There are two broad classes of analysis: classical methods and matrix methods. The distinction is based on theory: classical methods provide exact answers, but only for simple structural models; matrix methods can handle structures of any size and complexity, but are approximations. This distinction correlates with the calculation required, in that classical methods typcially require a few iterations of a few algebraic equations while matrix methods require the multiplication of matrices large enough to be impractical for hand calculation.
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 and cantilever analysis 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 secoind 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 matric 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.