Free vibration analysis of initially deflected stiffened plates for various boundary conditions

Abstract

Free vibration analysis of initially deflected stiffened plates subjected to uniformly distributed loading with different flexural boundary conditions involving simply supported and clamped ends and zero displacement in-plane boundary conditions has been presented. A domain decomposition technique depending on the number, orientation and location of the stiffeners is employed to ensure sufficient number of computation points around the stiffeners. Geometric nonlinearity arising out of large deflection is accounted for by consideration of non-linear strain-displacement relations. Mathematical formulation is based on a variational form of energy principle, and a solution technique, where static analysis serves as the basis for the subsequent dynamic study, is followed. The results are validated with the published results of other researchers. The dynamic behavior has been presented in the form of backbone curves in a dimensionless frequency-amplitude plane. Vibration mode shapes along with contour plots are provided in a few cases.

Effect of in-plane boundary conditions on forced vibration analysis of stiffened plates with a free edge

“…However it is found that in the majority of the dynamic studies, simply supported or clamped ends are considered as the flexural boundary conditions and only in few cases a combination of simply supported and clamped boundaries are taken into consideration. Mitra et al (2010) presented a free vibration analysis at a statically deflected position of eccentrically stiffened plates having simply supported, clamped and possible combinations of the two boundary conditions Few works involving stiffened plates with movable in-plane edges are available in the literature. Prathap and Varadan (1978) dealt with large amplitude, free flexural vibrations of clamped and simply supported thin elastic orthotropic stiffened plates with movable and immovable in-plane edge conditions.…”

“…However, the effect of torsion has been neglected in calculating the total strain. Detailed discussion on derivation of stiffener strain energy is provided in a previous work by Mitra et al (2010). So, only the final expression for the total stiffener energy is furnished in this paper.…”

“…Q p y ,Q q x are the first moment of area about the plate mid-plane and A p y , A q x are the cross-sectional areas of the p-th x-and q-th y-direction stiffeners, respectively. Strain energy of the plate (U p ) consists of two parts: strain energy due to pure bending and strain energy due to stretching of its mid-plane and the expressions for both these components are well known (Mitra et al, 2010). Expression for the plate strain energy (U p ) is provided below.…”

“…The kinetic energy (T) of the system is summation of plate kinetic energy (T p ) and stiffener kinetic energy (T s ), i.e., T ¼ T p þ T s , where, T s ¼ P ns x p¼1 T p sx þ P ns y q¼1 T q sy , where, T p sx , T q sy are kinetic energies of p-th stiffener along x-direction and q-th stiffener along y-direction, respectively. Substituting the expressions for the kinetic energy of plate and individual stiffeners (Mitra et al, 2010) the final expression of total kinetic energy is obtained as:…”

“…The backbone curve of the system is determined through a free vibration analysis at the deflected configuration of the system (Mitra et al, 2010). The present free vibration analysis of stiffened plates is carried out in two steps.…”