322 BEAMS WITH SHEAR DEFORMATION Table 7.2.Stiffnesses of orthotropic beams with shear deformation Tensile stiffness EA Sections 6.2-6.4 Bending stiffnesses El,Elyy.Ely Sections 6.2-6.4 Torsional stiffness Gi Sections 6.5.1-6.6.2 Warping stiffness E Section 6.6.4 Shear stiffnesses n,33,s Sections 7.2.1-7.2.3 7.2.1 Shear Stiffnesses and Compliances of Thin-Walled Open-Section Beams To determine the shear stiffness and shear compliance matrices of thin-walled open-section beams we consider an element of length AL of a thin-walled open- section beam(Fig.7.5).On the two faces of the element there are equal and opposite shear forces equal and opposite shear forcesequal and opposite torques T(=+T),and axial stresseso,which are different at the two faces. We neglect the Saint-Venant torque;hence,=T. The shear stresses are represented by the shear flow g, 9=Vyqy +V:q:+Toqo (7.41) where g and g are shear flows(per unit force)and are calculated by setting either V=1 and V.=0or V =0 and V.=1 in Eq.(6.281)and go is the shear flow(per unit torque)introduced by a unit torque T;q is evaluated by replacing Eh by 1/(see page 264)in the expression of givend for the corresponding isotropic beams. In the following we take into account only shear deformation caused by shear. Thus,the strain energy is(Eq.2.200) U-i (csnn）ddnd5, (7.42) where is the coordinate perpendicular to the wall,n is the coordinate along the circumference of the cross section,and=x.We assume that the shear flow g(=frd)does not vary along the ALlong element and the shear strain y is uniform across the thickness of the wall.Thus,the strain energy is U= )d△L (7.43) where is the shear strain in the wall's reference surface and is related to the shear flow in the composite wall byr=q(see Eq.6.195),whereis given by 6=i6- (see Eq.6.196).The superscript indicates that the compliances 88 4 T.H.G.Megson,Aircraft Structures for Engineering Students.3rd edition.Halsted Press,John Wiley Sons,New York,1999,pp.465-472.322 BEAMS WITH SHEAR DEFORMATION Table 7.2. Stiffnesses of orthotropic beams with shear deformation Tensile stiffness EA Sections 6.2–6.4 Bending stiffnesses EI zz, EI yy, EI yz Sections 6.2–6.4 Torsional stiffness GI t Sections 6.5.1–6.6.2 Warping stiffness EI ω Section 6.6.4 Shear stiffnesses
Syy,
Szz,
Sωω,
Syz,
Syω,
Szω Sections 7.2.1–7.2.3 7.2.1 Shear Stiffnesses and Compliances of Thin-Walled Open-Section Beams To determine the shear stiffness and shear compliance matrices of thin-walled open-section beams we consider an element of length L of a thin-walled open￾section beam (Fig. 7.5). On the two faces of the element there are equal and opposite shear forces V
y, equal and opposite shear forces V
z, equal and opposite torques T
(= T
sv + T
ω), and axial stresses σx, which are different at the two faces. We neglect the Saint-Venant torque; hence, T
= T
ω. The shear stresses are represented by the shear flow q, q = V
yq∗ y + V
zq∗ z + T
ωq∗ ω, (7.41) where q∗ y and q∗ z are shear flows (per unit force) and are calculated by setting either V
y = 1 and V
z = 0 or V
y = 0 and V
z = 1 in Eq. (6.281) and q∗ ω is the shear flow (per unit torque) introduced by a unit torque T
ω; q∗ ω is evaluated by replacing Eh by 1/α 11 (see page 264) in the expression of q∗ ω given4 for the corresponding isotropic beams. In the following we take into account only shear deformation caused by shear. Thus, the strain energy is (Eq. 2.200) U = 1 2 ))) (τξ ηγξ η) dζdηdξ , (7.42) where ζ is the coordinate perpendicular to the wall, η is the coordinate along the circumference of the cross section, and ξ = x. We assume that the shear flow q(=  τξ ηdζ ) does not vary along the L long element and the shear strain γξ η is uniform across the thickness of the wall. Thus, the strain energy is U = 1 2 )  γ o ξ ηq  dη L, (7.43) where γ o ξ η is the shear strain in the wall’s reference surface and is related to the shear flow in the composite wall by γ o ξ η = αν 66q (see Eq. 6.195), where αν 66 is given by αν 66 = α 66 − (β 66)2 δ  66 (see Eq. 6.196). The superscript  indicates that the compliances 4 T. H. G. Megson, Aircraft Structures for Engineering Students. 3rd edition. Halsted Press, John Wiley & Sons, New York, 1999, pp. 465–472.