# Shape of a Bending Beam with P Applied at the Axis of Symmetry

• Keegi Suvaline
In summary, the conversation discusses the shape of a beam that is bent between two supports. It is mentioned that the shape is a half period of a sine according to the Euler column. A force P is then applied at the axis of symmetry and it is questioned what the shape of the liner will be. The conversation also mentions modifying a case from a textbook and applying boundary conditions to keep the length of the liner constant. Finally, the conversation addresses the issue of the ratio h/d and whether there is a solution for both low and high values.

## Homework Statement

A beam (e.g. a steel liner) of a length L is bent between two supports, as shown in fig (a).
According to the Euler column, the shape is a half period of a sine.
Now, a force P is applied at the axis of symmetry - fig (b).
What is the shape of the liner? y=f(x)
Instead of the general case, let's choose P so weak that the liner remains always convex (h1=0.9*h or so ...).

## The Attempt at a Solution

I followed the "Strength of Materials and Structures" of Case-Chilver-Ross (ISBN:0340719206)
and tried to modify the 18.9 - "Strut with uniformly distributed lateral loading"
to a "... with concentrated lateral load" .
How to apply the boundary conditions so that the length of the liner (arc length) L remains constant?

What value is the ratio h/d for your problem? If the ratio is very large, your sinusoidal shape is not correct.

Agreed. But this way the drawing is more clear.
Is there a solution when h/d is:
a) so low that sine shape stands?
b) so large that it is not a sine any more?

## 1. What is the shape of a bending beam with a force applied at the axis of symmetry?

The shape of a bending beam with a force applied at the axis of symmetry is a parabolic curve, also known as a "cantilever" shape. This means that the beam will bend downwards at the center and gradually flatten out towards the ends.

## 2. How does the applied force affect the shape of a bending beam?

The applied force directly affects the magnitude of the bending in the beam. The greater the force, the more the beam will bend and the flatter the parabolic curve will become.

## 3. What is the significance of the axis of symmetry in a bending beam?

The axis of symmetry is the line that divides the bending beam into two equal halves. When a force is applied at this axis, it creates a symmetrical bending pattern in the beam.

## 4. Can the shape of a bending beam be modified by changing the location of the applied force?

Yes, the location of the applied force can greatly affect the shape of a bending beam. If the force is applied closer to one end of the beam, the resulting curve will be steeper and the beam will bend more at that end.

## 5. How does the shape of a bending beam with a force at the axis of symmetry differ from a beam with a force at a different location?

A bending beam with a force at the axis of symmetry will have a symmetrical parabolic curve, while a beam with a force at a different location will have a non-symmetrical curve. The magnitude and direction of the bending will also vary depending on the location of the applied force.