A 4m long circular bar was found to deflect 2cm, when it was used as a
simply supported beam and subjected to a load of 10kg at its center. If
this bar is now used as column, with both ends hinged, determine the
crippling load it can carry
To determine the crippling load that the column can carry, we need to calculate the critical buckling load. Buckling is a phenomenon that occurs when a slender member, like a column, fails due to excessive compressive forces.
The critical buckling load can be calculated using the Euler's formula:
P_critical = (π² * E * I) / (l_effective²)
Where:
- P_critical is the critical buckling load
- π is a constant approximately equal to 3.14159
- E is the elastic modulus of the material
- I is the area moment of inertia of the cross-section
- l_effective is the effective length of the column
Given information:
- The bar is circular, so the cross-sectional area moment of inertia I is (π * r^4) / 4, where r is the radius of the circular bar.
- The bar deflects 2cm when used as a simply supported beam with a load of 10kg at its center.
- The length of the bar is 4m.
To calculate the effective length, we can use the following relation for simply supported beams:
l_effective = 0.7 * l
Substituting the given values:
l_effective = 0.7 * 4m = 2.8m
To proceed with the calculation, we need the elastic modulus (E) of the material. The elastic modulus is a material property. Different materials have different elastic moduli. If you know the material of the bar, you can look up its elastic modulus in a materials table. Common materials used in construction include steel, aluminum, and concrete.
Let's assume the bar is made of steel. The elastic modulus of steel is approximately 200 GPa (200,000 MPa).
Substituting the values into the formula:
P_critical = (π² * 200,000 MPa * (π * r^4) / 4) / (2.8m)²
Now, you can substitute the desired radius of the circular bar into the formula to calculate the crippling load it can carry.