq 0\}\) represents the failure domain of the structure. If the structure has two independent random variables, \(R\) and \(S\), with their corresponding probability density functions denoted as \(f_{R}(r)\) and \(f_{S}(s)\), and their probability distribution functions represented as \(F_{R}(r)\) and \(F_{S}(s)\), respectively, the structural limit state function can be expressed as shown in Equation (5). \[Y=g(R,S)=R-S \tag{5}\]  
Then, the failure probability of the structure can be expressed using Equation (6). \[P=P(Y<0)=\int_{0}^{+\infty}[\int_{0}^{s}f_{R}(r)\mathrm{d}r]f_{S}(s)\mathrm{d }s=\int_{0}^{+\infty}F_{R}(s)f_{S}(s)\mathrm{d}s \tag{6}\]  
In the analysis of structural reliability, \(R\) is generally used to represent the resistance of the structure, while \(S\) represents the effect of the action on the structure. The resistance \(R\) can either be a single basic random variable or a comprehensive random variable represented by several basic random variables. In practical structural analysis, there are often numerous random variables, and moreover, the limit state function may be nonlinear, making the problem relatively complex. As a result, solving the failure probability through numerical integration becomes very difficult, or even impossible, in such cases. To address this challenge, the concept of the reliability index \(\beta\) is introduced. Assuming that \(R\) and \(S\) both follow a normal distribution, their means and standard deviations are \(\mu_{R}\), \(\mu_{S}\), \(\sigma_{R}\), and \(\sigma_{S}\), respectively. Then, the limit state function \(Y=R-S\) also follows a normal distribution, with its mean and standard deviation denoted as \(\frac{\mu_{S}+\mu_{S}}{2}\) and \(\sigma_{Y}=\sqrt{\sigma_{R}^{2}+\sigma_{S}^{2}}\), respectively. The probability of \(Y<0\) is referred to as the failure probability. At this point, the failure probability of the structure can be expressed as shown in Equation (7).