Tham khảo tài liệu 'aircraft structures 1 2011 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 306 Open and closed thin-walled beams Fig. Closed section beam of Example . Eq. simplifies to s fs ĩí -7 I O ds i Jxx Jo in which J 2 rlOa 8 2 I-I7a 8 2 4 dsi L 4 7 2 di2 0 10 Jo 17 Evaluating this expression gives Ixx 1152a3 . The basic shear flow distribution qb is obtained from the first term in Eq. i . Thus for the wall 41 . _ sy 9b 41 1152a3 Jo i 8 V -sy 2 2 io 1 d 1 ĩĩsẫds4 ii In the wall 12 9b 12 1152 a3 17a-s2 p-ds2 40a2 .Jo 17 which gives sy qbn 1152 a3 4 - . - 2 8a52 40a2 in The qb distributions in the walls 23 and 34 follow from symmetry. Hence from Eq. 9 48 54 7152J Jo 5S dS1 Jo -ĩ Ổ So 2 40 2U2 giving iv Torsion of closed section beams 307 Taking moments about the point 2 we have S . s 9 2 or 10 741 17í sin ớ 0 We may replace sinớ by sin ớ ớj sinớ COSỚỊ - cosớ sinớ2 where sin ớ 15 17 COSỚỊ 8 10 cosớ 8 17 and sinớ2 6 10. Substituting these values and integrating Eq. v gives s which means that the shear centre is inside the beam section. Torsion of closed section beams A closed section beam subjected to a pure torque T as shown in Fig. does not. in the absence of an axial constraint develop a direct stress system. It follows that the equilibrium conditions of Eqs and reduce to dq ds 0 and dq dz 0 respectively. These relationships may only be satisfied simultaneously by a constant value of q. We deduce therefore that the application of a pure torque to a closed section beam results in the development of a constant shear flow in the beam wall. However the shear stress T may vary around the cross-section since we allow the wall thickness t to be a function of s. The relationship between the applied torque and this constant shear flow is simply derived by considering the torsional equilibrium of the section shown in Fig. . The torque produced by the shear flow acting on an element 6s of the beam wall is pq6s. Hence T pq di or since q is constant and f p 2 1 as before T 2Aq Fig. Torsion of