Explanation Given information: The column is made up of W10 × 30 structural A992 steel. The ends of the column are pinned. The eccentricity of loading ( e ) is 8 in . The length ( L ) on the column is denoted by 15 ft . The load on the column is denoted by P . Calculation: Show the section properties of W10 × 30 as follows: The cross-sectional area ( A ) is 8.84 in . 2 . The moment of inertia ( I x ) is 170 in . 4 The moment of inertia ( I y ) is 16.7 in . 4 The radius of gyration ( r x ) is 4.38 in . The overall depth of cross-section ( d ) is 10.47 in . Consider the buckling of column about y-y axis: Thecolumn is pinned, thus the value of effective length factor ( K ) is 1. The Young’s modulus ( E ) of the steel column is 29 × 10 3 ksi . Show the expression for Euler’s formula as follows: P cr = π 2 E I y ( K L ) y 2 (1) Here, E is the Young’s modulus, I is the moment of inertia, K is the effective length factor, L is the length of the member. Substitute 15 ft for L , 29 × 10 3 ksi for E , 16.7 in . 4 for I , and 1 for K in Equation (1). P c r = π 2 ( 29 × 10 3 ) ( 16.7 ) ( 1 × 15 ft × 12 in . 1 ft ) 2 = 4779.849 × 10 3 32 , 400 = 147.5 kip Thus, the load on the column based on the buckling criteria is 147.5 kip _ . Check the validity of Euler’s formula: Consider the yielding stress of the steel as σ Y = 50 ksi . Calculate the critical stress using the relation; σ cr = P cr A (2) Substitute 147.5 kip for P cr and 8.84 in . 2 for A in Equation (2). σ cr = 147.5 kip 8.84 in . 2 = 16.67 ksi < ( σ Y = 50 ksi ) Thus, the Euler’s formula is valid. Consider the yielding of column about x-x axis. Show the expression for secant formula as follows: σ Y = P A [ 1 + e c r 2 sec ( K L 2 r P E A ) ] (3) Here, P is the vertical load, σ max is the maximum elastic stress, e is the eccentricity of the load P , c is the distance of centroidal axis to the outer fiber of the column, A is the cross-sectional area of the column, K is the effective length factor, L is the length of the column, E is modulus of elasticity of the material, and r is the radius of gyration

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Buckling of columns

The column is described as a vertical member of the structure that carries an axial compressive load, and buckling is a lateral deformation of the column under the application of load.

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Chapter 13.5, Problem 13.63P

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The maximum load supported by the column.

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