Copyrighted Materials Cay2 Cr家yPress rt女gd CHAPTER SEVEN Beams with Shear Deformation Frequently,beams undergoing small deformations may be analyzed by the Bernoulli-Navier hypothesis,namely,by the assumptions that planes of the cross section remain plane and perpendicular to the axis(Fig.7.1,left). In this chapter we treat beams for which the Bernoulli-Navier hypothesis is invalid.In the analysis we employ an x,y,z,coordinate system with the origin in the centroid.Furthermore,to simplify the notation,we use single subscripts y and z describing shear in the x-y and x-z planes. In the x-z plane the deflection of the beam's axis w is related to the rotation of the cross section xa by dw no shear deformation dx =Xz (71) x-z plane. For thick solid cross-section beams,sandwich beams,and thin-walled beams the first assumption (that planes of the cross section remain plane)is reasonable. The second assumption may no longer be valid because cross sections do not nec- essarily remain perpendicular to the axis(Fig.7.1,right).In this case the deflection of the beam is dw with shear deformation dx =X:十y (7.2) x-z plane, where ya is the transverse shear strain in the x-z plane.The theory,based on the assumption that cross sections remain plane but not perpendicular to the axis is frequently called first-order shear theory.A beam,in which shear deformation is taken into account is called a Timoshenko beam. Similarly,in the x-y plane,we have dv with shear deformation dx =Xy+Yy (7.3) x-y plane, 313
CHAPTER SEVEN Beams with Shear Deformation Frequently, beams undergoing small deformations may be analyzed by the Bernoulli–Navier hypothesis, namely, by the assumptions that planes of the cross section remain plane and perpendicular to the axis (Fig. 7.1, left). In this chapter we treat beams for which the Bernoulli–Navier hypothesis is invalid. In the analysis we employ an x, y, z, coordinate system with the origin in the centroid. Furthermore, to simplify the notation, we use single subscripts y and z describing shear in the x–y and x–z planes. In the x–z plane the deflection of the beam’s axis w is related to the rotation of the cross section χz by dw dx = χz no shear deformation x–z plane. (7.1) For thick solid cross-section beams, sandwich beams, and thin-walled beams the first assumption (that planes of the cross section remain plane) is reasonable. The second assumption may no longer be valid because cross sections do not necessarily remain perpendicular to the axis (Fig. 7.1, right). In this case the deflection of the beam is dw dx = χz + γz with shear deformation x–z plane, (7.2) where γz is the transverse shear strain in the x–z plane. The theory, based on the assumption that cross sections remain plane but not perpendicular to the axis is frequently called first-order shear theory. A beam, in which shear deformation is taken into account is called a Timoshenko beam. Similarly, in the x–y plane, we have dv dx = χy + γy with shear deformation x–y plane, (7.3) 313
314 BEAMS WITH SHEAR DEFORMATION X主 dw dw B B y Figure 7.1:Deformation of a beam in the x-z plane without shear deformation(left)and with shear deformation(right). The rotations of the cross sections x=,xy are caused by the bending moments My,M,and the transverse shear strains y Yy are caused by the transverse shear forces V.,V.The relationships between x and M and betweeny and Vare pre- sented in Section 7.1.2 for orthotropic beams. In torsion,when shear deformation of the wall is neglected,the rate of twist B is related to the twist of the cross section about the beam's axis by(Fig.7.2, left) dψ dx =B no shear deformation. (7.4) When shear deformation is not negligible,there is an additional rate of twist of the cross section s,as shown in Figure 7.2,right.Thus,in the presence of shear deformation,the rate of twist of the beam is! d地=B+9s d with shear deformation (7.5) For open-section beams,B is the rate of twist due to warping when the shear strain y is zero,and s is the rate of twist due to the shear deformation when warping is zero(Fig.7.2).For closed-section beams the interpretation ofB and s is more complicated and is not given here. 7.1 Governing Equations The response of a beam to the applied forces is described by the strain- displacement,force-strain,and equilibrium equations.These equations are given in this section for orthotropic beams,including the effect of restrained warping. 1 X.Wu and C.T.Sun,Simplified Theory for Composite Thin-Walled Beams.AlAA Journal,Vol.30, 2945-2951,1992
314 BEAMS WITH SHEAR DEFORMATION w γz χz dx dw dx dw x z B A x z B A χz A′ A′ B ′ B ′ Figure 7.1: Deformation of a beam in the x–z plane without shear deformation (left) and with shear deformation (right). The rotations of the cross sections χz, χy are caused by the bending moments My, Mz, and the transverse shear strains γz, γy are caused by the transverse shear forces V z, V y. The relationships between χ and M and between γ and V are presented in Section 7.1.2 for orthotropic beams. In torsion, when shear deformation of the wall is neglected, the rate of twist ϑB is related to the twist of the cross section about the beam’s axis ψ by (Fig. 7.2, left) dψ dx = ϑB no shear deformation. (7.4) When shear deformation is not negligible, there is an additional rate of twist of the cross section ϑS, as shown in Figure 7.2, right. Thus, in the presence of shear deformation, the rate of twist of the beam is1 dψ dx = ϑB + ϑS with shear deformation. (7.5) For open-section beams, ϑB is the rate of twist due to warping when the shear strain γ is zero, and ϑS is the rate of twist due to the shear deformation when warping is zero (Fig. 7.2). For closed-section beams the interpretation of ϑB and ϑS is more complicated and is not given here. 7.1 Governing Equations The response of a beam to the applied forces is described by the strain– displacement, force–strain, and equilibrium equations. These equations are given in this section for orthotropic beams, including the effect of restrained warping. 1 X. Wu and C. T. Sun, Simplified Theory for Composite Thin-Walled Beams. AIAA Journal, Vol. 30, 2945–2951, 1992
7.1 GOVERNING EQUATIONS 315 Figure 7.2:The rate of twist due to warping when the shear strain is zero(left)and the rate of twist s due to shear deformation when warping is zero(right). 7.1.1 Strain-Displacement Relationships There are seven independent displacements,of which the following four are iden- tical to the displacements of beams without shear deformation(Fig.6.3):the axial displacement u,the transverse displacements v and w in the y and z directions, respectively,and the twist of the cross section v.When the shear deformation is taken into account,there are three additional displacements,namely,the rotations of the cross section x,xy in the x-z and x-y planes,respectively,and the rate of twist due to warping,B,when the shear strain y is zero.We define the following generalized strain components(hereafter referred to as strain): ou 1=-w 1=-% =- 80B dψ 0= P ax dx (7.6) We note that p:and py are not the radii of curvatures of the beam's axis;they are the radii of curvatures only when shear deformation is neglected.Equations (7.2),(7.3),and(7.5)yield dw dv 收=-X y=衣- dx -9B (7.7) 7.1.2 Force-Strain Relationships The force-strain relationships are first presented in detail for an orthotropic I-beam with doubly symmetrical cross section subjected to bending moment My, shear force acting through the plane of symmetry V,and torque T(Fig.7.3).The relationships thus obtained are then generalized to beams with arbitrary cross sections. Figure 7.3:Thin-walled I-beam
7.1 GOVERNING EQUATIONS 315 γ ϑ S ϑ B Figure 7.2: The rate of twist ϑB due to warping when the shear strain is zero (left) and the rate of twist ϑS due to shear deformation when warping is zero (right). 7.1.1 Strain–Displacement Relationships There are seven independent displacements, of which the following four are identical to the displacements of beams without shear deformation (Fig. 6.3): the axial displacement u, the transverse displacements v and w in the y and z directions, respectively, and the twist of the cross section ψ. When the shear deformation is taken into account, there are three additional displacements, namely, the rotations of the cross section χz, χy in the x–z and x–y planes, respectively, and the rate of twist due to warping, ϑB, when the shear strain γ is zero. We define the following generalized strain components (hereafter referred to as strain): o x = ∂u ∂x 1 ρz = −∂χy ∂x 1 ρy = −∂χz ∂x = −∂ϑB ∂x ϑ = dψ dx (7.6) We note that ρz and ρy are not the radii of curvatures of the beam’s axis; they are the radii of curvatures only when shear deformation is neglected. Equations (7.2), (7.3), and (7.5) yield γz = dw dx − χz γy = dv dx − χy ϑS = dψ dx − ϑB. (7.7) 7.1.2 Force–Strain Relationships The force–strain relationships are first presented in detail for an orthotropic I-beam with doubly symmetrical cross section subjected to bending moment My, shear force acting through the plane of symmetry V z, and torque T (Fig. 7.3). The relationships thus obtained are then generalized to beams with arbitrary cross sections. hf d bf hf T z y x My Vz hw Figure 7.3: Thin-walled I-beam
316 BEAMS WITH SHEAR DEFORMATION X: Figure 7.4:Displacement u of point C in the axial direction. B I-beam.In the absence of twist,the displacement u in the x direction at a point located at distance z from the centroid is (Fig.7.4,X2 tan x:=-) u=-ZXz, (7.8) where x:is the rotation of the cross section in the x-z plane. The strain-displacement relationship (Eq.2.2),together with Eq.(7.8),gives the axial strain du ex二 dx =一Z1 dx (7.9) The bending moment about the y-axis is defined as (7.10) For an isotropic material ox =Eex,and we have 成=人5(0）A=-Ean器 (7.11) Recalling the analyses in Sections 6.3 and 6.4,we replace EI by EIfor com- posite beams and write My =Elyy orthotropic beam. (7.12) dx 1/py where Ely is the replacement bending stiffness.Replacement bending stiffnesses for beams are given in Tables 6.3-6.8(page 231)and A.1-A.4. The shear strain varies linearly with the shear force.Thus,formalistically,we write V.=Sav: orthotropic beam (7.13) where S is the shear stiffness. Next,we consider an orthotropic beam subjected to a torque T(Eq.6.240): T=Tsv To. (7.14)
316 BEAMS WITH SHEAR DEFORMATION x z B A χz C z −u A′ B′ C ′ Figure 7.4: Displacement u of point C in the axial direction. I-beam. In the absence of twist, the displacement u in the x direction at a point located at distance z from the centroid is (Fig. 7.4, χz ≈ tan χz = −u z ) u = −zχz, (7.8) where χz is the rotation of the cross section in the x–z plane. The strain–displacement relationship (Eq. 2.2), together with Eq. (7.8), gives the axial strain x = du dx = −z dχz dx . (7.9) The bending moment about the y-axis is defined as My = ) A zσxdA. (7.10) For an isotropic material σx = Ex, and we have My = ) A zE −z dχz dx dA= −E ) A z2 dA % &' ( Iyy dχz dx . (7.11) Recalling the analyses in Sections 6.3 and 6.4, we replace EI by EI for composite beams and write My = EI yy −dχz dx % &' ( 1/ρy orthotropic beam. (7.12) where EI yy is the replacement bending stiffness. Replacement bending stiffnesses for beams are given in Tables 6.3–6.8 (page 231) and A.1–A.4. The shear strain varies linearly with the shear force. Thus, formalistically, we write V z = Szzγz orthotropic beam, (7.13) where Szz is the shear stiffness. Next, we consider an orthotropic beam subjected to a torque T (Eq. 6.240): T = T sv + T ω. (7.14)
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 derived 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 Mf for an orthotropic flange in the presence of shear deformation as Mf = 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) Mω = Mfd. (7.22) 2 X. Wu and C. T. Sun, Simplified Theory for Composite Thin-Walled Beams. AIAA Journal, Vol. 30, 2945–2951, 1992.