Computational Dynamics CPMA.COMUNIDADES.NET

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Overview
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Chapter 1: OVERVIEW 1–2
§1.1 WHERE THE COURSE FITS
The field of Mechanics can be subdivided into four major areas:
Mechanics
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T heoretical
Applied
Computational
Experimental
Theoretical Mechanics deals with fundamental laws and principles of mechanics studied for their
intrinsic value. Applied Mechanics transfers this theoretical knowledge to scientific and engineering
applications, especially as regards the construction of mathematical models of physical phenomena.
Computational Mechanics solves specific problems by combining mathematical models through
numerical methods implemented on digital computers, a process called simulation. Experimental
Mechanics puts physical laws and mathematical methods under the ultimate test of observation. In
this course we shall be concerned with the third branch.
§1.1.1 Computational Mechanics
Several branches of computational mechanics can be distinguished depending on the focus of
attention:
Computational Mechanics
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Particle
Micromechanics
Solids and Structures
Fluids
Coupled Systems
Computational particle mechanics, which tackles phenomena at the molecular, atomic and sub-
atomic levels of matter, falls largely in the realm of physics and chemistry. Micromechanics,
which falls in between particle and continuum mechanics, looks primarily at the molecular through
crystallographic levels of matter.
Computational solid mechanics and structural mechanics are disciplines in engineering and applied
sciences. They are closely related because structures, for obvious reasons, are fabricated with solids.
Computational solid mechanics (CSM) emphasizes the more general applied-sciences approach
and is based on continuum mechanics, whereas structural mechanics emphasizes applications to
engineering design and analysis of structures.
Computational fluid mechanics (CFM)1 deals with problems that involve the motion and equilibrium
of liquid and gases. Well developed CFM subareas are hydrodynamics, aerodynamics, atmospheric
physics, and combustion.
1 More commonly abbreviated CFD for Computational Fluid Dynamics
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1–3 §1.1 WHERE THE COURSE FITS
Coupled systems is a more recent newcomer. This branch is meant to include mechanical systems
that transcend the classical boundaries of solid and fluid mechanics, as in interacting fluids and
structures. Phase change problems such as ice melting and metal solidification fit into this category.
§1.1.2 Statics vs. Dynamics
CSM and structural mechanics may be subdivided according to whether inertial effects are taken
into account or not:
Computational solid and structural mechanics
‰
Statics
Dynamics
In dynamics the time dependence is explicitly considered because the calculation of inertial (and/or
damping) forces requires derivatives respect to actual time to be taken.
Problems in statics may also be time dependent but the inertial forces are ignored or neglected.
Static problems may be classified into strictly static and quasi-static. For the former time need not
be considered explicitly; any historical time-like parameter will do. In quasi-static problems such
as creep deformation, rate-dependent plasticity or fatigue cycling, a more realistic estimation of
time is required but inertial forces are still neglected.
§1.1.3 Linear vs. Nonlinear
A classification of CSM statics that is particularly relevant to this course is
CSM static analysis
n Linear
Nonlinear
Linear static analysis deals with static problems in which the structural response is linear in the
cause-and-effect sense. For example: doubling the applied forces doubles the displacements and
internal stresses. Problems outside this domain are classified as nonlinear.
§1.1.4 Discretization methods
A final classification of CSM static analysis is based on the discretization method by which the
continuous mathematical model is discretized in space, i.e., converted to a discrete model of finite
number of degrees of freedom:
Spatial discretization method
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Finite elements
Boundary elements
Finite differences
For linear problems finite element methods currently dominate the scene, with boundary element
methods posting a strong second choice (and steadily gaining ground in some application areas).
For nonlinear problems the dominance of finite element methods is overwhelming.
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Chapter 1: OVERVIEW 1–4
Finite difference methods in solid and structural mechanics have virtually disappeared from practical
use except for a few one-dimensional problems. This statement is not true, however, for fluid
mechanics, where finite difference discretization methods still dominate.
§1.2 COMPUTATIONAL NONLINEAR STATICS
The topic of this course is the solution of nonlinear static problems by the finite element method.
As in the case of linear finite element methods, we can distinguish several steps:
1. Mathematical modeling. (Also called idealization.) The formulation of the set of mathematical
equations that simulates the physical problem within the scale and accuracy required by the
application.
2. Discretization. The reduction of the mathematical model to a computational model with finite
number of degrees of freedom.
3. Computer implementation. The numerical solution of the computational model on a digital
computer.
4. Result interpretation. The interpretation of the numerical results in terms of their mathematical
and physical significance.
The last aspect is especially important in nonlinear analysis, and cannot be overemphasized. Non-
linear analysis demands a persistent attention to the underlying physics to avoid getting astray as
the “real world” is covered by layer upon layer of mathematics and numerics.
§1.3 THE SOLUTION MORASS
Why is concern for physics of paramount importance? A key component of finite element nonlinear
analysis is the solution of the nonlinear algebraic systems of equations that arise upon discretization.
FACT
The numerical solution of nonlinear systems in “black box” mode is much
more difficult than in the linear case.
The key difficulty is tied to the essentially obscure nature of general nonlinear systems, about which
very little can be said in advance. And you can be sure that Murphy’s law2 works silently in the
background.
2 If something can go wrong, it will go wrong.
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1–5 §1.4 HISTORICAL BACKGROUND
One particularly vexing aspect of dealing with nonlinear systems is the solution morass. A deter-
minate system of 1, 1000, or 1000000 linear equations has, under mild conditions, one and only
one solution. The computer effort to obtain this solution can be estimated fairly accurately if the
sparseness (or denseness) of the coefficient matrix is known. Thus setting up linear equation solvers
as “black-box” stand-alone subroutines is a perfectly sensible thing to do.
By way of contrast, a system of 1000 cubic equations has 31000 … 10300 solutions in the complex
plane. This is much, much larger than the number of atoms in the Universe, which is merely 1050
give or take a few. Suppose just several billions or millions of these are real solutions. Which
solution(s) have physical meaning? And how do you compute those solutions without wasting time
on the others?
This combinatorial difficulty is overcome by the concept of continuation, which engineers also call
incremental analysis. Briefly speaking, we start the analysis from an easily computable solution
— for example, the linear solution — and then try to follow the behavior of the system as actions
applied to it are changed by small steps called increments. The previous solution is used as a starting
point for the iterative solution-search procedure. The underlying prescription: follow the physics.
This technique is interwined with the concept of response explained in Chapter 2.
REMARK 1.1
Not surprisingly, incremental analysis was used by theaerospace engineers that first used the finite element
method for geometrically nonlinear analysis in the late 1950s. Techniques have been considerably refined
since then, but the underlying idea remains the same.
We conclude this overview with a historical perspective on nonlinear finite element methods in
solid and structural mechanics, along with a succint bibliography.
§1.4 HISTORICAL BACKGROUND
In the history of finite element methods the year 1960 stands out. The name “finite element method” appears
for the first time in the open literature in an article by Clough [52]. And Turner, Dill, Martin and Melosh [187]
publish a pioneering paper in nonlinear structural analysis. The then-five-year-old “direct stiffness method”
(what we now call displacement-assumed finite element method) was applied to
“problems involving nonuniform heating and/or large deflections : : : in a series of linearized steps. Stiffness
matrices are revised at the beginning of each step to account for changes in internal loads, temperatures, and
geometric configuration.”
Thirty years and several thousand publications later, computerized nonlinear structural analysis has acquired
full adult rights, but has not developed equally in all areas.
The first fifteen years (1960-1975) were dominated by formulation concerns. For example, not until the
late 1960s were correct finite-deflection incremental forms for displacement models rigorously derived. And
interaction of flow-like constitutive behavior with the spatial discretization (the so called “incompressibility
locking” effects) led to important research into constitutive equations and element formulations.
While the investigators of this period devoted much energy to obtaining correct and implementable nonlinear
finite-element equations, the art of solving such equations in a reliable and efficient manner was understandably
neglected. This helps to explain the dominance of purely incremental methods. Corrective methods of Newton
type did not get much attention until the early 1970s, and then only for geometrically nonlinear problems. At
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Chapter 1: OVERVIEW 1–6
the time of this writing, progress in numerical solution techniques has been uneven: well developed for certain
problems, largely a black art in others. To understand the difference, it pays to distinguish between smooth
nonlinearities and rough nonlinearities.
§1.4.1 Smooth Nonlinearities
Problem with smooth nonlinearities are characterized by continuous, path-independent nonlinear relations at
the local level. Some examples:
1. Finite deflections (geometric nonlinearities). Nonlinear effects arise from strain-displacement equations,
which are well behaved for all strain measures in practical use.
2. Nonlinear elasticity. Stresses are nonlinear but reversible functions of strains.
3. Follower forces (e.g., pressure loading). External forces are smooth nonlinear functions of displacements.
A unifying characteristic of this problem is that nonlinearities are of equality type, i.e., reversible, and these
relations are continuous at each point within the structure. Mathematicians call these smooth mappings.
It is important to point out, however, that the overall structural behavior is not necessarily smooth; as witnessed
by the phenomena of buckling, snapping and flutter. But at the local level everything is smooth: nonlinear
strain-displacement equations, nonlinear elasticity law, follower pressures.
Methods for solving this class of problems are highly developed, and have received a great deal of attention
from the mathematical and numerical analysis community. This research has directly benefitted many areas
of structural analysis.
Let us consider finite deflection problems as prototype. Within the finite element community, these were
originally treated by purely incremental (step-by-step) techniques; but anomalies detected in the mid-1960’s
prompted research into consistent linearizations. A good exposition of this early work is given in the book by
Oden [123]. Once formulation questions were settled, investigators had correct forms of the “residual” out-
of-balance forces and tangent stiffness matrix, and incremental steps began to be augmented with corrective
iterations in the late 1960s. Conventional and modified Newton methods were used in the corrective phase.
These were further extended through restricted step (safeguarded Newton) and, more recently, variants of the
powerful conjugate-gradient and quasi-Newton methods.
But difficulties in detecting and traversing limit and bifurcation points still remained. Pressing engineering
requirements for post-buckling and post-collapse analyses led to the development of displacement control,
alternating load/displacement control, and finally arclength control. The resultant increment-control methods
have no difficulty in passing limit points. The problem of reliably traversing simple bifurcation points without
guessing imperfections remains a research subject, while passing multiple or clustered bifurcation points
remains a frontier subject. A concerted effort is underway, however, to subsume these final challenges.
These reliable solution methods have been implemented into many special-purpose finite element programs,
and incorporation into general-purpose programs is proceeding steadily.
REMARK 1.2
As noted above, incremental methods were the first to be used in nonlinear structural analysis. Among the pre-1970
contributions along this line we may cite Argyris and coworkers [9, 10], Felippa [62], Goldberg and Richard [82], Marcal,
Hibbitt and coworkers [89,108,109], Oden [122], Turner, Martin and coworkers [110,187,188],
REMARK 1.3
The earliest applications of Newton methods to finite element nonlinear analysis are by Oden [122], Mallet and Marcal [107],
and Murray and Wilson [117,118]. During the early 1970s Stricklin, Haisler and coworkers at Texas A&M implemented
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1–7 §1.4 HISTORICAL BACKGROUND
and evaluated self-corrective, pseudo-force, energy-search and Newton-type methods and presented extensive comparisons;
see Stricklin et. al. [175–178], Tillerson et. al. [185], and Haisler et. al. [85]. Almroth, Brogan, Bushnell and coworkers
at Lockheed began using true and modified Newton methods in the late 1960s for energy-based finite-difference collapse
analysis of shells; see Brogan and Almroth [34], Almroth and Felippa [5], Brush and Almroth [37], and Bushnell [38–39].
By the late 1970s Newton-like methods enjoyed widespread acceptance for geometrically nonlinear analysis.
REMARK 1.4
Displacement control strategies for finite element post-buckling and collapse analysis were presented by Argyris [11] and
Felippa [62] in 1966, and generalized in different directions by Sharifi and Popov [165,166] (fictitious springs), Bergan et.
al. [25,26], (current stiffness parameter), Powell and Simons [142] and Bergan and Simons [27] (multiple displacement
controls). A modification of Newton’s method to traverse bifurcation points was described by Thurston [184]. Arclength
control schemes for structural problems may be found in the following source papers: Wempner [194], Riks [154], Schmidt
[160], Crisfield [48,49], Ramm [146], Felippa [65–66], Fried [73], Park [133], Padovan [130,131], Simo et.al. [167], Yang
and McGuire [199], Bathe and Dvorkin [20]. Other articles of particular interest are Bathe and Cimento [18], Batoz and
Dhatt [22], Bushnell [39], Bergan [27], Geradin et al. [79,80]. Meek and Tan [112], Ramm [146,147], Riks [155–157], Sobel
and Thomas [170], Zienkiewicz [200,201,203]. Several conferences have been devoted exclusively to nonlinear problems
in structural mechanics, for example [12,28,17,125, 179,180,198]. Finite element textbooks and monographs dealing rather
extensively with nonlinear problems are by Oden [123], Bathe [19], Bushnell [40], White [196] and Zienkiewicz [202].
REMARK 1.5
In the mathematical literaturethe concept of continuation (also called imbedding) can be traced back to the 1930s. A survey
of the work up to 1950 is given by Ficken [70]. The use of continuation by parameter differentiation as a numerical method
is attributed to Davidenko [54]. Key papers of this early period are by Freudenstein and Roth [72], Deist and Sefor [58] and
Meyer [114], as well as the survey by Wasserstrom [191]. This early history is covered by Wacker [190].
REMARK 1.6
Arclength continuation methods in the mathematical literature are generally attributed to Haselgrove [87] and Klopfestein
[99] although these papers remained largely unnoticed until the late 1970s. Important contributions to the mathematical
treatment are by Abbott [1], Anselone and Moore [8], Avila [14], Brent [31], Boggs [29], Branin [30], Broyden [35,36],
Cassel [43], Chow et. al. [45], Crandall and Rabinowitz [47], Georg [77,78], Keller and coworkers [44,56,57,93–96],
Matthies and Strang [111], Moore [115,116], Po¨nish [139,140], Rheinboldt and coworkers [59,113, 149–150], Watson
[192] and Werner and Spence [195]. Of these, key contributions in terms of subsequent influence are [45,94,149]. For
surveys and edited proceedings see Allgower [2,3], Byrne and Hall [42], Ku¨pper [104,105], Rall [145], Wacker [190], and
references therein. Textbooks and monographs dealing with nonlinear equation solving include Chow and Hale [46], Dennis
and Schnabel [61], Kubı´cˇek and Hlava´cˇek [102], Kubı´cˇek and Marek [103], Ortega and Rheinboldt [128], Rabinowitz [143],
Rall [144], Rheinboldt [153], and Seydel [164]. Of these, the book by Ortega and Rheinboldt remains a classic and an
invaluable source to essentially all mathematically oriented work done prior to 1970. The book by Seydel contains material
on treatment of conventional and Hopf bifurcations not readily available elsewhere. Nonlinear equation solving is interwined
with the larger subject of optimization and mathematical programming; for the latter the textbooks by Gill, Murray and
Wright [81] and Fletcher [71] are highly recommended.
§1.4.2 Rough Nonlinearities
Rough nonlinearities are characterized by discontinuous field relations, usually involving inequality constraints.
Examples: flow-rule plasticity, contact, friction. The local response is nonsmooth.
Solution techniques for these problems are in a less satisfactory state, and case-by-case consideration is called
for. The local and overall responses are generally path-dependent, an attribute that forces the past response
history to be taken into account.
The key difficulty is that conventional solution procedures based on Taylor expansions or similar differential
forms may fail, because such Taylor expansions need not exist! An encompassing mathematical treatment
is lacking, and consequently problem-dependent handling is presently the rule. For this class of problems
incremental methods, as opposed to incremental-iterative methods, still dominate.
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Chapter 1: OVERVIEW 1–8
REMARK 1.7
Earliest publications on computational plasticity using finite element methods are by Gallagher et. al. [74], Argyris [9,10],
Marcal [108], Pope [138] and Felippa [62]. By now there is an enormous literature on the numerical treatment of inelastic
processes, especially plasticity and creep. Fortunately the survey by Bushnell [40], although focusing on plastic buckling,
contains over 300 references that collectively embody most of the English-speaking work prior to 1980. Other important
surveys are by Armen [13] and Willam [197]; see also the chapter by Willam in this volume. For contact problems, see
Oden [126], Bathe and Chaudhary [19], Kikuchi and Oden [97,98], Simo et. al. [167] Stein et. al. [173], Nour-Omid and
Wriggers [121], and references therein.
§1.4.3 Hybrid Approach
What does an analyst do when faced with an unfamiliar nonlinear problem? If the problem falls into the
smooth-nonlinear type, there is no need to panic. Robust and efficient methods are available. Even if the
whizziest methods are not implemented into one’s favorite computer program, there is a wealth of theory and
practice available for trouble-shooting.
But what if the problem include rough nonlinearities? A time-honored general strategy is divide and conquer.
More specifically, two powerful techniques are frequently available: splitting and nesting.
Splitting can be used if the nonlinearities can be separated in an additive form:
Smooth + Rough
This separation is usually done at the force level. Then the smooth-nonlinear term is treated by conventional
techniques whereas the rough-nonlinear term is treated by special techniques. This scheme can be particularly
effective when the rough nonlinearity is localized, for example in contact and impact problems.
Nesting may be used when a simple additive separation is not available. This is best illustrated by an actual
example. In the early 1970s, some authors argued that Newton’s method would be useless for finite-deflection
elastoplasticity, as no unique Jacobian exists in plastic regions on account of loading/unloading switches. The
argument was compelling but turned out to be a false alarm. The problem was eventually solved by “nesting”
geometric nonlinearities within the material nonlinearity, as illustrated in Fig. 1.1.
In the inner equilibrium loop the material law is “frozen”, which makes the highly effective Newton-type meth-
ods applicable. The non-conservative material behavior is treated in an outer loop where material properties
and constitutive variables are updated in an incremental or sub-incremental manner.
Another application of nesting comes in the global function approach (also called Rayleigh-Ritz or reduced-
basis approach), which is presently pursued by several investigators. The key idea is to try to describe the
overall response behavior by a few parameters, which are amplitudes of globally defined functions. The small
nonlinear system for the global parameters is solved in an inner loop, while an external loop involving residual
calculations over the detailed finite element model is executed occasionally.
Despite its inherent implementation complexity, the global function approach appears cost-effective for smooth,
path-independent nonlinear systems. This is especially so when expensive parametric studies are involved, as
in structural optimization under nonlinear stability constraints.
REMARK 1.8
For geometric-material nesting and subincremental techniques see Bushnell [38–41], and references therein. The global-
function approach in its modern form was presented by Almroth, Stern and Brogan [7] and pursued by Noor and coworkers
under the name of reduced-basis technique; see Noor and Peters [119] and Noor [120] as well as the chapter by Noor in this
volume. For perturbation techniques see the survey by Gallagher [75].
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1–9 §1.4 HISTORICAL BACKGROUND
§1.4.4 Summary of Present Status
Solution techniques for smooth nonlinearities are in a fairly satisfactory state. Although further refinements
in the area of traversing bifurcation points can be expected, incremental-iterative methods implemented with
general increment control appear to be as reliable as an engineer user may reasonably expect.
For rough nonlinearities, case-by-case handling is still necessary in view of the lack of general theories and
implementation procedures. Separation or nesting of nonlinearities, when applicable, can lead to signifi-
cant gains in efficiency and reliability, but at the cost of programming complexity and problem-dependent
implementations.
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2
A Tour of
Nonlinear
Analysis
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Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–2
§2.1 INTRODUCTION
This chapter reviews nonlinear structural problems by looking at the manifestation and physical
sources of nonlinear behavior.
We begin by introducing response as a pictorial characterization of nonlinearity of a structuralsystem. Response is a graphical representation of the fundamental concept of equilibrium path.
This concept permeates the entire course because of both its intrinsic physical value and the fact
that incremental solution methods (mentioned in Chapter 1) are based on it.
Finally, nonlinearities are classified according to their source in the mathematical model of con-
tinuum mechanics and correlated with the physical system. Examples of these nonlinearities in
practical engineering applications are given.
§2.2 EQUILIBRIUM PATH AND RESPONSE DIAGRAMS
The concept of equilibrium path plays a central role in explaining the mysteries of nonlinear
structural analysis. This concept lends itself to graphical representation in the form of response
diagrams. The most widely used form of these pictures is the load-deflection response diagram.
Through this representation many key concepts can be illustrated and interpreted in physical,
mathematical or computational terms.
§2.2.1 Load-deflection response
The gross or overall static behavior of many structures can be characterized by a load-deflection
or force-displacement response. The response is usually drawn in two dimensions as a x-y plot
as illustrated in Figure 2.1. In this figure a “representative” force quantity is plotted against a
“representative” displacement quantity. If the response curve is nonlinear, the structure behavior is
nonlinear.
REMARK 2.1
We will see below that a response diagram generally depicts the relationship between inputs and outputs. Or,
in more physical terms, between what is applied and what is measured. For structures the most common inputs
are forces and the most common outputs are displacements or deflections1
REMARK 2.2
The qualifier “representative” implies a choice among many possible candidates. For relatively simple struc-
tures the choice of load and deflection variables is often clear-cut from considerations such as the availability
of experimental data. For more complex structures the choice may not be obvious, and many possibilities may
exist.
REMARK 2.3
This type of response should not be confused with what in structural dynamics is called the response time
history. A response history involves time plotted usually along the horizontal axis, with either inputs or outputs
plotted vertically.
1 A deflection is the magnitude or amplitude of a displacement. Displacements are vector quantities whereas deflections
are scalars.
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2–3 §2.2 EQUILIBRIUM PATH AND RESPONSE DIAGRAMS
Equilibrium path
Representative
 load
Representative 
 deflection
Reference state
Figure 2.1. A load-deflection response diagram.
§2.2.2 Terminology
A smooth curve shown in a load-deflection diagram is called a path.2 Each point in the path
represents a possible configuration or state of the structure. If the path represents configurations
of static equilibrium it is called an equilibrium path. Each point in an equilibrium path is called an
equilibrium point. An equilibrium point is the graphical representation of an equilibrium state or
equilibrium configuration.
The origin of the response plot (zero load, zero deflection) is called the reference state because it is
the configuration from which loads and deflections are measured. However, the reference state may
be in fact chosen rather arbitrarily, and this freedom is exploited in some nonlinear formulations
and solution methods, as we shall see later.
For problems involving perfect structures3 the reference state is also unstressed and undeformed,
and is also an equilibrium state. This means that an equilibrium path passes through the reference
state, as in Figure 2.1.
The path that crosses the reference state is called the fundamental equilibrium path or fundamental
path for short. (Some authors also call this a primary path.) The fundamental path extends from
the reference state up to special states called critical points informally described in §2.3. Any path
2 The terms branch and trajectory are also used. “Branch” is commonly used in the treatment of bifurcation phenomena,
whereas “trajectory” has temporal connotations.
3 A concept to be explained later in connection with stability analysis. A perfect structure involves some form of
idealization such as perfectly centered loads or perfect fabrication. An imperfect structure is one that deviates from
that idealization in measurable ways.
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Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–4
 Critical point
Fundamental or
 primary path
 Secondary path
Reference state
Representative
 load
Representative 
 deflection
Initial linear
 response
Figure 2.2. Fundamental (primary) and secondary equilibrium paths.
that is not a fundamental path but connects with it at a critical point is called a secondary path. See
Figure 2.2.
§2.3 SPECIAL EQUILIBRIUM POINTS
Certain points of an equilibrium path have special significance in the applications and thus receive
special names. Of interest to our subject are critical, turning and failure points.
§2.3.1 Critical points
Critical points are characterized mathematically in later chapters. It is sufficient to note here that
there are two types:
1. Limit points, at which the tangent to the equilibrium path is horizontal, i.e. parallel to the
deflection axis, and
2. Bifurcation points, at which two or more equilibrium paths cross.
At critical points the relation between the given characteristic load and the associated deflection
is not unique. Physically, the structure becomes uncontrollable or marginally controllable there.
This property endows such points with engineering significance.
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2–5 §2.4 LINEAR RESPONSE
Linear fundamental path
Representative
 load
Representative 
 deflection
goes on forever
Figure 2.3. The response diagram for a purely linear structural model.
§2.3.2 Turning points
Points at which the tangent to the equilibrium path is vertical, i.e. parallel to the load axis, are called
turning points. These are not critical points and have less physical significance, but are of interest
for some structures. They have some computational significance, however, because they can affect
the performance of certain solution methods.
§2.3.3 Failure points
Points at which a path suddenly stops or “breaks” because of physical failure are called failure
points. The phenomenon of failure may be local or global in nature. In the first case (e.g, failure of
a noncritical structure component) the structure may regain functional equilibrium after dynamically
“jumping” to another equilibrium path. In the latter case the failure is catastrophic or destructive
and the structure does not regain functional equilibrium.
In the present exposition, bifurcation, limit, turning and failure points are often identified by the
letters B, L, T and F, respectively.
Equilibrium points that are not critical are called regular.
§2.4 LINEAR RESPONSE
A linear structure is a mathematical model characterized by a linear fundamental equilibrium path
for all possible choices of load and deflection variables. This is shown schematically in Figure 2.3.
The consequences of such behavior are not difficult to foresee:
1. A linear structure can sustain any load whatsoever and undergo any displacement magnitude.
2. There are no critical, turning or failure points.
3. Response to different load systems can be obtained by superposition.
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Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–6
4. Removing all loads returns the structure to the reference position.
The requirements for such a model to be applicable are:
† Perfect linear elasticity for any deformation
† Infinitesimal deformations
† Infinite strength
These assumptions are not only physically unrealistic but mutually contradictory. For example, if
the deformations are to remain infinitesimal for any load, the body must rigid ratherthan elastic,
which contradicts the first assumption. Thus, there are necessarily limits placed on the validity of
the linear model.
Nonetheless, the linear model can be a good approximation of portions of the nonlinear response. In
particular, the fundamental path response in the vicinity the reference state; see for instance Figure
2.2. Because for many structures this segment represents the operational or service range, the linear
model is widely used in design calculations. The key advantage of this is that the superposition-
of-effects principle applies. Practical implications of the failure of the superposition principle are
further discussed in Chapter 3.
§2.5 TANGENT STIFFNESS AND STABILITY
The tangent to an equilibrium path may be informally viewed as the limit of the ratio
force increment
displacement increment
This is by definition a stiffness or, more precisely, the tangent stiffness associated with the repre-
sentative force and displacement. The reciprocal ratio is called flexibility or compliance.
The sign of the tangent stiffness is closely associated with the question of stability of an equilibrium
state. A negative stiffness is necessarily associated with unstable equilibrium. A positive stiffness
is necessary but not sufficient for stability.4
If the load and deflection quantities are conjugate in the virtual work sense, the area under a
load-deflection diagram may be interpreted as work performed by the system.
§2.6 GENERALIZED RESPONSE
It is often useful to be able to generalize the load-displacement curve of Figure 2.1 in the following
way.
A control-state response involves two ingredients:
1. A control parameter, called ‚, plotted along the vertical axis versus
4 These sign criteria would be sufficient for a one-degree-of-freedom system.
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2–7 §2.7 RESPONSE FLAVORS
Equilibrium path
State parameter µ or u 
Control parameter λ 
Figure 2.4. A control-state response diagram.
2. A state parameter, called u or „, plotted along the horizontal axis.5
We shall see in following Chapters that ‚ and u (or „) characterize in some way the actions applied
to the structure and the state of the structure, respectively.
A diagram such as that shown in Figure 2.4 is called a control-state response. Throughout this
exposition the abbreviated term response is frequently used in this generalized sense. In practice the
control parameter is often a load amplitude whereas the state parameter is a displacement amplitude.
Thus the usual load-deflection response is a particular case of the control-state response.
REMARK 2.4
The interpretation of the tangent-to-the-path as stiffness discussed in §2.5 does not necessarily carry over
to more general control-state diagrams. Similarly, the interpretations of the sign of the tangent and of the
enclosed-area in terms of stability indicator and stored work, respectively, do not necessarily hold. This is
because control and state are not necessarily conjugate in the virtual work sense.
§2.7 RESPONSE FLAVORS
The response diagrams in Figure 2.5 illustrate three “monotonic” types of response: linear, hard-
ening, and softening. Symbols F and L identify failure and limit points, respectively.
A response such as in (a) is characteristic of pure crystals, glassy, and certain high strength composite
materials.
A response such as in (b) is typical of cable, netted and pneumatic (inflatable) structures, which may
be collectively called tensile structures. The stiffening effect comes from geometry “adaptation”
to the applied loads. Some flat-plate assemblies also display this behavior initially.
5 We shall use the symbol „ primarily for dimensionless state quantities.
2–7
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–8
F F L
F
R
 R R
(a) (b) (c)
Figure 2.5. Basic flavors of nonlinear response: (a) Linear until brittle failure,
(b) Stiffening or hardening, (c) Softening.
L
L
T
T
L
L
F
F
F
F
T
L
BB
B
(d) (e) (f) (g) 
R R R R
Figure 2.6. More complex response patterns: (d) snap-through,
(e) snap-back, (f) bifurcation, (g) bifurcation combined with
limit points and snap-back.
A response such as in (c) is more common for structure materials than the previous two. An almost
linear regime is followed by a softening regime that may occur slowly or suddenly. More “softening
flavors” are given in Figure 2.6.
The diagrams of Figure 2.6 illustrate a “combination of basic flavors” that can complicate the
response as well as the task of the analyst. Here B and T denote bifurcation and turning points,
respectively.
The snap-through response (d) combines softening with hardening following the second limit point.
2–8
2–9 §2.9 SOURCES OF NONLINEARITIES
The response branch between the two limit points has a negative stiffness and is therefore unstable.
(If the structure is subject to a prescribed constant load, the structure “takes off” dynamically when
the first limit point is reached.) A response of this type is typical of slightly curved structures such
as shallow arches.
The snap-back response (e) is an exaggerated snap-through, in which the response curve “turns
back” in itself with the consequent appearance of turning points. The equilibrium between the
two turning points may be stable and consequently physically realizable. This type of response is
exhibited by trussed-dome, folded and thin-shell structures in which “moving arch” effects occur
following the first limit point; for example cylindrical shells with free edges and supported by end
diaphragms.
In all previous diagrams the response was a unique curve. The presence of bifurcation (popularly
known as “buckling” by structural engineers) points as in (f) and (g) introduces more features. At
such points more than one response path is possible. The structure takes the path that is dynamically
preferred (in the sense of having a lower energy) over the others. Bifurcation points may occur in
any sufficiently thin structure that experiences compressive stresses.
Bifurcation, limit and turning points may occur in many combinations as illustrated in (g). One
striking example of such a complicated response is provided by thin cylindrical shells under axial
compression.
§2.8 ENGINEERING APPLICATIONS
Nonlinear Structural Analysis is the prediction of the response of nonlinear structures by a com-
bination of mathematical modeling, discretization methods and numerical techniques. As noted in
Chapter 1, finite element methods dominate the discretization step.
Table 2.1 summarizes the most important applications of nonlinear structural analysis.
§2.9 SOURCES OF NONLINEARITIES
A response diagram characterizes only the gross behavior of a structure, as it might be observed
simply by conducting an experiment on a mechanical testing machine. Further insight into the source
of nonlinearity is required to simulate such physical behavior with mathematical and computational
models.
For structural analysis there are four sources of nonlinear behavior. The corresponding nonlinear
effects are identified by the terms material, geometric, force B.C. and displacement B.C., in which
B.C. means “boundary conditions.” In this course we shall be primarily concerned with the last
three types of nonlinearity, with emphasis on the geometric one.
The four sources are discussed in more detail in following sections. To remember where the nonlin-
ear terms appear in the governing equations, it is useful to recall the fields that continuum mechanics
deals with, and the relationships among these fields. For linear solid continuum mechanics infor-
mation is presented in Figures 2.7 and 2.8.6
6 There is a close analog in structural mechanics, in which one or more fieldsare replaced by generalized quantities;
for example stresses may be replaced by stress resultants such as bending moments.
2–9
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–10
Table 2.1 Engineering Applications of Nonlinear Structural Analysis
Application Explanation
Strength analysis How much load can the structure support before
global failure occurs?
Deflection analysis When deflection control is of primary importance
Stability analysis Finding critical points (limit points or bifurcation
points) closest to operational range
Service configuration analysis Finding the “operational” equilibrium form of certain
slender structures when the fabrication and service
configurations are quite different (e.g. cables, inflat-
able structures, helicoids)
Reserve strength analysis Finding the load carrying capacity beyond critical
points to assess safety under abnormal conditions.
Progressive failure analysis A variant of stability and strength analysis in which
progressive deterioration (e.g. cracking) is consid-
ered.
Envelope analysis A combination of previous analyses in which multiple
parameters are varied and the strength information
thus obtained is condensed into failure envelopes.
In linear solid mechanics or linear structural mechanics the connecting relationships shown in Figure
2.8 are linear, and so are the governing equations obtained by eliminating all fields but one.
Any of these relations, however, may be nonlinear. Tracing this fact back to physics gives rise to
the types of nonlinearities depicted in Figure 2.9. Relations between body force and stress (the
equilibrium equations) and between strains and displacements (the kinematic equations) are closely
linked in a “duality” sense, and so the term geometric nonlinearities applies collectively to both sets
of relations. The force BC nonlinearities couple displacements and applied forces (surface tractions
and/or body forces) and thus bring the additional links drawn in Figure 2.8.
In the following sections these sources of nonlinearities are correlated to the physics in more detail.
2–10
2–11 §2.10 GEOMETRIC NONLINEARITY
Strains Stresses
Traction
 B.C.s
Constitutive
 equations
 Strain-
displacement
 equations
Prescribed
 displacem.
 Internal
displacem.
 Body
 forces
Equilibrium
 equations
Prescribed
 surface
 tractions
Displacement
 B.C.s
Figure 2.7. Fields in solid continuum mechanics
and connecting relationships.
d u
e
b
σ
^ u = d
^
in V on St
t^
= E eσ ^
on Sd
σn = t
e = D u
in V
D + b = 0Tσ
in V
Figure 2.8. Same as Figure 2.7, with symbols and equations
written down for the linear case.
§2.10 GEOMETRIC NONLINEARITY
Physical source
Change in geometry as the structure deforms is taken into account in setting up the strain-
displacement and equilibrium equations.
Applications
Slender structures in aerospace, civil and mechanical engineering applications. Tensile structures
such as cables and inflatable membranes. Metal and plastic forming. Stability analysis of all types.
2–11
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–12
 Material
nonlinearities
Displacement B.C.
 nonlinearities
 Geometric
nonlinearities
 Force B.C.
nonlinearities
d^ u
e
b
σ t^
Figure 2.9. Graphical depiction of sources of nonlinearities
in solid continuum mechanics.
Mathematical source
Strain-displacement equations:
e D Du .2:1/
The operator D is nonlinear when finite strains (as opposed to infinitesimal strains) are expressed
in terms of displacements. Internal equilibrium equations:
b D ¡D⁄¾ .2:2/
In the classical linear theory of elasticity, D⁄ D DT is the formal adjoint of D, but that is not
necessarily true if geometric nonlinearities are considered.
REMARK 2.5
The term geometric nonlinerities models a myriad of physical problems:
Large strain. The strains themselves may be large, say over 5%. Examples: rubber structures (tires, mem-
branes), metal forming. These are frequently associated with material nonlinearities.
Small strains but finite displacements and/or rotations. Slender structures undergoing finite displacements
and rotations although the deformational strains may be treated as infinitesimal. Example: cables, springs,
arches, bars, thin plates.
Linearized prebucking. When both strains and displacements may be treated as infinitesimal before loss of
stability by buckling. These may be viewed as initially stressed members. Example: many civil engineering
structures such as buildings and stiff (non-suspended) bridges.
2–12
2–13 §2.12 FORCE BC NONLINEARITY
§2.11 MATERIAL NONLINEARITY
Physical source
Material behavior depends on current deformation state and possibly past history of the deformation.
Other constitutive variables (prestress, temperature, time, moisture, electromagnetic fields, etc.)
may be involved.
Applications
Structures undergoing nonlinear elasticity, plasticity, viscoelasticity, creep, or inelastic rate effects.
Mathematical source
The constitutive equations that relate stresses and strains. For a linear elastic material
¾ D Ee .2:3/
where the matrix E contains elastic moduli. If the material does not fit the elastic model, general-
izations of this equation are necessary, and a whole branch of continuum mechanics is devoted to
the formulation, study and validation of constitutive equations.
REMARK 2.6
The engineering significance of material nonlinearities varies greatly across disciplines. They seem to occur
most often in civil engineering, that deals with inherently nonlinear materials such as concrete, soils and
low-strength steel. In mechanical engineering creep and plasticity are most important, frequently occurring
in combination with strain-rate and thermal effects. In aerospace engineering material nonlinearities are
less important and tend to be local in nature (for example, cracking and “localization” failures of composite
materials).
REMARK 2.7
Material nonlinearities may give rise to very complex phenomena such as path dependence, hysteresis, local-
ization, shakedown, fatigue, progressive failure. The detailed numerical simulation of these phenomena in
three dimensions is still beyond the capabilities of the most powerful computers.
§2.12 FORCE BC NONLINEARITY
Physical Source
Applied forces depend on deformation.
Applications
The most important engineering application concerns pressure loads of fluids. These include
hydrostatic loads on submerged or container structures; aerodynamic and hydrodynamic loads
caused by the motion of aeriform and hydroform fluids (wind loads, wave loads, drag forces). Of
2–13
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–14
more mathematical interest are gyroscopic and non-conservative follower forces, but these are of
interest only in a limited class of problems, particularly in aerospace engineering.
Mathematical source
The applied forces (prescribed surface tractionsbt and/or body forces b) depend on the displacements:
bt Dbt.u/; b D b.u/; .2:4/
the former being more important in practice.
§2.13 DISPLACEMENT BC NONLINEARITY
Physical source
Displacement boundary conditions depend on the deformation of the structure.
Applications
The most important application is the contact problem, in which no-interpenetration conditions
are enforced on flexible bodies while the extent of the contact area is unknown. Non-structural
applications of this problem pertain to the more general class of free boundary problems, for
example: ice melting, phase changes, flow in porous media. The determination of the essential
boundary conditions is a key part of the solution process.
Mathematical source
For the contact problem: prescribed displacementsbd depend on internal displacements u:
bd Dbd.u/ .2:5/
More complicated dependenciescan occur in the free-boundary problems mentioned above.
2–14
2–15 Exercises
Homework Exercises for Chapter 2
A Tour of Nonlinear Analysis
EXERCISE 2.1
Explain the difference, if any, between a load-deflection response and a control-state response.
EXERCISE 2.2
Can the following occur simultaneously: (a) a limit and a bifurcation point, (b) a bifurcation and a turning
point, (c) a limit and a turning point, (d) two bifurcation points coalescing into one. If you answer “yes” to an
item, sketch a response diagram to justify that reply.
EXERCISE 2.3
In §2.10–13, nonlinearities are classified according to physical source into geometric, material, force boundary
conditions, and displacement boundary conditions. For each of the following mechanical systems indicate the
source(s) of nonlinearity that you think are significant; note that there may be more than one. (If you are not
familiar with the underlying concepts, read those sections.)
(a) a long, slender elastic pipe bent under end couples while the pipe material stays elastic. See Figure E2.1.
(b) an inflating balloon. See Figure E2.2.
(c) a cable deflecting under action of wind forces while its material stays elastic. See Figure E2.3.
(d) a forming process in which hot metal is extruded through a rigid die. See Figure E2.4.
(e) a metal anchor is drilled into the soil to serve as a cable support; the hole is then filled with concrete.
See Figures E2.5 and E2.6. The question refers to the soil-drilling process, ignoring dynamics.
(f) a hefty bird — say a condor — sucked into an aircraft jet engine. Ignore dynamics; engine is the structure,
bird the load.
EXERCISE 2.4
Can you think of a mechanical component that has the load-deflection response diagram pictured in Figure
E2.7? (Explain why). Hint: Think of a helicoidal spring.
2–15
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–16
Slender tube bent
by end couples
Figure E2.1. Slender elastic pipe bent under end couples for Exercise 2.3(a).
Figure E2.2. Inflating balloon for Exercise 3(b).
2–16
2–17 Exercises
Wind
Cable
wind load
Figure E2.3. Cable deflecting under wind forces for Exercise 2.3(c).
Hot metal
Die
Figure E2.4. Hot metal extruded trough a rigid die for Exercise 2.3(d).
2–17
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–18
Figure E2.5. Drill element of a cable anchor, for Exercise 2.3(e).
���
���
���
���
���
���
���
���
���
���
concrete grouting
hole
soil
�
�
�
�
�
(a) (b)
Figure E2.6. Configuration of cable anchor after drilling in the soil, for Exercise 2.3(e).
Axial force
Axial deflection (shortening)
B
R
III
II
I
Figure E2.7. A “mystery” response diagram for Exercise 2.4.
2–18
2–19 Solutions to Exercises
Homework Exercises for Chapter 2
Solutions
EXERCISE 2.1
A control-state response is a generalization of the ordinary load-deflection response which plots an input
parameter (the control) versus an output parameter (the state).
EXERCISE 2.2
(a) Yes, it is easy to construct actual examples. See Figure E2.8(a). (b) Yes: examples are rare but response
easily pictured. See Figure E2.8(b). (c) Only if the response curve is allowed to have “corner” points, as
in contact problems, and the two tangents are perpendicular and parallel to the load axis, respectively. See
Figure E2.8(c). As this is a highly unlikely occurrence, answer “no” is taken as correct. (d) Yes: it is called a
multiple bifurcation point. See Figure E2.8(d).
B&T
B&T
B(multiple)
B&L
(a) (b) (c) (d) 
Corner needed
(highly unlikely)
L&T
Figure E2.8. Sketches for Exercise 2.2.
EXERCISE 2.3
(a) main source: geometric (change in pipe geometry is important).
(b) geometric (change in geometry important); material (rubber is elastic but nonlinear); force B.C. (pressure
is normal to deforming balloon wall). Displacement B.C. nonlinearities might occur for some inflation
devices, but need not be mentioned.
(c) geometric (change in cable geometry important), force B.C. (aerodynamic loading).
(d) geometric (change of geometry important); material (constitutive equations of flowing metal are highly
nonlinear); force B.C. (boundary forces change as metal displaces); displacement B.C. (contact condi-
tions).
(e) geometric (change of geometry as drilling progresses); material (soil behavior is nonlinear to allow drill
penetration); force B.C. (friction forces change direction as drill moves); displacement B.C. might be
present (contact conditions).
2–19
Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–20
(f) geometric (turbines blades undergo large rotations); force B.C. (aerodynamic loads); displacement B.C.
(bird contacts engine blades). Material nonlinearity in engine blades possible depending on impact
velocity and mass/stiffness of bird.7
EXERCISE 2.4
A compressed helicoidal spring. See sketch in Figure E2.9. In state I, the spring behaves, well, as a linear
spring. In state II the spring has “bottomed out” and is much stiffer. In state III the spring has buckled as an
Euler column and its load capacity rapidly diminishes.
PP
P
P
P
P
I II III
Figure E2.9. A structure that exhibits the “mystery response” of Figure E2.7.
7 The FAA has actually a “chicken cannon” that shoots whole chickens (dead, of course) into running jet engines or
other artifacts, such as high speed train mockups, to test what happens. There is a story that once by mistake a frozen
chicken was used : : :
2–20
-
3
Residual Force
Equations
3–1
Chapter 3: RESIDUAL FORCE EQUATIONS 3–2
Chapters 3 through 6 describe basic properties of systems of algebraic nonlinear equations that
depend on one or more control parameters. Algebraic means that these systems contain a finite
number of equations and unknowns.
Those systems result from the discretization of the continuum models of nonlinear structures. The
most widely used discretization method is the displacement-based Finite Element Method, or FEM.
The description of FEM discretization techniques is deferred until Chapter 8 and following. For
the moment it is assumed that the discretization has been carried out.
Physically the algebraic systems represent equilibrium of forces at the discrete level. More specifi-
cally, if the discrete model comes from FEM, the equilibrium of nodal forces. These are collectively
known as force residual equations or residual equations for short. In the present Chapter, associ-
ated differential forms of use in later Chapters are presented, and the important concept of staging,
through which multiple control parameters are reduced to one, is introduced.
§3.1 EQUILIBRIUM EQUATIONS
Discrete equilibrium equations encountered in nonlinear static structural analysis formulated by
the displacement method may be presented in the compact force residual form
r.u;⁄/ D 0: .3:1/
Here u is the state vector containing the degrees of freedom that characterize the configuration of
the structure, r is the residual vector that contains out-of-balance forces conjugate to u, and ⁄ is
an array of assignable control parameters.
In structural mechanics, control parameters are commonly mechanical load levels, but may also be
prescribed physical or generalized displacements, temperature variations, imperfection amplitudes
and even (in design and optimization) geometric dimensions or material properties. The degrees
of freedom collected in u are usually physical or generalized unknown displacements. The names
behavior or configuration variables are also used, however, to stress a more general significance in
other applications. In a general mathematical context, u and ⁄ are called the active and passive
variables, respectively.
The dependenceof r on u and ⁄ is assumed to be piecewise smooth so that first and second
derivatives exist except possibly at isolated critical points. If the system is conservative,1 r is the
gradient of the total potential energy 5.u;⁄/ for fixed ⁄:
r D @5
@u
; or ri D @5
@ui
: .3:2/
Then (3.1) expresses that equilibrium is associated with a energy-stationarity condition.
An alternative version of (3.1) that displays more physical meaning is the force-balance form:
p.u/ D f.u;⁄/: .3:3/
1 A property studied more carefully in Chapter 6.
3–2
3–3 §3.1 EQUILIBRIUM EQUATIONS
In this form p denotes the configuration-dependent internal forces resisted by the structure whereas
f are the control-dependent external or applied forces, which may also be configuration dependent.
The residual (3.2) is either r D p ¡ f or r D f ¡ p, the two versions being equivalent except for
sign. This expression states that if (3.2) is verified, internal forces p balance the applied forces f.
If a total potential energy exists, the decomposition associated with (3.2) is
p D @U
@u
; f D @P
@u
; .3:4/
where U and P are the internal and external energy components, respectively, of 5 D U ¡ P .
As noted at the start of this Chapter, the formulation of these discrete equations using the Finite
Element Method is treated in Chapter 7 and following. For the moment it is simply assumed that,
given u and ⁄, a computational method exists that returns r. In addition, most solution methods
require derivative information of the type discussed in §3.3.
EXAMPLE 3.1
Consider the following residual equilibrium equations
r1 D 4u1 ¡ u2 C u2u3 ¡ 631 D 0;
r2 D 6u2 ¡ u1 C u1u3 ¡ 332 D 0;
r3 D 4u3 C u1u2 ¡ 332 D 0:
.3:5/
The vector form of (3.5) is
r D
"
r1.u;⁄/
r2.u;⁄/
r3.u;⁄/
#
D
" 4u1 ¡ u2 C u2u3 ¡ 631
6u2 ¡ u1 C u1u3 ¡ 332
4u3 C u1u2 ¡ 332
#
D 0; with u D
"
u1
u2
u3
#
; ⁄ D
h
31
32
i
: .3:6/
The force-balance vector form of (3.5) is p D f, in which
p D
"
p1.u/
p2.u/
p3.u/
#
D
" 4u1 ¡ u2 C u2u3
6u2 ¡ u1 C u1u3
4u3 C u1u2
#
; f D
" f1.⁄/
f2.⁄/
f3.⁄/
#
D 3
" 231
32
32
#
: .3:7/
In this particular case f does not depend on u.
REMARK 3.1
The function u.⁄/ characterizes the equilibrium surface of the structure in the space spanned by u and ⁄. In
a general mathematical context the set of fu,⁄g pairs that satisfies (3.1) is called a solution manifold.
REMARK 3.2
The usefulness of the residual equation (3.1) is not restricted to static problems. It is also applicable to nonlinear
dynamical systems
r.u.¿ /;⁄.¿ // D 0; .3:8/
which have been discretized in time ¿ by implicit methods.2 In this case a system of nonlinear equations such
as (3.3) arises at each time station.
2 ¿ is used throughout this course for real time in lieu of t , which is used more extensively to denote pseudo-time.
3–3
Chapter 3: RESIDUAL FORCE EQUATIONS 3–4
REMARK 3.3
Equation (3.1) or its alternative version (3.3) are restricted in that no account for historical effects is made.
It is sufficient, however, for the class of problems considered here. In addition to geometric and force
B.C. nonlinearities, these forms are applicable to nonlinear elasticity and several types of displacement B.C.
nonlinearities.
REMARK 3.4
A more general form would include the history of past deformations:
r.u; 4;⁄/ D 0;
where 4 is a functional of the past history of the deformation gradients. This generalized form is needed for
the treatment of inelastic path-dependent materials.
§3.2 STIFFNESS AND CONTROL MATRIX
Varying the vector r with respect to the components of u while keeping ⁄ fixed yields the Jacobian
matrix K:
K D @r
@u
; with entries Ki j D @ri
@uj
: .3:9/
This is called the tangent stiffness matrix in structural mechanics applications. The inverse of K, if
it exists, is denoted by F D K¡1; a notation suggested by the name flexibility matrix used in linear
structural analysis for the reciprocal of the stiffness. If system (3.1) derives from a potential, both
K and F are symmetric matrices (see Exercise 3.5).
Varying the negative of r with respect to ⁄ while keeping u fixed yields
Q D ¡ @r
@⁄
; with entries Qi j D ¡ @ri
@3j
: .3:10/
REMARK 3.5
There is no agreed upon name for Q. In the sequel it is called the control matrix, although the name “loads
matrix” is also appropriate. The specialization of Q to the usual incremental load vector q is discussed in
Chapter 4.
§3.3 PARAMETRIC REPRESENTATIONS AND RATE FORMS
Parametric representations of u and ⁄ are useful in the description of solution methods as pseudo-
dynamical processes. The general form is
u D u.t/; ⁄ D ⁄.t/ ; .3:11/
where t is a dimensionless time-like parameter. Derivatives with respect to t will be denoted by
superposed dots, as in real dynamics. The first two t-derivatives of the residual in component form
are (with summation convention implied):
3–4
3–5 §3.4 REDUCTION TO SINGLE CONTROL PARAMETER: STAGING
Pri D @ri
@uj
Puj C @ri
@3j
P3j ; .3:12/
Rri D @ri
@uj
Ruj C
h @2ri
@uj@uk
Puk C @
2ri
@uj@3k
P3k
i
Puj C @ri
@3j
R3j C
h @2ri
@3j@uk
Puk C @
2ri
@3j@3k
P3k
i
P3j : .3:13/
In matrix form,
Pr D K Pu¡Q P⁄; .3:14/
Rr D K RuC PK Pu¡Q R⁄¡ PQ P⁄: .3:15/
Note that both PK and PQ are matrices. Their .i; j/ components are shown in square brackets in
(3.13). On the other hand, terms such as @2ri=@uj@uk , are three-dimensional arrays that may be
visualized as “cubic matrices.” Matrices PK and PQ are projections of those arrays on the subspace
spanned by the directions Pu and P⁄. Often these matrices can be more expediently formed by direct
time-differentiation:
PK D dK
dt
; PQ D dQ
dt
: .3:16/
§3.4 REDUCTION TO SINGLE CONTROL PARAMETER: STAGING
Multiple control parameters are quite common in real-life nonlinear problems. They are the analog
of multiple load conditions in linear problems. But in the linear world, multiple load conditions can
be processed independently because any load combination is readily handled by superposition. In
nonlinear problems, however, control parameters are not varied independently. This aspect deserves
an explanation, as it is rarely mentioned in the literature.
Typically the analysis proceeds as follows. The user defines the control parameters to the computer
program during a model preprocessing phase. To illustrate the process, let us assume that for the
analysis of a suspension bridge (Figure 3.1) there are six control parameters
⁄ D [31;32; : : : ; 36 ]T
where parameter31 is associated with own weight,32 and33 with live loads,34 with temperature
changes, 35 with foundation settlement and 36 with wind velocity.
Suppose that 31 D 10 corresponds to full own weight. The first nonlinear analysis involves going
from
⁄re f D [ 0; 0; 0; 0; 0; 0 ]T
to
⁄W D [ 10; 0; 0; 0; 0; 0 ]T
3–5
Chapter 3: RESIDUAL FORCE EQUATIONS 3–6
Figure 3.1. The Brooklyn Bridge in 1876, drawn by F. Hildebrand for the Library of Congress.
Next, assume that the effect of a temperature drop of¡20–C is to be investigated. If a unit increment
of 34 corresponds to 1–C, then the next nonlinear analysis corresponds to going from ⁄W to
⁄T D [ 10; 0; 0; ¡20; 0; 0 ]T
A live load analysis might entail going from ⁄W to
⁄L L D [ 10; ¡20; 6; 0; 0 ]T
and so on.
Each of these processes is called an analysis stage or simply stage. A stage can be defined as
“advancing the solution” from
⁄A to ⁄B
when the solution uA is known. Furthermore, if we assume that the components of ⁄ will vary
proportionally, we can introduce a single control parameter ‚ that varies from 0 through 1 according
to
⁄ D .1¡ ‚/⁄A C ‚⁄B .3:17/
3–6
3–7 §3.4 REDUCTION TO SINGLE CONTROL PARAMETER: STAGING
This ‚ is called the stage control parameter. The nonlinear equation to be solved in.A! B/ is
r.u; ‚/ D 0 .3:18/
with the initial condition u D uA at ‚ D 0. The solution curve u D u.‚/ is called the response of
the structure in the .A! B/ stage.
The importance of staging in nonlinear static analysis arises from the inapplicability of the super-
position principle of linear analysis. For example, the sequences
⁄A ! ⁄B ! ⁄C ⁄A ! ⁄C .3:19/
⁄A ! ⁄B ! ⁄A ⁄A ! ⁄C ! ⁄A .3:20/
do not necessarily produce the same final solution. Of course, this phenomenon is especially
important in intrinsically path-dependent problems.
REMARK 3.6
In mathematical circles, (3.17) receive names such as linear homotopy imbedding and piecewise linear imbed-
ding.
REMARK 3.7
The vector ure f for ⁄re f D 0 is called the reference or initial configuration. Commonly ure f D 0, i.e.,
the reference configuration is the origin of displacements. This is not necessarily a physically attainable
configuration. In the suspension bridge example, ure f is fictitious because bridges are not erected in zero
gravity fields; on the other hand the own-weight configuration uW is physically relevant.
REMARK 3.8
Many analysis sequences do start from the same configuration, for example the own-weight solution uW in
the suspension bridge analysis. A comprehensive analysis system must therefore provides facilities for saving
selected solutions on a permanent database, and the ability to restart the analysis from any saved solution.
REMARK 3.9
The definition of ‚ as a .0! 1/ parametrization of the .A! B/ stage is somewhat artificial for the following
reasons. First, there is no guarantee that a solution at ⁄B exists, so ‚ D 1 may not be in fact attainable.
Second, in stability analysis (3.20) defines only a direction in control parameter space; in this case the user
wants to know the smallest ‚ (or largest ¡‚) at which a critical point occurs, and the analysis stops there.
REMARK 3.10
As the number of control parameters grows, the number of possible analysis sequences increases combinato-
rially. Given the substantial costs usually incurred in these analyses, the experience and ability of the engineer
can play an important role in weeding out unproductive paths. Selective linearization can also reduce the
number of cases substantially, as being able to invoke the superposition principle “factors out” certain param-
eters. For example, if the bridge response to live loads is essentially linear about the own-weight solution,
parameters 32 and 33 may be removed from the nonlinear analysis, and the dimension of ⁄ is reduced from
6 to 4.
3–7
Chapter 3: RESIDUAL FORCE EQUATIONS 3–8
Homework Exercises for Chapter 3
Residual Force Equations
EXERCISE 3.1
For the example system (3.5) find K and Q.
EXERCISE 3.2
For the same example system and assuming the parametric representation (3.11) for u and ⁄, write down Pr,
PK, PQ and Rr in matrix/vector form.
EXERCISE 3.3
Continuing the above exercise: if 31 D 2‚ and 32 D ‚, write down r.u; ‚/ and Pr.u; ‚/ in vector form.
EXERCISE 3.4
Is (3.5) derivable from a potential energy function so that r can be represented as (3.2)? Can you guess by
inspection what the potential 5 is?
EXERCISE 3.5
Show that both K and F (assuming the latter exists) are symmetric if the residual is derivable from a potential
energy function.
3–8
3–9 Solutions to Exercises
Homework Assignment for Chapter 3
Solutions
EXERCISE 3.1
K D @r
@u
D
" 4 u3 ¡ 1 u2
u3 ¡ 1 6 u1
u2 u1 4
#
; Q D ¡ @r
@⁄
D
" 6 0
0 3
0 3
#
: .E3:1/
EXERCISE 3.2
Pr D K Pu¡Q P⁄ D
" 4 u3 ¡ 1 u2
u3 ¡ 1 6 u1
u2 u1 4
#" Pu1
Pu2
Pu3
#
¡
" 6 0
0 3
0 3
#h P31P32 i
D
" 4 Pu1 C .u3 ¡ 1/ Pu2 C u2 Pu3 ¡ 6 P31
.u3 ¡ 1/ Pu1 C 6 Pu2 C u1 Pu3 ¡ 3 P32
u2 Pu1 C u1 Pu2 C 4 Pu3 ¡ 3 P32
#
:
.E3:2/
Rr D K RuC PK Pu¡Q R⁄¡ PQ P⁄; .E3:3/
where K and Q are given above, and
PK D
" Pk11 Pk12 Pk13Pk21 Pk22 Pk23Pk31 Pk32 Pk33
#
D
" 0 Pu3 Pu2
Pu3 0 Pu1
Pu2 Pu1 0
#
; PQ D
" Pq11 Pq12
Pq21 Pq22
Pq31 Pq32
#
D
" 0 0
0 0
0 0
#
: .E3:4/
Hence
Rr D
" 4 u3 ¡ 1 u2
u3 ¡ 1 6 u1
u2 u1 4
#" Ru1
Ru2
Ru3
#
C
" 0 Pu3 Pu2
Pu3 0 Pu1
Pu2 Pu1 0
#" Pu1
Pu2
Pu3
#
¡
" 6 0
0 3
0 3
#h R31R32 i : .E3:5/
EXERCISE 3.3
r.u; ‚/ D
" 4u1 ¡ u2 C u2u3 ¡ 12‚
6u2 ¡ u1 C u1u3 ¡ 3‚
4u3 C u1u2 ¡ 3‚
#
: .E3:6/
Pr.u; ‚/ D
" 4 Pu1 C .u3 ¡ 1/ Pu2 C u2 Pu3 ¡ 12P‚
.u3 ¡ 1/ Pu1 C 6 Pu2 C u1 Pu3 ¡ 3P‚
u2 Pu1 C u1 Pu2 C 4 Pu3 ¡ 3P‚
#
: .E3:7/
EXERCISE 3.4
Yes:
5 D 2u21 C 3u22 C 2u23 ¡ u1u2 C u1u2u3 ¡ 631u1 ¡ 332.u2 C u3/C C .E3:8/
where C is an arbitrary function independent of u1, u2, u3 (in particular, a constant).
EXERCISE 3.5
If r derives from a potential 5, the entries of K are Ki j D @25=@ui@uj D @25=@uj@ui D Kji . Hence K is
symmetric. Because the inverse of a symmetric matrix is symmetric, so is F D K¡1 if it exists.
3–9
-
4
One-Parameter
Residual Equations
4–1
Chapter 4: ONE-PARAMETER RESIDUAL EQUATIONS 4–2
§4.1 INTRODUCTION
This Chapter continues on the topic of residual equations introduced in Chapter 3. The general
residual force equation presented there is specialized, through the concept of staging introduced in
§3.4, to the one-parameter form in which r is a function of u (the state) and ‚ (the control). Together
these form the control-state space. The separable case in which u and ‚ can be segregated to both
sides of the residual equations, is described.
Further insight into the structural response may be achieved with the help of constant-residual
incremental flows. Paths and orthogonal hypersurfaces are introduced and interpreted geometrically.
Finally, the concepts of arclength and scaling are discussed.
§4.2 RATE FORMS AND INCREMENTAL VELOCITY
In this section we study further the one-parameter residual equation (3.17), reproduced below for
convenience:
r.u; ‚/ D 0: .4:1/
The corresponding residual-derivative equations are
Pr D K Pu¡ qP‚; .4:2/
Rr D K RuC PK Pu¡ qR‚¡ PqP‚; .4:3/
K D @r
@u
; q D ¡ @r
@‚
.4:4/
where K is the tangent stiffness matrix introduced in §3.3, and q is the incremental load vector. The
latter is the specialization of the control matrix Q defined in §3.3, to the one-parameter case. These
equations will be used in the sequel instead of the more general (3.14)–(3.15) unless otherwise
noted.
Rate forms of r.u; ‚/ D 0 are obtained by equating the above derivatives to zero:
Pr D 0; or K Pu D qP‚; .4:5/
Rr D 0; or K RuC PKu D qR‚C PqP‚: .4:6/
At regular points of the (u,‚) space the tangent stiffness K is nonsingular. If so, we can solve the
first-order rate form (4.5) for Pu:
Pu D K¡1qP‚ D vP‚; or @u
@‚
D u0 D v; .4:7/
where
v D K¡1q: .4:8/
This vector is called the incremental velocity vector and is an important component of all solution
methods based on continuation.
4–2
4–3 §4.4 RESPONSE VISUALIZATION BY INCREMENTAL FLOW
§4.3 SEPARABLE RESIDUALS AND PROPORTIONAL LOADING
The force-balance equivalent of (3.3) for a one-parameter residual equation is
p.u/ D f.u; ‚/: .4:9/
If the right hand side, which represents the external force vector, does not depend on the state
parameters u, that is
p.u/ D f.‚/; .4:10/
the system of equations (4.1) or (4.9) is called separable. Furthermore, if f is linear in ‚ the loading
is said to be proportional. Obviously q D @f=@‚ is then a constant vector.
REMARK 4.1
The more general system (3.3) containing multiple control parameters is said to be separable if
p.u/ D f.⁄/: .4:11/
In this case the loading is called proportional if f is linear in all control parameters, thus giving a constant
control matrix Q.
REMARK 4.2
If a separable system derives from a total potential energy 5 D U ¡ P , then the external work potential P
must be linear in the state parameters ui . Furthermore for the loading to be proportional, P must also be linearin ‚.
§4.4 RESPONSE VISUALIZATION BY INCREMENTAL FLOW
§4.4.1 Diagrams for One Degree of Freedom
As discussed in Chapter 2, the solution of the one-parameter residual form
r.u; ‚/ D 0; .4:12/
is often plotted on the u versus ‚ plane, where u is a representative component of u.
One such diagram is illustrated in Figure 4.1. If ‚ is a load amplitude, this is called a load-
displacement response curve or simply a response curve. It is common practice to make the
curve pass through the origin ‚ D 0, u D 0. More general terms for this geometrization are
equilibrium path or equilibrium trajectory. The path passing through the origin is called the
primary or fundamental path because it usually represents the operation of the structure under
normal service conditions.
A path can, of course, be traversed in two directions. These are identified as positive or C sense,
and negative or ¡ sense. As illustrated in Figure 4.2, we shall use the convention that the positive
sense is associated with increasing values of the pseudo-time t when the path is parametrically
described as u D u.t/ and ‚ D ‚.t/.
4–3
Chapter 4: ONE-PARAMETER RESIDUAL EQUATIONS 4–4
r D0Response curve
u
λ
Figure 4.1. Typical response diagram showing primary equilibrium path.
‚
Pseudo-time t decreasing
Pseudo-time t increasing
C
¡
u
Figure 4.2. Positive and negative traversal senses on a path.
A diagram such as that in Figure 4.1 gives of course only a partial picture of the structural behavior
unless there is only a single degree of freedom. For a better understanding of the way numerical
solution procedures work (or fail to) it is instructive to “look around” the equilibrium path by
considering the perturbed residual equation
r.u; ‚/ D rc; .4:13/
where rc is a constant vector. This is the general solution of Pr D 0. Additional information can
be conveyed by drawing the solutions of (4.13) for various values of the right-hand side near zero.
4–4
4–5 §4.4 RESPONSE VISUALIZATION BY INCREMENTAL FLOW
r D0
tC
positive tangent at P
P(u, λ )
λ
u
Figure 4.3. The incremental flow field as a family of constant-residual trajectories.
r D0
tC
positive tangent at P
normal "hyperplane" at P P(u, λ )
λ
u
Figure 4.4. Incremental flow (full curves) and the flow-orthogonal envelope
(dashed curves). This envelope reduces here to a family of curves
because there is only one degree of freedom u.
4–5
Chapter 4: ONE-PARAMETER RESIDUAL EQUATIONS 4–6
This produces constant-residual paths as illustrated in Figure 4.3. Collectively these paths form the
incremental flow whose differential equation is either Pr D 0, or, if we take ‚ · t :
r0 D @r
@‚
D 0; .4:14/
where primes denote derivatives with respect to ‚. This can also be presented as
r0 D K@u
@‚
C @r
@‚
D Ku0 ¡ q D 0: .4:15/
If K is nonsingular, solving (4.15) yields u0 D K¡1q D v. The incremental solution methods
covered later exploit these forms, which explains the qualifier “incremental” applied to the flow.
Figure 4.3 also illustrates the construction of the tangent vector tC at an arbitrary point P.u; ‚/.
This procedure is described more precisely in §4.5.
Figure 4.4 depicts a set of curves whose trajectories are orthogonal to the incremental flow. This
set is called the flow-orthogonal envelope. It will be explained later in §4.4 that this set generally
consists of a family of hypersurfaces. For a system with one degree of freedom, however, the
envelope reduces to a family of curves, as in Figure 4.4. This concept will be useful later in
explaining how incremental-iterative solution methods work.
EXAMPLE 4.1
For simple one-degree of freedom systems it is easy to plot the incremental flow using standard graphic
packages. As an example consider the following residual equation, which is obtained as solution of one of the
Exercises of Chapter 6:
r.„; ‚/ D fi3„.1¡ „/.2¡ „/¡ ‚: .4:16/
Here „ is a dimensionless state parameter and fi an angle in radians characterizing the reference position of
the structure. The following Mathematica program produces the incremental flow plot for fi D 30– using the
ContourPlot function:
alpha = Pi/6; r = alpha^3*mu*(1-mu)*(2-mu)-lambda;
ContourPlot[r,{mu,0,2},{lambda,-.1,.1},PlotPoints->30];
0 0.5 1 1.5 2
-0.1
-0.05
0
0.05
0.1
Figure 4.5. Incremental flow plot for the residual (4.16) produced
by Mathematica via its ContourPlot function.
4–6
4–7 §4.5 INTRINSIC GEOMETRY OF INCREMENTAL FLOW
Examination of Figure 4.5 shows that the r D const curves are simply translations of each other along the ‚
axis because ‚ appears simply as ¡‚. This is typical of proportional loading situations.
§4.4.2 Diagrams for Multiple Degrees of Freedom
If the number of degrees of freedom increases to N > 1 the incremental flow still remains a family
of curves in the N C 1-dimensional control-state space space .u; ‚/. Visualization, however, is
restricted to N D 2 as illustrated in Figure 4.6. For three or more degrees of freedom, only cross
sections of the control-state space can be displayed, in which one or two representative degrees of
freedom or functions of such are plotted. This “projection” requires some ingenuity and experience.
The flow-orthogonal envelope becomes a family of ordinary surfaces if N D 2, as illustrated in
Figure 4.7. For three or more degrees of freedom, the envelope becomes a family of hypersurfaces.
§4.5 INTRINSIC GEOMETRY OF INCREMENTAL FLOW
§4.5.1 Tangent Vector
At a generic regular point P of coordinates .u; ‚/, not necessarily on the equilibrium path, we can
construct an unnormalized tangent vector t defined by
t D
•
u0
‚0
‚
D
•
v
1
‚
; .4:17/
where v D K¡1q is the incremental velocity vector (4.8). Tangent vectors are illustrated in Figures
4.3 and 4.8 for one and two degrees of freedom, respectively.
The tangent vector normalized to unit length is
tu D
•
v= f
1= f
‚
; .4:18/
where f is the scaling factor
f D jtj D C
p
jjtjj2 D C
p
1C vT v: .4:19/
The positive tangent direction and the positive unit tangent are defined as
tC D §
•
v
1
‚
; tCu D
tC
f D §
•
v= f
1= f
‚
: .4:20/
The positive tangent direction points in the positive sense of path traversal, as defined in §4.2 and
Figure 4.2.
4–7
Chapter 4: ONE-PARAMETER RESIDUAL EQUATIONS 4–8
r D0
1u
2u
λ
Figure 4.6. An incremental flow response diagram for two degrees of freedom.
The plane paths of Figure 4.3 now become space curves.
Only a few paths are shown to reduce clutter.
r D0
1u
2u
λ
Figure 4.7. A response diagram for two degrees of freedom, showing some
members of the flow-orthogonal envelope. Only the primary
equilibrium path r D 0 is shown to reduce clutter.
4–8
4–9 §4.5 INTRINSIC GEOMETRY OF INCREMENTAL FLOW
r D0
tC
tC
v T PuC P‚ D 0
C
C
sense of increasing t
P
Normal hyperplane
NP
1u
2u
λ
Figure 4.8. Illustrating the tangent vector and normal hyperplane in
an incremental flow diagram for two degrees of freedom.
Point P is on the primary equilibrium path but NP is generic.
§4.5.2 Normal Hyperplane and Flow-Orthogonal Envelope
The hyperplane NP normal to t at P.u; ‚/ has the equation
vT 1uC1‚ D 0; .4:21/
where 1u D u¡ uP and 1‚ D ‚¡ ‚P are increments from P . Dividing these increments by 1t
and passing to the limit one obtains
vT PuC P‚ D 0: .4:22/
For a one degree of freedom u the hyperplane reduce to a line in .u; ‚/ space, as illustrated in
Figure 4.4. For two degrees of freedom the normal hyperplane is an ordinary plane in the 3D space
.u1; u2; ‚/, as illustrated in Figure 4.8.
For one degree of freedom (4.22) is the differential equation of a flow orthogonal to the incremental
flow, as illustrated in Figure 4.3; this flow is the envelope of the normals.