Elasticity/Torsion of triangular cylinder

Example: Equilateral triangle
The equations of the three sides are


 * $$\begin{align}

\text{side}~\partial S^{(1)} ~: & f_1(x_1,x_2) = x_1 - \sqrt{3} x_2 + 2a = 0 \\ \text{side}~\partial S^{(2)} ~: & f_2(x_1,x_2) = x_1 + \sqrt{3} x_2 + 2a = 0\\ \text{side}~\partial S^{(3)} ~: & f_3(x_1,x_2) = x_1 - a = 0 \end{align}$$

Let the Prandtl stress function be



\phi = C f_1 f_2 f_3 \, $$

Clearly, $$\phi = 0\,$$ at the boundary of the cross-section (which is what we need for solid cross sections).

Since, the traction-free boundary conditions are satisfied by $$\phi\,$$, all we have to do is satisfy the compatibility condition to get the value of $$C\,$$. If we can get a closed for solution for $$C\,$$, then the stresses derived from $$\phi\,$$ will satisfy equilibrium.

Expanding $$\phi\,$$ out,

\phi = C (x_1 - \sqrt{3} x_2 + 2a)(x_1 + \sqrt{3} x_2 + 2a)(x_1 - a) $$

Plugging into the compatibility condition

\nabla^2{\phi} = 12 C a = -2\mu\alpha $$ Therefore,

C = -\frac{\mu\alpha}{6a} $$ and the Prandtl stress function can be written as

\phi = -\frac{\mu\alpha}{6a} (x_1^3+3ax_1^2+3ax_2^2-3x_1x_2^2-4a^3) $$

The torque is given by

T = 2\int_S \phi dA = 2\int_{-2a}^{a} \int_{-(x_1+2a)/\sqrt{3}}^{(x_1+2a)/\sqrt{3}} \phi dx_2 dx_1 = \frac{27}{5\sqrt{3}} \mu\alpha a^4 $$

Therefore, the torsion constant is

\tilde{J} = \frac{27 a^4}{5\sqrt{3}} $$

The non-zero components of stress are
 * $$\begin{align}

\sigma_{13} = \phi_{,2} & = \frac{\mu\alpha}{a}(x_1-a)x_2 \\ \sigma_{23} = -\phi_{,1} & = \frac{\mu\alpha}{2a}(x_1^2+2ax_1-x_2^2) \end{align}$$

The projected shear stress

\tau = \sqrt{\sigma_{13}^2+ \sigma_{23}^2} $$ is plotted below The maximum value occurs at the middle of the sides. For example, at $$(a,0)$$,

\tau_{\text{max}} = \frac{3\mu\alpha a}{2} $$

The out-of-plane displacements can be obtained by solving for the warping function $$\psi$$. For the equilateral triangle, after some algebra, we get

u_3 = \frac{\alpha x_2}{6a} (3x_1^2 - x_2^2) $$ The displacement field is plotted below