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Consider a beam of some material with Young modulus $E$, the axial stiffness of the beam is given by the expression

$$k = \frac{A E}{L}$$

where $A$ is cross-sectional area of the beam, and L is the length of the beam

Now, if we increase all the dimensions of the beam by a factor $\lambda$, we get that

$$A' \rightarrow \lambda^2 A $$ $$L' \rightarrow \lambda L$$

the Young modulus, being an intensive property of the material, stays unaffected by the scaling

The above implies that the stiffness scales as

$$k' \rightarrow \lambda k $$

Which is a counterintuitive result. Structurally, one sees that building very large buildings (higher than 500 meters) becomes increasingly difficult and demands stronger materials

How does one reconcile the apparently linear increase of stiffness with dimension, with the practical difficulty in building very large structures?

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  • $\begingroup$ Shouldn't the stiffness be directly proportional to (lambda^2)*A, and indirectly proportional to lambda*L? This means that as length increases, stiffness decreases, but as cross section increases, stiffness increases by the factor squared. Even though Young's modulus is unaffected by scaling, the numerator of the equation increases by the square of lambda, requiring greater cross section to preserve the same stiffness for each unit of length added, which sounds right. $\endgroup$
    – Ernie
    Commented Jul 11, 2015 at 18:57
  • $\begingroup$ the factor $\lambda$ affects all dimensions proportionally, that is the point. So what it means is that a rod with all its dimensions multiplied by ten, will have a stiffness multiplied by ten. So a larger rod is always stiffer than a smaller rod, and I'm hearing you giggle, please don't, this is serious ;) The point is that this scaling behaviour seems counterintuitive $\endgroup$
    – diffeomorphism
    Commented Jul 11, 2015 at 19:03

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The dominant load that a large structure has to face it's its own weight. The weight load scales as volume, which means that it goes as $\lambda^3$

The canonical strain $\epsilon$ upon a given load $F$ is

$$ \epsilon = \frac{F}{k} = \frac{ F L }{ A E} $$

so the strain will scale as $\lambda^3 \lambda^{-1} = \lambda^2$, which means that scaling up the structure, the load will grow faster than the rigidity, meaning that the strain will eventually dominate and buckle or collapse the structure

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  • $\begingroup$ Your expression for strain is incorrect. Strain is $F/(AE)$. If you just look at the units of what you wrote, you'll notice that they have units of length. You wrote what the deflection would be, given an axial load F. The strain, then, would scale with $\lambda$ under gravitational loading. Buckling is a separate issue entirely and is highly dependent on the distribution of the area (via it's 2nd moment), not just the total area. $\endgroup$
    – Tyler Olsen
    Commented Jul 18, 2015 at 15:10

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