WORLD FOR CIVIL

Topic 7.1b: Euler Buckling - Example 2

A 8 ft. long southern pine 2" x 4" is to be used as a column. The yield stress for the wood is 6,500 lb/in², and Young’s modulus is 1.9 x 10⁶ lb/in². For this column, determine the slenderness ratio, the load that will produce Euler buckling, and the associated Euler buckling stress.

The slenderness ratio = L_e / r . To determine the slenderness ratio in this problem, we first have to find the radius of gryration (smallest), which we may do from the relationship: Radius of Gyration: r_xx = (I_xx/A)^1/2 , where this is the radius of gyration about an x-x axis, and where I = (1/12)bd³ for a rectangular cross section. Or r_xx = [(1/12)bd³/bd]^1/2 , where we have substituted A = bd. We now simplify and obtain:

r_xx = [(1/12)d²]^1/2 = .5774(d/2) We want the smallest radius of gyration, so we use d =2". That is, buckling will first occur about the x-x axis shown is the diagram, and r = .5774 in.

Then Slenderness ratio is given by: L_e / r = ( 8 ft x 12"/ft)/.5774" = 166 which puts the beam in the long slender category.

The Euler Buckling Load is then give by: , and after substituting values, we obtain:

Pcr = [(3.14)²*1.9x106 lb/in² * (4*2³/12) in⁴/(8’x12"/ft)²] = 5,420 lb.

c) The Euler Stress is then easily found by Stress = Force/Area = 5,420 lb/(2"*4") in² = 678 lb/in² . Notice that this stress which will produce buckling is much less than the yield stress of the material.