How Do You Determine Displacement and Rotation at Point A in a Beam?

[kasse]

A beam AB has constant EI and u(x) describes the deformation:

http://www.badongo.com/pic/625051

The first task here is to show that there's no displacement or rotation at A. For the case of the displacement, I guess I can do that simply by replacing x with 0 in the u(x)-equation?

But how about the rotation? What is meant by that? Is it the same as slope?

 

[Pyrrhus]

A beam AB has constant EI and u(x) describes the deformation:

http://www.badongo.com/pic/625051

The first task here is to show that there's no displacement or rotation at A. For the case of the displacement, I guess I can do that simply by replacing x with 0 in the u(x)-equation?

But how about the rotation? What is meant by that? Is it the same as slope?

Yes and yes the first derivative du(x)/dx is the rotation or slope (Remember that for small angles $\tan \theta \approx \theta$)

 

[piercas]

I can provide a response to the topic of displacement and rotation in the context of a beam with constant EI and a deformation described by u(x). Firstly, displacement refers to the change in position of a point on the beam from its original position. In this case, since the beam has a constant EI, the displacement at point A can be determined by substituting x=0 into the u(x) equation. This will give us the displacement at point A, which is zero, indicating that there is no change in position at this point.

Rotation, on the other hand, refers to the change in orientation or angle of the beam at a particular point. It is not the same as slope, which refers to the change in elevation or height of the beam at a point. To determine the rotation at point A, we can use the slope of the beam at that point. This can be calculated by taking the derivative of the u(x) equation and substituting x=0. If the slope is zero, then the rotation at point A is also zero, indicating that there is no change in orientation at this point.

In summary, for a beam with constant EI and a deformation described by u(x), the displacement and rotation at point A can be determined by substituting x=0 into the u(x) equation and taking the derivative, respectively. This information is important in understanding the behavior and stability of the beam under different loading conditions.