Chapter 5 Approximate

Chapter 5 Approximate - Chapter 5 Approximate Methods of...

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Unformatted text preview: Chapter 5 Approximate Methods of Analysis of Plates 1. Validity of Classical Plate Theory As in the case of the classical beam theory, there are obvious discrepancies in the classical theory of plates that has been presented with regard to transforming a three-dimensional object into two- dimensional entity. Most notably, we have employed a plane stress assumption in which zz xz yz = = = . From equilibrium considerations of an isolated plate free-body, we expressed the vertical flexural shear stresses as functions of vertical bending shearing forces, and x y Q Q . It is, therefore useful to assure that our approach will yield meaningful results despite some obvious discrepancies. Recall an equilibrium equation of a 3D element given in terms of stresses x xy xx xz y z + + = (2-12a) where 3 3 , /12 /12 xy x xx xy M z M z h h = = Hence, 3 3 /12 /12 xy x xz M M z z h x h y z + + = 3 3 /12 /12 xy xz x x M M z z Q z h x y h = - + = - Now integrating with respect to z, gives ( 29 2 3 / 2 , /12 xz x z Q f x y h = - + Know xz = at / 2 z h = , which leads 1 2 3 / 8 /12 x h Q f h = - + 3 2 x f hQ = Therefore, 2 2 3 1 4 2 x xz Q z h h =- Similarly, 2 2 3 1 4 2 y yz Q z h h =- From Eqs. (2-8) and (2-12c), we have 2 2 2 2 3 3 1 4 1 4 2 2 xy y xz x zz Q Q z q z z x y h h x y h h = -- = -- + =- Integrating with respect to z, we have ( 29 3 2 3 4 , 2 3 zz q z z g x y h h =- + The boundary conditions for zz are (1) at / 2 and (2) 0 at / 2 zz zz q z h z h = - = - = = From condition (1), we have ( 29 3 2 3 4 , 2 2 24 q h h q g x y h h - =- + + ( 29 1 , 2 g x y q = - Hence, 3 3 1 3 2 2 2 zz z z q h h = -- + 2 We thus obtained a set of stresses , , and xz yz zz from rigorous equilibrium considerations using results from a simplified theory that neglected these very stresses. If it turns out that the above stresses which are assumed to be zero in the simplified theory are indeed small, we have inner consistency of our theory. In order to accomplish this, we shall make an order-of- magnitude study (adapted from Shames and Dym, Energy and Finite Element Methods in Structural Mechanics, Hemisphere, 1985) of all six stresses for comparison. From Fig. 5-1 showing inner portion of the plate, it becomes obvious Fig. 5-1 which implies that the order of magnitude of the total force is equal to the order of magnitude of the average applied load intensity, q , times the order of magnitude of the area of the inner portion. If we denote an average vertical shearing force around the perimeter of the inner portion, Q v , we may write the following order-of-magnitude equation: ( 29...
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Chapter 5 Approximate - Chapter 5 Approximate Methods of...

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