# Overview of FEM-Based Resistance Models for Local Buckling of Welded Steel Box Section Columns
## 3 Finite Element Model
### Imperfections and Residual Stresses
#### 3.4.3 Geometrical Imperfection  
If residual stress is considered in the computational model, the initial geometrical imperfection of the element should be considered in a realistic manner. The values of this imperfection can be taken from different academic sources, such as from [PERSON] and [PERSON] [42], where the plate slenderness and yield stress are taken into account to determine the imperfection value; this work considered different local imperfection values for normal- and high-strength steel using Equations (12) and (13), respectively. [PERSON] et al. [50] studied the partial factor to the application of the Winter-type buckling curve in accordance with the safety requirements of Eurocode and made a differentiation between normal-strength steel and high-strength steel. In their studies, the imperfection based on Equation (11) was considered, defined based on experimental measurements. Since two imperfection models from the same authors will be used, this model is managed under the name \"2021 model\". [PERSON] et al. [51] assessed welded thin-walled hollow sections under a cyclic axial load. Equation (14) shows the proposed expression as a function of the section size. Another resource for the proposed value of local imperfection is the BSI manual [52], which presents unique value of b/125, Equation (15). Additionally, one more expression is used that is in research under publication at the moment of writing this work. It considers  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**H/t** & **Welding Type** & \(\sigma_{\text{r}}\)**/f\({}_{\text{y}}\)** & \(\sigma_{\text{rc}}\)**/f\({}_{\text{y}}\)** & **Tensile Zone** & **Transition Zone** \\ \hline
10 & — & 1 & \(-0.6\) & 0 & from equilibrium \\
20 & Heavy weld & 1 & \(-0.85\) & 3t & 3t \\
20 & Light weld & 1 & \(-0.29\) & 1.5t & 1.5t \\
40 & Heavy weld & 1 & \(-0.29\) & 3t & 3t \\
40 & Light weld & 1 & \(-0.13\) & 1.5t & 1.5t \\ \hline \hline \end{tabular}
\end{table}
Table 2: Residual stresses for welded box sections from ECCS model.  
Figure 6: Application of residual stress distribution on a plate of the computational model.  
local imperfection as a function of the b/t ratio, taking its value as an average of b/68t Equation (16), and an upper limit of b/32t. This model is referred later under the name of \"2023 model\". In the given equations parameter b refers to the plate width, t refers to the thickness, and e to the imperfection magnitude. f\({}_{\text{y}}\) is the yield strength of the material, f\({}_{\text{cr}}\) refers to the critical buckling stress, and \(\lambda_{\text{p}}\) refers to the relative slenderness of the plate defined by EN 1993-1-5 [4].  
\[\frac{\text{b}}{\text{e}}=\max\left(\frac{-0.0513\times\left(\frac{\text{b}}{ \text{t}}\right)+2.6131}{1000},\frac{0.8176}{1000}\right) \tag{11}\]  
\[\frac{\text{b}_{\text{NSS}}}{\text{e}}=\left\{\begin{aligned} & 675-0.53\text{f}_{\text{y}}, \quad\overline{\lambda_{\text{p}}}<1.2\\ &\\ &\\ &\frac{\text{b}_{\text{NSS}}}{\text{e}}=\left\{\begin{aligned} & 675-0.53\text{f}_{\text{y}}, \quad\overline{\lambda_{\text{p}}}<1.2\\ &\\ &\\ &\frac{\text{b}_{\text{NSS}}}{\text{e}}=0.53\text{f}_{\text{y}}, \quad\overline{\lambda_{\text{p}}}\;\geq 1.4\\ &\\ &\frac{\text{b}_{\text{HSS}}}{\text{e}}=\left\{\begin{aligned} & 424,& \overline{\lambda_{\text{p}}}<1\\ &\\ & 420\,\overline{\lambda_{\text{p}}}^{-5.5}+4&\\ &\\ &\frac{\text{b}}{\text{e}}=0.034\times\text{t}\left(\frac{\text{f}}{ \text{f}_{\text{cr}}}\right)^{0.5}\\ &\\ &\frac{\text{b}}{\text{e}}=125\\ &\\ &\frac{\text{b}}{\text{e}}=68\times\text{t}\end{aligned}\right. \tag{13}\]  
Figure 7 shows an example of the general shape of square section, which includes initial geometrical imperfection in an over-scaled manner.