For axisymmetric plane stress problems in polar coordinates $(r, \theta)$ without body forces, the radial stress $\sigma_r$ and circumferential hoop stress $\sigma_\theta$ derived from an Airy stress function $\phi(r)$ satisfy $\sigma_r = \frac{1}{r} \frac{d\phi}{dr}$ and $\sigma_\theta = \frac{d^2 \phi}{dr^2}$. The general solution of the biharmonic equation $\nabla^4 \phi = 0$ for purely axisymmetric stress fields is $\phi(r) = A \ln r + B r^2 \ln r + C r^2 + D$. For a hollow circular cylinder subjected to uniform internal pressure with zero external pressure, the stress component that varies as $1/r^2$ is:

(A) Both radial stress $\sigma_r$ and circumferential stress $\sigma_\theta$ have a $\pm A/r^2$ term (Lamé solution)

(B) Only the radial stress $\sigma_r$ varies as $1/r^2$

(C) Only the circumferential stress $\sigma_\theta$ varies as $1/r^2$

(D) Neither stress varies as $1/r^2$; both vary linearly with $r$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In axisymmetric 2D elasticity (Lamé problem for thick-walled cylinders): $$\sigma_r(r) = \frac{A}{r^2} + 2C$$ $$\sigma_\theta(r) = -\frac{A}{r^2} + 2C$$ where the term $B r^2 \ln r$ is discarded because single-valued displacements require $B = 0$. ### 2. Analytical Verification Both the radial stress $\sigma_r$ and the circumferential hoop stress $\sigma_\theta$ contain the $1/r^2$ hyperbolic variation with equal magnitude and opposite signs, superimposed on the uniform hydrostatic component $2C$. ### 3. Distractor Forensics - **Option A:** Correct Lamé stress distribution formulation. - **Option B/C:** Considers only one stress component. ### 4. Exam Trap & Standard Reference - **Trap Alert:** In thick cylinders, $\sigma_r + \sigma_\theta = 4C = \text{constant}$ across the entire wall thickness. - **Standard Reference:** Timoshenko, S.P. & Goodier, J.N., *Theory of Elasticity*, Chapter 4.

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