ABSTRACT
On the basis of the hypotheses of the linear elastic behavior of a material and of small displacements, the problem of elastic plates and shells, and more generally of elastic solids, may be resolved, as we have found in Chapter 3, by means of Lamé’s equation, where the operator [L] is always linear. If, that is, {F} is the vector of the external forces and {η} is the corresponding displacement vector, obtained by resolving Lamé’s equation, the loads should be multiplied by a constant c, also the displacements, and hence the deformations and the static characteristics will be multiplied by the same constant:
L F[ ]{ } = −{ }c cη (10.1)
Furthermore, if {Fa}, {Fb} are two different vectors of the external forces and {ηa}, {ηb} the corresponding displacement fields, in the case of the superposition of the forces, the principle of superposition will also hold good for displacements:
L F F[ ] +{ } = − +{ }η ηa b a b (10.2) A first case of nonlinearity was examined in Chapter 7, where it was shown how the exter-
nal loads do not always increase proportionally to the induced displacements (Figures 7.1b, 7.2b, 7.8b, and 7.9b) in cases where such displacements cannot be considered small. A second fundamental case of nonlinearity will be examined in this chapter, where the ductile behavior of a material will be considered. The first is the case of geometrical nonlinearity, while the second is the case of constitutive nonlinearity of the material.
