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Curve c approaches the horizontal asymptote

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Unformatted text preview: of these cases is that for prac>cal purposes the cri>cal load represents the maximum load- carrying capacity of the column. ☞ And besides… large deflec>ons are not acceptable in most applica>ons 4/17/13 M. Mello/Georgia Tech Aerospace OpFmum shapes of columns NONPRISMATIC COLUMNS ☞ Reinforcement achieved by varying shape such that cross sec>on is larger where the bending moment is largest 00 M ☞ Think of ⇢ = v = EI ☞ ideal op>miza>on (a) ☞ (b) is a prac>cal aeempt to realize (a) 4/17/13 WHICH IS THE OPTIMUM CROSS SECTION FOR A PRISMATIC COLUMN? ☞ Now let’s think about a prisma>c column with pinned ends that is free to buckle in any lateral direc>on. ☞ QUESTION: For a given cross- sec>onal area, which shape ⇡ 2 EI Pcr = results in the largest cri>cal load? L2 ☞ ANSWER: THE EQUILATIERAL TRIANGLE! M. Mello/Georgia Tech Aerospace EXAMPLE 9- 1 GIVEN: ☞ Long, slender column is pin- supported At the ends and compressed by axial load P. ☞ Lateral support is provided at midpoint B (constrained from buckling within the plane of the figure) ☞ lateral support perpendicular to plane of figure is provided only at the ends. COLUMN PROPERTIES: ☞ steel wide- flange sec>on (IPN 220) ☞ E = 200GPa and ☞ σpl = 300 MPa (typo in text book) ☞ column length L = 8m PROBLEM STATEMENT: Determine allowable load Pallow using a safety factor n=2.5 with respect to Euler buckling of the column 4/17/13 M. Mello/Georgia Tech Aerospace EXAMPLE 9- 1 (cont.) SOLUTION: Column might buckle either in- plane or out- of- plane ☞ for in- plane buckling (about the 2- 2 axis) the distance between the lateral supports is L/2 = 4m. ☞ for out- of- plane buckling (about the 1- 1 axis) the distance between the lateral supports is L = 8m. ☞ consult tables (appendix E) I1 = 3060cm4 I1 = 162cm4 A = 39.5cm2 4/17/13 M. Mello/Georgia Tech Aerospace EXAMPLE 9- 1 (cont.) L/2 L/2 x x 1- 1 (strong axis) out- of- plane buckling 4/17/13 M. Mello/Georgia Tech Aerospace In- plane buckling about axis 2- 2 (2- 2 axis perpendicular to plane on column diagram on leT) Pcr ⇡ 2 EI2 = (L/2)2 x 2- 2 (weak axis) In- plane buckling (n = 1) EXAMPLE 9- 1 (cont.) ☞ If column buckles in- plane (about weak ax...
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