## 4 Observations through a Numerical Example of Steel Diagrid Grillages
### Finite Element Analysis
Based on the incremental formulation of the FEM equations [39; 81; 82], the following approximation of the displacement increment vector \(\Delta\mathbf{u_{e}}\) (\(\xi\)) was considered in the finite element:  
\[\Delta\mathbf{u_{e}}(\xi)=\mathbf{N}(\xi)\Delta\mathbf{q_{e}} \tag{29}\]  
where \(\mathbf{N}\) represents the shape functions matrix, \(\Delta\mathbf{q_{e}}\) denotes the increments of the displacements vector and \(\xi\) is the dimensionless coordinate of the section location. The global deformation was computed in the serviceability limit state (SLS) combination (\(G+Q\)), the stresses--in the ultimate limit state (ULS) combination (\(1.15\times 0.85G+1.5Q\)) obtained in accordance with the formulas contained in [75]. The second-order global analysis P-Delta was used taking into account the influence of deformation on the statics of the system--geometric nonlinearity--to more realistically predict the structural behavior [77]. The incremental method was applied using a modified Newton-Raphson algorithm to solve the nonlinear problem. Therefore, the following FEM matrix equilibrium equations series [83] were used for each test necessary for the response function method recovery:  
\[\mathbf{K}(\mathbf{q}_{n})\Delta\mathbf{q}_{n+1}^{i}=\Delta\mathbf{Q}_{n+1}^ {i} \tag{30}\]  
where \(\mathbf{K}\) denotes the stiffness matrix of the system changing under the influence of deformation, resulting classically from an aggregation process over all the finite elements, \(\mathbf{q}\) is the generalized displacements vector, \(\Delta\) denotes the increment of a given quantity, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Nodal Points** & **Supports** & **Bars** & **Panels** & **Basic Panel Size** & **Glass Panel** & **Steel Weight** \\  & **Number** & **Number** & **Number** & **Number** & **(m)** & **(m)** & **(kg)** \\ \hline OB & 24 & 16 & 38 & 15 & \begin{tabular}{c} 3.00 \(\times\) 3.12 \\ (rectangle sides) \\ \end{tabular} & 2 \(\times\) 19 + 8 & 7325 \\ OS & 40 & 22 & 67 & 28 & \begin{tabular}{c} 2.25 \(\times\) 2.23 \\ (rectangle sides) \\ \end{tabular} & 2 \(\times\) 12 + 8 & 8557 \\ RB & 24 & 16 & 53 & 30 & 3.00 \(\times\) 3.12 (legs) & 2 \(\times\) 10 + 8 & 9983 \\ RS & 40 & 22 & 95 & 56 & 2.25 \(\times\) 2.23 (legs) & 2 \(\times\) 8 + 8 & 11,388 \\ TB & 32 & 20 & 73 & 42 & 3.00 (equilateral  triangle side) & 2 \(\times\) 10 + 8 & 10,090 \\ TS & 50 & 26 & 121 & 72 & 2.25 (equilateral  triangle side) & 2 \(\times\) 8 + 8 & 11,789 \\ LB & 24 & 16 & 53 & 30 & \
\begin{tabular}{c} 3.46 (equilateral  triangle side) \\ \end{tabular} & 2 \(\times\) 10 + 8 & 9582 \\ LS & 38 & 22 & 89 & 52 &
\begin{tabular}{c} 2.60 (equilateral  triangle side) \\ \end{tabular} & 2 \(\times\) 8 + 8 & 10,841 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Properties of the models. \(\mathbf{Q}\) is equivalent to the external loads vector, \(n\) denotes the increment number and superscript \(i\) indicates the iteration number. The stiffness matrix \(\mathbf{K}\) was updated only after each subdivision--not after each iteration--as a result of its correction with the algorithm of the BFGS method. The following parameters were set to complete these experiments:  
* number of load increments: 5;
* the maximum number of iterations per increment: 40;
* number of reductions for the increment length: 3;
* coefficient of the increment length reduction: 0.5;
* the maximum number of BFGS corrections: 10;
* tolerance factor of the relative norm for the residual forces: 0.0001;
* tolerance factor for the relative norm of displacements: 0.0001. The finite element method-based modal analysis was performed, too. Higher modes were also computed, but only the first mode was verified. The subspace iteration algorithm was used for solving the eigenvalue problem. The maximum number of iterations was set as 40 with a tolerance factor of 0.0001. Yield strength was assumed as 235 MPa and the structural members were made of rectangular hollow sections (Figure 16).