# A Probable Approach to Reliability Assessment of Reinforced Plates
## 3 Failure Loading Probability Distribution Function  
The probability distribution density of the random variable \(l\) was chosen in accordance with the results of the development of the statistical theory of strength [23], in the exponential law form  
\[f(l)=e^{-l/h}/h \tag{13}\]  
Here, \(h\) is a distribution parameter with the dimension of the random variable \(l\). The physical content of (13) consists of reducing the probability of the quantity \(l\) presence in case of its increase. In articles [24; 25], the study was based on the laws of size distribution of defects during the failure of materials. The integral probability distribution function depending on the parameter \(l\) can be defined as follows:  
\[F(l)=1-e^{-l/h}. \tag{14}\]The failure loading distribution function of a plate element with one crack can be determined by applying formula [22]  
\[F_{1}(|P|,\eta)=\int\limits_{D_{\alpha}}f(\alpha)(1-F(l))\,d\alpha\ \ (i=1,2). \tag{15}\]  
According to Expressions (8)-(12), from (15), one can obtain  
\[F_{1}(|P|,\eta)=\int\limits_{D_{\alpha}}f(\alpha)\bigg{(}1-F\bigg{(}\frac{K_{ 0}{}^{2}\Phi_{i}{}^{2}(\alpha,\eta,\varkappa)}{P^{2}}\bigg{)}\bigg{)}\,d\alpha\ \ (i=1,2). \tag{16}\]  
\(D_{\alpha}\) is a set of the parameter \(\alpha\) possible values in the case when the condition is satisfied under the given loading \(P\) and \(Q\)  
\[0\leq\frac{K_{0}{}^{2}\Phi_{i}{}^{2}(\alpha,\eta,\varkappa)}{P^{2}}<\infty.\]  
For the distribution of inclusion half-lengths in the form of an exponential law (13), according to Formulas (14) and (16), we obtain  
\[F_{1}(|P|,\eta)=\int\limits_{-\pi/2}^{\pi/2}f(\alpha)\,e^{-\frac{K_{0}{}^{2} \Phi_{i}{}^{2}(\alpha,\eta,\varkappa)}{P^{2}h}}d\alpha\ \ (i=1,2). \tag{17}\]  
Since we suppose the material of the plate to be isotropic, we set the probability distribution density of the random variable \(\alpha\) by the uniform law \(f(\alpha)=1/\pi\). We substitute this expression into Formula (17) and the relation for the function \(\Phi_{i}(\alpha,\eta,\varkappa)\) (5)-(7); according to Formulas (8)-(12), we obtain the following expressions. For the case \(P>0\) (biaxial tension, or tension-compression):  
* at \(\frac{3-\varkappa}{1+\varkappa}\leq\eta\leq 1\) \[F_{1}(P,\eta)=\frac{2}{\pi}\int\limits_{0}^{\pi/2}e^{-\frac{K_{0}{}^{2}(\varphi _{1}(\eta,\varkappa)+\varphi_{2}(\eta,\varkappa)\cos 2h)^{-2}}{P^{2}h}}d\alpha;\] (18)
* at \(-\infty<\eta\leq\frac{3-\varkappa}{1+\varkappa}\) \[F_{1}(P,\eta)=\frac{2}{\pi}\bigg{\{}\int\limits_{0}^{\pi/2}e^{-\frac{K_{0}{}^{ 2}(\varphi_{1}(\eta,\varkappa)+\varphi_{2}(\eta,\varkappa)\cos 2h)^{-2}}{P^{2}h}}d \varkappa+\int\limits_{\alpha_{1}}^{\pi/2}e^{-\frac{K_{0}{}^{2}(\varphi_{2}( \eta,\varkappa)+\varphi_{4}(\eta,\varkappa)\cos 2h)^{-2}}{P^{2}h}}d\alpha \bigg{\}}.\] (19) For the case \(P<0\), \(Q<0\) (biaxial compression):
* at \(1\leq\eta\leq\frac{1+\varkappa}{3-\varkappa}\) \[F_{1}(-P,\eta)=\frac{2}{\pi}\int\limits_{0}^{\pi/2}e^{-\frac{K_{0}{}^{2}( \varphi_{1}(\eta,\varkappa)+\varphi_{2}(\eta,\varkappa)\cos 2h)^{-2}}{P^{2}h}}d\alpha;\] (20)
* at \(\frac{1+\varkappa}{3-\varkappa}\leq\eta<\infty\) we obtain the Expression (20) with a replacement in its left part \(P\) by \(-P\). If \(\eta=1\), then Expressions (18) and (20) are appreciably simplified. From Formulas (6) and (7) one can obtain \(\varphi_{2}(1,\varkappa)=\varphi_{4}(1,\varkappa)=0\). In accordance:
* at \(P=Q>0\) (biaxial symmetric tension) \[F_{1}(P,1)=e^{-\frac{K_{0}^{2}}{P^{2}h}P_{1}{}^{2}(1,\varkappa)};\] (21)at \(P=Q<0\) (biaxial symmetric compression)  
\[F(l)=1-e^{-l/h}. \tag{14}\]0  
Using the obtained relations for the distribution function \(F_{1}(|p|,\eta)\) (18)-(22), it is possible to determine the statistical characteristics of the strength of the plate with randomly distributed rigid linear inclusions under the different types of stress state: the probability of failure, the most probable and mean value, the dispersion and the coefficient of variation of the failure loading, etc.