# Do scaled models reflect strength characteristics of full size

Lets say you want to build a bridge out of carbon fiber. This has never been never been done before, and it's super expensive stuff. So your goal is to use as little material as possible, but obviously your bridge still needs to have enough strength.

To start you look up how strong is carbon fiber, and the online data sheets tell you that a 10 oz cloth has a tensile strength of 800KSI.

After running the math this sounds perfect and should give your bridge enough strength for it's uses.

However, before you start a super expensive project that's never been done, based on nothing more than datasheets you found online and equations that might not fully apply to your material you run a sanity check.

Before you start building your full sized bridge, you build a small one that's 1/10th the size.

Assuming this small bridge is exactly the same as the large one, just using 1oz fiber instead of 10 oz fiber would your small bridge endure 1/10th the load of the full sized one?

I don't mean to apply this question to carbon fiber alone. With anything in engineering if you scale up a model perfectly using the same materials would the strength scale up proportionally, or is there some kind of real life nuance that blows all this mathematical beauty out of the water?

• Yes, it is part of Dimensional analysis and dynamic, kinematic and geometric similarity is looked for - some things need to be adjusted to get similar5ity - but that depends on what results you are looking for and which parameters can be ignored or adjusted. – Solar Mike Jul 13 '18 at 17:26
• area of objects scale differently for their volume for example.. So while you can do this its not just a simple scale up. – joojaa Jul 13 '18 at 19:06
• The short answer is you can never assume this to be true. You have to do the analysis to decide how to make the comparisons. Then you have to do validation on the model which will either prove or disprove your analysis and methods. Without validation, it's worthless. UCSD Shake tables – Phil Sweet Jul 13 '18 at 20:03
• Carbon fibre bridges already exist. Not that I think this changes the premise of your question. – AndyT Aug 7 '18 at 15:17

Things scale, but not always in a nice simple way.

For example consider the units of stiffness of part of the structure are $E$ (Young's Modulus) times some length, but the units of weight are $\rho$ (density) times some length cubed.

So a scaled-down model structure will deform much less under its own weight than the original. (This explains why elephants have thicker legs than ants, relative to the size of their bodies.)

The same sort of thing happens with other types of loading, for example wind loads would probably be proportional to a surface area, i.e. length squared.

In real life the joins between the various parts will be important for the strength of the complete structure, and you might not be able to scale those down by a factor of 10 easily. Think about the practical size of bolts and rivets or the thickness of adhesive films, welds, etc.

Geometrical tolerances usually don't scale the way you would like either. Neither do things like defects in the surface finish that might act as sites for crack propagation.

The above list is far from complete. All this doesn't mean that scale models are useless, but you have to think through what you are really trying to do when you use them, and they aren't a "magic solution" to validating a design.

• Last I checked stiffness was EI where E is in force per unit area (length squared) and I is in length^4, giving stiffness in units of force times length squared. Doesn't change your general point though. – AndyT Aug 7 '18 at 15:16

There has been some valid comments on the other answers, at the expense of being repetitive I mention just a few more.

In your analogy of the bridge for example. The cables' strength scales to the surface area of the cable, L^2 but the seismic forces are related to mass of the structure which is volume L^3.

Wind testing of a scaled model is very complex and depending on geometry and openings has to be individually element by element analyzed and scaled back up

In lab tests, say to test the shear strength of a manufacturer's proprietary assembly of a shear wall they do have scaled down models occasionally, but they are aware and calculate all these disparities.

City of Los Angeles accepts sometimes theses test results for performance of some products( LA RR reports) but they expect careful analysis and logic used to justify scaling.

However thanks to powerful simulation algorithms many projects can be directly simulated for analysis from the digital drawings.

According to J. E. Gordon in his book on structures, the answer is no. The reasons are simple: structural strength is a property related to material density. As you decrease the size of the structure you must proportionately decrease its material structural strength. This means you need to use a material that is $\frac 1{10}$ in scale strength also. When you scale size, scale material strength also. This means that its appearance may differ in the model but in performance metrics be similar to the full scale structure.

• Well, some experiments for aerofoils were completed in water tanks as water is approximately 1000 times the density of air so that other ratios could be achieved... – Solar Mike Jul 13 '18 at 19:53
• That may be but as the proportions of stone decrease, is relative strength increases by the square-cube rule. This means that to maintain the same structural strength of a model bridge as a full scale bridge, you must scale down the material strength in proportion of the square-cube rule. Where you intend to use carbon fibre you must use a model material that is a tenth of its strength to simulate the final product behaviour under load – Rhodie Aug 6 '18 at 15:10