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
deformation, internal
forces, and
stress. 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
behavior.
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 is
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
deformation 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.
Methods Using Numerical Approximation
It is common practice to use approximations the solution of
differential equations as the basis for structural analysis. This
is usually done using numerical approximiation techniques. The most
commonly used numerical approximation in structural analysis is the
Finite Element Method.
The finite element method approximates 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 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.
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