Can someone define Section modulus and area moment of inertia "verbally as opposed to mathematically". I am very confused on there relationship to each other. For instance, Youngs Modulus simply put would be the strength of a beam as a function of the material, however, which term would define stength as a function of shape? I am attempting to teach some of this at the high school level without a whole lot of background in this area. Can anyone shed some light on these 2 terms and there relationship to the strength of a beam. I can use all the formulas to calculate deflection etc, however, without understanding the terms it is somewhat meaningless. Thanks!!!
Taking your definition of Young's Modulus as a cue: Moment of inertia is the stiffness (you misuse "strength") of a beam as a function of shape. Section modulus is the maximum stress under a given load as a function of shape. So for a given cross section, the moment of inertia tells us what kind of stiffness to expect, and the section modulus tells us what kind of strength to expect.
This is all horribly general, but it should suit your purposes. You may want to take a quick look at a reference site or a mechanics of materials text to make sure you're solid on the terminology. Beer & Johnston is pretty widely used, although I'm partial to Gere.
I'll just address Young's modulus. This is not about strength - it's about stiffness of a material. One was of visualising the stiffness co efficient is to think of it as the stress that would be needed to double the length of the material. Now its true that most materials fracture at a few percent extension or strain, but that is a function of their strength. Think of an elestic band: what stress is needed to double its length?
Not much! That means it has a small Youngs modulus in the region of just 100's of psi to stretch it to 2X length. But think of a steel rod. To stretch that to double length would take 29 million psi
Both the moment of inertia and the section modulus are measurements of the relative stiffness of a crossection.
The difference is that I (moment of inertia) is used for more general calculations. When calculating the stress in a beam, for example, the formula using I is
stress = M*y / I
Where M is the bending moment at a point on the beam and y is the vertical distance from the bending axis at the middle (centroid) of the crossection. This is a general formula because you can determine what the stress is at any point in the crossection by plugging in a value for y.
But, for a lot of engineering work, you don't really give a damn what the stress is at a quarter-inch from the core of the beam--you just want to know when it will yield. In that case, you'd want the section modulus, Z. It's just had the y divided out of it, so
Z = I/y
Why does this make it more useful? Because if you switch this around it also means that
I = Z*y
If you substitute this into the stress formula you get
stress = M*y / Z*y
The y's cancel out and you're left with
stress = M/Z
This is the stress at the *extreme fiber* of the beam, which is the worst case. And the worst case is what we're usually aiming at when we're designing a beam for strength.
I guess the concise answer is that I is used when you want to know the stress at *any* point and Z is used when you just want to know when a beam yields.
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