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Unformatted text preview: Chapter 4 Plates Subjected to Transverse Loads 1. Equations of Plates g Assumptions (1) At the boundary, the plates are assumed to move freely in the plane of the plate; thus the reactive forces at the edges are normal to the plate. (2) The deflections are assumed to be small in comparison with the thickness of the plate. (3) The normal to the middle plane is assumed to remain straight and perpendicular to the middle surface as plate is deformed (justified if the plate is thin in comparison with lateral dimensions of the plate). Thus, the shear deformation is not considered. (4) Strains in the middle plane is assumed negligible. (5) The stress normal to the middle plane, z σ , is assumed equal to zero everywhere. Thus, the thin plate is assumed to be a planestress problem. Fig. 41 Position of middle plane Let , , and u v w are the components of displacement of a point at a distance z from the middle plane. Then, these displacement components are expressed as w u z x ∂ =  ∂ (41a) w v z y ∂ =  ∂ (41b) y z x z /2 t /2 t middle plane in the  plane x y plate x 1 ( 29 , w w x y = (41c) Fig. 42 Displacement of a point It is obvious from Fig. 42 that a microgeometry relationship is adopted due to the assumption of the small displacement theory in establishing Eq. (41a). From Eqs. (41a) and (41b), the straincurvature relationships can be readily derived. 2 2 x u w w z z x x x x ε ∂ ∂ ∂ ∂ = = =  ∂ ∂ ∂ ∂ Likewise 2 2 y v w w z z y y y y ε ∂ ∂ ∂ ∂ = = =  ∂ ∂ ∂ ∂ 2 2 2 2 xy v u w w w z z z x y x y x y x y γ ∂ ∂ ∂ ∂ ∂ = + =  =  ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ z ε ≡ g Strainstress or Stressstrain Relationship ( 29 ( 29 1 1 x x y y y x E E ε σ μσ ε σ μσ = = , y v , z w , x u z /2 t /2 t x g g z z u u w x ∂ ∂ w x ∂ ∂ 2 ( 29 ( 29 2 2 1 1 x x y y y x E E σ ε με μ σ ε με μ = + = + ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 x y xy xy xy E w w z x y E w w z y x E E w G z x y σ μ μ σ μ μ τ γ γ μ μ ∂ ∂ =  +  ∂ ∂ ∂ ∂ =  +  ∂ ∂ ∂ = = =  + + ∂ ∂ (42) g In the linear elasticity, stresses are linearly proportional to the applied load. Fig. 43 Stresses in plate element When a stress is written with two subscripts such as αβ τ , the first subscript α indicates the direction of the normal of the plane which the stress acts while the second subscript β represents the direction of the stress. Due to Maxwell’s reciprocal theorem, the magnitude of αβ τ is taken equal to βα τ ; the direction of the stress is 90 degrees apart. , : bending (transverse) shear stresses xz yz τ τ z y x yx τ yz τ yy σ xy τ xz τ xx σ , : bending stresses xx yy σ σ , : inplane (twisting) shear stresses (parallel to middle plane) xy yx τ τ 3 g Stress Resultants Fig. 44 Resisting moment and shear The subscript convention for moment is similar to that applied to stresses. The subscript convention for moment is similar to that applied to stresses....
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 Summer '10
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 Sin, Trigraph, Hyperbolic function, nπ mπ

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