Virtual work in finite plane bending of Euler-Bernoulli beam

[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 [itex]\delta u_n[/itex] and [itex]\delta u_t[/itex] as functinos of arc length s.

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

[itex]\delta \theta = \frac{d\delta u_n}{ds}[/itex]

And the incremental strain would be:

[itex]\delta \epsilon = \frac{d\delta u_t}{ds}[/itex]

I understand that part.

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

[itex]\delta \theta = \frac{d\delta u_n}{ds} + \frac{d\theta}{ds}\delta u_t[/itex]

[itex]\delta \epsilon = \frac{d\delta u_t}{ds} - \frac{d\theta}{ds}\delta u_n[/itex]

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

 

[Achy47]

The extra terms in the equations for incremental rotation and strain account for the effect of curvature on the displacement components. In a straight beam, the displacement components are solely determined by the virtual displacements \delta u_n and \delta u_t. However, in a curved beam, the virtual displacements are also influenced by the curvature of the beam, which is represented by \frac{d\theta}{ds}. This is because the curvature of the beam causes the virtual displacements to change as the arc length s changes.

To understand this better, let's consider a simple example. Imagine a straight beam with a small curvature at a specific point. If we apply a virtual displacement in the normal direction at that point, it will result in a small rotation, as expected. However, if we apply the same virtual displacement at a point with a larger curvature, the resulting rotation will be larger. This is because the curvature affects the rotation caused by the virtual displacement.

In the equations for incremental rotation and strain, the extra terms \frac{d\theta}{ds}\delta u_t and -\frac{d\theta}{ds}\delta u_n take into account the influence of curvature on the virtual displacements \delta u_t and \delta u_n, respectively. These terms can be thought of as correction factors that adjust the virtual displacements to account for the curvature of the beam.