06_PlateTheory_07_InPlaneForces

# 06_PlateTheory_07_InPlaneForces - Section 6.7 6.7 In-Plane...

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Section 6.7 Solid Mechanics Part II Kelly 169 6.7 In-Plane Forces and Plate Buckling In the previous sections, only bending and twisting moments and out-of-plane shear forces were considered. In this section, in-plane forces are considered also. The in-plane forces will give rise to in-plane membrane strains, but here it is assumed that these are uncoupled from the bending strains. In other words, the membrane strains can be found from a separate plane stress analysis of the mid-surface and the bending of the plate does not affect these membrane strains. The possible effect of the in-plane forces on the bending strains is the main concern here. 6.7.1 Equilibrium for In-plane Forces Start again with the equations of equilibrium, Eqns. 6.4.6. Integrating the first and second through the thickness of the plate (this time without multiplying first by z ), and using the definitions of the in-plane forces 6.1.1-6.1.2, leads to 0 0 = + = + y N x N y N x N y xy xy x (6.7.1) 6.7.2 The Governing Differential Equation Consider an element of the deflected plate, Fig. 6.7.1. Only a deflection in the y direction, y / ω , is considered for clarity. Resolving the components of the in-plane forces into horizontal and vertical components: y x y w N x y w N y y w N x x y w N y y w N x y w N F y x x N N y N x y y N N x N F xy xy xy y y y V xy xy xy y y y H Δ Δ +

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06_PlateTheory_07_InPlaneForces - Section 6.7 6.7 In-Plane...

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