## 2 Theoretical Background
### Response Function Method  
The response function method consists in approximating the real dependence of the structure response \(f(b)\) with an analytical metamodel--the response function \(f_{\mathrm{a}}(b)\). To complete this task, it is necessary to determine such a function using a multiple solution of an investigated boundary value problem about the mean value of the random parameter, for \(b_{i}\) (\(i=1,\,\ldots,\,N\)) belonging to the interval \(b^{0}-\Delta b,b^{0}+\Delta b\). As a result, we obtain a set of discrete values \(f_{\mathrm{o}}\) (\(b_{i}\)) with an error in the numerical procedure, e.g., the finite element method. There are several ways to choose the \(\Delta b/b^{0}\) ratio [57; 60] as well as this interval's discretization both in terms of uniformity and the number of trial points [29; 61]. Here, uniform interval subdivision with \(n=11\) trial points and the ratio \(\Delta b/b^{0}=0.05\) was applied (Figure 2) as the most frequently used. This choice was confirmed using the numerical experiments included in [28; 58]. Each unknown response function was approximated here using composite functions based on polynomials (Table 1) computed using the classic least squares method [62] as well as its weighted version (WLSM) to modulate the importance of computational analysis results [63]. In the latter method, the values computed for the expectation of random variables were recognized as crucial. Their weights (10) were assumed as the sum of the rest ten weights of equivalent results (equal to 1) [57; 64; 65]--the Dirac-type distribution of the weights. However, in connection with the conclusions contained in [28], a uniform weight scheme was adopted as the basic one. The weighed one is only auxiliary. Figure 2: Graphical representation of the response function method for \(n=11\) and \(\Delta b/b^{0}=0.05\) (blue points—discrete data, red curve—approximation). \begin{table} \begin{tabular}{c c c c c c c} \hline \hline \(y\) & \(y_{1}=w(x)\) & \(y_{2}=w(1/x)\) & \(y_{3}=w(\sqrt{x})\) & \(y_{4}=w(1/\sqrt{x})\) & \(y_{5}=w(1(x))\) & \(y_{6}=w(\exp(x))\) \\ \(1/y\) & \(y_{7}=1/w(x)\) & \(y_{8}=1/w(1/x)\) & \(y_{9}=1/w(\sqrt{x})\) & \(y_{10}=1/w(1/\sqrt{x})\) & \(y_{11}=1/w(\ln(x))\) & \(y_{12}=1/w(\exp(x))\) \\ \(y^{2}\) & \(y_{13}=\sqrt{w(x)}\) & \(y_{4}=\sqrt{w(1/x)}\) & \(y_{5}=\sqrt{w(x)}\) & \(y_{6}=\sqrt{w(1/x)}\) & \(y_{77}=\sqrt{w(x)}\) & \(y_{8}=\sqrt{w(x)}\) \\ \(1/y^{2}\) & \(y_{93}=1/\sqrt{w(x)}\) & \(y_{83}=1/\sqrt{w(x)}\) & \(y_{12}=1/\sqrt{w(x)}\) & \(y_{13}=1/\sqrt{w(x)}\) & \(y_{14}=1/\sqrt{w(x)}\) & \(y_{15}=1/\sqrt{w(x)}\) \\ \(1/y^{2}\) & \(y_{93}=1/\sqrt{w(x)}\) & \(y_{83}=1/\sqrt{w(x)}\) & \(y_{12}=1/\sqrt{w(x)}\) & \(y_{23}=1/\sqrt{w(1/x)}\) & \(y_{24}=1/\sqrt{w(x)}\) \\ \(\ln(y)\) & \(y_{83}=\exp(w(x))\) & \(y_{83}=\exp(w(1/x))\) & \(y_{83}=\exp(w(x))\) & \(y_{83}=\exp(w(1/x))\) & \(y_{83}=\exp(w(1/x))\) & \(y_{83}=\exp(w(x))\) \\ \(\exp(y)\) & \(y_{93}=\ln(w(x))\) & \(y_{93}=\ln(w(1/x))\) & \(y_{93}=\ln(w(1/x))\) & \(y_{94}=\ln(w(1/x))\) & \(y_{95}=\ln(w(1/x))\) & \(y_{96}=\ln(w(x)(x))\) \\ \(\sinh(y)\) & \(y_{97}=\pi\sinh(w(x))\) & \(y_{98}=\pi\sinh(w(1/x))\) & \(y_{99}=\pi\sinh(w(1/x))\) & \(\pi\sinh(w(1/x))\) & \(y_{94}=\pi\sinh(w(1/x))\) & \(y_{92}=\pi\sinh(w(\exp(x)))\) \\ \(\pi\sinh(y))\) & \(y_{93}=\sinh(w(x))\) & \(y_{94}=\sinh(w(1/x))\) & \(y_{95}=\sinh(w(\sqrt{x}))\) & \(y_{96}=\sinh(w(1/x))\) & \(y_{96}=\sinh(w(1/x))\) & \(y_{97}=\pi\sinh(w(x))\) \\ \hline \hline \end{tabular}   
* Special cases exponential functions when the base polynomial is of the first order as well as Gaussian functions when the base polynomial is of the second order. * A special case power functions when the base polynomial is of the first order. * B special case power functions when the base polynomial is of the first order.