# How do I apply the finite difference method to a torsion equation?

So I've previously used the method to determine temperature across a thin plate. But my textbook doesn't what to do when the second derivates of space (x and y) are equal to a function.

I'm trying to use the first order aproximation to determine tortion through a prismatic bar. Here is the specific question:

Problems of elastic torsion of prismatic bars are reprsented mathematicallly by the equation:

$$T_{xx}$$ is the second order partial derivative of T with respect to x.

$$T_{xx} - T_{yy} = -2G\theta$$

Where $$G$$ is the elastic shear modulus and $$\theta$$ is the angle of twist for each section. The function T is the stress function, defined as T = 0 on the boundaries.

Consider a bar of rectangular cross section with dimensions 30cm by 30cm. Use the finite difference method to find the value of the stress functionT at all internal nodal points of the bar, adopting a step length $$\Delta x=\Delta y=h=10cm$$ assume values G = 0.01 and $$\theta =2x$$. Generate a system of equations and solve for each internal node.

This is a laplace example I'd been given previously + my attempt for a Poisson equation. I'm particularly unsure about the h**2 coefficient. There aren't examples of solutions to: second orders space derivatives = function.

https://i.imgur.com/imA1Spr.png

Would appreciate any help. Even if it's a, "Yep keep doing that".

• Yeah, seems fine Apr 30 '20 at 23:28
• @David thanks!! May 1 '20 at 0:31
• @RoryMcDonald I just want to point out a typo in your *png file (sign of T_{yy})
– Albe
May 1 '20 at 13:29
• @Albe Oh yeah fixed now. ty May 1 '20 at 15:05