I'm reading through Beam Theory and finding it an excellent description an ultra novice can follow.

But then I came to this and it has me held up:

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Bending – if we take a little section of the beam and put a bending moment on each end of the section, the little section of beam becomes curved (a tinyamount) and stresses build up across the section in relation to how far we are from the neutral axis.** And because plane cross sections remain plane, even in the deflected beam, the stresses are linear with distance from the neutral axis. **And so the resistance to bending goes with E (we know that strain and stress are related linearly by Young’s Modulus, E) and with I, the Second Area Moment of Inertia of the section. So now we have: Bending stiffness at any point along a beam is EI.

.......... *******

I'm good up to "because plane sections remain plane" Can someone maybe state that in a different phrase or further explain please?

I think I follow "stresses are linear with distance from neutral" - it simply means the increase in stress is a linear function as you move outward (from neutral axis) and not an exponential function??

**Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is.**

Perhaps I'm too inexperienced for your thread? Sorry.

The analogy of the book being fanned open is a good one. The spacing (or gap) between the pages is linear from the binding to the outer edge of the pages. The gap is equivalent to the strain (material stretch) and therefore directly related to the stress by Young's modulus which is a material dependent constant.

If the pages were to bend or curve in the radial direction, they would no longer define (or be contained within a plane) a "plane" and the distance between the pages (stress) would not be a linear function of the distance from the binding.

Since the material is uniform the relation between stress and strain is constant (Young's modulus). Therefore, the force increases in a linear (straight line) fashion with the distance to the book binding. In the case of the beam the book binding is replaced by the neutral axis (the part that does not stretch).

That is a lot of words to express a fairly simple concept. Sometimes it is necessary to stand back and look at the overall picture. It reminds me of stepping back to look at the forest and ignoring the trees.

Now if you will return to the quote reference above and click to area to expand the quote you will find the next bold print that reads "

**Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is.** "

The part here that I have found most confusing is where the "

**second moment of inertia**" is mentioned.

OK, here we need to go back into the forest and touch a few of the trees.

We have just determined that the resisting force within the beam increases linearly with the distance from the neutral axis. We also know that a moment or torque (twisting force) is the product of a force times a distance.* Therefore, the resisting moment to bending of the beam has the distance to the neutral axis (x) included twice or what is commonly known as x squared (x*x).

It just so happens that these equations for force and bending are the exact same formula for the moment of inertia (and second moment of inertia) about the neutral axis. These are common engineering values that are frequently needed in calculations. These terms are used as a "shorthand" for the equations. The moment of inertia of common shapes are frequently used and are relatively easy to calculate or measure so that term is applied in general. The values (or formulas) for various sections are frequently contained in tables with those terms (

*first*** moment of inertia and

*second* moment of inertia) are used as the column headings in the table.

* If the nut is too tight to loosen with the wrench you are using, you need more torque. Get a longer wrench and you can apply more torque without having to (pull harder) apply more force.

** The first moment is implied if "first" is not mentioned.

I hope that helps.