Everything about Structural Analysis totally explained
Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. The subjects of structural analysis are engineering artifacts whose integrity is judged largely based upon their ability to withstand loads; they commonly include buildings, bridges, aircraft, and ships. Structural analysis incorporates the fields of mechanics and dynamics as well as the many failure theories. From a theoretical perspective the primary goal of structural analysis is the computation of
deformations, internal
forces, and
stresses. In practice, structural analysis can be viewed more abstractly as a method to drive the engineering design process or prove the soundness of a design without a dependence on directly testing it.
Analytical Methods
To perform an accurate analysis a structural engineer must determine such information as
structural loads,
geometry, support conditions, and materials properties. The results of such an analysis typically include support reactions,
stresses and
displacements. This information is then compared to criteria that indicate the conditions of failure. Advanced structural analysis may examine
dynamic response,
stability and
non-linear behaviour.
There are three approaches to the analysis: the
mechanics of materials approach (also known 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 formulations which apply mostly to simple linear elastic models, lead to closed-form solutions, and can often be solved by hand. The finite element approach is actually a numerical method for solving differential equations generated by theories of mechanics such as elasticity theory and strength of materials. However, the finite-element method depends heavily on the processing power of computers and is more 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.
Limitations
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's always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations.
Strength of materials methods (classical 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 infinitesimal elasticity and Euler-Bernoulli beam theory. 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. 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 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 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 applications 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 the most popular by far 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 isn't 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, doesn't imply that the computed solution will automatically be reliable because much depends on the model and the reliability of the data input.
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