tailieunhanh - Aircraft Structures 3E Episode 13

Tham khảo tài liệu 'aircraft structures 3e episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 466 Structural constraint a b Fig. a Torsion of l-section beam b plan view of beam showing undistorted shape of flanges. Fig. . Obviously the beam still twists along its length but the rate of twist is no longer constant and the resistance to torsion is provided by the St. Venant shear stresses unrestrained warping plus the resistance of the flanges to bending. The total torque may therefore be written T Tj Tr where Tj GJ àd áz from the unconstrained torsion of open sections but in which dớ dz is not constant and Tr is obtained from a consideration of the bending of the flanges. It will be instructive to derive an expression for Tr for the I-section beam of Fig. before we turn our attention to the case of a beam of arbitrary section. Suppose that at any section z the angle of twist of the I-beam is Ớ. Then the lateral displacement u of the lower flange is Fig. Bending effect of axial constraint on flanges of l-section beam subjected to torsion. Constraint of open section beams 467 Ĩ Fig. Torsion of l-section beam fully built-in at one end. and the bending moment Mp in the plane of the flange is given by d2u Mf -EIf see Section for sign convention dzz where Zp is the second moment of area of the flange cross-section about they axis. It is assumed here that displacements produced by shear are negligible so that the lateral deflection of the flange is completely due to the self-equilibrating direct stress system crr set up by the bending of the flange. We shall not however assume that the shear stresses in the flange are negligible. The shear in the flange is then - dAA r dJ Sf dz Elriỉ or substituting for u in terms of 0 and h F2d Similarly there is a shear force in the top flange of the same magnitude but opposite in direction. Together they form a couple which represents the second part Tf of the total torque thus m h2 d30 Tr Sph -EIfn- and the expression for the total torque may be written dớ h2 d30 T GJdz EIf 2 dJ The insight