# Deflection of an elastic curve

1. Jul 10, 2013

### hihiip201

Hi:

In mechanics of materials. I learned that the deflection of a beam can be characterized by its elastic curve which is the deformed neutral axis.

But I am bothered by the fact that, if two end of a beam is fixed, and the elastic curve is continuous in between, then it must mean that the length of the neutral axis has elongated.

However, this seems to be inconsistent with the fact that neutral axis should experience no longitudinal strain or stress by definition.

can anyone care to explain this to me?

thanks

2. Jul 10, 2013

### Baluncore

There are two separate issues here.

If one end of the beam rests on a roller then the length of the neutral axis will not change.

If two fixed points are tied by an elastic cable, that cable will sag in a catenary until the elongation tension counters the gravitational force.

3. Jul 10, 2013

### hihiip201

for 1st case: but this is because the neutral axis now span a shorter x distance, so how would one "track" the infinitesimal elements in the before and after deformation given that their x location has likely moved?

2nd : what if the two fixed points are tied by the beam?

Last edited: Jul 10, 2013
4. Jul 10, 2013

### Baluncore

You can model the beam: Either 1. As an elastic deflection that pulls the two endpoints together.
Or 2. As an elastic catenary tension member.

If you model it as both then you will need to define the method of the attachments and the rigidity of the endpoints.

The infinitesimal elements remain in their assigned order in the object but they can be distorted and move in x, y and z as a result of deflection.

5. Jul 10, 2013

### hihiip201

In the textbook they model deflection as change in y and longitudal displacement as change in x as longitudinal displacement.

which bothers me because it doesn't seem like they take into the account that the neutral axis would span a shorter x distance after deflection if its length does not change. Because all the say is that : longitudinal displacement for any x (in original beam) equals y * theta ( theta is the angel the local slope makes with horizontal). In this equation they are assuming that every point on the neutral axis stays on the same x.

Last edited: Jul 10, 2013
6. Jul 10, 2013

### Baluncore

"In the textbook they model deflection as change in y and longitudal displacement as change in x as longitudinal displacement."

You are extending their deflection model beyond its assumptions.
Those simple assumptions apply to very minor deflections, not to major distortions.

7. Jul 10, 2013

### hihiip201

understood, thanks!