where one has reused the notations of Section 12.1.6 for the terms _.4 The longitudinal displacement u(x,y,z)then takes the form: u(x,y,z)=uo(x,y)+20,(x,y)+nx(x,y,2) with: In effect,one can obtain starting from this expression: the integrals disappearing due to antisymmetry in 2: (y型）nzd …+,+面面2 In the second member,the first integral disappears due to midplane symmetry.In addition,taking into account the definition of 0,written above,the second integral is also nil. Translation along the y direction:This is vo(,y)such that: 】bM2 。=J(x,八2) -b12 Rotation about the x axis:This is e such that: w(x,y,z)×zdz The longitudinal displacement u(,y,z)then takes the form: v(x,y)=vo(x,y)-z0(x,y)+n(x,y,z) Recall that (Section 12.1.6): ply k=l ply 5 The coefficient of 0,is 1 because one can note that: 原+)+ 品面：Gnca-GGca-G1 2003 by CRC Press LLCwhere one has reused the notations of Section 12.1.6 for the terms .4 The longitudinal displacement u(x, y, z) then takes the form: with: In effect, one can obtain starting from this expression: the integrals disappearing due to antisymmetry in z 5 : In the second member, the first integral disappears due to midplane symmetry. In addition, taking into account the definition of qy written above, the second integral is also nil.
Translation along the y direction: This is v0(x,y) such that:
Rotation about the x axis: This is qx such that: The longitudinal displacement v(x,y,z) then takes the form: 4 Recall that (Section 12.1.6): 5 The coefficient of qy is 1 because one can note that: 1 EIij ----- 1 EI ----- [ ] C –1 , where Cij E ij k zk 3 zk-1 3 – 3 -------------------- Ë ¯ Ê ˆ k=I st ply nth ply = = Â u x( ) , y, z = u0( ) x, y + zqy( ) x, y + hx( ) x, y, z E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú = 0 u zd –h/2 h/2 Ú h u¥ o + qy z z hx dz –h/2 h/2 Ú d + –h/2 h/2 Ú = E11 EI11 -------- E12 EI12 + -------- Ë ¯ Ê ˆ z2 dz –h/2 h/2 Ú C11 EI11 -------- C12 EI12 + -------- C11C22 C11C22 C12 2 – ---------------------------- C12 2 C11C22 C12 2 – == = – ---------------------------- 1 E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ uz zd –h/2 h/2 Ú uo E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ z zd º –h/2 h/2 Ú = … qy E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú + + v0 1 h -- v x( ) , y, z dz –h/2 h/2 Ú = qx E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ v x( ) , y, z ¥ z dz –h/2 h/2 Ú = – v x( ) , y = v0( ) x, y – zqx( ) x, y + hy( ) x, y, z TX846_Frame_C17 Page 321 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC