Question

A prismatic vertical column of cross-section \( a \times 0.5a \) and length \( l \) is rigidly fixed at the bottom and free at the top. A compressive force \( P \) is applied along the centroidal axis at the top surface. The Young’s modulus of the material is 200 GPa and the uniaxial yield stress is 400 MPa. If the critical value of \( P \) for yielding and for buckling of the column are equal, the value of \( \frac{l}{a} \) is \_\_\_\_ (rounded off to one decimal place).
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Hint:

When solving for the critical loads in column problems, use both the yielding load and the buckling load formulas. Set them equal to each other to solve for the unknown ratio.

Answer 1

Step 1: Critical Load for Yielding \[ P_{{yield}} = \sigma_y \times A = 400 \times 0.5a^2 = 200a^2 \, {N} \] Step 2: Critical Load for Buckling Effective length for fixed-free column: \[ L_e = 2l \] Moment of inertia (weak axis): \[ I = \frac{a^4}{96} \] Euler's buckling load: \[ P_{{buckle}} = \frac{\pi^2 E I}{L_e^2} = \frac{\pi^2 \times 200 \times 10^3 \times a^4}{384l^2} \] Step 3: Equate Yielding and Buckling Loads \[ 200a^2 = \frac{\pi^2 \times 200 \times 10^3 \times a^4}{384l^2} \] Simplify: \[ \frac{l}{a} = \pi \sqrt{\frac{10^3}{384}} \approx 5.06 \approx 5 \] Final Answer \[ \boxed{5} \]