Plane StressYI: Consider the domain bounded two stress-freeplanes z=h, where h is small in comparison2hwith other dimensions in the problem.? Since the region is thin in the z-direction,there can be little variation in the stressRcomponents O., Tzx, Ta, through the thickness,7xand thus they will be approximately zerothroughout the entire domaina=a(x,y)? Finally since the region is thin in the z-x.y)a=adirection it can be argued that the other non-(x,y)tTxyzero stresses will have little variation with z0oTT-3: Under these assumptions, the stress field canbe simplified as11
Plane Stress • Consider the domain bounded two stress-free planes z=h, where h is small in comparison with other dimensions in the problem. • Since the region is thin in the z-direction, there can be little variation in the stress components σz , τzx, τzy through the thickness, and thus they will be approximately zero throughout the entire domain. • Finally since the region is thin in the zdirection it can be argued that the other nonzero stresses will have little variation with z. • Under these assumptions, the stress field can be simplified as ( , ) ( , ) ( , ) 0 x x y y x y x y z z x z y x y x y x y 11
Plane Stress Field EquationsIsotropic Hooke's Law: . = ass2EE: Displacement-strain relation:9avauavawauaxaya22ayax12
Plane Stress Field Equations 1 , , , , 0 , 0 2 x y z x y y z x z u v w u v x y z y x 1 1 , , 1 1 , 0 2 , 2 , 2 , 0 x x y y y x z x y x y x y x y z x z y x x y z x y x y z y x y x y z z x z y E E E E G G G 12 • Displacement-strain relation: 1 2 ; . i j k k i j i j i j i j k k i j G E E • Isotropic Hooke’s Law: , , 1 2 i j i j j i u u