Modified Kirchhoff's theory of plates including transverse shear deformations

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Abstract

A primary flexure problem defined by Kirchhoff theory of plates in bending is considered. Significance of auxiliary function introduced earlier in the in-plane displacements in resolving Poisson–Kirchhoff's boundary conditions paradox is reexamined with reference to reported sixth order shear deformation theories, in particular, Reissner's theory and Hencky's theory. Sixth order modified Kirchhoff's theory is extended here to include shear deformations in the analysis.

Introduction

In Kirchhoff's theory of plates in bending, gradients of an auxiliary harmonic function are earlier introduced in the in-plane displacements (Vijayakumar, 2009) through which a proper resolution of Poisson–Kirchhoff boundary conditions paradox is presented. These gradients of auxiliary function correspond to self-equilibrating transverse shear stresses in the interior due to specified zero shear stresses along top and bottom faces of the plate. This modified (sixth order) theory (which reduces to Kirchhoff's theory if in-plane tangential displacement is zero all along the edge of the plate) is extended here to include transverse shear deformation effects on vertical deflection. They are governed by a second order system giving correction to vertical deflection of mid-plane of the plate.

Section snippets

Primary bending problem

For simplicity in presentation, a rectilinear domain 0 ≤ x ≤ a, 0 ≤ y ≤ b, −h ≤ z ≤ h with reference to Cartesian coordinate system (x, y, z) is considered. Thickness 2h of the plate is small compared to its lateral dimensions a and b. Material of the plate is homogeneous and isotropic with elastic constants E (Young's modulus), ν (Poisson's ratio) and G (modulus of rigidity) that are related to one other by E = 2(1 + ν)G.

Face and edge conditions are specified such that in-plane displacements u,v and bending

Analysis

In-plane displacements u1 and v1 in the assumed displacements[w,u,v]=[w0(x,y),zu1,zv1]are treated as unknown functions so that zero shear conditions[u1+w0,x,v1+w0,y]=[0,0]in Eq. (1) are expressed in the formw0=−∫(u1dx+v1dy)u1,y−v1,x=0in which suffix after “,” denotes partial derivative operator.

In the earlier work (Vijayakumar, 2009), u1 = −w0,x and v1 = −w0,y in Kirchhoff's theory are modified by addition of gradients of an auxiliary harmonic function φ(x,y) in the form[u1,v1]=−[(w0,x−φ,y),(w0,y+φ,

Example

Consider a simply supported square plate of side length ‘a’ subjected to sinusoidal vertical load q = q0 sin(πx/a)sin(πy/a). Here, φ is identically zero and ψ is vertical deflection from Kirchhoff's theory. In the case of a square plate, wmax is at mid-point (1/2, 1/2, 0) in neutral plane due to coupling with torsion. With h/a = 1/6 and ν = 0.3, (E/2hq0)wmax(1/2, 1/2, 0) from Kirchhoff “s theory is 2.27 and correction due to coupling with torsion is about 1.45 from the present theory and 1.42 from

Concluding remarks

In higher order theories to approach exact 3-D solution of primary flexure problem, zero shear conditions in Eq. (2) are satisfied through z-distributions like in Eq. (13). They are to nullify residual errors between the present eighth order 2-D problem and 3-D problem. They do not effect top and bottom face values of vertical deflection ψ from present analysis. If φ is zero, ψ from Kirchhoff's theory represents exact values of top and bottom face deflections. ψ* represents a correction to

Acknowledgement

The author expresses thanks to his granddaughter M. Tanya, University of Southern California, Los Angeles, USA for her help in typing and online submission of this manuscript.

References (6)

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