# Ch1sol - y y x B P x 20 y 20 A 20 B(a 15 P 25(b y M B M 1.1...

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y y x B y M B M B M B M B M B C B M Py+ x M Q M/ P+Q M/ P x y A B P 20’ 20’ 20’ 15’ 25’ 1.1 (a) Using 4th order DE, determine the lowest three critical loads. (b) Determine the lowest two critical loads. Fig. P1-1 (a) a. b. c. , from b, A =0 d. , since A = C =0, eigenvalue with n =1,2,3, . . . eigenvector with n =1,2,3, . . . 1

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(b) Using 2nd order DE Taking the two spans, AB and BC separately, assume nonzero moment is present at support B. To balance this moment, the continuity shear must be developed in each freebody. For span AB 2
For span BC From the boundary conditions of span AB 3

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From the boundary conditions of span BC Equating two slopes at support B gives From the problem statement, . Let , then . Hence, The two roots of this transcendental equation are by either BISECT or Maple ® . 4
Corresponding eigen-mode shapes are given below. Using 4th order DE For span AB 5

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For span BC 6
From the continuity consideration Equating these two slopes gives 7

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Let , then . Note: the unknown moment, , assumed does not enter the stability equation. 8
1.2 Two rigid bars are connected with a linear rotational spring of stiffness as shown in Fig. P1-2. Determine the critical load of the structure in terms of the spring constant and the bar length. Fig. P1-2 The strain energy stored: The loss of potential of external load: The necessary condition for stability: 9

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1.3 For the structure shown in Fig. P1-2, plot the load vs. transverse deflection in a qualitative sense when (a) the transverse deflections are large, (b) the load is applied eccentrically, and (c) the model has an initial transverse deflection . (a) ,,. (By L’Hopital rule) (b) From the geometry, . Hence, the additional loss of potential of the external load is: or For small displacement theory, For large displacement theory, 10
11

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(c) Neglecting the higher order terms, For large displacement theory, 12
1.4 Determine the critical load of the structure shown in Fig. P1-4. Assume the continuity shear Q that develops in the beam due to the uneven moment is negligibly small compared to the load P . Fig. P1-4 Taking the coordinate axes as shown in the figure, moment equilibrium for the column segment yields where The solution of the nonhomogeneous differential equation is given as B.C.: Hence 13

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Know the rotation of the column and the beam should be the same as the joint is rigidly connected. The slope (rotation) of the column at is The rotation of the beam is obtained from the ordinary slope-deflection equation. Since the moment at the right-hand end is zero, the rotation at the right-hand side is equal to . Equating the two slope yields
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## This note was uploaded on 09/24/2011 for the course CIVL 3110 taught by Professor Melville during the Spring '08 term at Auburn University.

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Ch1sol - y y x B P x 20 y 20 A 20 B(a 15 P 25(b y M B M 1.1...

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