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### Tech Briefs  A "Twist" in the Buckling of Beams

Three-dimensional beam models are widely used to represent structural behaviors. Bridges, buildings, rotor blades, components of motorcars, and many other slender structures can frequently be modeled, in the first instance, as beam structures. However, due to their slenderness these beam models are prone to buckling instabilities, and need to be analyzed carefully for their buckling and post-buckling response.

In this News we focus on the buckling analysis of a beam loaded with a concentrated force in the x-y plane at its center, see Figure 1. The beam has, in the first analysis, the cross-section shown in Figure 2a, and then, in a second analysis, the cross-section shown in Figure 2b. These thin open cross-sections could be the actual sections of steel beams used in the construction of a building, or may be the result of modeling a complex structure as a beam. In particular, the section shown in Figure 2b could well be the result of modeling essential load carrying parts of a bridge or an automobile.

We are interested in solving for the buckling and large displacement post-buckling response of the beam. The key is to capture accurately the lateral buckling of the beam involving torsion.

This lateral instability occurs because beyond a certain level of applied load, further bending in the x-y plane becomes energetically unfavorable when compared to an out-of-plane deformation that bends the beam out of the x-y plane while twisting it about the x-axis. Figure 1  Schematic of the beam problem considered (a) Cross-section in Analysis 1 (b) Cross-section in Analysis 2

Figure 2  Cross-sections of the beam in Figure 1

We model the complete beam using 20 Hermitian beam elements that include warping degrees of freedom. The boundary conditions are that the rotation about the x axis is fixed at both supports while the other rotations are free and all the displacements at both supports are fixed except the displacement in the x direction at the right support. The warping displacements are free at both ends. In both analyses using different cross-sections, we consider two loading conditions: the concentrated load is once applied in the upward direction and once in the downward direction. In both cases, the load is applied at the centroid of the cross-section. At first sight, it may appear that the direction of loading will not change the absolute value of the buckling load, whichever beam cross-section is used, but there is a “twist”.

Figures 3 to 5 below show the calculated response when considering the doubly-symmetric I-beam section of Figure 2a. The beam bends initially only in the x-y plane but at a load level of P = 135 kN buckles out of that plane and twists about the x-axis. As expected, the response of the beam for upward loading is the same as for downward loading, apart from a sign change of the displacements. Figure 3  Load vs. downward displacement of doubly-symmetric I-beam Figure 4  Load vs. z-displacement of doubly-symmetric I-beam Figure 5  Load vs. x-rotation of doubly-symmetric I-beam

Figures 6 to 8 below show the response of the beam with the mono-symmetric I-beam section of Figure 2b. An interesting phenomenon is observed, namely, the buckling load and post-buckling response significantly depend on the direction of load application, because of the Wagner effect.

To predict this load-direction dependent phenomenon accurately, it is necessary to include in the beam formulation all kinematic nonlinearities regarding warping, in particular also the Wagner effect. Figure 6  Load vs. y-displacement of the beam with cross-section of Fig. 2b Figure 7  Load vs. z-displacement of the beam with cross-section of Fig. 2b Figure 8  Load vs. x-rotation of the beam with cross-section of Fig. 2b

Figures 6 to 8, and the above movies, show also the response calculated when modeling the beam with MITC4 shell elements. We see that the shell model response is very close to the beam model solution.

This beam element is available for modeling many different types of beam structures, and indeed many complex structures modeled as beams. The element can be used with various cross-sections, and for general linear and large displacement nonlinear analyses. It is remarkable for how many different types of analyses the beam element is useful.

Keywords:
Thin-walled open-section beam, warping, Wagner effect, bending-torsion coupling, lateral buckling, large displacement, stability, steel structures