the rectangular steel bar shown supports a concentric load of 13000lbf. Both ends are fixed. If te modulus of elasticity is 29x10^6 lbf/in^2, what is the maximum length the rod can be without experiencing buckling failure?
P=13,000 lbf
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_________________ .75 in
| _______
| | |
| | | 1.25 in
L | | |
| -------
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| cross section
_________________
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P=13,000 lbf
To calculate the maximum length of the rod without experiencing buckling failure, we can use the Euler's buckling formula.
The Euler's buckling formula for a fixed-fixed rod is:
L = (π² * E * I) / (4 * P * A)
Where:
L = Maximum length of the rod without buckling failure
E = Modulus of Elasticity (29x10^6 lbf/in²)
I = Area moment of inertia of the cross-section (for a rectangular cross-section: I = (b * h³) / 12, where b is the width and h is the height of the cross-section)
P = Load applied to the rod (13,000 lbf)
A = Cross-sectional area of the rod (A = b * h)
Given the dimensions of the cross-section provided in your diagram, we can calculate the values needed for the equation.
Cross-sectional area (A) = 0.75 in * 1.25 in = 0.9375 in²
Area moment of inertia (I) = (0.75 in * (1.25 in)³) / 12 = 0.24414 in^4
Plugging the values into the Euler's buckling formula:
L = (π² * 29x10^6 lbf/in² * 0.24414 in^4) / (4 * 13,000 lbf * 0.9375 in²)
L = (9.87 * 29x10^6 * 0.24414) / (4 * 13,000 * 0.9375)
L = 0.3545
Therefore, the maximum length of the rod without experiencing buckling failure is approximately 0.3545 inches.
To determine the maximum length of the rectangular steel bar before buckling failure occurs, we can use Euler's buckling formula. Euler's formula helps to calculate the critical buckling load for a slender column:
P_critical = (π^2 * E * I) / L^2
Where:
P_critical is the critical buckling load,
E is the modulus of elasticity of the material (29x10^6 lbf/in^2 in this case),
I is the moment of inertia of the cross-section,
L is the length of the column.
To find the maximum length, we rearrange the formula to solve for L:
L = sqrt((π^2 * E * I) / P_critical)
To calculate the moment of inertia for a rectangular cross-section, we use the formula:
I = (b * h^3) / 12
Where:
b is the width of the rectangular cross-section (0.75 in in this case),
h is the height of the rectangular cross-section (1.25 in in this case).
Plugging in the given values, we can calculate the maximum length:
I = (0.75 * (1.25^3)) / 12 = 0.7399 in^4 (approx.)
P_critical = 13,000 lbf
L = sqrt((π^2 * (29x10^6) * 0.7399) / 13000)
Calculating this equation, we find that the maximum length the rod can be without experiencing buckling failure is approximately 24.9 inches.