Chapter4 - Chapter 4 Plates and Shells 1. Characteristic...

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Chapter 4 1 Chapter 4 Plates and Shells 1. Characteristic Actions Examples of applications: • Slender beam, frame elements • Slabs, plates subject to twisting or transverse loading • Shells In general, these systems may be subject to both membrane stresses and bending stresses . The two types of actions, in beams and plates, act independently of each other unless when the deflections become large. In shells they occur simultaneously. Membrane action in a flat element may be modeled by the plane-stress elements of previous chapters. Bending action is the primary focus of this chapter. 1.1 Common Characteristics: • The plate or shell thickness is much smaller than other dimensions (Fig.1.1), and thus, a single bending element may span across the thickness. The element's nodes lie in the midplane. This model is efficient but possible only if bending action is built into the element. x z y x z (b) Plate element (a) Beam element t w (c) Cylindrical shell element Figure 1.1 Types of elements
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Chapter 4 2 • Applied loading is usually perpendicular to the plane of the element. • The basic assumptions: i. All points across the thickness have the same transverse deflection. ii. (a) Kirchhoff theory: a plane section perpendicular to the midplane remains plane and perpendicular to the midplane after deformations. This is the classical thin plate theory , which neglects transverse shear deformations. Most existing analytical solutions for plate bending fall into this category. (b) Mindlin-Reissner theory (Fig.1.2): a plane section (perpendicular to the midplane) remains plane but not necessarily perpendicular to the midplane. This theory takes into account transverse shear deformations, and hence is suitable for the analysis of thick plates. . a b a ' b' w ( x ) x z Original position (a) Deformations at a section u o z - z u o (x,y) (b) Displacements w , x a ' b ' Mid-plane θ x θ x w ,x θ x w , x Figure 1.2 Bending deformations - Mindlin theory Thus, the displacements of a point ( x , y , z ) any where in a flat plate can be expressed as: w ( x , y , z ) = w ( x , y ) u ( x , y , z ) = u o ( x , y ) z θ x ( x , y ) v ( x , y , z ) = v o ( x , y ) z θ y ( x , y ) (1.1) where w = transverse deflection (perpendicular to element surface) u o , v o = in-plane displacements at the point ( x , y ) of the midplane ( z = 0). These displacements are zero if the plate is not subject to in-plane forces, and if transverse loading is not too great to cause stretching due to arching action. θ x = rotation of the x -section at the point ( x , y ). In the case of Kirchhoff theory where the section a ' b ' remains perpendicular to the mid plane
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Chapter 4 3 θ x ( x , y ) = w ( x , y ) x , θ y ( x , y ) = w ( x , y ) y (1.2) The assumptions leading to Eq.1.1 reduce a 3-D problem to 2-D: the deflection function and section rotations are functions of x , y not z . 1.2 Strain-Displacement Relations By assumption (i) the normal stress in z-direction is zero: ε z = 0.
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This note was uploaded on 02/08/2010 for the course MECHANICAL 6371 taught by Professor Ha during the Winter '10 term at Concordia Canada.

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Chapter4 - Chapter 4 Plates and Shells 1. Characteristic...

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