4.6 Comparison of elements
5.1.3 Finite element model
The Scandinavium Arena will in this section be analysed with finite elements, as a comparison to the original calculations. As a demonstration, the comparison will be performed for only one load case: uniformly distributed dead load of −0.6 kN/m2 and snow load of−0.75 kN/m2 on the whole roof. Due to symmetry in both structure and load case only a quarter of the structure had to be modelled for this case, Figure 5.7.
y x
Figure 5.7: One quarter of the Scandinavium Arena.
The finite element model of a quarter of the structure is shown in Figure 5.8. The beam nodes on the symmetry lines x = 0 and y = 0 have the following boundary
130
5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA
conditions: θy = 0, θz = 0, and ux = 0. Cable nodes on x = 0 are prevented to move in the y-direction and vice versa for the other symmetry line. All columns are pin-jointed to the ground and to the ring beam, while the pylon is rigidly connected at both ends.
Figure 5.8: Element model.
All cables are modelled using the elastic catenary element presented in Chapter 4.
This element was chosen instead of the bar element to avoid the problems with local mechanisms and numerical instability due to slackening cables. The ring beam and the pylon were modelled by non-linear three-dimensional beam elements based on cubic shape functions. The co-rotational approach is used to compute the tangent stiffness matrix and internal force vector [90]. The Matlab routines for the beam element have been developed by Costin Pacoste and have been used to analyse other structures [85]. Since the ring beam is relatively stiff, the results using the non-linear beam elements were compared with results using linear three-dimensional beam elements. The differences in the results were very small. Despite the small difference, the non-linear beam element was used for the analyses in this chapter.
It should be mentioned that some of the beam elements are very short and stiff, and therefore do not fit into the beam assumption. Still, the beam model is used for these elements. A more accurate analysis, which avoids the short elements but keeps the cable spacing would require the use of solid elements. However, some difficulties in connecting the solid elements and the cable elements may arise since the solids cannot cope with high concentrated load in the same way as the beam elements. The columns were modelled with straight bar elements. The magnitudes of the prestressing forces are computed according to subsection 5.1.2. The radially oriented tension rod, shown in Figure 5.5, was not included in the finite element model of the structure.
As mentioned earlier, the main difference in analysis between cable structures and other structures, such as frames and trusses, is that the initial configuration is generally unknown for cable structures. According to Møllmann [76], the following iterative procedure is used for a cable structure with an elastic boundary structure (arches or beams):
CHAPTER 5. STATIC ANALYSIS
1. Assuming that the boundary joints are fixed in the positions corresponding to the un-stressed state of the arch, the shape of the cable net is determined corresponding to cables in vertical planes.
2. The cable forces at the boundary joints obtained from the previous stage are now re-garded as external loads acting on the arch. The arch is then analysed separately for these forces and for the weight of the arch members.
3. Keeping the boundary joints fixed in the positions obtained from stage 2, the shape of the net is now recalculated (cables in vertical planes or geodesic net).
4. Return to 2.
With this procedure, the boundary structure and the cable net are calculated sep-arately until the displacement changes of the joint coordinates of the cable net and boundary are sufficiently small. A somewhat similar procedure was used in the analysis of the Scandinavium Arena. The only difference is that at step 2 the pre-tensioned cable net is numerically attached to the unstressed ring beam. This means that the stiffness of the whole structure is used to compute the displacements of the ring beam. Some cables will be unloaded during step 2, but since the ring beam is quite stiff only 3–4 iterations are needed to get a deviation in the horizontal com-ponent of the cable forces of a most 0.5 %. What this error corresponds to in the unstrained length of a cable will now be checked. For a bar the following equation holds:
T = AE∆L
L0 = AEL − L0
L0 . (5.11)
This equation can be written as:
L0 = L
T /AE + 1. (5.12)
The cables for the Scandinavium Arena have AE = 343 MN, T ≈ 145.8·4 = 583 kN.
Assuming a cable length of 108 m, the error in unstrained length is 0.92 mm. This tolerance can not be reached in practice. However, one should bear in mind that a small error in unstrained length may give large errors in force; for example, an error in unstrained length of 0.05 % gives an error in force of 30 % with the cable data given here. For the form-finding of the cable net the grid method (section 3.2.2) was used. Nevertheless, equation (3.7) cannot be used in this case due to a non-equidistant mesh. Instead, the following equation expressing the vertical equilibrium is used:
in this case, Figure 3.1.
Material and cross-sectional properties for pylons, the ring beam, columns and cables are given in Figure 5.10. More data, including coordinates for beam elements, node loads, prestressing forces, etc., can be found in Appendix A.
132
5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA
(a) Initial equilibrium procedure—step 1
(b) Initial equilibrium procedure—step 2
(c) Initial equilibrium procedure—step 3
Figure 5.9: Numerical procedure to find the initial equilibrium configuration sug-gested by Møllmann [76]: (a) the shape of the cable net is obtained by assuming fixed nodes, (b) the cable forces are regarded as external loads on the ring beam (step 1 and 2 are repeated until convergence), and (c) the two structures are connected and the whole structure should now be
CHAPTER 5. STATIC ANALYSIS
x y
z
0 0
00
0
3
45
5
5 6
Pylon E = 32 GPa G = 12.8 GPa A = 4.55 m2 Ix = 0.566 m4 Iy = 4.645 m4 Iz = 17.021 m4
x y
z
00
0 0
12
35
Ring beam E = 32 GPa G = 12.8 GPa A = 4.2 m2 Ix = 1.581 m4 Iy = 0.504 m4 Iz = 4.288 m4
0 0 8
Column E = 32 GPa G = 12.8 GPa A = 0.503 m2 Ix = 0.040 m4 Iy = 0.020 m4 Iz = 0.020 m4
Cable
E = 162 GPa A = 2.12 · 10−3 m2
Figure 5.10: Cross-sectional and material properties for finite elements. Drawn di-mensions in millimetres.
134
5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA