# The Impact of Support Type on the Reliability of Steel Trusses Subjected to the Action of a Fire
## 2 Methods
### Reliability Analysis  
The reliability analysis constitutes the main part of the article; therefore, the manner of its performance will be discussed in more detail. The following assumptions were made in order to simplify the issue:  
* the adopted random variables included geometrical characteristics of the cross-section area (\(A\)), mechanical properties of material (yield point \(f_{y}\)), and the value of the effect of actions (\(E\)),
* resistance of single elements was approximated by normal distribution,
* random variables (\(x\)) were characterized by normal distribution with average value (\(\overline{x}\)) and standard deviation (\(\sigma_{x}\)),
* randomness in the calculations of the temperature of elements and combustion gases were not taken into account,
* it was assumed that random variables were not correlated and that they had a normal distribution,
* the single mechanisms were separated. The paper uses the system method. It allows estimating the reliability of the entire structure based on knowledge about the reliability of individual elements (in the analyzed examples, of rods in a truss). In order to estimate the reliability of an \(i\)-th element, it is necessary to know the average value of the effect of actions (\(\overline{E}_{i}\)) and the average value of bearing capacity (\(\overline{N}_{i}\)). This should be followed by the calculation of standard deviations, i.e., \(\sigma_{E_{i}}\) and \(\sigma_{N_{i}}\). It was assumed in the article that the respective coefficients of variation for the effect of actions is equal to \(\
u_{E}\) = 0.06 [34]. A coefficient of variation for bearing capacity was appointed in the following way: the resistance of the element in compression/ tension in fire situation is defined as follows:  
\[N_{b,fi/c,fi}=\left(\chi_{fi}\right)\cdot A\cdot f_{y}\]  
where \(\chi_{fi}\) is the reduction factor for flexural buckling in the fire design situation. This value is bracketed because it is used only during the calculation of buckling resistance of members in compression (\(N_{b,fi}\)). The resistance of elements in tension in fire situation (\(N_{c,fi}\)) is the product of only two variables: cross-section area (\(A\)) and yield point (\(f_{y}\)). Only these values are random (reduction factor \(\chi_{fi}\) is deterministic), and according to [34], their coefficients of variation are equal to: \(\
u_{A}=0.06\) and \(\
u_{fg}=0.08\). The resistance coefficient of variation can be approximated as follows:  
\[\
u_{N}=\sqrt{\
u_{fg}^{2}+\
u_{A}^{2}}=\sqrt{0.08^{2}+0.06^{2}}=0.1 \tag{1}\]  
Therefore, for the \(i\)-th element:  
\[\sigma_{E_{i}}=\
u_{E}\cdot\overline{E}_{i}=0.06\cdot\overline{E}_{i} \tag{2}\]  
\[\sigma_{N_{i}}=\
u_{N}\cdot\overline{N}_{i}=0.1\cdot\overline{N}_{i} \tag{3}\]  
Subsequently, the average values of safety margin (\(\overline{Z}_{i}\)) and its standard deviation (\(\sigma_{Zi}\)) are to be calculated for each element:  
\[\overline{Z}_{i}=\overline{N}_{i}-\overline{E}_{i} \tag{4}\]  
\[\sigma_{Z_{i}}=\sqrt{\sigma_{N_{i}}+\sigma_{E_{i}}} \tag{5}\]  
The quotient of the average value of the safety margin (\(\overline{Z}_{i}\)) and standard deviation (\(\sigma_{Zi}\)) constitutes a reliability index of the \(i\)-th element (\(\beta_{i}\)):  
\[\beta_{i}=\frac{\overline{Z}_{i}}{\sigma_{Z_{i}}} \tag{6}\]  
On this basis, it is possible to calculate the probability of failure for a given element (\(p_{fi}\)), using the Laplace function:  
\[p_{f_{i}}=\Phi(-\beta_{i}) \tag{7}\]  
where \(\Phi\) is cumulative distribution function of standard normal distribution. Reliability (\(R_{i}\)) is the complement of the probability of failure added to make one:  
\[R_{i}=1-p_{f_{i}} \tag{8}\]  
Knowing the reliability of all elements in a structure, one can commence the performance of a system analysis. Basic structural systems include the serial, the parallel, and the mixed system. All statically determinate structures are characterized by undergoing failures when even a single element of this arrangement is destroyed.