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Aircraft Structures 1 2011 Part 6

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. 6.22 Column section of Example 6.1. The second moments of area of the cross-section about the centroidal axes Cxy are see Section 9.1 respectively Ixx 2 X 37.5 X 2.5 X 37.52 2.5 X 753 12 3.52 X 105 mm4 Iyy 2 X 2.5 X 37.53 12 0.22 X 105mm4 The polar second moment of area zo is Io IXX Iyy - s J s see derivation of Eq. 6.82 i.e. Io 3.52 X 105 0.22 X 105 3.74 X 105mm4 The torsion constant J is obtained using Eq. 9.59 which gives J 2 X 37.5 X 2.53 3 75 X 2.53 3 781.3 mm4 Finally r is found using the method of Section 11.5 and is r 2.5 X 37.53 X 752 24 30.9 X 106mm6 Substituting the above values in Eqs 6.90 we obtain CR 6.5 X 104 N Pcr 0.41 X 104 N PCR 9 2.22 X 104 N The column will therefore buckle in bending about the Cy axis when subjected to an axial load of 0.41 X 104N. Equation 6.86 for the column whose buckled shape is defined by Eqs 6.84 may be rewritten in terms of the three separate buckling loads given by Eqs 6.90 . Thus 0 p pCR xx Px P PcR yy 0 Pys Pys -Pxs IO P PCR gỵ A 0 6.91 6.12 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 6.91 thereby reduces to p - PcR xx -Px Px A -P CR e M 0 6.92 The roots of the quadratic equation formed by expanding Eqs 6.92 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 6.2 A column of length 1 m has the cross-section shown in Fig. 6.23. If the ends of the column are pinned and free to warp calculate its buckling load E 70000N mm2 G 30000N mm2. Fig. 6.23 Column section of Example 6.2. In this case the shear centre s is positioned on the Cx axis so that ys 0 and Eq. 6.92 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 .