318 BEAMS WITH SHEAR DEFORMATION The last two equations give 2 dx (7.23) where the term in the first parentheses EE)is the warping stiffness, which for an isotropic beam is given by Eq.(6.238)and for an orthotropic l-beam byEq.(6.244) The shear force in the flange is related to Mr by =d dx (7.24) By referring to Eq.(7.13),we write Vi as =(y)h(y), (7.25) where (Syy)is the shear stiffness of the flange in the x-y plane.The warping- induced torque is(Eq.6.235) T。=d. (7.26) Equations (7.22),(7.24),and (7.26)result in 立= dx (7.27) From Eqs.(7.26),(7.25),and (7.20)we obtain To= d2 restrained-warping- (7.28) induced torque, where S is the rotational shear stiffness defined as 及=区，h号 (7.29) Equations (7.12),(7.13),(7.15),(7.23),and (7.28)are the force-strain relation- ships for an orthotropic l-beam with doubly symmetrical cross section subjected to a bending moment M,a shear force acting in the plane of symmetry V,and a torque T(=Tsv +T). Arbitrary cross-section beams.We now consider orthotropic beams of arbi- trary cross sections with internal forces N,My,M,V.V,andM.T. The relationship between the axial force(acting at the centroid)and the axial strain is (Egs.6.7 and 6.8) N=EAe. (7.30)318 BEAMS WITH SHEAR DEFORMATION The last two equations give M
ω = d2 2 EI f  % &' ( EI
ω  −dϑB dx  % &' (  , (7.23) where the term in the first parentheses EI ω(= d2 2 EI f) is the warping stiffness, which for an isotropic beam is given by Eq. (6.238) and for an orthotropic I-beam by Eq. (6.244). The shear force in the flange is related to M
f by V
f = dM
f dx . (7.24) By referring to Eq. (7.13), we write V
f as V
f = (
Syy)f (γ )f , (7.25) where (
Syy)f is the shear stiffness of the flange in the x–y plane. The warping￾induced torque is (Eq. 6.235) T
ω = V
fd. (7.26) Equations (7.22), (7.24), and (7.26) result in T
ω = dM
ω dx . (7.27) From Eqs. (7.26), (7.25), and (7.20) we obtain T
ω = (
Syy)f d2 2 ! % &' ( S
ωω ϑS restrained–warping￾induced torque, (7.28) where
Sωω is the rotational shear stiffness defined as
Sωω = (
Syy)f d2 2 . (7.29) Equations (7.12), (7.13), (7.15), (7.23), and (7.28) are the force–strain relation￾ships for an orthotropic I-beam with doubly symmetrical cross section subjected to a bending moment M
y, a shear force acting in the plane of symmetry V
z, and a torque T
(= T
sv + T
ω). Arbitrary cross-section beams. We now consider orthotropic beams of arbi￾trary cross sections with internal forces N
, M
y, M
z, V
y, V
z, and M
ω, T
. The relationship between the axial force (acting at the centroid) and the axial strain is (Eqs. 6.7 and 6.8) N
= EAo x . (7.30)