A laboratory compression test is performed on an imperfect slender column. Lateral deflection $\delta$ is measured at various axial load levels $P$. The experimental data is linearized using the **Southwell plot**, where $\delta / P$ is plotted on the vertical $y$-axis against $\delta$ on the horizontal $x$-axis ($\\frac{\delta}{P} = \frac{1}{P_{cr}} \delta + \frac{\delta_0}{P_{cr}}$). From this plot, the critical Euler buckling load $P_{cr}$ is determined from:

(A) The reciprocal of the slope of the straight line $\left(P_{cr} = \frac{1}{\text{Slope}}\right)$

(B) The slope of the straight line directly

(C) The $y$-intercept of the straight line

(D) The area under the linear regression line

Answer Key:

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### 1. Strategy & Theoretical Principles For an elastic column with an initial geometric imperfection $\delta_0$ under axial compressive load $P$: The total lateral deflection $\delta$ is amplified according to: $$\delta = \frac{\delta_0}{1 - P / P_{cr}}$$ Rearranging algebraically: $$\delta \left(1 - \frac{P}{P_{cr}}\right) = \delta_0 \implies \delta - \frac{P \delta}{P_{cr}} = \delta_0$$ Dividing through by $P$: $$\frac{\delta}{P} = \frac{1}{P_{cr}} \delta + \frac{\delta_0}{P_{cr}}$$ ### 2. Analytical Interpretation of the Plot Comparing with the standard linear equation $y = m x + c$: - $y = \frac{\delta}{P}$ - $x = \delta$ - Slope $m = \frac{1}{P_{cr}}$ - Intercept $c = \frac{\delta_0}{P_{cr}}$ Therefore, the critical Euler buckling load is the **reciprocal of the slope**: $$P_{cr} = \frac{1}{\text{Slope}}$$ (The initial imperfection is obtained from $\delta_0 = c / m = c P_{cr}$). ### 3. Distractor Forensics - **Option A:** Correct Southwell plot property. - **Option B:** Confuses reciprocal of slope with slope. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Southwell plots allow determining the critical buckling load $P_{cr}$ non-destructively without having to buckle the column to destruction. - **Standard Reference:** Timoshenko, S.P. & Gere, J.M., *Theory of Elastic Stability*, Chapter 2.

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