Chapter 3 Plates Subjected to nplane Forces3章板在平面力.docVIP

Chapter 3 Plates Subjected to In-plane Forces 1. Derivation of Airy’s Stress Function, 2-D From equations of equilibrium, we have (by crossing out the third row and column) (3-1) From the 2-D compatibility equation in terms of stress, we have (3-2) Assume arbitrary functions R and S, such that (3-3) Since (3-4) Assume (3-5) From Eqs. (3-4) and (3-5), it follows (3-6) From Eqs. (3-3), (3-4), and (3-5), it follows (3-7) where is called Airy’s stress function after George Biddell Airy (1801-1892). Substituting Eq. (3-7) into Eq. (3-2), yields (3-8) Eq. (3-8) is called a biharmonic equation, sometimes written as . Boundary Forces and Airy’s Stress Function where =boundary traction (externally applied surface load)/unit area (unit width, see Timoshenko Theory of Elasticity, 2nd ed., page 13). Consider equilibrium of the infinitesimal triangular prism. (3-9) Dividing Eq. (3-9) by , gives (3-10) In traversing from 0 to s (counter clockwise positive for the right hand coordinates system), x decreases in the first and second quadrants and dx will be negative. Substituting Eq. (3-7) into Eq. (3-10), yields (3-11) The x and y and the perimeter coordinate s are related as (3-12) Recalling the following chain-rule expansion: Similarly, Therefore (see Timoshenko, Theory of Elasticity, 2nd. ed., page 190), (3-13) Let h be the thickness of the plate, then (remember ). Then Likewise, Assume the integral constant to be zero as it does not affect the stresses whether it is a constant or zero. Then (3-14) Likewise, (3-15) Recall the following chain-rule expansion plus integration by parts: Hence, Recall (3-16) Eq. (3-16) is used to determine the Airy’s stress function once the surface traction is known. It is in general very tedious and time consuming operation. Frame Analogy In order to facilitate the application of Eq. (3-16), a handy analogy called, Frame Analogy, is introduced. Recall Eqs. (3-14) and (3-15).  where Remember are surface loads (tractions) p