How Beams resist bending?

Beams are in every home and in one sense are quite simple. You put weight on them and they bend but don’t break.

As shown in the diagram below, when bending, the top portion is resisting a desire to shorten while the bottom portion is resisting a desire to lengthen. Notice the center portion has no desire to change length.

For this reason the fibers at the top and bottom play a much larger role in resisting bending which is why modern I- beams are designed with all the material away from the neutral axis whether built of steel, wood or concrete.

So how do we quantify the strength of a particular shape of a beam?

Engineers call this moment of inertia.

Solution to How Moment of Inertia Works:

Consider a cross-sectional slice out of a beam. Let’s name the width of the slice (b) and the height (d).

Moment of inertia (I) is a term engineers use to quantify a beams ability to resist bending based on its shape.

We said earlier that the fibers that are farther from the neutral axis have a larger effect on the beams ability to resist bending and it turns out that they have an effect proportional to the square of their distance from the neutral axis.

If we look at a slice of this slice representing all the fibers of a certain distance (x) from the neutral axis, we can add up all their strengths … which will require integral calculus since there are infinitely many infinitely thin slices.

The area of the rectangular slice is b * dx and it is a distance of x from the neutral axis. (note that dx just means a “little bit of x”)

풅 ퟐ ퟐ (1) 푰 = ퟐ ∫ퟎ 풃풙 풅풙 … this adds up all the rectangular slices above the neutral axis and the coefficient of 2 doubles it to include the bottom.

풃풙ퟑ 푑 (2) 푰 = ퟐ … which must be evaluated at x = ퟑ 2

풃풅ퟑ (3) 푰 = this is the commonly used moment of inertia formula ퟏퟐ

This simple algebra formula numerically expresses a beams ability to resist bending and notice that the height of the beam has a cubed effect on its strength when compared to the width. Hence a beam laid on its side will bend more than when stood vertically.

See the page on how beam deflection works … moment of inertia is one variable in the deflection equation.

Also notice moment of inertia is listed in the Versa-lam design chart and now you can find it for yourself: