The Prandtl stress function for torsion works the same way the Airy stress function works for plane elasticity problems: shear stresses are written as derivatives of a single scalar field so that one of the field equations is met without any extra work. Going through each statement against that idea gives the answer.
- It satisfies the equilibrium equation: true. Because $\tau_{xz}=\partial \phi/\partial y$ and $\tau_{yz}=-\partial \phi/\partial x$, plugging into the equilibrium equation gives $\partial^2\phi/\partial x\partial y - \partial^2\phi/\partial y\partial x$, which is always zero for a smooth $\phi$. Equilibrium holds for any choice of $\phi$, not just the correct one.
- It is zero on the lateral surfaces of the bar: true. The curved outer surface of the bar carries no applied traction, and enforcing that condition on the shear stresses forces $\phi$ to be constant all along the boundary curve; for a solid cross-section that constant is taken as zero.
- It satisfies the compatibility equation: true. $\phi$ is not free to be any function; it must also obey $\nabla^2 \phi = -2G\theta$, which is the compatibility statement rewritten in terms of $\phi$. Only functions solving this Poisson equation, with $\phi=0$ on the boundary, are valid Prandtl stress functions.
- It does not satisfy the equilibrium equation: false. This directly contradicts point 1 above; equilibrium is satisfied automatically by the way $\phi$ is defined.
So three of the four statements, that $\phi$ satisfies equilibrium, is zero on the lateral surface, and satisfies compatibility, are all correct descriptions of the Prandtl stress function.
Let's summarize:
- Equilibrium is built into the definition of $\phi$ and holds automatically.
- Compatibility shows up as the Poisson equation $\nabla^2\phi=-2G\theta$ that $\phi$ must solve.
- The stress-free lateral surface forces $\phi=0$ on the boundary of a solid section.
The correct statements are (A), (B) and (C).