# Development of Modern Structural Mechanics

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Development of Structural Mechanics

Structural Mechanics

is a branch of structural engineering concerned with applying Newtonian
mechanics to the
analysis

of deformations, internal forces and stresses within
framed and/or
continuum
structural elements and

sub
-
assemblages, either for design or for performance
evaluation of existing structures.
Structural mechanics
limits the scope of study
from
solid

mechanics
which deals with the deformation and motion of solids including
microscopic analysis
o
f viscoelas
tic and hereditary materials. Advanced structural
analysis

may include the effects of
dynamic, stability
,

and non
-
linear behavior.

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
w
o

methods make use of
analytical formulation
s

-
form solutions. The third, a numerical method of
solving field problems such as displacement field and stress field, etc., is very widely used for
str
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,
structural analysis

is based on the same fundamental relations: equilibrium, constitutive,
compatibility, and strain
-
deformation. The

so
lutions are approximate when any of these
relations are only approximately satisfied, or only
represent
an approximation of reality

(Wikipedia, 2006).

Strength of materials
approach

-

Being
the
simplest of the three discussed, the strength of
materials
app
roach

Except for
the
moment distribution
method
developed by Cross (193
0
), the basic strength of

2

materials methods were available in their current forms

in the second half of the n
ineteenth
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
-
Bernoulli
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

the members
are ideal
ized with one
-
dimensional entit
ies
.
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
gerous) the
result

becomes
.

Elasticity methods

-

Elasticity methods are available for an elastic body of any
general
shape.
Individual members such as beams, columns, rods, plates and shells may be modeled. The
solutions are derived from the equations of

linear elasticity

(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
upon
or initiated by Stephen P. Timoshenko

(1951, 1959,
1961)
. Vasilii Z. Vlasov (1961) generalized many of Timoshenko
's

problems
,

bridging the
theoret
ical gap between the
elementary beam theory of Euler
-
Bernoulli and the modern theory of
thin
-
walled members including non
-
uniform torsion.

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

3

Table 1. Equations of linear elasticity

3D

2D

1D

Strain
-
displacement equations

6

3

1

Stress
-
strain equations (constitutive equations)

6

3

1

Equations of equilibrium (motion)

3

2

1

Total

15

8

3

elements interconnected at
a
finite number of nodes. The behavior of individual elements is
characterized by the
element's stiffness

or flexibility

relation
s
, which altogethe

to the
system's stiffness

or flexibility relation
s
. To establish the element's stiffness or flexibility
relation
s
, one can use the strength of materials approach for simple one
-
dimensional bar
elements, and the elasticity

appr
o
ach for more complex two
-

and three
-
dimensional elements.

he analytical and computational development
s

are best handled throughout by means of matrix
algebra.

Early application of matrix methods were
formulated
for articulated frameworks with
four b
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
ctly
b
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.

Later
,

more advanced matrix methods, referred to as "finite e
lement analysis," model the
structure with one
-
, two
-
, and three
-
dimensional elements to analyze more complex systems such
as pressure vessel
s
, 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
s available

having the number of natural
coordinates exactly equal to the number of kinematic degrees of freedom

for each of these

4

he accuracy of the finite element solutions depends upon the grid
refinement.

Commercial computer codes for structural analysis typically use matrix finite
-
element analysis,
which can be further classified into two main approaches: the displacement or stif
fness method
and the force or flexibility method.
The stiffness method is, by far, more popular thanks to its
ease of

implementation as well as
its

F
inite
-
element
technology is now sophisticated enough to handle jus
t about any system as long as sufficient
computer 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
in nature
.
finite
-
element analysis is gradually extended into exotic territories such as non
-
linear visco
-
elasticity and rheologic
al

material
.
-
element analysis of stru
ctural
system
s has progressed
so much that one can eliminate a vast
number
of experiments.
Well
planed and executed finite
-
element analysis can simulate structural response under external
effects,
though

material properties are primarily determined by exp
eriments.

Time
-
line

-

It is believed that
the early
Egyptians had some empirical rules regarding the
strength of material
s

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
s

were great builders. Vitruvius
recorded

some of their
building method
s

in his book
, "Architecture" (French translation by De Bioul, Brussels, 1816).

5

Mo
st of the knowledge that the Greeks and Roman
s

accumulated was lost during the Middle
Age
s

and only since the Renaissance has it been rediscovered.
According to
Heyman (1999)
,
scientists use

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.

L
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
follows:

1452
-

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."

1667
-

1748: Daniel Bernoulli introduced the principle of virtual work.

1707
-

1783: Leonhard Euler developed the theory of buckling of column.

1750: Euler
-

Bern
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.

193
0
: 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.

6

1956: J. Turner, R.W. Clough, H.C. Marti
n
, and L.J. Topp presented a paper on
"Stiffness and Deflection of Complex Structures.
"

This

paper is recognized as the first
comprehensive treatment of the matrix method/finite element method as it is known
today.

1980
-

present: Implementation and refinement of the general non
-
linear analysis
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
experiment u
p until the middle to late nineteenth century
(Heyman, 1999)
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
ntal results
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
ding
universal testing machines and

strain
-
gages (including vibrating

wire
-
gages) are
being developed
and refined and are readily available. It would be fair to say that this trend may undergo yet
another turn
-
around. As material properties can never be determined by purely analytical means
alone, experimental investigati
ons will continue to be needed.
C
omplex structural response
presently
be analyzed
(MacNeal, 1994)
for systems
including
,

but not
limited to
,

non
-
linearity, hyperelasicity, visco
-
elasticity as affected by environmental

parameters
(such as temperature, atmospheric pressure, moisture contents)
, residual stresses,

and geometric
imperfections. Hence, the majority of expensive and usually time
-
consuming experimental

7

investigations can be eliminated and only selected few tests need to b
e performed to verify and
calibrate the analytically extracted values.

References

Clough, R.W. (1960). "The Finite Element Method in Plane Stress Analysis," Proc. 2nd ASCE
Conf. on Electronic Computation, Pittsburgh, pp. 345
-
378.

Cross, H.

(1930). "Analys
is of Continuous Frames by Distributing Fixed End Moments," Proc.

ASCE, May.

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

York, NY.

Timoshenko, S.P. (1983). History of Strength of Materials, Dover Publications, Inc., New York,
NY.

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
-
Hill,
New York, NY.

Timoshenko, S.P., and Woinowsky
-
Krieger (1959). Theory of Plates and Shells, Second Edition,
McGraw
-
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
-
23.

Vlasov, V.Z. (1961). Thin
-
Walled Elastic Beams, Second Edition, Revised and Augmented,
Israel Program for Scientific Translation, Jerusalem.

Wiki
pedia (2006). Structural Analysis
-

GNU Free Documentation

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(http://en.wikipedia.org/wiki/Structural_analysis).