Beams in Bending: Bending stress / strain distrution

Question

first post so bear with me please!

Studying Mech Eng and in all notes provided and I have come across online the stress/strain distribution is shown as I have sketched belo (see pic). Surely this is only a representation of the right hand side of the beam (for hogging as shown) as the left would be as I have sketched below.

Am I correct in assuming this? Why do they always just show the first distribution? Why only show the 2 triangles and not the full square? What is the distribution in the middle of the beam (assuming that is where a single point load is applied)?

Hope this is clear enough to understand!

Thanks for you help!

EDIT: Additionally, I feel this is where my confusion orginated from:

I now understand the stress distribution is constant throughout the length of the beam (although unsure as to why it's shown in that particular direction). But can someone please explain why the above image then makes sense. Is this only valid on the right hand side?

Answer 1

Before discussing any such matters, we must define our internal force sign conventions. It is common to define them thusly:

So, tension is positive, compression is negative. Shear forces are positive upwards when on the left side of the cut, but downwards on the right side of the cut (a good way to remember is that positive shear rotates the beam clockwise). Bending moment is positive if it generates tension on the bottom of the section.

When an external bending moment is applied to a beam, it resists deformations by generating an internal stress state. This stress state is such that it can be described by an internal bending moment which is equal and opposite to the external bending moment, cancelling it out and allowing the beam to remain at rest in this new deformed configuration. It also generates zero total axial force (under pure bending).

As can be trivially seen from the deflection, your beam is under negative bending moment. This means the top fibers of the beam are elongated and the bottom fibers are shortened, meaning the internal stress state goes from tension at the top to compression at the bottom.

So, if we cut the beam at midspan, we can choose to look at either the left or right side of the cut, which will give us:

It is obvious that both sides must be equivalent, since they represent the exact same things in the exact same spot, only looking at different sides of it. And indeed, they are exactly the same, once we remember the sign convention.

• Both sides show tension (positive) in the top, compression (negative) in the bottom. Good.
• On the left side of the cut, the tension on top and compression in the bottom would generate a clockwise rotation, which is negative. On the right side of the cut, they would generate a counter-clockwise rotation, which is also negative. Good.
• I didn't represent shear in the image because it isn't relevant to the current question, but we'd expect the forces to either side to be equal and opposite.

Therefore, there is no benefit to displaying the stress distribution on both sides. If you have one, it is all you need and the other is entirely redundant. Now, this also means that one is not more correct than the other. So, if you're going to write a diagram, should you do it with positive axial forces going to right (as in the right side of the cut and in the example you gave in your edit) or to the left? It's entirely up to you. It is customary to have positive axial forces going to the right because, well, usually positive values go to the right.

Answer 2

You seem to be confusing the stress distribution in the beam with the forces on either side of a cut in the beam.

The axial stress component is positive (tension) on the outside of the curve and negative (compression) on the inside.

If you cut the beam, the stress creates equal and opposite forces on the two cut faces. This is the same basic idea as tension in a string, which is equivalent to equal and opposite forces acting on the ends of the string.

Drawing two separate diagrams for the forces doesn't add any more information to the one diagram for the stress. For example you could cut the beam at any angle, and the same find the normal and shear forces on the cut surface from the stress distribution (using Mohr's circle, for example). The single diagram showing the stress distribution contains all the information about the state of the structure.

Answer 3

1. understand that the force diagram is just a representation and depends on many presuppositions, on the other hand if you consider it in an actual practice the results are different than that of standard diagrams, but we can say that the fluctuations are very minimal. Still, it has a different output every time you run it.
2. showing one diagram is enough to explain what is happening throughout the beam, more or less with a factor of safety.
3. now if you want to remember the simple concept of sagging and hogging and to clear the positive and negative confusions, you go through this video this will sure take exactly 2 min 5sec of your life, but trust me you will remember this sign conventions for the rest of your life.