WORLD FOR CIVIL

Topic 7.1a: Euler Buckling - Example 1

An 16 ft. long ASTM-A36 steel, W10x29 I-Beam is to be used as a column with pinned ends. For this column, determine the slenderness ratio, the load that will result in Euler buckling, and the associated Euler buckling stress. The beam characteristics may be found in the I-Beam Table, and are also listed below.

-	-	-	Flange	Flange	Web	Cross	Section	Info.	Cross	Section	Info.
Designation	Area	Depth	Width	thick	thick	x-x axis	x-x axis	x-x axis	y-y axis	y-y axis	y-y axis
-	A-in2	d - in	wf - in	tf - in	tw - in	I - in4	S -in3	r - in	I - in4	S -in3	r - in
W 10x29	8.54	10.22	5.799	0.500	0.289	158.0	30.8	4.30	16.30	5.61	1.38

The slenderness ratio = L_e / r = 16 ft. * 12 in./ft./ 1.38 in = 139

Notice that we must use the smallest radius of gyration, with respect to the y-y axis, as that is the axis about which buckling will occur. We also notice that the slenderness ratio is large enough to apply Euler’s buckling formula to this beam. To verify this we use the relationship for the minimum slenderness ratio for Euler’s equation to be valid.

Or, after finding for ASTM-A36 Steel, E = 29 x 10⁶ lb/in², and yield stress = 36,000 lb/in², we can solve and determine that L_e/r = 89.
The Euler Buckling Load is then give by: , and after substituting values, we obtain:
Pcr = [(3.14)²*29x10⁶ lb/in² * 16.30 in⁴/(16’x12"/ft)²] = 126,428 lb

c) The Euler Stress is then easily found by Stress = Force/Area = 126,428 lb/8.54 in² = 14,800 lb/in². Notice that this stress which will produce buckling is much less than the yield stress of the material. This means that the column will fail in buckling before axial compressive failure.