A qualified plate theory for rigid rotation in postcritical nonlinear analysis

Abstract

A new, simple method is presented for deriving the instability potential of thin plates, based on the equilibrium conditions and constitutive law for incremental forces considering the geometric nonlinear effect. First, the equations of equilibrium are established for the plate at the last calculated C-1 and current deformed C-2 states, from which the equilibrium equations for the incremental forces at C-2 are derived. Then, the incremental forces are derived as the summation of two parts: (1) the part from the increase of the Cauchy stresses at C-1 to the 2nd Piola-Kirchhoff stresses at C-2, which are related to the strain increments and derivable from the linear constitutive; and (2) the part by the difference between the 2nd Piola-Kirchhoff and 1st Piola-Kirchhoff stresses, which are related to the cross-sectional forces at C-1 and rigid displacements. Based on the previous concept, the rigid body rule and coordinate indifference of tensor expressions, the constitutive law for incremental forces including the geometric nonlinear effect is derived. Further, by the variational principle, the following instability potentials are derived: delta U-GB caused by the balance conditions for initial forces, delta U-GC caused by the constitutive law for incremental forces, and delta U-GR caused by the rotation of boundary moments. Consequently, the total instability potential is presented for the plate in the virtual work form. The present approach is self-explanatory, which requires only simple integration operations. Moreover, the two essential conditions are satisfied: equilibrium conditions for incremental forces and rigid body rule. Numerical examples are presented to demonstrate the rationality of the present plate theory compared with the conventional one.