Three pinnedend columns of the same material have the same l

Three pinned-end columns of the same material have the same length and the same cross-sectional area (see figure). The columns are free to buckle in any direction. The columns have cross sections as follows: a circle, a square, and an equilateral triangle. Determine the ratios P1: P2: P3 of the critical loads for these columns.

Solution

>> As, Critical Load, P is directlyproportoinal to Moment of Inertia about the axis it is monimum

So, P1:P2:P3 = I1 : I2 : I3

As, I1 = R⁴/64

I2 = a⁴/12

and, I3 = b⁴/36 [ \"b\"= Side of Triangle ]

Now, As, Cross Sectional Area is Same

=> A1 = A2 = A3

=> R² = a² = 0.433*b²

=> a = 1.772*R

and, b = 1.5197*a = 2.693*R

>> As, I1 = R⁴/64 = 0.0491*R⁴

I2 = a⁴/12 = 1.772⁴*R⁴/12 = 0.8216*R⁴

and, I3 = b⁴/36 = (2.693*R)⁴/36 = 1.461*R⁴

=> I1 : I2 : I3 = 0.0491*R⁴ : 0.8216*R⁴ : 1.461*R⁴

=> I1 : I2 : I3 = 1 : 16.733 : 29.76