Tham khảo tài liệu 'aircraft structures 1 2011 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 186 Structural instability Fig. Column section of Example . The second moments of area of the cross-section about the centroidal axes Cxy are see Section respectively Ixx 2 X X X X 753 12 X 105 mm4 Iyy 2 X X 12 X 105mm4 The polar second moment of area zo is Io IXX Iyy - s J s see derivation of Eq. . Io X 105 X 105 X 105mm4 The torsion constant J is obtained using Eq. which gives J 2 X X 3 75 X 3 mm4 Finally r is found using the method of Section and is r X X 752 24 X 106mm6 Substituting the above values in Eqs we obtain CR X 104 N Pcr X 104 N PCR 9 X 104 N The column will therefore buckle in bending about the Cy axis when subjected to an axial load of X 104N. Equation for the column whose buckled shape is defined by Eqs may be rewritten in terms of the three separate buckling loads given by Eqs . Thus 0 p pCR xx Px P PcR yy 0 Pys Pys -Pxs IO P PCR gỵ A 0 Flexural-torsional buckling of thin-walled columns 187 If the column has say Cx as an axis of symmetry then the shear centre lies on this axis and ys 0- Equation thereby reduces to p - PcR xx -Px Px A -P CR e M 0 The roots of the quadratic equation formed by expanding Eqs are the values of axial load which will produce flexural-torsional buckling about the longitudinal and X axes. If PcR jy is less than the smallest of these roots the column will buckle in pure bending about the y axis. Example A column of length 1 m has the cross-section shown in Fig. . If the ends of the column are pinned and free to warp calculate its buckling load E 70000N mm2 G 30000N mm2. Fig. Column section of Example . In this case the shear centre s is positioned on the Cx axis so that ys 0 and Eq. applies. The distance X of the centroid of area c from the web of the section is found by taking first moments of area about the web. Thus 2 .
