Structural analysis consists in the verification of the resistance of the sections of a structure, which requires knowing the distribution of internal forces in the structure. The internal forces of a structure isostatic can be found with the equations of static equilibrium. In an indeterminate structure static equilibrium equations are not enough and must meet geometric conditions under load. The internal forces of a structure can be determined by means of elastic analysis or plastic overall analysis, although the latter serves only when certain conditions are satisfied. The internal forces are calculated using different methods, as you can disregard or not the effect of the deformation in the structure. In the first-order theory, calculations relate to the initial geometry of the structure (small deformations).
The forces acting on the bars vary not just with displacement. In the theory of second-order calculations refer to geometry deformed under load of the structure. The displacements of the structure and its effect on forces in the bars are not negligible. The theory of second order, more generally, is used for all cases without restrictions. If we apply the theory of first order, and structure materials satisfy Hooke’s law, we are in a case of linear calculation, where all movements vary linearly with the applied forces.
In this case, the superposition principle can be applied to tensions, deformations, internal forces and displacements due to different actions. The superposition principle says that displacement due to multiple loads acting simultaneously is equal to the sum of the displacements due to the action of each charge separately. This principle cannot be applied if the material stress-strain relationship is not linear, or if the structure (although the material obeys Hooke’s law) does not behave linearly due to geometry changes caused by the applied loads.