11-01 Chap Gere - 11 # Columns Chapter Title Idealized...

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# Chapter Title 671 Idealized Buckling Models Problem 11.2-1 through 11.2-4 The figure shows an idealized structure consisting of one or more rigid bars with pinned connections and linearly elastic springs. Rotational stiffness is denoted b R and translational stiffness is denoted b . Determine the critical load P cr for the structure. 11 Columns A B L b R P L b P C B a A L 2 L 2 A C B b R P L 2 L 2 A C B b R b R P Prob. 11.2-1 Prob. 11.2-2 Prob. 11.2-3 Prob. 11.2-4
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Solution 11.2-1 Rigid bar AB 672 CHAPTER 11 Columns P cr 5 b R L P ( u L ) 2 b R u 5 0 a M A 5 0 A B P u u L M 5 b R u Solution 11.2-2 Rigid bar ABC P cr 5 b a 2 L P u L 2 bu a 2 5 0 a M A 5 0 A C P u L F 5 bu a a u Solution 11.2-3 Two rigid bars with a pin connection Shows that there are no horizontal reactions at the supports. F REE - BODY DIAGRAM OF BAR BC P cr 5 6 b R L b R (2 u ) 1 b R u 5 PL u 2 a M B 5 0 Ê M B 1 M C 2 P u ¢ L 2 5 0 M B 5 b R u ) M C 5 b R u a M A 5 0 A C B M 5 b R u M 5 b R u P u u C B M C 5 b R u P P u M C ( L 2 ) u M B Solution 11.2-4 Two rigid bars with a pin connection F REE - BODY DIAGRAM OF BAR BC P cr 5 H u 5 b R L a M B 5 0 Ê H ¢ L 2 2 P ¢ u L 2 5 0 H 5 b R u L a M A 5 0 Ê HL 2 b R u 5 0 A C B M 5 b R u P P u u H H P P C B u H H
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Critical Loads of Columns with Pinned Supports The problems for Section 11.3 are to be solved using the assumptions of ideal, slender, prismatic, linearly elastic columns (Euler buckling). Buckling occurs in the plane of the figure unless stated otherwise. Problem 11.3-1 Calculate the critical load P cr for a W 8 3 35 steel column (see figure) having length L 5 24 ft and E 5 30 3 10 6 psi under the following conditions: (a) The column buckles by bending about its strong axis (axis 1-1), and (b) the column buckles by bending about its weak axis (axis 2-2). In both cases, assume that the column has pinned ends. Solution 11.3-1 Column with pinned supports SECTION 11.3 Critical Loads of Columns with Pinned Supports 673 1 C 1 2 2 W 8 3 35 steel column L 5 24 ft 5 288 in. E 5 30 3 10 6 psi I 1 5 127 in. 4 I 2 5 42.6 in. 4 A 5 10.3 in. 2 (a) B UCKLING ABOUT STRONG AXIS P cr 5 p 2 EI 1 L 2 5 453 k (b) B UCKLING ABOUT WEAK AXIS N OTE : [ Solution is satisfactory if s PL $ 44 ksi s cr 5 P cr A 5 453 k 10.3 in. 2 5 44 ksi P cr 5 p 2 EI 2 L 2 5 152 k Probs. 11.3-1 through 11.3-3 Problem 11.3-2 Solve the preceding problem for a W 10 3 60 steel column having length L 5 30 ft. Solution 11.3-2 Column with pinned supports W 10 3 60 steel column L 5 30 ft 5 360 in. E 5 30 3 10 6 psi I 1 5 341 in. 4 I 2 5 116 in. 4 A 5 17.6 in. 2 (a) B UCKLING ABOUT STRONG AXIS P cr 5 p 2 EI 1 L 2 5 779 k (b) B UCKLING ABOUT WEAK AXIS N OTE : [ Solution is satisfactory if s PL $ 44 ksi s cr 5 P cr A 5 779 k 17.6 in. 2 5 44 ksi P cr 5 p 2 EI 2 L 2 5 265 k Problem 11.3-3 Solve Problem 11.3-1 for a W 10 3 45 steel column having length L 5 28 ft. Solution 11.3-3 Column with pinned supports W 10 3 45 steel column L 5 28 ft 5 336 in. E 5 30 3 10 6 psi I 1 5 248 in. 4 I 2 5 53.4 in. 4 A 5 13.3 in. 2 (a) B UCKLING ABOUT STRONG AXIS P cr 5 p 2 EI 1 L 2 5 650 k (b) B UCKLING ABOUT WEAK AXIS N OTE : [ Solution is satisfactory if s PL $ 49 ksi s cr 5 P cr A 5 650 k 13.3 in. 2 5 49 ksi P cr 5 p 2 EI 2 L 2 5 140
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This note was uploaded on 03/19/2008 for the course E M 316 taught by Professor Korkolis during the Spring '08 term at University of Texas at Austin.

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11-01 Chap Gere - 11 # Columns Chapter Title Idealized...

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