A "Twist" in the Buckling of Beams
Threedimensional 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 postbuckling response.
In this News we focus on the buckling analysis of a beam loaded with a concentrated force in the xy plane at its center, see Figure 1. The beam has, in the first analysis, the crosssection shown in Figure 2a, and then, in a second analysis, the crosssection shown in Figure 2b. These thin open crosssections 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 postbuckling 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 xy plane becomes energetically unfavorable when compared to an outofplane deformation that bends the beam out of the xy plane while twisting it about the xaxis.
Figure 1 Schematic of the beam problem considered
(a) Crosssection in Analysis 1 

(b) Crosssection in Analysis 2 
Figure 2 Crosssections 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 crosssections, 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 crosssection. At first sight, it may appear that the direction of loading will not change the absolute value of the buckling load, whichever beam crosssection is used, but there is a “twist”.
Figures 3 to 5 below show the calculated response when considering the doublysymmetric Ibeam section of Figure 2a. The beam bends initially only in the xy plane but at a load level of P = 135 kN buckles out of that plane and twists about the xaxis. 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 doublysymmetric Ibeam
Figure 4 Load vs. zdisplacement of doublysymmetric Ibeam
Figure 5 Load vs. xrotation of doublysymmetric Ibeam
Figures 6 to 8 below show the response of the beam with the monosymmetric Ibeam section of Figure 2b. An interesting phenomenon is observed, namely, the buckling load and postbuckling response significantly depend on the direction of load application, because of the Wagner effect.
To predict this loaddirection 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. ydisplacement of the beam with crosssection of Fig. 2b
Figure 7 Load vs. zdisplacement of the beam with crosssection of Fig. 2b
Figure 8 Load vs. xrotation of the beam with crosssection 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 crosssections, 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:
Thinwalled opensection beam, warping, Wagner effect, bendingtorsion coupling, lateral buckling, large displacement, stability, steel structures
