High School

Determine the length \( L \) for which the pin-ended rectangular column will buckle at the same time that the material fails (when stress reaches the yield strength \( F_y \)). The column is braced at mid-height in the weak direction.

Given:
- \( b = 2 \text{ in} \)
- \( d = 5 \text{ in} \)
- \( F_y = 50 \text{ ksi} \)
- \( E = 29000 \text{ ksi} \)

Answer :

Final Answer:

To determine the length L for which the pin-ended rectangular column will buckle at the same time that the material fails, you can use the formula for the critical buckling load of a pinned-end column, which is given by:

Explanation:

[tex]P_critical = (π² * E * b * d²) / (4 * L²)[/tex]

Where:

P_critical = Critical buckling load

E = Young's Modulus (29000 ksi)

b = Width of the column (2 in)

d = Depth of the column (5 in)

L = Length of the column (to be determined)

Now, we know that the critical buckling load P_critical is equal to the yield strength Fy (50 ksi), as the column will buckle when stress reaches the yield strength. So, we can set up the equation as:

[tex]50 ksi = (π² * 29000 ksi * 2 in * 5 in²) / (4 * L²)[/tex]

Now, solve for L:

[tex]L² = (π² * 29000 ksi * 2 in * 5 in²) / (4 * 50 ksi)[/tex]

L² = 226194.548

L ≈ 475.50 inches

So, the length L for which the pin-ended rectangular column will buckle at the same time that the material fails is approximately 475.50 inches.

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