## 3 Proposed Stress-Strength Method
### Numerical Application of the Proposed Method
#### 3.3.1 Stress Distribution Based on the Time-Dependent Stress  
In this case, the beam is supposed to be continuously braced, and for the analysis, the variables are the clear length \(L=9.00\) m, and the uniform load \(W=49.25\) kN/m. The selected beam is the W33X291, and in the analysis, its weight was added. Now, based on Figure 2, the reaction force at point \(A\) is \(Ay=443.22\) kN. As the flexural members are subjected to fatigue, by using the static analysis for beams, the maximum bending moment at point \(A\) is \(M_{max}=\)1994.50 kN\(\cdot\)m. Using the static analysis values, the required inertia moment \(I_{x}=\)18,222.33 in [4] was compared against the selected W33X291 beam inertia moment \(I_{x-AISC}=\)17,700 in [4]. Additionally, the principal variables to validate the selected W beam were the AASHTO normative, from which the generated beam deflection should not be higher than \(\frac{L\ (in)}{360}\), and the design specification for flexural members as per the LRFD method [6]. Thus, the deflection analysis is given next. _Step 2._ Based on the application case, the deflection of the bending beam is determined as  
\[f(\delta)=\frac{WL^{4}}{8 EI} \tag{23}\]  
By using the estimated values in step 1 of Equation (23) with \(I_{x}=18,222.33\) in [4], the maximum deflection for the beam is \(\delta_{max}=0.0227\) m. Thus, based on the AISC design of flexural members [6] and the W beam weight-cost criterion, the selected beam is considered to be right. Now, let us determine the generated normal stress values that are acting on the cross-sectional area as follows. _Step 3._ For the application case, the formulated analysis for the normal and shear stresses are  
\[I=\frac{1}{12}\Big{[}bh^{3}-bh_{1}^{3}+th_{1}^{3}\Big{]} \tag{24}\]  
\[\sigma_{x}=\frac{My}{I} \tag{25}\]  
\[\sigma_{y}=\Big{[}\frac{bt_{f}+ht_{w}}{12I_{z}}\Big{]}\Big{(}h^{3}W-h^{2}Wy \Big{)} \tag{26}\]  
\[\tau_{xy}=\Big{(}\frac{bt_{f}}{t_{w}}+\frac{h}{6}\Big{)}\frac{h^{2}t_{w}W}{2I _{z}} \tag{27}\]  
From Equation (25), the normal stress value is \(\sigma_{x}=105.74\) MPa; from Equation (26), the normal stress value is \(\sigma_{y}=67.47\) MPa [26]; from Equation (27), the shear stress value is \(\tau_{xy}=1.93\) MPa. _Step 4._ Based on data from step 3, from Equation (5), \(\sigma_{y}=86.60\) MPa; from Equation (2), \(\sigma_{1}=105.84\) MPa; from Equation (3), \(\sigma_{2}=67.37\) MPa, and; from Equation (4), \(\tau_{max}=19.23\) MPa. As a summary of the previous analysis and based on the 98% safety factor given on the LRFD method for design [5; 6], the W33X291 beam is selected to be analyzed by using the viscoelastic approach. However, due to the addressed principal stress \(\sigma_{1}\) value being static, let us now, based on the linear viscoelastic approach, determine it depending on time as \(\sigma_{1i}\). _Step 5._ By incorporating the value of the maximum principal stress \(\sigma_{1}\) as \(\sigma_{1i}\) in Equation (8), the effect of the elastic modulus through time is determined. Similarly, by using the time-dependent elastic modulus (\(E_{i}\)) values in Equation (13), the corresponding time-dependent stress \(\sigma_{1i}\) value is determined. In Equation (8), the used viscoelastic  
Figure 2: Static/elastic analysis: (**a**) Left end fixed-right end free and uniformly loaded \(W\) beam; (**b**) Loads, shear forces, and bending moments for a given cross-section. parameter was \(\eta=9.5\times 10^{10}\) GPa-s, which according with [PERSON] et al. [16] corresponds to 20 \({}^{\circ}\)C. The corresponding analysis is given in the Table 2.  
In Table 2, elastic modulus is given in GPa, time is measured in seconds s, the corresponding deflections \(\delta_{i}\) are given in meters m, and the strain values are dimensionless.