# Structural analysis

Structural analysis is the mathematical calculation of forces, stresses, and deflections within structures, either as part of the design of those structures or as a tool in understanding the performance of existing structures. Pre-defined loads that are applied to a structure (such as its own weight, or the pressure of wind against a building's facade) are force inputs in the calculations; reactions (such as the forces applied to a building's foundations) are force outputs. The calculated stresses in various members (such as beams and columns in a building or wing struts in an airplane) are compared to the allowable stresses for the materials used. The calculated deflections (static movements) are compared to various standards for serviceability. In advanced analysis, vibrations (dynamic movements) may be analyzed as well.

There are two broad classes of analysis: classical methods and matrix methods. The distinction is based on theory: classical methods provide answers by means of analytical formulation, but only for simple structural models; matrix methods can handle structures of any size and complexity, and are computer-oriented using matrix computations. Both approaches, however, are based on the same three fundamental relations: equilibrium, constitutive, and compatibility. The solutions are approximate when any of these relations are only approximately satisfied.

Classical methods for individual members include beam and column formulas. Classical methods for entire structures include the method of sections and method of joints for truss analysis, moment distribution for small rigid frames, and portal frame and cantilever method for large rigid frames. Except for moment distribution, which came into use in the 1930s, these methods were developed in their current forms in the second half of the nineteenth century. They are still used for small structures and for schematic design of large structures.

Matrix methods model a structure as an assembly of small elements with varying forms of connection between elements. The first matrix methods were frame analyses with individual beams and columns used as elements; more advanced matrix methods, usually referred to as "finite element analysis" break an entire structure into small elements and can be used on structures (such as a pressure vessel) with no inherent divisions. Commercial frame analysis computer software typically uses matrix methods.

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