with: (b/2) E2+ ny z dz =0 -b/2 Translation along the z direction:This is wo(x,y)such that: 1(b/2) o(x,y）= w(x,y,z)dz The vertical displacement takes the form: w(x,y,z)=wo(x,y)+n2(x,y,z) In summary,one obtains for the elastic displacement field: w=lo+z6,+刀x(x,y,z) v=vo-zex +n(x,y,z) (17.3) w Wo nz(x,y,z) nx,ny,n antisymmetric in z. (17.4) E+ (17.5) 17.3 STRAINS One deduces from the previous displacements the strains: dey onx e=ex+派+0派 + ∂8x,any 6=0y-z+历 + ∂， a0x Y=g+苏- + (17.6) d +8，+梁+架 dx wo-0x+ ny+ z Yy= dz dy 17.4 CONSTITUTIVE RELATIONS 17.4.1 Membrane Equations Recall the method that was already used in Section 12.1.1. 2003 by CRC Press LLCwith:
Translation along the z direction: This is w0(x,y) such that: The vertical displacement takes the form: In summary, one obtains for the elastic displacement field: 17.3 STRAINS One deduces from the previous displacements the strains: (17.6) 17.4 CONSTITUTIVE RELATIONS 17.4.1 Membrane Equations Recall the method that was already used in Section 12.1.1. E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hy z zd –h/2 ( ) h/2 Ú = 0 w0( ) x, y 1 h -- w x( ) , y, z dz –h/2 ( ) h/2 Ú = w x( ) , y, z = w0( ) x, y + hz( ) x, y, z u u = 0 + + zqy hx( ) x, y, z v v = 0 – zqx + hy( ) x, y, z (17.3) w w = 0 + hz( ) x, y, z hx, hy, hz antisymmetric in z. (17.4) E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hy z zd –h/2 h/2 Ú = = 0 (17.5) ex e 0x z ∂qy ∂x -------- ∂hx ∂x = + + -------- e y e 0y – z ∂qx ∂y -------- ∂hy ∂y = + -------- g xy g 0xy z ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂hx ∂y -------- ∂hy ∂x = + + + -------- g xz ∂w0 ∂x --------- qy ∂hx ∂z -------- ∂hz ∂x = ++ + -------- g yz ∂w0 ∂y --------- – qx ∂hy ∂z -------- ∂hz ∂y = + + -------- TX846_Frame_C17 Page 322 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC