Virtual work in finite plane bending of Euler-Bernoulli beam

1. Jul 17, 2015

c0der

Please refer to the following image, which shows a portion of the deformed centerline of a beam in its equilibrium configuration with a uniformly distributed load.

The stress resultants are the axial forces T, transverse shears Q, and bending moments M at sections 1 and 2, with the rotations being relative to the horizontal.

We apply infinitesimal virtual displacements from the equilibrium configuration, with normal and tangential components $\delta u_n$ and $\delta u_t$ as functinos of arc length s.

If the beam were straight, then the incremental rotation is:

$\delta \theta = \frac{d\delta u_n}{ds}$

And the incremental strain would be:

$\delta \epsilon = \frac{d\delta u_t}{ds}$

I understand that part.

Now, given that the beam is not straight in its equilibrium position:

$\delta \theta = \frac{d\delta u_n}{ds} + \frac{d\theta}{ds}\delta u_t$

$\delta \epsilon = \frac{d\delta u_t}{ds} - \frac{d\theta}{ds}\delta u_n$

It's not intuitive for me how curvature multiplied by displacement gives the extra terms. How are they obtained?

2. Jul 22, 2015