Development of Structural Mechanics
is a branch of structural engineering concerned with applying Newtonian
mechanics to the
of deformations, internal forces and stresses within
structural elements and
assemblages, either for design or for performance
evaluation of existing structures.
limits the scope of study
which deals with the deformation and motion of solids including
tic and hereditary materials. Advanced structural
may include the effects of
There are three approaches to structural analysis: the strength of materials approach, the
elasticity theory approach, and t
he finite element approach.
The first t
methods make use of
leading to closed
form solutions. The third, a numerical method of
solving field problems such as displacement field and stress field, etc., is very widely used for
uctural analysis. The equations
needed for the formulation of the finite element method are
based on theories of mechanics such as elasticity theory and strength of materials. Analytical
formulations apply mostly to simple linear elastic problems, while
the finite element method is
computer oriented, applicable to structures of great complexity. Regardless of approach,
is based on the same fundamental relations: equilibrium, constitutive,
compatibility, and strain
lutions are approximate when any of these
relations are only approximately satisfied, or only
an approximation of reality
Strength of materials
simplest of the three discussed, the strength of
is applicable to simple structural members subjected to specific loadings.
developed by Cross (193
), the basic strength of
materials methods were available in their current forms
in the second half of the n
century. They are still widely used for small structures and for preliminary design of large
structures. The solutions are based on linear isotropic infinitesimal elasticity and Euler
elementary beam theory. The simplifying assumptio
ns include that the materials 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
ized with one
Solutions for special cases exist for certain
structures such as thin
walled pressure vessels. As with any simplifying assumptions in
engineering, the more the model deviates from reality, the less useful (and more dan
Elasticity methods are available for an elastic body of any
Individual members such as beams, columns, rods, plates and shells may be modeled. The
solutions are derived from the equations of
(see Table 1)
. 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 necessar
y. Many of the developments in the strength of materials and elasticity
approaches have been expounded
or initiated by Stephen P. Timoshenko
. Vasilii Z. Vlasov (1961) generalized many of Timoshenko
ical gap between the
elementary beam theory of Euler
Bernoulli and the modern theory of
walled members including non
Finite element methods
Finite element methods (Clough, 1960) model a structure as an
assembly of elements or compon
ents 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
Table 1. Equations of linear elasticity
strain equations (constitutive equations)
Equations of equilibrium (motion)
elements interconnected at
finite number of nodes. The behavior of individual elements is
characterized by the
, which altogethe
or flexibility relation
. To establish the element's stiffness or flexibility
, one can use the strength of materials approach for simple one
elements, and the elasticity
ach for more complex two
he analytical and computational development
are best handled throughout by means of matrix
Early application of matrix methods were
for articulated frameworks with
asic elements: planar truss, beam, planar frame, and grid elements; later planar elements
were extended to space elements by multiplying appropriate transformation matrices. The
stiffness/flexibility relations for these basic elements can be expressed exa
y means of the
homogeneous solutions of the respective governing differential equations. Hence, the analysis
results from these basic elements are exact regardless of the grid refinement.
more advanced matrix methods, referred to as "finite e
lement analysis," model the
structure with one
, and three
dimensional elements to analyze more complex systems such
as pressure vessel
, plates, shells, and three dimensional solids. As these advanced elements
used for the finite element analysis a
re derived based on assumed displacement field functions
(there are no exact displacement field function
having the number of natural
coordinates exactly equal to the number of kinematic degrees of freedom
for each of these
advanced elements), t
he accuracy of the finite element solutions depends upon the grid
Commercial computer codes for structural analysis typically use matrix finite
which can be further classified into two main approaches: the displacement or stif
and the force or flexibility method.
The stiffness method is, by far, more popular thanks to its
implementation as well as
formulation for advanced applications.
technology is now sophisticated enough to handle jus
t about any system as long as sufficient
computer power is available. Its applicability includes, but
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
element analysis is gradually extended into exotic territories such as non
elasticity and rheologic
The advancement in finite
element analysis of stru
s has progressed
so much that one can eliminate a vast
planed and executed finite
element analysis can simulate structural response under external
material properties are primarily determined by exp
It is believed that
Egyptians had some empirical rules regarding the
strength of material
so that rules for determining safe dimensions of structural members could
be drawn, for without them it would have been impossible to
erect their great monuments,
temples, pyramids, and obelisks, some of which still exist. The Greeks further advanced the art
of building. Archimedes (287
212 B.C.) established the conditions of equilibrium of a lever and
outlined methods of determining
centers of gravity of objects. The Greeks used his theory in
developing hoisting devices. The Roman
were great builders. Vitruvius
some of their
in his book
, "Architecture" (French translation by De Bioul, Brussels, 1816).
st of the knowledge that the Greeks and Roman
accumulated was lost during the Middle
and only since the Renaissance has it been rediscovered.
scientific and mathematical tools to deepen understanding of the
ir own subject,
while engineers use the same tools in order to do (build) something. In the early days, this
distinction was not possible or needed.
eonardo da Vinci (1452
1519) was a truly outstanding man during the Renaissance. He was
not only the lea
ding artist of his time but also a great scientist and engineer. He studied strength
of materials experimentally
Although Archimedes is occasionally credited to be the father of
statics, modern structural engineering is truly indebted to many brilliant
minds over the long
journey of human civilization
. Significant milestones in this regard may be summarized as
1519: Leonardo da Vinci made many contributions.
1638: Galileo Galilei published the book "Two New Sciences."
1660: Hooke's law b
y Robert Hooke.
1687: Issaac Newton published "Philosophiae Naturalis Principia Mathematica."
1748: Daniel Bernoulli introduced the principle of virtual work.
1783: Leonhard Euler developed the theory of buckling of column.
oulli beam equation.
1783: Alberto Castigliano presented his dissertation "Introno ai sistemi elastici."
1826: L.M. Navier published a treatise on the elastic behavior of structures.
: Hardy Cross published
the moment distribution method.
1941: A. Hrem
mikoff submitted his D.Sc thesis (MIT) dealing with discretization of
plane elasticity problems using a lattice framework.
1956: J. Turner, R.W. Clough, H.C. Marti
, and L.J. Topp presented a paper on
"Stiffness and Deflection of Complex Structures.
paper is recognized as the first
comprehensive treatment of the matrix method/finite element method as it is known
present: Implementation and refinement of the general non
capability into commercially available computer soft
ware such as NASTRAN,
ABAQUS, ADINA, SAP, etc. Exotic new elements are currently being developed and
incorporated into existing computer software.
It would be interesting to note that structural engineers seemed to have preferred analysis to
p until the middle to late nineteenth century
mainly due to the
difficulty associated with creating large enough scale
model test specimens. However, this
situation reversed around
the middle to late nineteenth century
as important experime
were obtained for problems where theory either did not exist or had been insufficiently
developed. This trend continues even today as new
effective experimental tools inclu
universal testing machines and
gages (including vibrating
and refined and are readily available. It would be fair to say that this trend may undergo yet
around. As material properties can never be determined by purely analytical means
alone, experimental investigati
ons will continue to be needed.
omplex structural response
under a given loading can
linearity, hyperelasicity, visco
elasticity as affected by environmental
(such as temperature, atmospheric pressure, moisture contents)
, residual stresses,
imperfections. Hence, the majority of expensive and usually time
investigations can be eliminated and only selected few tests need to b
e performed to verify and
calibrate the analytically extracted values.
Clough, R.W. (1960). "The Finite Element Method in Plane Stress Analysis," Proc. 2nd ASCE
Conf. on Electronic Computation, Pittsburgh, pp. 345
is of Continuous Frames by Distributing Fixed End Moments," Proc.
Heyman, J. (1999). The Science of Structural Engineering, Imperial College Press, London.
MacNeal, R. H. (1994). Finite Elements: Their Design and Performance, Marcel Dekker, New
Timoshenko, S.P. (1983). History of Strength of Materials, Dover Publications, Inc., New York,
Timoshenko, S.P., and Gere, J.M. (1961). Theory of Elastic Stability, Second Edition, McGraw
Hill, New York, NY.
Timoshenko, S.P., and Goodier, J.
N. (1951). Theory of Elasticity, Second Edition, McGraw
New York, NY.
Timoshenko, S.P., and Woinowsky
Krieger (1959). Theory of Plates and Shells, Second Edition,
Hill, New York, NY.
Turner, M.J., Clough,
R.W., Martin, H.C., and Topp, L.J. (19
56). "Stiffness and Deflection
Analysis of Complex Structures," J. Aeronautical Science, Vol. 23, pp. 803
Vlasov, V.Z. (1961). Thin
Walled Elastic Beams, Second Edition, Revised and Augmented,
Israel Program for Scientific Translation, Jerusalem.
pedia (2006). Structural Analysis
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