A hollow steel tube of 2.5cm external diameter and 2.0cm internal
diameter is used as a column 3m long with both ends hinged.
Determine the Euler’s crippling load, if elasticity is 2.0 x 106 kg / cm2
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mana jawabanya
To determine the Euler's crippling load for the hollow steel tube used as a column, we need to make use of Euler's formula. Euler's formula calculates the critical buckling load or the maximum load that a slender column can bear before it buckles under compression.
The formula is given by:
P = (π² * E * I) / (L)²
Where:
P = Euler's crippling load
E = Modulus of Elasticity of the material
I = Moment of inertia of the section (in this case, the hollow tube)
L = Length of the column
1. First, we need to calculate the moment of inertia of the hollow tube. The moment of inertia can be found by subtracting the moment of inertia of the internal circle (Ii) from the moment of inertia of the external circle (Ie).
Moment of inertia of a circle, I = π * r^4 / 4
For the external circle:
r = external diameter / 2 = 2.5 cm / 2 = 1.25 cm = 0.0125 m
Ie = π * (0.0125)^4 / 4
For the internal circle:
r = internal diameter / 2 = 2.0 cm / 2 = 1.0 cm = 0.01 m
Ii = π * (0.01)^4 / 4
Now subtract Ii from Ie to get the moment of inertia of the hollow tube, I:
I = Ie - Ii
2. Next, substitute the values into the Euler's formula:
P = (π² * E * I) / (L)²
Given:
E = 2.0 x 10^6 kg/cm² = 2.0 x 10^11 kg/m² (1 kg/cm² = 10^6 kg/m²)
L = 3 m
Substitute the values and calculate P:
P = (π² * 2.0 x 10^11 * I) / (3)²
3. Calculate the value of I and substitute it back into the formula:
I = Ie - Ii
4. Finally, calculate P to obtain the Euler's crippling load.