Ch6sol - (3) Substituting Eq. (1) into Eq. (2) gives 3 or...

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6.1 Develop the buckling strength envelope for a simply supported column of light gage channel section shown in Fig. P6-1. Material properties and dimensions of the cross section are as follows: Fig. P6-1 Solution SECP computes the following section properties: where Length (in) (kips) (kips) 80 362.7 274.1 61.0 54.1 160 90.7 68.5 16.8 14.7 1
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Fig. PS6-1 6.2 Develop the buckling strength envelope for a simply supported column of an unequal leg angle shown in Fig. P6-2. Material properties and dimensions of the cross section are as follows: Fig. P6-2 Solution SECP computes the following section properties: Fig. PS6-2 6.3 For the coupled system of differential equations given by Eqs. (6.4.23) and (6.4.24), prove why the solution eigenfunctions are sine functions for a simply supported beam of a doubly symmetric section. In other words, state the reason why the eigenfunctions do not include hyperbolic functions or polynomials. Solution 2
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(6.4.23) (6.4.24) For the problem,. Hence, they are simplified to (1) (2) Integrating twice , Eq. (1) yields
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Unformatted text preview: (3) Substituting Eq. (1) into Eq. (2) gives 3 or (4) where . Equation (4) is an ordinary 4 th order linear homogeneous differential equation with constant coefficients. According to the general theory, assume the solution to be of the form (5) Taking appropriate derivatives of Eq. (5) and substituting yields (6) Let Hence, (7) 4 Equation (7) shows a total of four distinct roots as expected since Eq. (4) is a 4 th order differential equation. Examination of Eq. (7) reveals that two of the roots are imaginary and the other two are real. Converting the imaginary value by multiplying with and The both are positive, real values. Then the general solution of Eq. (4) is (8) Since , and For nontrivial solution for A and C , the determinant for A and C must vanish. Hence, 5 (9) Since the sum of the terms in the parenthesis are nonzero, (10) . The smallest value of satisfying Eq. (10) is Hence, Substituting into above gives , Finally, 6 QED 7...
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Ch6sol - (3) Substituting Eq. (1) into Eq. (2) gives 3 or...

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