Theory of models

It is common practice in engineering to use models in solving complicated mechanical problems. Models of such structures as bridges and dams are sometimes used to investigate the strength of those structures under actual conditions. Likewise models of ships are tested under various speeds to establish the necessary power of engines. Models are also widely used in airplane design.

In the case of statical problems, the relations between the mechan­ical properties of the model and of the prototype are comparatively simple. For example, if we are interested in stresses produced in a structure by live load and use the known formulas of strength of materials, we find that the stresses are proportional to the applied forces and inversely proportional to the squares of linear dimensions. This indicates that if the model is geometrically similar to the proto­ type with the ratio 1/ of linear dimensions to corresponding actual dimensions, then the stresses in the model will be the same as in the actual structure provided the forces applied to the model are in the ratio 1/ to the actual forces. If the materials of model and prototype are the same, equal stresses produce equal strain and the deformed shape of the model will be geometrically similar to that of the pro­totype. Multiplying the deflections measured on tie model by  we obtain the actual deflections of the structure.

If we are interested in elastic stability of compressed members of a structure, we observe that the magnitude of the critical stress, given by Euler's formula, is

s_cr=²Er²/ l² (37)

This stress is independent of linear dimensions and, for materials with equal module E, is the same for model and prototype. Equations simi­lar to eq. (37) can be established in other cases of instability, which indicates that from model tests, we can find the load at which lateral buckling of members of the actual structure may occur.

If we wish to find from model tests the stresses produced in the prototype by dead load, such as the proper weight of a structure, we find that this problem is more complicated. When the linear dimensions are reduced in the ratio 1/X, the weight is reduced in the ratio 1/³ which indicates that the stresses produced in the model by proper weight will be  times smaller than in the prototype if the materials are the same. To have²⁸ the same stresses in the model as in the prototype, we must use for the model a material the density of which is  times larger than that in the actual structure. This requirement is difficult to realize; and in practical model tests, the insufficiency of proper weight is usually compensated by dead loads distributed over the model.

If we are investigating the behavior of a structure only within the elastic limit of the material, we do not need to compensate for insuf­ficiency of dead load and can dispense with the requirement of equal stresses in model and prototype. It² is only necessary to measure the dead load stresses in the model and then multiply by  to obtain the actual stresses in the structure. In such cases, however, the dead load stresses and deformations in the model will usually be very small and will require very sensitive measuring instruments. To remove this difficulty, the use of model materials having a small modulus of elasticity has been recommended. For example, in studying stresses in arch dams, models made of rubber have been used.

To investigate the behavior of a structure beyond the elastic limit and establish its ultimate strength under dead load, we have to use for the model the same material as for the actual structure. To have stresses of the same magnitude as in the prototype, the model must then be put in a force field, the intensity of which is  times larger than that of the gravitational field. This can be accomplished by revolving the model in a centrifuge. The centrifugal force field can be assumed as approximately uniform if the radius of the centrifuge is large in comparison with the dimensions of the model. Such experi­ments have proved very useful in solving problems encountered in mine structures and in the construction of tunnels. Failure of such structures may occur due to dead load alone; and since the material does not follow Hooker’s law, a theoretical solution becomes very involved. In such cases, a reliable value of the ultimate strength can be obtained from model tests alone.