Four identical pin-pin ended solid concrete cylindrical columns that are L=10m tall need to be able to withstand 6.8×108N of force before Euler buckling. What should be the diameter, d, in meters, of each column if the force is uniformly distributed? The Young’s modulus, E, for concrete is 17GPa, and I=πr44. Assume n=1.
d (in meters):
1.2 m
To determine the diameter of the columns, we can use the Euler buckling formula for pin-pin ended columns. The formula is:
F = (π² * E * I) / (L)²
Where:
F is the critical buckling force,
E is the Young's modulus,
I is the moment of inertia, and
L is the length of the column.
First, let's calculate the moment of inertia, I. We are given the diameter, d, so we can calculate the radius, r, as d/2.
I = (π * r^4) / 4
Now, let's substitute the values into the Euler buckling formula:
6.8 × 10^8 = (π² * 17 × 10^9 * ((π * (d/2)^4) / 4)) / (10^2)
Simplifying the equation:
d^4 = (32 * 6.8 × 10^8 * 10^2) / (π² * 17 × 10^9 * π)
Now, let's solve for d:
d^4 ≈ 3.7482 × 10^-4
Taking the fourth root of both sides:
d ≈ (3.7482 × 10^-4)^(1/4)
Simplifying further:
d ≈ 0.0824 meters
Therefore, the diameter, d, of each column should be approximately 0.0824 meters.