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 LLC
with: 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
stress resultant Ns =dx:from [17.2]et [17.6] b/2 u6+A+d+斯 stress resultant N=dz: Ny A2Eox+A22E0y stress resultant T=dz: =+(-++梁 Txy =A3sYox In summary,one finds again the relations already established in Chapter 12 (Equations 12.5)as: Nx A11 A12 0 Eox Ny A21 A22 0 0 0 Yoxy or,in inverse form,by using the notations in Equation 12.9: Eox N 1/Eg -Vjax/Ey N 1 Eoy b[A] N -Vxy/Ex 1/E, 0 N (17.7) T 0 0 1/Go 17.4.2 Bending Behavior One has again the already known moment resultants (see Section 12.2.1). Moment resultant M,=dz: with[17.2and[17.5: any dz +(-+ 6/2 dy dy a+C2×- My=Cu dx + 历+axJb2 + E2nyzdz The simplifications are due to the antisymmetry of the integrated functions (midplane symmetry). 2003 by CRC Press LLC
stress resultant 6 : stress resultant stress resultant In summary, one finds again the relations already established in Chapter 12 (Equations 12.5) as: or, in inverse form, by using the notations in Equation 12.9: (17.7) 17.4.2 Bending Behavior One has again the already known moment resultants (see Section 12.2.1). Moment resultant with [17.2] and [17.5]: 6 The simplifications are due to the antisymmetry of the integrated functions (midplane symmetry). Nx Ú–h/2 h/2 = sxdx: from 17.2 [ ] et 17.6 [ ] Nx E11 e 0x z ∂qy ∂x -------- ∂hx ∂x + + -------- Ë ¯ Ê ˆ z E 12 e 0y – z ∂qx ∂y -------- ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú d + –h/2 h/2 Ú = Nx A11e 0x A12e 0y ∂ ∂x ------ E11hx dz ∂ ∂y ----- E12hy dz –h/2 h/2 Ú + –h/2 h/2 Ú = + + Ny Ú–h/2 h/2 = sy dz: Ny = A21e 0x + A22e 0y Txy Ú–h/2 h/2 = txy dz: Txy E33 g oxy z ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂hx ∂y -------- ∂hy ∂x + + + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = Txy = A33g oxy Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ A11 A12 0 A21 A22 0 0 0 A33 e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ h A[ ]–1 1 h ¥ -- Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ 1 h -- 1/Ex nyx – /Ey 0 nxy – /Ex 1/Ey 0 0 01/Gxy Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = = My Ú–h/2 h/2 = sx z zd : My E11 ze ox z2 ∂qy ∂x -------- z ∂hx ∂x + + -------- Ë ¯ Ê ˆ dz… –h/2 h/2 Ú = º E12 ze oy z2 – ∂qx ∂y -------- z ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú + My C11 ∂qy ∂x -------- C12 ∂qx ∂y – -------- ∂ ∂x ¥ + ------ E11hx z z ∂ ∂y ----- E12hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + TX846_Frame_C17 Page 323 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
In the last two terms there appear the nonzero integrals of even functions.If one neglects the contribution of the rates of variation along the x and y direction,respectively,of these terms,the previous equation is reduced to' a0+C× My =Cu ax d0. dy Moment resultantM b/2 Oz dz b/2 dz.. 6M2 -b12 which is reduced to: -Ms=Gi2dx and in neglecting the contribution of the last two terms': a8+C2×- -Ms Cdx aθ dy 。Moment resutantM=-」2g2tk b/2 which is reduced to: Essnyzdz and in neglecting the contribution of the variations of the differences nx and ny: -M=C- The existence of such approximation does not appear if one neglects apriori the increments nx nn:in Equation 17.3. 2003 by CRC Press LLC
In the last two terms there appear the nonzero integrals of even functions. If one neglects the contribution of the rates of variation along the x and y direction, respectively, of these terms, the previous equation is reduced to7 Moment resultant : which is reduced to: and in neglecting the contribution of the last two terms7 : Moment resultant which is reduced to: and in neglecting the contribution of the variations of the differences hx and hy 7 : 7 The existence of such approximation does not appear if one neglects a priori the increments hx, hy, hz in Equation 17.3. My C11 ∂qy ∂x -------- C12 ∂qx ∂y = + ¥ –-------- Mx sy z zd –h/2 h/2 Ú = – –Mx E12 ze ox z2 ∂qy ∂x -------- z ∂hx ∂x + + -------- Ë ¯ Ê ˆ dzº –h/2 h/2 Ú = º E22 ze oy z2 – ∂qx ∂y -------- z ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú + –Mx C12 ∂qy ∂x -------- C22 ∂qx ∂y ¥ –-------- ∂ ∂x ------ E12hx z z ∂ ∂y ----- E22hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + + –Mx C12 ∂qy ∂x -------- C22 ∂qx ∂y = + ¥ – -------- Mxy txy z zd –h/2 h/2 Ú = – –Mxy E33 zg oxy z2 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ z ∂hx ∂y -------- z ∂hy ∂x + + + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = –Mxy C33 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂ ∂y ----- E33hx z z ∂ ∂x ------ E33hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + –Mxy C33 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ = TX846_Frame_C17 Page 324 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC