SOCIOECONÓMICO

In order to verify the 3D shell- and 2D frame models a comparison of the bending moment have been done for the self-weight. Two strips in the 3D shell model have been selected according to figure 3.12. The result from the comparison is presented in figure 3.15 and 3.16.

Figure 3.15- Comparison between the main bending moments from the3D shell and the2D frame analysis for the middle strip, for the slab frame bridge subjected to self- weight -80 -60 -40 -20 0 20 40 60 80 0 1 2 3 4 5 6 7 8 9 Original mesh density Doubled mesh density Length [m] Moment [kNm/m] -80 -60 -40 -20 0 20 40 60 80 0 1 2 3 4 5 6 7 8 9 10 3D shell 2D frame Length [m] Moment [kNm/m]

Figure 3.16- Comparison between the main bending moments from the3D shell and the2D frame analysis for the end strip, for the slab frame bridge subjected to self- weight.

The correlation for the middle strip is good. For the end strip the values obtained from the 3D shell analysis are larger, which depends on redistribution of load to the edges. This effect is not reflected in the 2D frame analysis.

The reason why the graph obtained from 3D shell analysis does not extend along the length axis as far as the graph obtained from 2D frame analysis is due to how the finite element method produces the results. The computational software Strip step 2is based on two-dimensional beam theory as described in chapter 2.7, which means that the slab is subdivided into a number of beam elements connected by their nodes. This means that the starting- and ending coordinate of the slab will end up in a node, from where the results will be extracted. The results obtained from the finite element software Nastran produces the results by Gauss integration between the nodes. This means that the values are obtained from the middle of the element; it is due to that the results are obtained from half the element size that the graph from the 3D shell analysis is shorter; see Figure 3.17.

-80 -60 -40 -20 0 20 40 60 80 0 1 2 3 4 5 6 7 8 9 10 3D shell 2D frame Length [m] Moment [kNm/m]

4

Loads

In reality a structure is loaded with many different loads. In this study, the load types on the bridges studied were limited and a representative amount of loads were selected in order to obtain a credible comparison. The loads included in the analysis are the self-weight of the structure and the pavement, thermal actions, traffic loads, shrinkage, earth pressure and support settlement.

4.1

Self-weight and pavement

The self-weight was calculated from the geometry of the bridge section. The density of concrete was assumed to be 25 kN/m3; see Appendix A.

The pavement was assumed to be equal for both the slab bridge and the slab frame bridge. This assumption was made to simplify the calculations of temperature loads which depend on the pavement thickness. Since the purpose was not to design a bridge, but rather to perform a comparison, this assumption will not have any influence. The pavement consisted of a 10 mm bituminous sealing placed in two layers, covered with three layers asphaltic concrete with a total thickness of 140 mm; see Appendix A. The pavement was assumed to have the density of 24 kN/m3.

4.2

Thermal Actions

The thermal actions applied on the bridge structures can be subdivided into two types, uniform temperature and temperature gradient. The uniform temperature component of the thermal action determines the expansion and contraction of the bridge, which results in change of length. The temperature gradient does in turn determine the variation of temperature between the superstructure´s upper and lower surfaces, resulting in curvature changes.

In statically determinate structures the need for movement due to thermal actions does not result in sectional forces but only in translations and rotations of the structure. In statically indeterminate structures, on the other hand, the need for movement due to thermal actions results in translations, rotations and sectional forces. As the magnitude of these sectional forces depends of the stiffness of the structure, the forces will be dramatically reduced when the concrete cracks. This is due to the reduced stiffness of the concrete section in cracked state. Furthermore, in concrete structures the stiffness is influenced by creep, resulting in further reduction of the sectional forces due to thermal actions.

For ultimate limit state design of concrete structures the sectional forces due to thermal actions are of minor importance, provided that the ductility and rotational capacity of the structural elements are sufficient and the vertical stability of members is not compromised. This is due to that concrete structures experience a severe cracking under ultimate loads and, hence, the stiffness is dramatically reduced. Furthermore the reinforcement will yield, leading to plastic deformation and redistribution of internal forces and moments. In serviceability limit state, especially for control of crack widths, the sectional forces due to thermal actions should be taken into account. In this case, a gradual evaluation of cracking should be considered, (Eurocode 1 CEN (2002). This is however a difficult task as the precise solution requires a non-linear analysis. A simplified and more practical approach is to calculate the sectional forces due to thermal actions assuming reduced stiffness corresponding to cracked sections and an effective modulus of elasticity. A common practice in

Sweden is to reduce the stiffness of the structure with about 40% and to employ a creep coefficient of about 1.3 for the calculation of sectional forces due to thermal actions

For the purpose of this thesis, the uniform temperature components are calculated to ∆T567= −27.6 ∘C and ∆T=>? = 23.5 ∘C; see Appendix A.In the case of the slab

frame bridge no thermal actions were applied to the bottom slab, due to the fact that the bottom slab lies protected from outside environmental changes. Because the bottom slab does not expand or contract, thus providing a restraint for the rest of the structure, forces in the integral abutment walls are obtained and not only deformations. In the case of the slab bridge, the boundary conditions are such that the superstructure is statically determined in its own plan; thus no sectional forces arise from the uniform thermal action.

The temperature gradient load is calculated to ∆T@AAB= −6.5C and ∆TCDEF = 6.0 ∘C;

see Appendix A. Due to the fact that mainly the roadway will be exposed to solar heating, application of the temperature gradient load is limited to the bridge deck slab in both models.