Structural analysis
In contrast to theory of elasticity, the models used in structural analysis are often differential equations in one spatial variable.
Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, aircraft and ships.
Important examples related to Civil Engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important.
To design a structure, an engineer must account for its safety, aesthetics, and serviceability, while considering economic and environmental constraints.
For example, columns, beams, girders, the floor slab, roofing, walls, windows, plumbing, electrical fixtures, and other miscellaneous attachments.
Advanced structural analysis may examine dynamic response, stability and non-linear behavior.
The first two make use of analytical formulations which apply mostly simple linear elastic models, leading to closed-form solutions, and can often be solved by hand.
However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary size and complexity.
The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems.
It does, however, make it generally possible to solve these equations, even with highly complex geometry and loading conditions, with the restriction that there is always some numerical error.
The simplest of the three methods here discussed, the mechanics of materials method is available for simple structural members subject to specific loadings such as axially loaded bars, prismatic beams in a state of pure bending, and circular shafts subject to torsion.
The solutions can under certain conditions be superimposed using the superposition principle to analyze a member undergoing combined loading.
In other words, they contain the assumptions (among others) that the materials in question are elastic, that stress is related linearly to strain, that the material (but not the structure) behaves identically regardless of direction of the applied load, that all deformations are small, and that beams are long relative to their depth.
This method is used by introducing a single straight line cutting through the member whose force has to be calculated.
Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries.
The analytical and computational development are best effected throughout by means of matrix algebra, solving partial differential equations.
Early applications of matrix methods were applied to articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referred to as "finite element analysis", model an entire structure with one-, two-, and three-dimensional elements and can be used for articulated systems together with continuous systems such as a pressure vessel, plates, shells, and three-dimensional solids.
Its applicability includes, but is not limited to, linear and non-linear analysis, solid and fluid interactions, materials that are isotropic, orthotropic, or anisotropic, and external effects that are static, dynamic, and environmental factors.
