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the deformation is suп¬ѓciently small to enable the eп¬Ђect of changes in the
geometrical conп¬Ѓguration of the solid to be ignored, whereas in the nonlinear
case the magnitude of the deformation is unrestricted.
Practical stress analysis of solids and structures is unlikely to be served
by classical methods, and currently numerical analysis, predominately in the

1
2 INTRODUCTION

form of the п¬Ѓnite element method, is the only route by which the behav-
ior of a complex component subject to complex loading can be successfully
simulated. The study of the numerical analysis of nonlinear continua using
a computer is called nonlinear computational mechanics, which, when ap-
plied speciп¬Ѓcally to the investigation of solid continua, comprises nonlinear
continuum mechanics together with the numerical schemes for solving the
resulting governing equations.
The п¬Ѓnite element method may be summarized as follows. It is a pro-
cedure whereby the continuum behavior described at an inп¬Ѓnity of points
is approximated in terms of a п¬Ѓnite number of points, called nodes, located
at speciп¬Ѓc points in the continuum. These nodes are used to deп¬Ѓne regions,
called п¬Ѓnite elements, over which both the geometry and the primary vari-
ables in the governing equations are approximated. For example, in the
stress analysis of a solid the п¬Ѓnite element could be a tetrahedra deп¬Ѓned by
four nodes and the primary variables the three displacements in the Carte-
sian directions. The governing equations describing the nonlinear behavior
of the solid are usually recast in a so-called weak integral form using, for
example, the principle of virtual work or the principle of stationary total
potential energy. The п¬Ѓnite element approximations are then introduced
into these integral equations, and a standard textbook manipulation yields
a п¬Ѓnite set of nonlinear algebraic equations in the primary variable. These
equations are then usually solved using the NewtonвЂ“Raphson iterative tech-
nique.
The topic of this book can succinctly be stated as the exposition of the
nonlinear continuum mechanics necessary to develop the governing equations
in continuous and discrete form and the formulation of the Jacobian or
tangent matrix used in the NewtonвЂ“Raphson solution of the resulting п¬Ѓnite
set of nonlinear algebraic equations.

1.2 SIMPLE EXAMPLES OF NONLINEAR STRUCTURAL
BEHAVIOR

It is often the case that nonlinear behavior concurs with oneвЂ™s intuitive
expectation of the behavior and that it is linear analysis that can yield the
nonsensical result. The following simple examples illustrate this point and
provide a gentle introduction to some aspects of nonlinear behavior. These
two examples consider rigid materials, but the structures undergo п¬Ѓnite
displacements; consequently, they are classiп¬Ѓed as geometrically nonlinear
problems.
3
1.2 SIMPLE EXAMPLES OF NONLINEAR STRUCTURAL BEHAVIOR

L
FL/K

20
K
ПЂ/2
F
15
M=K
nonlinear
10
linear
F 5
F

0
Free body diagram 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
(a) (b)

FIGURE 1.1 Simple cantilever.

1.2.1 CANTILEVER
Consider the weightless rigid bar-linear elastic torsion spring model of a
cantilever shown in Figure 1.1(a). Taking moments about the hinge gives
the equilibrium equation as,
FL cos Оё = M (1.1)
If K is the torsional stiп¬Ђness of the spring, then M = KОё and we obtain the
following nonlinear relationship between F and Оё,

FL Оё
= (1.2)
K cos Оё
If the angle Оё в†’ 0, then cos Оё в†’ 1, and the linear equilibrium equation
is recovered as,

K
F= Оё (1.3)
L
The exact nonlinear equilibrium path is shown in Figure 1.1(b), where clearly
the nonlinear solution makes physical sense because Оё < ПЂ/2.

1.2.2 COLUMN
The same bar-spring system is now positioned vertically (see Figure 1.2(a)),
and again moment equilibrium about the hinge gives,

PL Оё
PL sin Оё = M or = (1.4)
K sin Оё
4 INTRODUCTION

PL/K
P P 5
4.5
nonlinear
4
3.5
3
linear
L
2.5
2
1.5
K 1
0.5
M=K bifurcation
0
P 0 1 2 3
в€’3 в€’2 в€’1
Free body diagram

(a) (b)

FIGURE 1.2 Simple column.

The above equilibrium equation can have two solutions: п¬Ѓrstly if Оё = 0, then
sin Оё = 0, M = 0, and equilibrium is satisп¬Ѓed; and secondly, if Оё = 0, then
PL/K = Оё/ sin Оё. These two solutions are shown in Figure 1.2(b), where
the vertical axis is the equilibrium path for Оё = 0 and the horseshoe-shaped
equilibrium path is the second solution. The intersection of the two solutions
is called a bifurcation point. Observe that for PL/K < 1 there is only one
solution, namely Оё = 0 but for PL/K > 1 there are three solutions. For
instance, when PL/K в‰€ 1.57, either Оё = 0 or В±ПЂ/2.
For very small values of Оё, sin Оё в†’ Оё and (1.4) reduces to the linear (in
Оё) equation,

(K в€’ PL)Оё = 0 (1.5)

Again there are two solutions: Оё = 0 or PL/K = 1 for any value of Оё, the
latter solution being the horizontal path shown in Figure 1.2(b). Equation
(1.5) is a typical linear stability analysis where P = K/L is the elastic
critical (buckling) load. Applied to a beam column such a geometrically
nonlinear analysis would yield the Euler buckling load. In a п¬Ѓnite element
context for, say, plates and shells this would result in an eigenvalue analysis,
the eigenvalues being the buckling loads and the eigenvectors being the
corresponding buckling modes.
Observe in these two cases that it is only by considering the п¬Ѓnite dis-
placement of the structures that a complete nonlinear solution has been
achieved.
5
1.3 NONLINEAR STRAIN MEASURES

L
A

a

l

FIGURE 1.3 One-dimensional strain.

1.3 NONLINEAR STRAIN MEASURES

In the examples presented in the previous section, the beam or column re-
mained rigid during the deformation. In general structural components or
continuum bodies will exhibit large strains when undergoing a geometrically
nonlinear deformation process. As an introduction to the diп¬Ђerent ways in
which these large strains can be measured we consider п¬Ѓrst a one-dimensional
truss element and a simple example involving this type of structural com-
ponent undergoing large displacements and large strains. We will then give
a brief introduction to the diп¬ѓculties involved in the deп¬Ѓnition of correct
large strain measures in continuum situations.

1.3.1 ONE-DIMENSIONAL STRAIN MEASURES
Imagine that we have a truss member of initial length L and area A that is
stretched to a п¬Ѓnal length l and area a as shown in Figure 1.3. The simplest
possible quantity that we can use to measure the strain in the bar is the
so-called engineering strain ОµE deп¬Ѓned as,

lв€’L
ОµE = (1.6)
L

Clearly diп¬Ђerent measures of strain could be used. For instance, the change
in length в€†l = l в€’ L could be divided by the п¬Ѓnal length rather than the
initial length. Whichever deп¬Ѓnition is used, if l в‰€ L the small strain quantity
Оµ = в€†l/l is recovered.
An alternative large strain measure can be obtained by adding up all
the small strain increments that take place when the rod is continuously
stretched from its original length L to its п¬Ѓnal length l. This integration
6 INTRODUCTION

process leads to the deп¬Ѓnition of the natural or logarithmic strain ОµL as,

l
dl l
ОµL = = ln (1.7)
l L
L

Although the above strain deп¬Ѓnitions can in fact be extrapolated to
the deformation of a three-dimensional continuum body, this generalization
process is complex and computationally costly. Strain measures that are
much more readily generalized to continuum cases are the so-called Green
strain ОµG and Almansi strain ОµA deп¬Ѓned as,

l2 в€’ L2
ОµG = (1.8a)
2L2
l2 в€’ L2
ОµA = (1.8b)
2l2

Irrespective of which strain deп¬Ѓnition is used, a simple Taylor series anal-
ysis shows that for the case where l в‰€ L, all the above quantities converge
to the small strain deп¬Ѓnition в€†l/l. For instance, in the Green strain case,
we have,

(l + в€†l)2 в€’ l2
ОµG (l в‰€ L) в‰€
2l2
1 l2 + в€†l2 + 2lв€†l в€’ l2
=
l2
2
в€†l
в‰€ (1.9)
l

1.3.2 NONLINEAR TRUSS EXAMPLE
This example is included in order to introduce a number of features as-
sociated with п¬Ѓnite deformation analysis. Later, in Section 1.4, a small
FORTRAN program will be given to solve the nonlinear equilibrium equa-
tion that results from the truss analysis. The structure of this program is,
in eп¬Ђect, a prototype of the general п¬Ѓnite element program presented later
in this book.
We consider the truss member shown in Figure 1.4 with initial and
loaded lengths, cross-sectional areas and volumes: L, A, V and l, a, v re-
spectively. For simplicity we assume that the material is incompressible and
hence V = v or AL = al. Two constitutive equations are chosen based,
without explanation at the moment, on GreenвЂ™s and a logarithmic deп¬Ѓnition
7
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