6.1 GOVERNING EQUATIONS 209 Figure 6.6:The y,z'coordinate system in which ly is zero. By referring to Eqs.(6.10)-(6.12),we express the bending stiffnesses in the yz coordinate system in the forms Elyy+Ele (6.14) 2 +武 Elyy +Elz (色） +武 (6.15) Elyz=0. (6.16) 6.1.3 Compliance Matrix With respect to the x-y-z coordinate system attached to the centroid,the strain- force relationships are defined as W 0 0 Wi47 111 0 W22 W23 W24 应， (6.17) 0 W23 W33 W34 M. Wi4 W24 W34 W44 The W2i and W31 terms are zero because an axial force applied at the centroid does not cause bending,whereas Wi2 and Wi3 are zero because the compliance matrix is symmetrical.The compliance matrix [W]is the inverse of the stiffness matrix Wu 0 0 Wi4 P P2 3 0 W22 W23 W24 P2 P2 P3 P24 (6.18) 0 W23 W W34 P3 P3 P33 P34 Wi4 W24 W34 W44 P4 B4 P4 P44 We obtain the compliance matrix of an orthotropic beam by substituting the elements of the matrix [P]given in Eq.(6.8)into this expression.The6.1 GOVERNING EQUATIONS 209 z y ϕ z ′ y ′ Figure 6.6: The y , z coordinate system in which Iyz is zero. By referring to Eqs. (6.10)–(6.12), we express the bending stiffnesses in the y –z coordinate system in the forms EI y y = EI yy + EI zz 2 + 7889 - EI yy − EI zz 2 .2 + EI 2 yz (6.14) EI z z = EI yy + EI zz 2 − 7889 - EI yy − EI zz 2 .2 + EI 2 yz (6.15) EI y z = 0. (6.16) 6.1.3 Compliance Matrix With respect to the x–y–z coordinate system attached to the centroid, the strain– force relationships are defined as    o x 1 ρy 1 ρz ϑ    =      W11 0 0 W14 0 W22 W23 W24 0 W23 W33 W34 W14 W24 W34 W44         N
M
y M
z T
   . (6.17) The W21 and W31 terms are zero because an axial force applied at the centroid does not cause bending, whereas W12 and W13 are zero because the compliance matrix is symmetrical. The compliance matrix [W] is the inverse of the stiffness matrix      W11 0 0 W14 0 W22 W23 W24 0 W23 W33 W34 W14 W24 W34 W44      =      P11 P12 P13 P14 P12 P22 P23 P24 P13 P23 P33 P34 P14 P24 P34 P44      −1 . (6.18) We obtain the compliance matrix of an orthotropic beam by substituting the elements of the matrix [P] given in Eq. (6.8) into this expression. The