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4.6 Steel girder with holes In the last example we discuss the stability behaviour of a beam with holes in a thin web subjected to a vertical load P at the center. Fig. 13 shows the ultimate buckling state in a similar experiment and the essential geometrical data. Experimental results are not available. 31 2 2 3 34.5 1 y z x 10.5 1 0.2 10 1 10.5 21.5 P/2 8 8 25.5 25.5 94 21.5 [cm] Figure 13: Steel girder with holes: a) similar experiment, b) geometrical data A constant load distribution over the web is assumed. The web is modelled in the range (94 < x < 128.5 cm) as a rigid plate. The following boundary conditions are taken into account: symmetry conditions at x=0, fully clamped at x=128.5 cm. Furthermore no deflections in y-direction are permitted at x=y=0. E = 21000 kN/cm 2 and ν =0.3 are chosen as elastic material properties. Within a pure elastic solution the load deflection curves for three selected points are depicted in Fig. 13. Here, an arc–length scheme and a branch switching procedure at the first buckling point have been used. Since the web is very thin the first buckling mode is governed by local buckles around the holes in the web, see Fig. 15. Again the developed element allows large load steps in the post–critical range. In a second calculation we assume an ideal plastic material behaviour described by the model according to appendix E with a yield stress y 0 =36kN/cm 2 . Results for the associated load– deflection behaviour of the defined points are depicted in Fig. 16. Here the arc–length scheme is mandatory due to the decreasing load deflection curve. A similar deformation pattern is obtained. Since the web is very thin yielding occurs in large ranges starting from the holes, see Fig. 17, where the von Mises stresses are plotted with respect to the deformed mesh. 22
200 175 150 Load P [kN] 125 100 u_z (Point 1) 75 -u_y (Point 2) u_y (Point 3) 50 u_z (Point 1) large steps 25 -u_y (Point 2) large steps u_y (Point 3) large steps 0 0,00 0,20 0,40 0,60 0,80 1,00 1,20 Displacement u i [cm] Figure 14: Load deflection curves in the elastic case -5.810E-01 min -4.681E-01 -3.551E-01 -2.422E-01 -1.293E-01 -1.634E-02 9.658E-02 2.095E-01 3.224E-01 4.354E-01 5.483E-01 6.612E-01 7.741E-01 8.871E-01 1.000E+00 max Figure 15: Displacement u y of first buckling mode 200 175 150 Load P [kN] 125 100 75 u_z (Point 1) el 50 -u_y (Point 2) el u_y (Point 3) el 25 u_z (Point 1) pl -u_y (Point 2) pl u_y (Point 3) pl 0 (P i t 1) l 0,00 0,20 0,40 0,60 0,80 1,00 1,20 Displacement u i [cm] Figure 16: Load deflection curves in the elasto–plastic case 23
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Page 13 and 14: To avoid numerical difficulties the
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Page 17 and 18: 4 Examples The derived element form
Page 19 and 20: 4.3 Hemispherical shell with a 18
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Page 23: Load P in kN 20,0 18,0 16,0 14,0 12
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