7.1 GOVERNING EQUATIONS 317 The torque Tsv(Saint-Venant torque,Fig.6.56,top)is(Eq.6.240) Tsy GId Saint-Venant (7.15) torque. The torque(restrained-warping-induced torque,Fig 6.56,bottom)is de- rived below following the reasoning used for an l-beam without shear deformation (Section 6.5.5). The displacement of the flange vr is(Fig.6.57) w=ψ2 (7.16) where is the twist of the cross section about the beam's axis and d is the distance between the midplanes of the flanges.The rate of twist is=d/dx (Eq.6.1), and we write dvr d =20 (7.17) On the basis of Eq.(7.3),the first derivative of the displacement is written as dvi (7.18) dx =(x)4+(y: where(x)is the rotation of the cross section of the flange about the z-axis(Fig.7.3), and (y)is the shear strain in the flange. We express the rate of twist in the form2(Eq.7.5) 0=0B+0s (7.19) The first term represents the rate of twist in the absence of shear deformation, and the second term is the rate of twist due to shear deformation.Equations (7.17)-(7.19)give =8s 4a时 (7.20) Recalling Eq.(7.12),we write the bending moment Mr for an orthotropic flange in the presence of shear deformation as M=立（ (7.21) where the second equality is written by virtue of Eq.(7.20),and E is the bending stiffness of the flange about the z-axis. For an I-beam the bimoment is (Eq.6.232) M.=Mrd. (7.22) 2 X.Wu and C.T.Sun,Simplified Theory for Composite Thin-Walled Beams.ALAA Journal,Vol.30. 2945-2951,1992.7.1 GOVERNING EQUATIONS 317 The torque T
sv (Saint-Venant torque, Fig. 6.56, top) is (Eq. 6.240) T
sv = GI tϑ Saint-Venant torque. (7.15) The torque T
ω (restrained–warping-induced torque, Fig 6.56, bottom) is de￾rived below following the reasoning used for an I-beam without shear deformation (Section 6.5.5). The displacement of the flange vf is (Fig. 6.57) vf = ψ d 2 , (7.16) where ψ is the twist of the cross section about the beam’s axis and d is the distance between the midplanes of the flanges. The rate of twist is ϑ = dψ/dx (Eq. 6.1), and we write dvf dx = d 2 ϑ. (7.17) On the basis of Eq. (7.3), the first derivative of the displacement is written as dvf dx = (χ)f + (γ )f , (7.18) where (χ)f is the rotation of the cross section of the flange about the z-axis (Fig. 7.3), and (γ )f is the shear strain in the flange. We express the rate of twist in the form2 (Eq. 7.5) ϑ = ϑB + ϑS. (7.19) The first term represents the rate of twist in the absence of shear deformation, and the second term is the rate of twist due to shear deformation. Equations (7.17)–(7.19) give (χ)f = d 2 ϑB (γ )f = d 2 ϑS. (7.20) Recalling Eq. (7.12), we write the bending moment M
f for an orthotropic flange in the presence of shear deformation as M
f = EI f  −d (χ)f dx  = −EI f d 2 dϑB dx , (7.21) where the second equality is written by virtue of Eq. (7.20), and EI f is the bending stiffness of the flange about the z-axis. For an I-beam the bimoment is (Eq. 6.232) M
ω = M
fd. (7.22) 2 X. Wu and C. T. Sun, Simplified Theory for Composite Thin-Walled Beams. AIAA Journal, Vol. 30, 2945–2951, 1992.