For a tall skyscraper, what are the requirements for young/shear modulus?

Question

I am teaching physics 101 for the first time and am discussing stress/strain. As an illustrative example to students, I'd like to make the broad statement, "The primary supports in skyscrapers need a high Young's modulus (to support the structure's own weight without deformation) but a low shear modulus (to allow for sway in case of wind/earthquake/etc)." However, I'm neither a structural engineer nor have I taught this material before so it's somewhat fresh to me. In a general sense, are those statements true?

Answer 1

You are confusing the modulus of a material, with the stiffness of an object made from that material.

Let's assume the skyscraper has a steel frame. Steel is isotropic, so like all isotropic materials, there is an equation linking its Youngs modulus, shear modulus, and Poisson's ratio: $$E = 2G(1 + \nu).$$

Since for most structural metals Poisson's ratio is about $0.3$, $G$ is about $2.6$ times smaller than $E$.

On the other hand, the vertical members of the skyscraper frame might be considered as a uniform square bar with side $a$ and length $L$. (This is greatly over simplified, but it illustrates the point). We can then find the cross section area of the bar, $A = a^2$, and its second moment of area $I = a^4/12$.

The stiffness of each bar under a vertical compressive load (i.e. the weight of the building) is $$k_{\text{axial}} = \frac{EA}{L} = \frac{Ea^2}{L}$$ and the bending stiffness for a wind load is $$k_{\text{bending}} = \frac{3EI}{L^3} = \frac{Ea^4}{4L^3}.$$

The ratio of the stiffnesses is $$\frac{k_\text{axial}}{k_{\text{bending}}} = \frac{4L^2}{a^2}.$$

So the huge difference in the two stiffness (or flexibility) terms actually has nothing to do with the material itself, but with the geometry of the bar.

In a real design, you would change the geometry to increase the bending stiffness, by replacing a single thick solid bar with an arrangement of thinner bars, held apart by a weaker (but cheaper and lighter) material such as concrete. This would make no difference to the axial stiffness, but the second moment of area would then depend on the distance separating the thinner bars, not on the thickness of each individual bar.

Incidentally, note that the formula for the bending stiffness is written in terms of $E$, not $G$. In fact, there is an additional stiffness term which does depend on $G$, but unless $L < 10a$, it is too small to bother about. The only "elementary" stiffness formulas that I can think of which directly involves $G$ are for the torsional stiffness of an object, where the deformation of the material is mainly in shear.

These formulas and basic ideas in this answer will be in any "strength of materials" textbook or website.