Engineers often make scale models of structures they plan to build in order to test function and stability. Sometimes, however, structures fail even when the models function, so engineers have developed extensive theory in order to determine how to build proper scale models and extrapolate reliable results from them.
At first we might assume that a model made of the same material as the intended final structure with each dimension scaled by a single factor will accurately reproduce the behavior of the final structure. That this is not so was known in antiquity by tragic observation, and it was first explained by Galileo around AD 1600. We will not present his detailed argument but will sketch some of the conclusions.
To summarize Galileo's conclusion on this point, when the linear dimensions of a structure are all increased by a factor, the load across any surface increases by the cube of that factor, whereas the strength, or the maximum force the structure can hold across any surface increases by the square of that factor. Therefore, as a structure gets larger, it tends to become unstable, more susceptible to failure.
To illustrate the point, let's consider a block of metal connected to a cylinder, which has much greater length than its diameter and is connected to the ceiling (part A in figure). The stress in the cylinder is the force per area across a cross section. Each material has a threshold stress, such that stress larger than the threshold causes the material to fail. If all the linear dimensions are increased by a factor (part B in figure), then the volume of the block increases by the cube of the factor, as well as the mass and weight of the block. The cross-sectional area of the cylinder increases by the square of the factor, so the stress increases as the factor itself.
This is the simplest example of the subtlety involved in model building.
The figure below depicts a human biceps, which when flexed, has a cross-sectional area at its center of 5 x 10–3 m2 and cross-sectional area at the forearm of 5 x 10–5 m2. If the force exerted at the forearm is F0, what is the force exerted at the shoulder?