## Structural Hot-Spot Stress Evaluation Approach
### ASME Through Thickness Stress Linearization (TTSL)  
The stress linearization methodology was first presented by the Pressure Vessel Research Council (PVRC) in its publication by [PERSON] and [PERSON] [20]. They defined preliminary guidelines to assess FEA stress results. The detailed information was published in the welding research council bulletin 429, \"3-D stress criteria guidelines for applications\". Later on, these guidelines became part of the ASME boiler and pressure vessel code. These guidelines provide recommendations for the post-processing of the results from an elastic finite element stress analysis. The details of this method is given in Annexure-5A of ASME boiler and pressure vessel code section VIII, division 2 [21].  
According to this method, stress results derived from the FEA of 3D brick elements may be processed using the stress integration method. The stress components are integrated along the stress classification lines (SCLs) through the wall thickness (\(t\)) to determine the membrane and the bending stress components, as shown in Figure 3. The SCLs are typically located at the gross structural discontinuities, but for the evaluation of local stresses, SCLs are positioned at the local structural discontinuities, i.e., the weld-toe or the weld-root of the RD joint. It is important to mention that, in this study, [PERSON]'s integration rule was applied for the stress linearization at the weld-toe. The stress components normal to the weld-toe were used for SHSS evaluation (i.e., normal stress \(\sigma_{xx}\) and shear stresses \(\tau_{xy},\tau_{xx}\)). The line of elements through the thickness of the plate at the weld-toe was used for  
Figure 2: Haibach method for evaluation of SHSS at the weld-toe with 2.0 mm CHEXA20 mesh.  
the calculation of SHSS. The membrane stress \(\sigma_{m}\) was calculated taking the average of each stress component or node along the SCL, as shown in Figure 3. Mathematically, it can be expressed as:  
\[\sigma_{m}=\frac{1}{t}\int_{0}^{t}\sigma_{xx}(y)dy+\frac{1}{t}\int_{0}^{t}\tau_{ xy}(y)dy+\frac{1}{t}\int_{0}^{t}\tau_{xx}(y)dy \tag{4}\]  
Additionally, for the calculation of the bending stress \(\sigma_{b}\), the difference of each stress component or node along the SCL was taken. Mathematically, we can write it as:  
\[\sigma_{b}=\frac{6}{t^{2}}\int_{0}^{t}\sigma_{xx}\Big{(}\frac{t}{2}-y\Big{)} dy+\frac{6}{t^{2}}\int_{0}^{t}\tau_{xy}\Big{(}\frac{t}{2}-y\Big{)}dy+\frac{6}{t^{2}} \int_{0}^{t}\tau_{xx}\Big{(}\frac{t}{2}-y\Big{)}dy \tag{5}\]  
If shell elements are used for SHSS evaluation, then the membrane and the bending stresses can be obtained directly from the stress resultants. The membrane and the bending stresses are developed on the cross-sections through the thickness of the plate; these sections are called stress classification planes (SCPs). The method to derive the membrane and the bending stress components from a linearized stress distribution is shown in Figure 3. The stresses used for the calculations are based on a local coordinate system defined by the orientation of the SCL. The stress results are derived from the FEA of 2D or 3D shell models of the plate structures. The membrane stress \(\sigma_{m}\) is the average of top \(\sigma_{top}\) and bottom stresses \(\sigma_{bot}\) along the SCL and mathematically can be expressed as:  
\[\sigma_{m}=\frac{\sigma_{top}+\sigma_{bot}}{2} \tag{6}\]  
The bending stress \(\sigma_{b}\) comprises the linear varying portion of the top \(\sigma_{top}\) and the bottom \(\sigma_{bot}\) stress along the SCL and mathematically can be written as:  
\[\sigma_{b}=\frac{\sigma_{top}-\sigma_{bot}}{2} \tag{7}\]  
Figure 3: American Society of Mechanical Engineers (ASME) through thickness stress linearization (TTSL) method for evaluation of SHSS at the weld-toe with 2.0 mm CHEXA20 mesh.  
After the calculation of the membrane stress (\(\sigma_{m}\)) and bending stress (\(\sigma_{b}\)) from a 3D solid or shell model, the SHSS can be determined by adding them together.  
\[\sigma_{SHSS}=\sigma_{m}+\sigma_{b} \tag{8}\]