Greeting Researchers
It is well known that within the linear range, the in-plane and transverse motion of a plate are independent of each other i.e. the equations governing the in-plane and transverse motions are uncoupled. The stiffening/softening effect of the in-plane loads on the transverse vibrations of the plates is then accounted for by considering the work done by the in-plane loads during the transverse motion. In FEM, the work done by the in-plane loads is used to obtain the geometric stiffness matrix. The FEM equations can be given as
Ma + (K + Kg)u = f ... (1)
where u is the vector of nodal displacements, a is the vector of nodal accelerations, K is the stiffness matrix and Kg is the geometric stiffness matrix.
When the range of motion is no longer linear, the equations of motion for the in-plane and transverse motion are inherently coupled. In this case, incorporating geometric nonlinearity in the Von Karman sense, the FEM equations may be written as
Ma + (K + Knl)u = f ... (2)
where Knl is the nonlinear stiffness matrix.
Now my question is whether it is necessary to include Kg in eq. (2) i.e. whether the correct dynamic equation of motion with the incorporation of nonlinearity is as shown below in eq. (3)
Ma + (K + Kg + Knl)u = f ... (3)
My opinion is that since in the nonlinear case, the equations of motion are inherently coupled and thus, there is no need for the inclusion of matrix Kg as done in eq. (3). The coupling is incorporated through the matrix Knl and eq. (2) is the correct dynamic equation of motion. In the linear case, since the equations of motion are uncoupled, it is necessary to add the matrix Kg to incorporate the effect of the in-plane loads on the dynamics of the transverse motion.
I would like to have your valuable opinions on the same.
Thank you for your time.
Best Regards,
Jatin
The nonlinear vibration of plates under in-plane loads is a complex and challenging problem that arises in the study of structural dynamics and mechanics. When a plate is subjected to in-plane loads (loads applied within the plane of the plate), its deformation and vibration behavior can become nonlinear due to large displacements, material nonlinearity, geometric effects, and other factors. Here's an overview of some key aspects involved:
Given the complexity of this problem, research in nonlinear vibrations of plates often involves advanced mathematical techniques, computational methods, and expertise in structural mechanics. The specific approach you take will depend on the level of complexity and accuracy required for your analysis.
The previous answerer is correct, there are risks with a simple yes/no answer to your question. But, for your specific question concerning buckling and vibration with geometric non-linear effects but with linear elastic material properties, I think such an answer is possible. I would say equation (2) is correct and equation (3) would incorrectly double the non-linear effect.
Thank you, Aziz Khan and Stephan D A Hannot for your valuable suggestions.
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