7.1 GOVERNING EQUATIONS 319 The bending moments and bimoment for a (symmetrical)l-beam are (see Eqs.7.12and7.23) M. dXy 0 0 d M 0 x: (7.31) 0 El d dx where the equation forM.is analogous to the equation for My.For an arbitrary cross-section orthotropic beam,the 12 and 21 elements are not zero.Hence,we write M. dxy 0 El 0 d (7.32) dx 0 0 dx By utilizing the definitions given in Eq.(7.6),these equations may be written as M. 1 Elyy 0 (7.33) P 0 The relationship between Tsv and the rate of twist is(Eq.7.15) Tsv =GIt. (7.34) The shear forces and the restrained-warping-induced torque for a(doubly symmetrical)l-beam is (see Egs.7.13 and 7.28) 0 (7.35) 0 0 "s where the first equality is analogous to the second one.Shear is introduced by T and by V and V.Therefore,the force-strain relationships are coupled. We now extend the preceding equations to include these couplings and(for arbi- trary cross-section beams)write the force-strain relationships as 屋美到周 (7.36)7.1 GOVERNING EQUATIONS 319 The bending moments and bimoment for a (symmetrical) I-beam are (see Eqs. 7.12 and 7.23)    M
z M
y M
ω    =    EI zz 0 0 0 EI yy 0 0 0 EI ω       −dχy dx −dχz dx −dϑB dx    , (7.31) where the equation for M
z is analogous to the equation for M
y. For an arbitrary cross-section orthotropic beam, the 12 and 21 elements are not zero. Hence, we write    M
z M
y M
ω    =    EI zz EI yz 0 EI yz EI yy 0 0 0 EI ω       −dχy dx −dχz dx −dϑB dx    . (7.32) By utilizing the definitions given in Eq. (7.6), these equations may be written as    M
z M
y M
ω    =    EI zz EI yz 0 EI yz EI yy 0 0 0 EI ω       1 ρz 1 ρy     . (7.33) The relationship between T
sv and the rate of twist ϑ is (Eq. 7.15) T
sv = GI tϑ. (7.34) The shear forces and the restrained–warping-induced torque for a (doubly symmetrical) I-beam is (see Eqs. 7.13 and 7.28)    V
y V
z T
ω    =   
Syy 0 0 0
Szz 0 0 0
Sωω       γy γz ϑS    , (7.35) where the first equality is analogous to the second one. Shear is introduced by T
ω and by V
y and V
z. Therefore, the force–strain relationships are coupled. We now extend the preceding equations to include these couplings and (for arbi￾trary cross-section beams) write the force–strain relationships as    V
y V
z T
ω    =   
Syy
Syz
Syω
Syz
Szz
Szω
Syω
Szω
Sωω       γy γz ϑS    = "
Si j#    γy γz ϑS    , (7.36)