Question

The deformation of an open-section bar subjected to pure torsion can be solved by choosing an appropriate Prandtl stress function. Which of the following statements is/are true about the Prandtl stress function?

• A. It satisfies the equilibrium equation
• B. It is zero on the lateral surfaces of the bar
• C. It satisfies the compatibility equation
• D. It does not satisfy the equilibrium equation

Hint:

Recall that shear stresses are defined as derivatives of \(\phi\); this makes equilibrium automatic, while compatibility gives \(\nabla^2\phi=-2G\theta\), and the stress-free lateral surface gives \(\phi=0\) on the boundary.

Answer 1

The Correct Option is A, B, C

Step 1: Recall why the Prandtl stress function is introduced.
In Saint-Venant torsion theory, instead of solving directly for the two shear stress components, Prandtl defined a single scalar function \(\phi(x,y)\) and wrote the shear stresses as \(\tau_{xz} = \dfrac{\partial \phi}{\partial y}\) and \(\tau_{yz} = -\dfrac{\partial \phi}{\partial x}\). The whole point of building the stresses this way is to make one of the governing conditions automatic.

Step 2: Check the equilibrium equation.
For a bar under pure torsion with no body forces, the equilibrium equation in the cross-sectional plane reduces to \(\dfrac{\partial \tau_{xz}}{\partial x} + \dfrac{\partial \tau_{yz}}{\partial y} = 0\). Substituting Prandtl's definitions gives
\[ \frac{\partial^2 \phi}{\partial x \partial y} - \frac{\partial^2 \phi}{\partial y \partial x} = 0 \]
which is true for any smooth function \(\phi\), since mixed partial derivatives are equal. So equilibrium is satisfied identically, by construction, no matter what \(\phi\) turns out to be. Statement (A) is TRUE, and statement (D), which says the opposite, is FALSE.

Step 3: Check the compatibility equation.
Equilibrium alone does not fix \(\phi\); a second condition, compatibility of strains (equivalently, that the warping displacement is single-valued), must also hold. Working this condition through leads to the governing Poisson equation
\[ \nabla^2 \phi = -2 G \theta \]
where \(G\) is the shear modulus and \(\theta\) is the twist per unit length. This equation is exactly the compatibility requirement written in terms of \(\phi\), so any valid Prandtl stress function must satisfy it. Statement (C) is TRUE.

Step 4: Check the boundary condition.
The lateral surface of the bar (its outer boundary, running along the length of the bar) is stress free, since no external traction acts there, only the two end faces carry the applied torque. Working out what this stress-free condition means for \(\phi\) shows that \(\phi\) must stay constant along the entire boundary curve. For a solid, simply connected cross-section (a single unbroken boundary curve, as an open-section bar has), that constant can always be chosen as zero without loss of generality. So \(\phi = 0\) on the lateral surface. Statement (B) is TRUE.

Final Answer:
The Prandtl stress function satisfies the equilibrium equation by construction, satisfies the compatibility equation through \(\nabla^2\phi = -2G\theta\), and equals zero on the lateral surface. \[ \boxed{\text{A, B, C}} \]