## 3 Three-Dimensional Finite Element Model
### Geometry, Material Properties, Constraints, and Loading Conditions  
A 6 mm thick flat plate is considered to perform the hole-drilling simulation. An increase of the thickness of the plate is studied and there is almost no (less than 1%) effect on the results. Only a quarter of the entire plate has to be considered due to symmetry. The plate is a thick plate according ASTM E837-13a since the thickness of the plate is larger than the mean diameter of the strain gauge rosette. The width of the model is 15 times the hole diameter [10]. The thickness of the modeled plate is also equal to the thickness of the longitudinal stiffener of the orthotropic steel deck. The steel grade of the orthotropic steel bridge deck is S235. This material is also used for the modeling, resulting in the material parameters specified in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rosette Type** & **D** & **GL** & **GW** & **R\({}_{1}\)** & **R\({}_{2}\)** \\ \hline CEA-06-062 UL-120 & 5.13 & 1.59 & 1.59 & 1.77 & 3.36 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Strain gauge rosette dimensions in mm; D: diameter gauge circle; GL: length gridlines; GW: width gridlines; R\({}_{1}\): radius 1; R\({}_{2}\): radius 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties S235** \\ \hline Mass density \(\rho\) & 7.85 \(\times\) 10\({}^{-6}\) kg/mm\({}^{3}\) \\ Young’s modulus E & 210,000 MPa \\ Poisson’s ratio \(\
u\) & 0.3 \\ Shear modulus D & 81,000 MPa \\ Yield stress \(\sigma_{\gamma}\) & 235 MPa \\ Ultimate tensile strength \(\sigma_{T}\) & 360 MPa \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties.  
Symmetry constraints are necessary because only a quarter of the plate is considered for the finite element modeling. These constraints are applied on the sides where the hole is situated. The displacements along x- and y-axes along these sides are set to zero. For the bottom surface, the displacement along the z-direction is set to zero [10].  
A difference in material behavior is specified with the specification of a different stress-strain curve. A linear stress-strain curve is used for linear-elastic material behavior with the yield stress as an upper limit (Figure 4). A bilinear stress-strain curve (Figure 5) is specified when the material behavior is elastic-plastic and an isotropic plasticity hardening model is used [10].  
For the linear-elastic stress-strain behavior, the finite element program follows the same linear curve for the post-critical stresses and strains. For the elastic-plastic stress-strain behavior, the post-critical behavior is determined by the second branch of the curve where it follows the same trend.  
Both uniform (Figure 6a) and non-uniform (Figure 6b) in-depth residual stress fields will be evaluated with linear-elastic and elastic-plastic stress-strain behavior. The presence of a residual stress field is simulated by imposing time dependent uniform or non-uniform pressure distributions on the sides not containing the drilled hole. These imposed pressures are assumed to be equal to the principal residual stresses before a hole was drilled [15]. The drilling of the hole is simulated by a time dependent deactivation of layers in the hole. For each time step that results in a removed layer, a corresponding pressure is applied on the sides at the same height of the hole depth. Figure 6 gives a representation of the residual stress fields for a drilled hole.  
Figure 4: Linear-elastic stress–strain behavior.  
Figure 5: Elastic-plastic stress–strain behavior.