At present, the research on the stability of the combined compression rod in academia mostly focuses on the static force, and the research on dynamic stability is relatively rare. The combined pressure rod can be divided into strip type and strip type . We use the energy method and the Hamilton principle to establish the Mathieu-type parametric vibration equations of the strip and plate composite compression bars under cyclic longitudinal loads. We solve for the dynamically unstable region surrounded by its periodic solution. At the same time, the paper discusses the dynamic stability of the parametric resonance of the two combined compression bars.
The combined compression rod comprises a split limb bearing axial load and a splicing piece connecting the split limb . The shear deformation is much larger than that of a solid web column. Additional deflection effects due to shear forces cannot be ignored. Therefore, we must consider the effect of shear deformation in the stability study of the combined compression rod. The cross-section of the composite strut after the structure is bent is not perpendicular to the deflection curve
The sheer force is:
The strip-type composite compression rod (Figure 1) can be calculated as a truss. The connection nodes between the column legs and the strips are hinge nodes. The rods in the truss cause only additional axial forces when the stability is lost. The axial force
In the formula,
Then the lateral shear deformation caused by unit shear force is:
The load potential energy is:
Substitute into equation (3) to get the expression of the total potential energy. The kinetic energy is:
Assume that the lateral rod displacement
Equation (10) can be further simplified as:
Equation (12) is the Mathieu-type parametric vibration equation of the strip-type composite compression bar under the action of longitudinal periodic loads . The elastic critical load of the strip-type composite compression rod considering the shear deformation is as follows:
The article computes unstable regions surrounded by periodic solutions. Assuming that equation (12) has periodicity, the periodic solution of 4
We substitute into equation (12) and take the second-order determinant of the critical frequency equation . The critical frequency approximation formula can be obtained by solving in this way:
A rigid frame can be taken as the plate-type combined pressing rod (Fig. 2). Its deformation state can be decomposed into two parts: the overall deformation as a rod and the local bending deformation between the nodes as a rigid frame . The latter can be seen as additional bending moments due to internode shear forces.
Under the action of unit shear force, the linear stiffness of the siding is generally greater than that of the column leg . We can use graph multiplication to obtain the lateral shear deformation:
The load potential energy is:
The kinetic energy is:
According to Hamilton's principle, we can obtain the dynamic differential equation of the combined pressure rod:
We discretize the equation using the Galerkin method. We assume that the lateral rod displacement
Equation (19) can be further simplified as:
In this way, we can determine the dynamic instability area in the same way as the tie-bar combination strut.
We make the following assumptions for the hinge support at both ends of the double-limb strip (plate) type combined pressure rod:
We give the theoretical value of the elastic critical load of the combined compression bar without considering the shear deformation as a comparison:
When the width of the compression bar is large, the elastic critical load without considering the shear deformation is much larger than the finite element calculation result . This shows that the closer the length and width of the beam or column, the greater the effect of shear deformation (Figure 3). The elastic critical load value of the composite compression rod considering shear deformation in this paper is in good agreement with the finite element calculation results.
At this time, the dynamic instability area expands rapidly. This shows that the greater the slenderness ratio, the greater the possibility of parametric resonance . Therefore, the slenderness ratio is an important factor in determining the dynamic instability region of the two combined struts (Fig. 4). With the increase in the area of the slatted combination strut, the stable critical bearing capacity will also increase. At this time, the frequency of the parametric resonance of the pressure rod increases, and the dynamic instability region narrows accordingly (Fig. 5). This shows that the larger the area of the slats, the less likely the rod is to suffer from instability failure caused by parametric excitation vibration. But the strip area has little effect on the dynamic instability region. With the increase of the rigidity of the affixed plate combined pressure rod, the stable critical bearing capacity of the pressure rod will increase . At this time, the frequency of parametric resonance of the structure also increases, and the dynamic instability region also narrows (Fig. 6). This shows that the greater the stiffness of the slat line, the less likely the rod is to suffer from instability failure caused by parametric excitation vibration. However, the effect of panel line stiffness on the dynamic instability region is not obvious.
The critical load of the composite compression bar is always smaller than that of the solid web column with the same moment of inertia, so the influence of shear deformation must be considered in the stability study of the composite compression bar. With the increase of the slenderness ratio of the combined strut, the frequency of the parametric resonance of the structure decreases, and the dynamic instability region expands rapidly. Therefore, the slenderness ratio is an important factor in determining the dynamic instability region of the two lattice columns. With the increase of the strip area, the frequency of parametric resonance of the strip-type combined compression bar increases. At this time, the structure is less likely to be unstable. However, it has little effect on the dynamic instability region. Similarly, with the increase of the stiffness of the slatted line, the stable critical bearing capacity of the slatted composite compression bar increases, and the frequency of the parametric resonance of the structure also increases. At this time, the possibility of parametric vibration of the rod becomes smaller, but its influence on the dynamic instability region is not obvious.
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