Mechanics of Thin-walled Open-section Beams
While the mechanics of thin-walled beams has been the subject of extensive studies for many years, most engineering practitioners’ intuition about the mechanics of these structural elements is based on experience with linear small deformation response of beams where it is assumed that different modes of deformations (axial, bending, torsion and warping) are uncoupled. However, in 3D problems this assumption is only valid in the small deformation regime and only if the cross-section of the beam is doubly symmetric. In the nonlinear large deformation regime, even for doubly symmetric sections, this assumption breaks down as there is always coupling between different deformation modes.
To take into account these coupling effects accurately, one needs to consider a number of phenomena, like the warping of the cross-section, the so-called Wagner effect, and the coupling caused by the eccentricity of the shear center with respect to the center of gravity of the beam cross-section.
The ADINA beam element takes into account all the coupling effects. The element includes warping DOFs and also accounts for the Wagner effect.
Previously, we presented results of the warping beam formulation for lateral buckling and post-buckling analysis of a mono-symmetric I-beam. In this News, we present several examples of the mechanics of thin-walled open-section beams analyzed using the new beam element. For each example we compare the response obtained using the warping beam element and the MITC4 shell element. The aim is to illustrate the behavior of thin-walled open-section beams and assess the accuracy of the new beam element.
We start with a linear vibration problem where the mode shapes and frequencies of a cantilever beam with a channel cross-section are studied (see Figure 1). At the clamped end of the beam, all displacement, rotation and warping DOFs are restrained.
The animations below show the first two mode shapes of the beam. At first glance, one might think that the mode shapes should only represent bending about the two principal axes of the section. Furthermore, the lowest modes should represent bending about the weak axis (y) and as the mode number increases, bending about the strong axis (z) should ensue. However, we see that while the 1st mode represents pure bending about the weak axis, the 2nd mode is not a pure bending mode; instead, bending about the strong axis and twisting occur simultaneously. There is bending-torsion coupling caused by the eccentricity of the shear center with respect to the center of gravity. Table 1 compares the vibration frequencies of a shell model with the warping beam model. As can be seen, the accuracy of the beam model is quite remarkable.
Table 1 Comparison of the vibration frequencies of a channel section using shell and warping beam
As expected, the frequencies of the beam model are slightly higher than those of the shell model as the beam model is slightly stiffer due to its kinematic assumptions.
A question may be: why are bending and twisting not coupled in the 1st mode, while they are coupled in the 2nd mode? The reason is that, when a beam bends in the plane of symmetry of the cross-section (x-z plane), symmetry prevents twist. However, when it bends out of the plane of symmetry (x-y plane), the response is coupled as simultaneous bending and twisting requires less strain energy compared with pure bending.
Now, let us consider a doubly symmetric section, a cruciform, as depicted in Figure 2. This problem shows the importance of considering the axial-torsional coupling. All the displacements, rotations and warping DOFs at both ends of the beam are fixed with the exception of the x-displacement DOF where the axial force is applied. Linearized buckling analysis of this beam is carried out using both the warping beam and MITC4 shell models. The animations below show the first three buckling modes of this structure using the MITC4 shell element discretization. Note that in these buckling modes the centerline of the cross-section remains a straight line (it doesn’t bend) while the cross-section twists about the centerline. Table 2 compares the buckling loads calulated using the warping beam model with the closed form solution [1,2].
Table 2 Linearized buckling loads obtained using the warping beam and closed-form solution
For doubly symmetric cross-sections, this mode of buckling — the axial-torsional buckling, can be dominant when beams are short. However, Euler buckling is dominant when beams are long.
Now let us consider the linearized buckling analysis of a mono-symmetric I-beam as depicted in Figure 3. All displacement DOFs are fixed at the two ends except the z-displacement at the end where the axial force is applied. Only the rotations about the z-axis are fixed at the two ends and the warping DOFs are free at both ends. The animation below shows the first buckling mode of this section. This example highlights the effect of axial-flexural-torsional coupling in the response of such beams. The buckling load obtained using the warping beam model is in excellent agreement with the closed-form solution .
As can be seen, the buckling mode is a combination of bending out of the plane of symmetry (y-z plane) and twisting. This is a characteristic of the response of mono-symmetric sections. As the asymmetry of the cross-section becomes more pronounced, the degree of coupling between the out of plane bending and twist increases.
In the next example we consider once again the channel section (see Figure 4), but now in large deformation analysis. A concentrated load is applied at the center of gravity of the section, once in the y-direction, and once in the z-direction. One end of the beam is clamped and all the DOFs including warping are fixed at the clamped end. Figures 5 and 6 show the load-displacement response of the structure using warping beam and MITC4 shell models when the load is applied in the y-direction. An excellent agreement is achieved between the two solutions. The animation at the very top shows the deformation of the beam when the load is applied in the y-direction. The animation just below it shows the response when the load is applied in the z-direction. As can be seen, when the load is applied in the y-direction, initially the beam moves in the y-direction without much lateral movement and only little twisting, but beyond a critical value, the section starts to twist significantly and the lateral movement of the beam becomes pronounced.
However, when the load is applied in the z-direction, the beam only bends in the x-z plane without twisting.
This example highlights the importance of considering the effect of eccentricity of the shear center with respect to the center of gravity in the kinematic formulation of the beam. It also shows the importance of bending-torsion-warping coupling and the so-called Wagner effect in the response of thin-walled open section beams, when subjected to large twists.
For the last example, we study the response of the same channel section beam, but this time the beam is subjected to a torque at its free end, which is applied at its center of gravity (see Figure 7).
All displacements, rotations and the warping DOFs at the clamped end of the beam are restrained. At first glance, engineering intuition might lead us to believe that the section only twists about its centerline and the beam will not bend. However, as can be seen in the animation below, due to the coupling between bending and torsion for this mono-symmetric cross-section, the twisting will cause this beam to initially displace only in the y-direction, and for larger twists, also in the z-direction.
Figure 8 shows the y-displacement of the tip of the beam as a function of the applied torque using the warping beam and MITC4 shell models. An excellent agreement between the two sets of results is observed.
These examples illustrate that a good beam formulation can mimic many 3D shell-like responses of slender structures at a small fraction of computational cost, provided that the kinematic assumptions used in the formulation of the beam element are general enough to take into account all the aforementioned coupling phenomena. Compared to shell models, the use of the beam element in ADINA not only significantly reduces the computational cost, but also reduces the time required for model preparation and interpretation of results.