# Equilibrium equation for forces in tangential direction

1. Oct 21, 2013

### roldy

1. The problem statement, all variables and given/known data
Derive the differential equilibrium equation in the tangential direction $\theta$

2. Relevant equations
$\sum F_\theta = 0$

$V = 1/2(2r + \Delta r)\Delta r \Delta \theta$

Small angle approximation:
$\cos(\frac{\Delta \theta}{2}) = 1$

$\sin(\frac{\Delta \theta}{2}) = \frac{\Delta \theta}{2}$

3. The attempt at a solution

$(\sigma_{\theta \theta} + \Delta \sigma_{\theta \theta})\cos(\frac{\Delta \theta}{2})\Delta r(1) - \sigma_{\theta \theta}\cos(\frac{\Delta \theta}{2})\Delta r(1) + (\sigma_{r\theta} + \Delta \sigma_{r\theta})\sin(\frac{\Delta \theta}{2})\Delta r(1) - \sigma_{r\theta}\sin(\frac{\Delta \theta}{2})\Delta r(1) + (\sigma_{r\theta} + \Delta\sigma_{r\theta})(r + \Delta r)\Delta \theta - \sigma_{r\theta}r\Delta \theta + \Theta V = 0$

Simplifying:
$\Delta \sigma_{\theta \theta} \Delta r + \Delta \sigma_{r \theta} \frac{\Delta \theta}{2} \Delta r + \sigma_{r \theta} \Delta r \Delta \theta + \Delta \sigma_{r \theta} r \Delta \theta + \Delta \sigma_{r \theta}\Delta r \Delta \theta + 1/2\Theta(2r + \Delta r)\Delta r \Delta \theta = 0$

Divide by $\Delta r \Delta \theta$

$\frac{\Delta \sigma_{\theta \theta}}{\Delta \theta} + \frac{\Delta \sigma_{r \theta}}{2} + \sigma_{r \theta} + \frac{\Delta \sigma_{r \theta}r}{\Delta r} + \Delta \sigma_{r \theta} + 1/2\Theta(2r + \Delta r) = 0$

Take the limit as $\Delta r$ approaches 0 and $\Delta \theta$ approaches 0

$\frac{\partial \sigma_{\theta \theta}}{\partial \theta} + (?) + \sigma_{r \theta} + \frac{\partial \sigma_{r \theta}r}{\partial r} + (?) + r\Theta = 0$

$\frac{1}{r} \frac{\partial \sigma_{\theta \theta}}{\partial \theta} + \frac{1}{r}(?) + \frac{1}{r} \sigma_{r \theta} + \frac{\partial \sigma_{r \theta}}{\partial r} + \frac{1}{r}(?) + \Theta = 0$

I searched online for the correction equation and it is as follows:

$\frac{1}{r} \frac{\partial \sigma_{\theta \theta}}{\partial \theta} + \frac{\partial \sigma_{r \theta}}{\partial r} + \frac{2\sigma_{r \theta}}{r} + \Theta = 0$

I'm not sure what to do about the terms with no $\Delta$ in the denominator when I take the limit. Attached is the figure I'm working from.

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