## 3 Spectra Determination for Various Models
### Numerical Simulation of a Prestressed Beam with I-Shaped Solid Cross-Section  
The first example concerns a simply supported prestressed concrete beam with a constant I-shaped cross-section and with a single equivalent prestressing strand with steel diameter of 36 mm. The cross-section geometry of the considered beam is showed in Figure 4a, the boundary and loading conditions in Figure 4b. In order to validate the numerical formulation of the rate type creep model and its implementation into a three-dimensional finite element code, the mid-span deflection in time obtained from the finite element analysis is compared with its analytical calculation resulting from the Effective Modulus Method (EMM) [36] applied to the type of load history considered (see Figure 4c). Using the EMM the mid-span deflection can be calculated with the Principle of Virtual Work as  
\[f(t,t_{0})=\int_{0}^{l}M^{\prime}(x)\frac{M(x)}{E_{m}(t,t_{0})I(x,t,t_{0})} \mathrm{d}x+\int_{0}^{l}V^{\prime}(x)\frac{V(x)\kappa(x,t,t_{0})}{G_{m}(t,t_{ 0})\,A(x,t,t_{0})}\mathrm{d}x \tag{68}\]  
where \(E_{ef}(t,t_{0})=1/l(t,t_{0})\) is the effective modulus calculated from the creep function \(J(t,t_{0})\), \(G_{ef}(t,t_{0})=E_{ef}(t,t_{0})/2(1+\
u)\) is the effective shear modulus, \(I(x,t,t_{0})\) is the cross-section moment of inertia, \(A(x,t,t_{0})\) is the cross-section area, and \(\kappa(x,t,t_{0})\) is the shear factor [37]. The geometric properties depend on time through the homogenization coefficient \(n=E_{s}/E_{ef}(t,t_{0})\). Assuming a constant cross-section and substituting the bending moment and the shear corresponding to the external uniform distributed load of 30 kN/m applied at time \(t_{0}\) (sketched in Figure 4c) the mid-span deflection, \(f_{q}(t,t_{0})\), is given by  
\[f_{q}(t,t_{0})=\frac{5}{384}\frac{q(t)\;l^{4}}{E_{ef}(t,t_{0})\;I(t,t_{0})}+ \kappa(t,t_{0})\frac{1}{8}\frac{q(t)\;l^{2}}{G_{ef}(t,t_{0})\;A(t,t_{0})} \tag{69}\]  
The prestressing tendon is modeled using beam finite elements connected rigidly to the nodes of the three-dimensional mesh (no slip). No regular reinforcing steel bars are considered in this application. The prestress force, \(N_{p}\), is applied in the tendon by assigning an initial equivalent thermal deformation which generates a stress which accounts for the initial elastic loss. The initial prestress force is of 1297 kN that is equal to prestressing force of \(N_{p}=1250\) kN after the initial elastic loss. Substituting the bending moment (\(M_{p}=N_{p}e\) with \(e=\) the eccentricity of the tendon) the mid-span deflection, \(f_{p}(t,t_{0})\), is given by  
\[f_{p}(t,t_{0})=\frac{1}{8}\frac{N_{p}\;e(t,t_{0})\;l^{2}}{E_{ef}(t,t_{0})\;I( t,t_{0})} \tag{70}\]It has been considered no relaxation of the steel and no shrinkage of concrete in this first example. The creep model adopted is the EC2 creep model assuming \(h=0.5\), \(h_{0}=20.445\) m\({}^{2}/5.397\) m \(=164.9\) mm, \(f_{ck}=55\) MPa, \(f_{cm}=63\) MPa, \(t_{0}=14\) days. The comparison between the mid-span deflection of the beam as calculated using the beam theory with the EMM (Equations (69) and (70)) and the deflection obtained with the 3D finite element analysis with rate type formulation is presented in Figure 5 for different loading conditions showing a coincidence of the calculated deflections with the two methods. The excellent agreement is also conformed by the evolution of the normal stresses and by the force in the steel tendon, as shown in the following. Figure 4: Simply supported prestressed concrete beam: (**a**) Cross-section geometry in millimeters; (**b**) boundary conditions and loading; (**c**) load history; (**d**,**e**) finite element mesh. Figure 5: Mid-span deflection of the beam in Figure 5 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.