Structural analysis

Structural analysis is the computation of deformations, deflections, and internal forces or stresses (Stress Eqiuivalents) within structures, either for design or for performance evaluation of existing structures. Structural analysis needs input data such as structural loads, the structure's geometry and support conditions, and the materials' properties. Output quantities may include support reactions, stresses and displacements. Advanced structural analysis may include the effects of vibrations, stability and non-linear behaviours.

There are three approaches to the analysis: the mechanics of materials approach (also know as strength of materials), the elasticity theory approach (which is actually a special case of the more general field of continuum mechanics), and the finite element approach. The first two make use of analytical formulation leading to closed-form solutions. The third, actually a numerical method for solving differential equations, is very widely used for structural analysis. The equations solved by the finite element method are generated by theories of mechanics such as elasticity theory and strength of materials. Analytical formulations apply mostly to simple linear elastic models, while the finite-element method is computer-oriented, applicable to structures of arbitrary size and complexity. Regardless of approach, the formulation is based on the same three fundamental relations: equilibrium, constitutive, and compatibility. The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality.

Each method has noteworthy limitations. The method of mechanics of materials is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory of elasticity allows the solution of structural elements of general geometry under general loading conditions, in principle. Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires the solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution of an ordinary differential equation. The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories (such as the other two discussed here) for equations to solve. 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. Effective and reliable use of this method requires a solid understanding of its limitations.

Mechanics of materials methods

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. Solutions for special cases exist for common structures such as thin-walled pressure vessels.

For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to 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 preliminary design of large structures.

The solutions are based on linear isotropic infinitessimal elasticity and Euler-Bernoulli beam theory. In other words, they contain the asssumptions (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. As with any simplifying assumption in engineering, the more the model strays from reality, the less useful (and more dangerous) the result.

Elasticity methods

Elasticity methods are available for generally for an elastic solid of any shape. Individual members such as beams, columns, shafts, plates and shells may be modeled. The solutions are derived from the equations of linear elasticity. The equations of elasticity of are a system of 15 partial differential equations. Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method such as the finite element method is necessary.

Many of the developments in the mechanics of materials and elasticity approaches have been expounded or initiated by Stephen Timoshenko.

Finite element methods

Finite element method models a structure as an assembly of elements or components with various forms of connection between them. Thus, a continuous system such as a plate or shell is modeled as a discrete system with a finite number of elements interconnected at finite number of nodes. The behaviour of individual elements is characterised by the element's stiffness or flexibility relation, which altogether leads to the system's stiffness or flexibility relation. To establish the element's stiffness or flexibility relation, we can use the mechanics of materials approach for simple one-dimensional bar elements, and the elasticity approach for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by means of matrix algebra.

Early application of matrix methods were for 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. Commercial computer software for structural analysis typically uses matrix finite-element analysis, which can be further classified into two main approaches: the displacement or stiffness method and the force or flexibility method. The stiffness method is, by far, more popular thanks to its ease of implementation as well as of formulation for advanced applications. The finite-element technology is now sophisticated enough to handle just about any system as long as sufficient computing power is available. 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. This, however, does not imply that the computed solution will automatically be reliable because much depends on the model and the reliability of the data input.

Time-line

1452-1519 Leonardo da Vinci made many contributions.
1638: Galileo Galilei published the book "Two New Sciences" in which he examined the failure of simple structures.
1660: Hooke's law by Robert Hooke.
1687: Issac Newton published " Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion.
1750: Euler-Bernoulli beam equation.
1667-1748: Daniel Bernoulli introduced the principle of virtual work.
1707-1783: Leonhard Euler developed the theory of buckling of columns.
1826: L.M. Navier published a treatise on the elastic bahaviors of structures.
1873: Alberto Castigliano presented his dissertation "Intorno ai sistemi elastici", which contains the theorem for computing displacement as partial derivative of the strain energy. This theorem includes the method of least work as a special case.
1936: Hardy Cross' publication of the moment distribution method which was later recognized as a form of the relaxation method applicable to the problem of flow in pipe-network.
1941: A. Hrennikoff sumitted his D.Sc thesis in MIT on the discretization of plane elasticity problems using a lattice framework.
1942: R. Courant divided a domain into finite subregions.
1956: J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp's paper on the "Stiffness and Deflection of Complex Structures". This paper introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today.