862_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

862_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: 11Ch11.qxd 9/27/08 856 2:21 PM Page 856 CHAPTER 11 Columns Problem 11.3-10 Three identical, solid circular rods, each of radius r and length L, are placed together to from a compression member (see the cross section shown in the figure). Assuming pinned-end conditions, determine the critical load Pcr as follows: (a) The rods act independently as individual columns, and (b) the rods are bonded by epoxy throughout their lengths so that they function as a single member. What is the effect on the critical load when the rods act as a single member? 2r Solution 11.3-10 Three solid circular rods (b) RODS ARE BONDED TOGETHER The x and y axes have their origin at the centroid of the cross section. Because there are three different centroidal axes of symmetry, all centroidal axes are principal axes and all centroidal moments of inertia are equal (see Section 12.9). From Case 9, Appendix D: R= Radius I L= Length p2EI Pcr L2 3p3Er 4 Pcr pr 4 4 (3) I p2EI Pcr (A) RODS ACT INDEPENDENTLY pr 4 5pr 4 + 2a b 4 4 IY 11pr 4 4 11p3Er 4 2 ; 4L2 L NOTE: Joining the rods so that they act as a single member increases the critical load by a factor of 11/3, ; or 3.67. ; 4L2 Problem 11.3-11 Three pinned-end columns of the same material have the same length and the same cross-sectional area (see figure). The columns are free to buckle in any direction. The columns have cross section as follows: (1) a circle, (2) a square, and (3) an equilateral triangle. Determine the ratios P1 : P2 : P{3} of the critical loads for these columns. Solution 11.3-11 Three pinned-end columns E,L and A are the same for all three columns. p EI L2 ‹ P1 : P2 : P3 I1 : I2 : I3 (1) CIRCLE Case 9, Appendix D I pd 4 64 (2) SQUARE I b4 12 A pd 2 4 ‹ I1 Case 1, Appendix D A b2 ‹ I2 A2 12 b 4 13 96 I P1 : P2 : P3 A2 4p (2) b 2 13 4 (3) A2 13 18 (3) EQUILATERAL TRIANGLE Case 5, Appendix D 2 Pcr (1) A I1 : I2 : I3 1.000 : 1.047 : 1.209 p 2p 13 : 3 9 ‹ I3 1: ; NOTE: For each of the above cross sections, every centroidal axis has the same moment of inertia (see Section 12.9) ...
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