## 3 Spectra Determination for Various Models
### Numerical Simulation of a Prestressed Box Girder  
The second example deals with a cantilever prestressed concrete box girder with variable cross-section and with two equivalent prestressing bars with steel diameter of 36 mm. In Figure 8a,b are shown the cross-section geometries at the fixed and at the free end, respectively, of the considered girder. The boundary and loading conditions are presented in Figure 8c. For this type of geometry, the drying process, which drives shrinkage and drying creep, causes nonhomogeneity of shrinkage and creep properties throughout the cross-section that are nonsymmetric with respect to neutral axis. This phenomenon can not be captured by a 1D beam analysis and it has often been one major cause of gross mispredictions of long-time deflections of structures [38; 39]. Therefore, only using a 3D analysis one can capture the different properties in the cross-section.  
Moreover, for this application, the 3D finite element numerical solution of the rate type creep model is compared with the analytical calculation using the beam theory (with the above-mentioned limitations) and the Effective Modulus Method (EMM) using the load history plotted in Figure 8c). Using the EMM the free end displacement can be easily calculated with the Principle of Virtual Work as in Equation (68) where the effective modulus and effective shear modulus are calculated as in previous example. The moment of inertia, \(I(x,t,t_{0})\), the area, \(A(x,t,t_{0})\), and the shear factor, \(\chi(x,t,t_{0})\), are calculated according to the current the cross-section, which varies from the cross-section at the fixed end in Figure 8a to the cross-section at the free end. No relaxation of the steel and no shrinkage of concrete is adopted for this second example. The creep model adopted is the EC2 creep model assuming \(h=0.6\), \(f_{ck}=55\) MPa, \(f_{cm}=63\) MPa, \(t_{0}=14\) days, and a cross-section value of \(h_{0}\) which is assumed varying lineally from 218.6 mm at the fixed end to 155.56 mm at the free end. The comparison between the displacements of the beam as calculated using the beam theory with the EMM (Equations (69) and (70)) and the displacements obtained with the 3D finite element analysis with rate type formulation is presented in Figure 9 for different loading conditions showing a perfect coincidence of the calculated deflections with the two methods. The excellent agreement is also confirmed by the evolution of the normal stresses and by the force in the steel tendon. The normal stresses in the concrete cross-section can be calculated using the expression in Equation (71) with the correction factors proposed by [40] as \(k_{\alpha 1}=1+4\tan(\alpha)^{2}\) and \(k_{\alpha 2}=1-4\tan(\alpha)^{2}\) for the straight and tapered side, respectively. In Figure 10a the time evolution of the axial force in the steel tendon is shown and compared with the analytical solution obtained from Equation (72) presenting a very good agreement between them. Figure 10b-d report the evolution of the normal stresses obtained with the analytical (Equation (71) with the above correction factors) and the numerical solution which basically correspond with each other.  
Figure 8: Cantilever prestressed concrete box girder with variable cross-section: (**a**) Cross-section geometry at the fixed end; (**b**) cross-section geometry at the free end; (**c**) boundary conditions and loading history; (**d**,**e**) finite element mesh. All the lengths are in millimeters.  
Figure 9: Mid-span deflection of the beam in Figure 4 in logarithmic scale for different loading conditions obtained by 3D finite element analysis (with the amplified deformation) and by the beam theory with the effective modulus method.