## 3 Dimensioning Aspects
### Occurring Deformations  
In the course of the vertical load application, a deformation of the simply supported girder occurs. The permissible deflections are defined by ISO 22986 and differ from \(1/250\) to \(1/1500\) of the span width with respect to the considered use case. In this context, the eccentrically (excentricity \(\epsilon\left[\text{mm}\right]\)) applied prestressing force \(N_{0}\) is of particular importance in the box section design. The force acting eccentrically to the centroid axis of the girder (flexural stiffness \((EI)_{22}\) regarding bending about \(x_{2}\)-axis) produces an initial bending moment \(M_{0}=N_{0}e\) which just counteracts the deflection of the structure. Here, the self-weight distribution is additionally taken into account by \(\mu=\rho\text{g}A\left[\text{N}/\text{mm}\right]\). Figure 17 illustrates the according mechanical model. Figure 16: Qualitative Haigh diagram for ductile materials based on ([31] p. 89). If one now assumes that a certain dimensionless ratio \(\delta\in[0,1)\) of deflection \(w(l_{0})\) to (half) span \(l_{0}\) is given, i.e., \(\delta=w(l_{0})/l_{0}\), the initial bending moment \(M_{0}\) necessary for this deflection can be determined considering Euler-Bernoulli beam theory which is usually applied in crane construction. In [26], it is assumed for simplicity that the load is evenly distributed on all four wheels of the trolley (usually the load is distributed unevenly, the exact load distribution can be found in the data sheet of the trolley; wheelbase \(2l_{\text{w}}\), ratio \(\alpha_{1}=l_{\text{w}}/l_{0}\)). Furthermore, it is assumed that the structure is always under pressure to presuppose the beam as a continuous structure. By utilizing the symmetry and considering the boundary conditions, the underlying differential equation can be solved and the bending moment is to be calculated by  
\[M_{0}=\frac{5}{12}\mu_{0}^{2}+\frac{2}{3}(\alpha_{1}^{3}-3\alpha_{1}^{2}+2)R_{ 3}l_{0}-(EI)_{22}\frac{2\delta}{l_{0}} \tag{4}\]  
to determine the initial bending moment necessary for obtaining a resulting deflection ratio \(\delta\) at \(x=l_{0}\). While in conventional structures the deformation can only be influenced by the geometry, prestressed structures offer the advantage of adjusting the deformation via the initial bending moment. However, the additional compressive stress must be taken into account when assessing the strength of the material; see Section 3.1. This property is of great interest for structures with large spans, since the resulting deflection is the dominant parameter in the design. For given geometric dimensions, Equation (4) can also be used to estimate the required compressive force \(N_{0}\) in relation to the entire payload \(4R_{3}\), since the moment is proportional to the product of compressive force and eccentricity \(\epsilon\):  
\[\frac{N_{0}}{4R_{3}}=\frac{1}{\epsilon}\bigg{\{}\frac{5}{48R_{3}}\mu_{0}^{2}+ \frac{1}{6}(\alpha_{1}^{2}-3\alpha_{1}^{2}+2)l_{0}-(EI)_{22}\frac{\delta}{2R_ {3}l_{0}}\bigg{\}}.\]  
Due to the geometry, the modeling of the crane bridge in box section design as a continuous beam is permitted. The crane bridge in truss design, on the other hand, requires a more detailed approach to determine the deflection. Usually, trusses are modeled as a composite of massless members that can only transmit tensile and compressive forces and are hinged together. Due to the dimensions of the components, such an approach is no longer applicable since the hollow profiles are capable of transmitting bending moments as well and thus cannot be interpreted as simple bars any longer. A first approximation for determining the resulting stiffness of the crane bridge can be made here by considering the cross-sectional area from Figure 4. With the geometric dimensions of the hollow U-profiles and their relative position to the centroid axis (parallel axis theorem), an equivalent area moment of inertia can be determined.