0521830540 cambridge university press non linear modeling and analysis of solids and structures aug 2009

1.1.2 Virtual work and potential energy 61.2 Simple non-linear solution methods 7 1.2.3 Modified Newton–Raphson method 13 2.5 Assembly of global stiffness and forces 31 2.6 Total or update

Non-linear Modeling and Analysis of Solids and

Structures

Steen Krenk

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,

São Paulo, Delhi, Dubai, Tokyo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-83054-6

ISBN-13 978-0-511-60413-3

© Cambridge University Press 2009

2009

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To Jette

1.1.2 Virtual work and potential energy 6

1.2 Simple non-linear solution methods 7

1.2.3 Modified Newton–Raphson method 13

2.5 Assembly of global stiffness and forces 31

2.6 Total or updated Lagrangian formulation 36

3.2 Rotation of a vector into a specified direction 53

3.3 The increment of the rotation variation 55

3.4 Parameter representation of an incremental rotation 60

3.5 Quaternion parameter representation 63

3.5.1 Representation of the rotation tensor 64

v

3.5.2 Addition of two rotations 65

3.5.3 Incremental rotation from quaternion parameters 67

3.5.4 Mean and difference of two rotations 68

3.6 Alternative representation of the rotation tensor 69

3.7 Summary of rotations and their virtual work 72

4.2 Virtual work, strain and curvature 78

4.3 Increment of the virtual work equation 81

4.5 Summary of ‘elastica’ beam theory 98

5.1 Co-rotating beams in two dimensions 101

5.1.1 Co-rotation form of the tangent stiffness 104

5.1.2 Element deformation stiffness 107

5.1.4 Finite element implementation 112

5.2 Co-rotating beams in three dimensions 117

5.2.1 Co-rotation form of the tangent stiffness 120

5.2.2 Element deformation stiffness 127

5.2.4 Finite element implementation 133

6.1.2 Decomposition into deformation and rigid body motion 151

6.2.2 Cauchy and Kirchhoff stresses 158

Contents vii

6.3.1 Equilibrium and residual forces 166

6.3.3 Finite element implementation 170

6.4.1 Transformation from total to updated format 174

6.4.2 Virtual work in the current configuration 176

6.4.3 Finite element implementation 180

6.5 Summary of non-linear motion of solids 185

7.1.2 Strain invariants and small strain elasticity 198

7.1.3 Isotropic elasticity at finite strain 200

7.2.2 Maximum plastic dissipation rate 207

7.2.4 Isotropic and kinematic hardening 216

7.3.1 Yield surface and flow potential 219

7.4 General aspects of plasticity models 229

7.4.1 Combined isotropic and kinematic hardening 230

7.4.2 Internal variables and non-associated flow 234

7.4.3 General computational procedure 237

7.5.1 Flow potential and yield surface 242

8.1 Iterative solution of equilibrium equations 257

8.1.1 Non-linear iteration strategies 259

8.1.2 Direction and step-size control 260

9.1 Newmark algorithm for linear systems 292

9.1.1 Energy balance and stability 295

9.1.2 Numerical accuracy and damping 300

9.3.2 Non-linear kinematics for Green strain 311

9.3.3 Energy-conserving algorithm 315

9.4.1 Spectral analysis of linear systems 323

9.4.2 Linear algorithm with energy dissipation 325

9.4.3 Non-linear algorithm with energy dissipation 327

The aim of this book is to take the reader on a concentrated tour of some

of the central issues of non-linear modeling and analysis of structures andsolids Traditionally, the non-linear theories of solids have been treated inbooks on continuum mechanics, while the questions of analysis have formedthe focus of books on finite element techniques The idea of the presentbook is to place the emphasis on modeling with a view to its numericalimplementation right from the outset Two guiding principles have deter-mined the main style of the book: the story should be told in the form ofconcentrated chapters, each giving the central ideas of a specific aspect such

as ‘finite rotations’ or ‘elasto-plastic solids’, and the reader should have thepossibility of getting a feel for the numerical implementation by access anduse of simple high-level implementations of the basic algorithms A textbased on these principles cannot provide exhaustive coverage, but aims atgiving an interesting introduction to the basic ideas, which can then be stud-ied elsewhere in greater detail as needed It is hoped that the combination

of a concise theoretical presentation in plain language supported by specificalgorithms will make the text of interest to graduate students as well asprofessionals

The book contains nine chapters: a brief introductory chapter setting thescene by use of elementary arguments, four chapters on structures, two chap-ters on non-linear deformation and material behavior of solids, and finallytwo chapters on numerical techniques for non-linear quasi-static and dy-namic analysis The theory is combined with demonstrations and exercisesusing a small Matlab toolbox FemFiles providing routines for creationand assembly of element matrices and permitting the solution of non-linearfinite element problems in a fairly simple script file format The toolboxFemFiles is available from the author via the internet Exercises that re-quire the use of a high-level program like FemFiles are marked ∗.

ix

The text started as a draft manuscript prepared for a short introductorycourse on non-linear aspects of the finite element method at Aalborg Univer-sity in the fall of 1992 A visit to Lund Institute of Technology sponsored

by NorFA provided an opportunity to include additional material on thenumerical aspects The text was later extended with material on finite ro-tations, co-rotating formulation of elements, potential theory of plasticitytheory and plasticity models for geotechnical materials, and conservationalgorithms for numerical integration of dynamic problems Several parts ofthis work have been sponsored by the Danish Technical Research Council.The work on bringing it all together was initiated during a visiting appoint-ment as Melchor Professor at the University of Notre Dame, Indiana, in thefall of 2001 The final stage has been combined with courses at Helsinki Uni-versity of Technology 2004, and at Aalborg University and Lund Institute

of Technology 2005

Introduction

Many problems of practical interest involve non-linear behavior of solidsand structures In the present context a solid means a body with a firmshape, as opposed to a fluid, while a structure refers to a solid composed

of slender elements such as beams, plates and shells Typical problemsare the motion of robots, collapse scenarios of structures, metal formingprocesses in industrial production, and material deformation and failure ingeotechnical engineering These problems typically involve a considerablechange of shape, often accompanied by non-linear material behavior.The finite element method is an important tool for the analysis of non-linear problems, such as geometrical and material non-linear behavior ofsolids and structures The solution of non-linear problems by the finiteelement method involves modeling, leading to the formulation of an appro-priate set of non-linear equations describing the problem, followed by anappropriate strategy for the numerical solution of these equations In con-trast to linear problems, where the solution strategy reduces to the solution

of a system of linear equations, the solution phase in a non-linear problemtypically involves an iterative procedure

Non-linear modeling and analysis is a very active research area with manyengineering applications The many different aspects involved are not cov-ered in any single text However, some central references to general textsshould be given here A brief introduction to some of the basic problems

of non-linear finite element analysis of solids and structures is included in

the book by Cook et al (1989) A general state-of-the-art presentation of

the finite element method, including the non-linear aspects of solids, tures and fluids, has been given in Zienkiewicz and Taylor (2000) A pre-sentation with main emphasis on incremental formulation of geometricallynon-linear problems, including details of implementation, has been given by

struc-Bathe (1996) The books by Crisfield (1991, 1997) and Belytschko et al.

1

(2000) are entirely devoted to non-linear analysis of solids and structures,combining illustrative examples with specific finite element procedures.The present text is an introduction to some of the central ideas of non-linear modeling and finite element analysis It covers theoretical aspects ofgeometric and material non-linearity and associated numerical techniques.The text proceeds from the elementary level to a fairly rigorous presenta-tion of ideas used in current research Only the main ideas can be covered,and the references should be consulted according to need This first chap-ter gives an illustration of geometric non-linear behavior with reference to

a simple two-element truss model The example serves as a vehicle for aninformal introduction to a non-linear load–displacement relation, the tan-gent stiffness, and the relation between the equilibrium and the virtual workapproach to the problem The example also provides a simple realistic non-linear equation on which to try different variants of the Newton–Raphsonsolution technique

1.1 A simple non-linear problem

The simple two-element truss model shown in Fig 1.1 has often been used

to illustrate some basic features of geometric non-linear behavior, see e.g.Bathe (1996, p 494) and Crisfield (1991, pp 2–13) The structure consists

of two identical truss elements, loaded with a vertical force f at the center

and simply supported at the other ends The vertical displacement at the

center is called u In the initial configuration the length of the bars is l0.

Fig 1.1 Two-element truss model.

Application of the load leads to a deformed state with vertical

displace-ment u of the central node, Fig 1.2 The structure is assumed to be shallow, i.e a  b This permits series expansion of the square roots defining the

original bar length l0 and the bar length l corresponding to the current

1.1 A simple non-linear problem 3deformed state:

Fig 1.2 Initial length l0 and current length l.

The deformation of the bars is described by their elongation A dimensional measure of deformation is the engineering strain, defined as theelongation relative to the original length,

non-linear function of the displacement component(s) If the displacement u is small relative to all characteristic lengths of the geometry – l0 and a – the

linear term will constitute a fair approximation, but if this approximation isused, some of the characteristic non-linear features of the problem are lost

1.1.1 Equilibrium

The two bars are assumed to be linear elastic with axial stiffness EA, where

E is the elastic modulus and A is the cross-section area Thus, the axial

force in each bar is expressed in terms of the strain as

u

l0

2

Equilibrium of the central node in the deformed state requires that the

external force f is equal to the internal force g(u) generated by deformation

of the structure Projection of the normal force gives

In non-dimensional form this is

u

a

2+ 12

u

a

3

where the normalized displacement is u/a The load–displacement relation

(1.6) is shown in Fig 1.3 corresponding to a downward load

Fig 1.3 Load–displacement curve for two-element truss.

From the unloaded state A an increasing downward load leads to a local maximum B In this state the structure cannot support a further increase

of the load Thus, further increase of the load from B would lead to through to F The snap-through is an unstable dynamic process, and thus

snap-the load–displacement curve in Fig 1.3 is not fully representative natively the structure may be loaded in displacement control, in which thecentral node is given a controlled downward displacement −u This would

Alter-require an increasing load from A to B, and then a decreasing load from B

to C, where u = −a and the two bars form a straight line An upward force

is now required to proceed to D and E, where the structure is stress-free,

forming an angle symmetric to the original configuration with respect to the

base line Further downward load leads through F with increasing stiffness

1.1 A simple non-linear problem 5differentiation of (1.6):

u

a

2

Although this expression defines the tangent stiffness K, it does not convey

the physics of the problem very clearly This is better accomplished bydifferentiation of the equilibrium equation (1.5):

l0

Here a + u is the height of the structure in the current state, while N is

the current value of the axial force The first term is due to changes in the

normal force N , while the second term is due to changes in the geometric configuration with constant normal force N Sometimes the first term is separated into a constant corresponding to u = 0 and the rest, whereby

(1.9) takes the form

where K0 is the linear stiffness, K u is the initial displacement stiffness, and

K σ is the initial stress stiffness In an incremental procedure, where the

geometry is updated, the current value of u is absorbed in the updated value of a, and in that case the initial displacement stiffness K u vanishes

Fig 1.4 Load–displacement curve for two-element truss with spring.

A family of load–displacement curves with different degrees of non-linearitycan be obtained by introducing a vertical linear elastic spring with stiffness

k at the central node of the structure The load–displacement relation (1.6)

u

a

2+ 12

u

a

2

+ k. (1.12)Figure 1.4 shows the load–displacement curve for different values of the

spring stiffness k For k ≥ EAa2/l30 the variation of load with displacement

is monotonic, corresponding to K ≥ 0.

1.1.2 Virtual work and potential energy

The load–displacement relations (1.6) and (1.11) were obtained from librium of the center node For structures with more degrees of freedom ormore complicated elements it is often convenient to make use of the principle

equi-of virtual work Essentially, the principle equi-of virtual work is a restatement

of a set of equilibrium equations, where each equation is multiplied by acorresponding infinitesimal virtual displacement component With an ap-propriate definition of the force and displacement components summation

of their products forms a scalar invariant, known as the virtual work

In the particular example of the two-element truss with an elastic springthe equilibrium equation can be written as

If, for the time being, the difference between l0and l is neglected, the virtual

work equation (1.14) can now be written as

δV  2 l0

0

N δε ds + (ku) δu − f δu = 0. (1.16)The integral is the internal virtual work of the bar elements, the second term

1.2 Simple non-linear solution methods 7

is the virtual work of the elastic spring, while the last term is the externalvirtual work

Apart from the factor l0/l that is somehow missing, the use of virtual work

in the present case where δε is constant within the elements is almost trivial.

However, for more general problems with more degrees of freedom and trivial displacement fields within the elements, the principle of virtual work

non-is an important tool for establnon-ishing the balance equations of the dnon-iscretized

model The question of the factor l0/l is discussed in Chapter 2, where the

theory of non-linear bar elements is discussed more rigorously Here, therelation between virtual work and potential energy is discussed briefly beforeturning to elementary numerical solution methods for non-linear equilibriumequations

When the internal forces such as the axial force N and the spring force

ku are functions of the state of displacement given by u, and the external

load is also a function of u, the virtual work δV can be considered as the differential of an energy function Φ(u) – the potential energy In the present

case (1.16) is written as

δΦ(u) = 2

l0 0

EA εδε ds + ku δu − f δu. (1.17)

This relation can be integrated with respect to the displacement u, giving

the following expression for the potential energy:

Φ(u) = 2

l0 0

1

2EAε2 ds + 12k u2 − fu. (1.18)The potential energy is the internal strain energy of the structure, including

the spring, minus the external work represented by f u For linear

elas-tic structures it may be simpler to derive the equilibrium equations from

the potential energy by considering an incremental change δu of the

dis-placements However, the principle of virtual work is valid irrespective ofthe specific material behavior, and thus the principle of virtual work hasbecome the method of choice for setting up equilibrium equations

1.2 Simple non-linear solution methods

For a system with only one degree of freedom non-linear behavior can often

be described explicitly as a function of the displacement u, and the problem

may then be considered as one of displacement control However, in the case

of several degrees of freedom the use of displacement control is non-trivial,and most problems are formulated in terms of a load history, for which

the corresponding displacement history is to be calculated This requiresthe solution of a system of non-linear equations Here some of the simplermethods for solving non-linear equations are briefly introduced, leaving morespecialized techniques to Chapter 8 The methods are illustrated for a singledegree of freedom and then generalized to matrix form.

1.2.1 Explicit incremental method

An explicit incremental method, often called the Euler explicit method, isobtained by replacing the differentials in the definition (1.7) of the tangent

stiffness with finite increments ∆f and ∆u:

The load–displacement history is described by a number of increments ∆f n,

∆u n , n = 1, 2, defining the states

f n = f n −1 + ∆f n , u n = u n −1 + ∆u n , n = 1, 2, (1.20)

In the explicit incremental method the tangent stiffness K corresponds to the

state at the beginning of the increment Thus, the precise form of (1.19) is

∆u n = K −1 (u

n −1 ) ∆f n , n = 1, 2, (1.21)This procedure is illustrated in Fig 1.5

Fig 1.5 Explicit incremental method.

It is seen that the computed states deviate more and more from the act load–displacement curve There are two reasons for this: the tangentstiffness of each increment is taken at the left end-point and in this particu-lar case overestimates the stiffness, and deviations from the exact curve are

ex-1.2 Simple non-linear solution methods 9added to a cumulative error While it is difficult to use an exact represen-tation for the stiffness corresponding to the full increment, the problem ofincreasing deviations can be countered by introducing equilibrium iterations

as discussed in the following

The explicit incremental method is easily generalized to multi-degree of

freedom systems Let the displacement vector be u and the corresponding load vector f The tangent stiffness matrix K is then defined by

The corresponding explicit incremental method is

∆un = K−1(u

n −1) ∆fn , n = 1, 2, (1.23)

The use of the inverse matrix K−1 in (1.23) should not be taken literally.

In practice the matrix K is factored and the product K−1∆f found by back

The first step consists in checking the equilibrium equation This is done

by forming the difference between the external load f and internal force

g(u),

where r(u, f ) is called the residual force In a state of equilibrium the nal force g(u) is equal to the external load f , and thus the residual vanishes.

inter-In practice, lack of equilibrium will be produced at the beginning of each

load increment, where the load f is increased, while no new displacement estimate u is yet available Thus, the need arises for obtaining an improved estimate of the state of displacement u.

In the absence of equilibrium an improved estimate of the displacement

u is obtained from a linearized form of the residual r(u + δu, f ) around the

known residual r(u, f ),

r(u + δu, f ) = r(u, f ) + δr(u, f ) + · · · = 0. (1.25)

The dots indicate higher-order terms, because δr is only a linearized form of

the increment of the residual In the classic form of equilibrium iterations

the load f is assumed fixed within the given load step, and thus the ment of the residual only depends on the internal force g(u) The linearized

incre-increment is then given by the first derivative of the internal force as

δr = − dg(u)

Here the tangent stiffness K, introduced in (1.7), has been introduced The

displacement increment is now obtained from the linearized form of (1.25)

by substitution of the tangent stiffness relation (1.26) When rearrangingthe terms in (1.25), the linearized equation becomes

In this equation the residual r(u, f ) is known, as it relates to the current state of load f and displacement u The tangent stiffness K(u) at the current displacement state u can also be calculated Thus, this equation permits determination of the displacement increment δu,

gram the iteration superscript i is not needed, as the register containing

u i −1 is simply overwritten by the new value u i according to the assignmentstatement

Here, : = is the assignment operator, implying that the variable u is assigned

a new value In this book many of the algorithms are presented in the form

of pseudocode – i.e a code format that appears like high-level programs such

as Matlab In the pseudocode presented here assignments are indicated bythe normal equality sign, as all equalities are assignment statements.The Newton–Raphson equilibrium iteration procedure is illustrated in

Fig 1.6 The figure shows load step n This load step starts from a state of equilibrium already established at the previous load f n −1with displacement

u n −1 The load step is initiated by increasing the load by ∆f n to f n This

generates the first residual r1

n = ∆f n This residual and the tangent stiffness

1.2 Simple non-linear solution methods 11

Fig 1.6 Newton–Raphson equilibrium iterations.

K(u n −1 ) lead to the displacement increment δu1n, shown in the figure At the

new displacement u n −1 + δu1n , the internal force g – represented by the curve

– is still smaller than the imposed load The difference forms the residual

r2n, and the procedure is continued It should be noted that the use of and superscripts to indicate load step and iteration number, respectively, ismerely for illustration in relation to the figure These indices are not neededwhen programming the algorithm

sub-The iteration process needs a termination criterion This may be taken

as the requirement that the current residual force r n should be less than a

prescribed fraction  of the load increment ∆f of the present load step,

The value of  could be on the order of say 10 −4–10−6. For structures

developing very small stiffness, the criterion (1.31) may be supplemented bythe displacement criterion

where ∆u is the total displacement increment accumulated in the present

load step

In the corresponding multi-component problem with displacement vector

u and load vector f , the residual force vector is

The tangent stiffness matrix is defined by the incremental change of the

Algorithm 1.1 Newton–Raphson method.

Load steps n = 1, 2, , nmax

Stop iteration when r n  <  ∆f n 

End of load step

In contrast to the one-dimensional case, the solution of these equations may

be a non-trivial part of the procedure The termination criteria will typicallymake use of the ‘length’ of the corresponding vectors, whereby

(rT nrn)1/2 <  (∆f T∆f )1/2 , (1.36)

(δu T δu) 1/2 <  (∆u T∆u)1/2 (1.37)

The Newton–Raphson procedure is summarized as Algorithm 1.1 Note that

in the iteration loop the computer overwrites quantities like rnby their new

value in the same register Therefore, the superscript i does not appear explicitly in the algorithm Similarly, the load step subscript n is only used

to avoid the explicit indication of storing the result of each completed loadstep The actual algorithm is conveniently programmed without the use ofindexed variables in the iteration loops

1.2 Simple non-linear solution methods 13

1.2.3 Modified Newton–Raphson method

In the original Newton–Raphson method the current tangent stiffness matrix

K(u) is computed and factored in each iteration For non-linear problems

with a single or a few degrees of freedom this is usually not a problem, but forproblems with many degrees of freedom the computational cost involved in

forming the stiffness matrix K(u) and solving the corresponding equations

for δu in each iteration may be considerable It is seen from Fig 1.6 and

Al-gorithm 1.1 that Knappears within the inner loop A simple modification of

the Newton–Raphson method consists in moving the stiffness matrix K side the iteration loop Then K = Kn −1 is only computed and factored once

out-for each load step on the basis of the previous state of displacement un −1,

Kn −1 = dg(u du n −1). (1.38)

This simplifies the iteration loop as shown in Fig 1.7 and Algorithm 1.2

Fig 1.7 The modified Newton–Raphson method.

The asymptotic convergence of the modified Newton–Raphson method isslower than that of the Newton–Raphson method, and this may offset some

of its computational efficiency A different, more refined type of

modifica-tion makes use of a secant approximamodifica-tion of K This requires a non-trivial

generalization of the secant concept to multi-degree of freedom systems.The corresponding methods, called quasi-Newton methods, are described inChapter 8

The classic Newton methods encounter problems at a load maximum.Several methods have been developed to deal with this problem A commonfeature of these methods is that the load increment is also subject to changesduring the iterations, e.g by linking load and displacement increments This

Algorithm 1.2 Modified Newton–Raphson method.

Load steps n = 1, 2, , nmax

Stop iteration when r n  <  ∆f n 

End of load step

introduces a kind of displacement control near the maximum, as discussed

in Chapter 8

1.3 Summary and outlook

Non-linear problems of structures and solids involve processes in whichneighboring states are connected by non-linear relations This is illustrated

by the simple example of a two-bar truss loaded by a quasi-static force

A model of the problem requires representation of the material behaviorand the formulation of suitable equilibrium equations In this introductorychapter these issues were dealt with in an ad hoc fashion, also introducingapproximations to simplify the presentation In the following chapters theseissues are dealt with in a rigorous way within the framework of finite elementanalysis

While the equilibrium equations of simple systems often can be lated directly, it is generally advantageous to use the principle of virtualwork, which retains its simplicity for larger and more complicated systems.Several aspects of this will appear in later chapters The idea of the principle

formu-of virtual work is to consider the work done by the actual forces through animagined – or virtual – displacement field This changes a multi-componentproblem into a similar number of scalar problems A necessary requirement

is that the product of the internal forces and the virtual measures of mation constitute work This property of the virtual work serves to identifysuitable internal force and stress measures, when a displacement and strain

defor-1.4 Exercises 15representation has been selected This is illustrated in the next chapter inconnection with a general discussion of bar elements, and is used to defineseveral stress measures for solids in Chapter 6 The virtual work also plays

a key role in defining the properties associated with rotations and moments

as discussed in Chapter 4

In order to obtain the solution to specific problems, the material behaviormust be represented in the form of a relation between the internal forcesand the corresponding measures of deformation In the problem of thetwo-bar truss the bars were assumed to be linear elastic, thus simplifyingthe presentation In many situations non-linear material behavior is animportant part of the problem The basic form of the equations of elasto-plastic material behavior is described in Chapter 7

Even the simple two-bar truss exhibits a non-monotonic relation betweenload and displacement The simple Newton-type solution methods brieflyoutlined in this chapter need to be modified in order to enable computation

of non-monotonic force–displacement relations This problem is dealt with

in Chapter 8 on numerical methods The chapter on numerical methods can

be read without reference to the chapters before, and indeed reading thischapter first will enable the reader to supplement the theory of the interme-diate chapters with non-trivial numerical examples In the final chapter thenumerical methods are extended to dynamic problems with inertial effects

1.4 Exercises Exercise 1.1 Consider the load–displacement curve of the two-elementtruss shown in Fig 1.3

(a) Determine the non-dimensional coordinates to the points B and D.

(b) Select a suitable load step magnitude and sketch the states produced

by the explicit incremental method, the Newton–Raphson method, andthe modified Newton–Raphson method Note in particular the passage

of the maximum at B.

Exercise 1.2 Consider the two-element truss shown in Fig 1.1 and add a

vertical spring at the central node with spring stiffness k = 1.2EAa2/l30

(a) Introduce the non-dimensional displacement v = −u/a and the

non-dimensional load p = −fl3

0/(EAa3) Give the relation between p and v

and the corresponding tangent stiffness

(b) Organize the explicit incremental method, the Newton–Raphson method,and the modified Newton–Raphson method in tabular form and com-

pute the p–v relation with a suitable load step, e.g 0.2–0.4 Sketch the

result for each of the three methods

Exercise 1.3* Implement the Newton–Raphson algorithm shown as

Algo-rithm 1.1 in Matlab for analysis of the two-bar truss treated in Section 1.1.Organize the implementation in four m-files:

(a) data.m: script file containing the model parameters and the load

in-crement, e.g b = 1.0, a = 0.1, EA = 1.0, k = 0, and ∆f = −0.0001.

(b) g bar.m: the internal force in (1.24), obtained as a function of the

displacement u from the expression (1.11).

(c) kt bar.m: the function K(u) in (1.12), giving the tangent stiffness as a function of the displacement u.

(d) nr bar: script file containing the Newton–Raphson algorithm from

Al-gorithm 1.1 (Use limited loops, e.g n = 1 : nmax and i = 1 : imax,

with nmax = 20 and imax= 8.)

Use the program to study the behavior of the Newton–Raphson algorithm

via plots of f as a function of displacement u, particularly near the turning point of the load–displacement curve Use e.g ∆f = −1.0 × 10 −4 and

∆f = −0.5 × 10 −4.

Exercise 1.4* Make a modified version mnr bar.m of the driver routine

nr-bar.m from Exercise 1.3 using the modified Newton–Raphson algorithm Usethe two routines to study the behavior of the algorithms for different values

of the spring constant k Note in particular that the modified Newton– Raphson algorithm has convergence problems for increasing stiffness Sketch

this problem

Exercise 1.5* Both the full and the modified form of the Newton–Raphson

algorithm are based on prescribed load increments, and therefore requiremonotonically changing load In the two-bar truss problem the load will

increase monotonically if a vertical spring with stiffness k > (EA/l0)(a/l0)2

is introduced The actual force transferred to the truss is feff = f − ku.

Introduce a spring of sufficient stiffness and solve the problem with the

program developed in Exercise 1.3 Plot the effective load feff against the

displacement u to demonstrate recovery of the full curve from Fig 1.3.

For a very stiff spring equal load increments correspond to equal ments, and the method is equivalent to the use of displacement control Thismethod is only directly applicable in the case of a single load component.General solution methods are discussed in Chapter 8

Non-linear bar elements

The finite element method has been the method of choice for modeling andanalysis of structures and solids for several decades The basic idea is thatthe structure (or solid) is considered as an assembly of elements, and thateach element is modeled in a standard format that permits repetitive use ofthe individual element formats Bar elements only contain a single internaldegree of freedom – the elongation – and they are therefore a convenientmeans for introducing and illustrating the basic features of geometricallynon-linear finite element analysis

In a geometrically non-linear problem the first question to arise is the inition of a non-linear measure of deformation, the strain This is addressed

def-in Section 2.1 When a structure is assembled from the def-individual elements,use is generally made of the principle of virtual work The principle of vir-tual work is a restatement of the equilibrium equations in scalar form Itturns out that once a non-linear strain definition has been adopted, the cor-responding definition of stress follows from the formulation of the principle

of virtual work This is the subject of Section 2.2 The tangent stiffnessmatrix of a geometrically non-linear bar element is derived in Section 2.3 inglobal coordinates

The derivation of the equilibrium and stiffness relations of the bar element

is quite simple because the strain is constant within the element In order

to illustrate the relation to more complex problems involving other types ofelements, the tangent stiffness matrix is re-derived by use of shape functions

in global coordinates This indicates the procedure followed in isoparametricsolid elements Another alternative makes use of a local coordinate systemrotating together with the element during displacement This so-called co-rotational approach is typically used for generalized continuum elementssuch as beams and shells, where the displacement representation depends

on the local orientation of the beam axis, middle plane, etc Co-rotating

17

beam elements are treated in Chapter 5 Bar elements have been given anextensive treatment by Crisfield (1991), and Mattiasson (1983) has describedthe co-rotational formulation of bar elements in detail.

The implementation of the finite element method involves the assembly ofelements into the global structure A brief sketch of the assembly procedure

is given in Section 2.5 The actual solution of the finite element relations canfollow one of two formulations In the first the initial configuration is used

as reference through the full analysis, and total displacements from this figuration are used This is the total Lagrangian formulation In the otherformulation the reference state is updated each time an equilibrium statehas been established Thus, the displacements in this formulation refer tothe last equilibrium state This is the updated Lagrangian formulation Thesimple bar element is used to illustrate these two formulations in Section 2.6

con-In spite of its simplicity the bar element contains the main features of ometrically non-linear structural and solid elements, and Section 2.7 sums

ge-up these main features in a general form These general features are countered repeatedly in various settings in the following chapters

en-2.1 Deformation and strain

In most finite element formulations for structures and solids the ments are the primary variables of the problem The displacements maylead to deformation of the elements, and this in turn to internal forces It

displace-is important to use deformation measures that vandisplace-ish identically for rigidbody displacements In a theory with finite displacements this requirementcan be satisfied by several different strain definitions The most common ofthese are briefly discussed here for a simple bar element to indicate their useand limitations

In a bar element the deformation is characterized by the elongation

Fig-ure 2.1 shows a bar element with initial length l0 Deformation introduces the elongation u, whereby the new length is

2.1 Deformation and strain 19

Fig 2.1 Bar element.

l0 Alternatively, the strain increment δε can be defined with reference to the current length l by the relation

This strain makes no distinction between initial and final length, and

inter-change of l0 and l merely changes the sign of ε L For small strains ε E and

ε L are nearly equal

Many problems involve large displacements but only small to moderatestrains In those problems the important point is to use a strain definitionwithout ‘self straining’, i.e a strain definition that does not produce strain-

ing for arbitrary rigid body motion It is then convenient to use l2 instead of

l as a basis for the strain definition One reason for the use of l2 is that the

square of the length a of a vector a = [a1 , a2, a3]T is the sum of the squared

coordinates, a2 = a2

1 + a2

2 + a2

3 Another is that a definition based on l2

is simpler to generalize to two- and three-dimensional continua A strain

definition based on l2 can be obtained by rewriting (2.2) in the form

ε E = (l − l0)(l + l0)

l0(l + l0) =

l2− l2 0

2 l2 0

Fig 2.2 The strain measures ε E , ε G and ε L.

The strains ε E , ε L and ε G are shown in Fig 2.2 as a function of u/l0 It is

seen that ε G > ε E > ε L For|u/l0| < 0.05 the deviation from ε E is of theorder 2–3% For larger strains it may be necessary to distinguish betweenthe strain to be used or account for the difference through the stress–strainrelation

2.2 Equilibrium and virtual work

Figure 2.3 shows a bar element with end-points A and B The coordinates

of A and B are referred to a global Cartesian coordinate system In the

initial configuration the coordinates are xA0 and xB0, respectively Boldfaceletters are used to denote vectors and matrices The dimension corresponds

to the dimension of the space, i.e 2 or 3 It turns out that this dimension isnot important in the formulation of the equilibrium and stiffness relationsexcept at the very end, when individual components are written down

Fig 2.3 Bar element AB in initial and displaced configurations.

The points A0 and B0 are now given the displacements uA and uB, spectively Only the direction and length of the vector −−→

re-AB influence the

2.2 Equilibrium and virtual work 21equilibrium and stiffness The initial position is

The strain of the bar can now be expressed in terms of l0 and l by any of

the strain measures defined in Section 2.1

The fact that the square of the length is given in simple form suggests the

use of the Green strain ε G From the definition (2.6) it follows that

ε G = l

2− l2 0

be helpful in the evaluation of ε G A slightly more general formulation isdiscussed in Exercise 2.1

Fig 2.4 Projection of displacement u on mean vector x1/2= 12(x0+ x).

The variation of the strain (2.13) is needed for the formulation of theprinciple of virtual work:

Note that the variation of the strain consists of the projection of the

dis-placement variation δu on the current vector x, scaled by l20

The universal method of establishing finite element relations consists inthe use of the principle of virtual work In the case of bar elements thestate of deformation within the element is homogeneous, and the relationscould therefore in principle be obtained directly, but in order to illustratethe general procedure and to identify precisely the internal force that isconjugate to the Green strain, the principle of virtual work is also usedhere According to the principle of virtual work an arbitrary displacementvariation must lead to identical internal and external work The internalvirtual work is expressed in terms of the internal force(s), while the externalvirtual work is expressed in terms of the external loads Thus, the principle

of virtual work establishes a relation between the internal and the externalforces

For a bar element with the external forces fA and fB acting at A and B,

respectively, the principle of virtual work is

together with the initial length l0 of the element With this choice and δε G

from (2.15) the principle of virtual work takes the form

(N x T )(δu B − δu A ) ds − f T

A δu A − f T

B δu B = 0. (2.17)Taking the transpose of (2.17) leads to changing the order of the factors,whereby

(N x) ds − f A



+ δu T B

l0 0

1

l2 0

2.2 Equilibrium and virtual work 23

δu A and δu B, and therefore the following two equations are obtained afterevaluating the integrals:

fA = − N 1

l0x, fB = N

1

l0x. (2.19)

These equations give a precise definition of the normal force N appearing in

the principle of virtual work as conjugate to the Green strain The vector

x/l0 is in the direction of the bar after displacement with length l/l0 Thus

N l/l0 is the actual force in the bar element For moderate strain l/l0  1,

and N gives the actual force directly For large strains the difference between

N and the actual force can be absorbed in the constitutive equation for

N The main point is that with this definition of N the use of the Green

strain and initial length in the principle of virtual work (2.16) is an exact

representation of the equilibrium equations The use of engineering strain

is discussed in Exercise 2.2

For a linear elastic bar, in which N is proportional to ε G, the constitutiverelation is

where E is the modulus of elasticity and A is the cross-section area The

internal forces qA , q Bgenerated by the deformation of the element can then

be expressed in terms of the displacements uA and uB by substitution of(2.20) into (2.19):

With the displacements uA and uB available the displacement vector u is

evaluated from (2.9), the current vector x from (2.10), and ε G from (2.14)

This enables the evaluation of the internal forces qA , q B in an iterative tion procedure In order to find the corresponding displacement incrementthe current stiffness is needed

solu-The formulae for strain and equilibrium derived so far have been expressed

in a compact form making use of the differences in position and displacementbetween the two ends of the bar This form is convenient for derivation anddiscussion of the physical meaning of the formulae However, in a finiteelement formulation it is important to have a convenient matrix format, in

which all coordinates, displacements and forces appear in vector format A

tilde is introduced to denote an extended vector, containing the vectors fromall element nodes In this notation the extended position, displacement andinternal force vectors of the bar element are

˜

xT = [ xT A , x T B ], ˜ uT = [ uT A , u T B ], ˜ qT = [ qT A , q T B] (2.22)

for coordinates, displacements and forces, respectively In terms of thisnotation the formula (2.11) for the initial length of the element is

where ε G is evaluated from (2.24) Note that the strain and the internal

forces ˜ q are non-linear functions of the displacements ˜ u.

2.3 Tangent stiffness matrix The tangent stiffness matrix gives the changes in the internal forces qA and

qB corresponding to infinitesimal changes in the displacements uA and uB

It is conveniently found from differentiation of (2.21a):

2.3 Tangent stiffness matrix 25For a linear elastic bar with the constitutive relation (2.20), differentiationgives

Note that the format of the equations (2.27) implies the identity dq B =

−dq A and dq B = dq A = 0 for a rigid body translation du B = du A

The first matrix in (2.30) represents the constitutive stiffness due to rial deformation, while the second matrix is the initial stress stiffness, oftencalled geometric stiffness The separation into these two parts can be illus-

mate-trated by considering the representation of a vector N by its length N and

a unit vector e = N/N giving the direction, i.e.

The corresponding incremental form is

The increments are illustrated in Fig 2.5 for the case where the length of

the vector e remains constant, i.e de corresponds to a rotation of e The rotation term N de corresponds to the initial stress term, while the change

of length dN e corresponds to the constitutive stiffness term.

Fig 2.5 Decomposition of increment into a rotation and a change of length.

In terms of the vector notation (2.22), the tangent stiffness relation (2.30)takes the form

d˜q = ∂˜q

The tangent stiffness matrix K consists of the following three contributions:

with the linear stiffness matrix

l3 0

The compact products in terms of x and u can be computed by formulae

similar to (2.23), but direct computation will be more efficient In the erence state the initial displacement is zero, and the initial displacementstiffness matrix vanishes

ref-2.4 Use of shape functions

The bar element is so simple that the full description could be made entirely

by reference to the vector x connecting the end-points A and B For most

other elements the notion of shape functions is needed Shape functionsdescribe the displacement field in the element, usually in terms of a set oflocal, or basic, coordinates In order to illustrate the use of shape functions

in a simple case, the equilibrium equations already treated in Section 2.2are re-derived

The position vector r(ξ) of an arbitrary point of the bar AB is defined by

interpolation between the end-points xAand xB In the initial configuration

r0(ξ) = h A (ξ)x A0 + h B (ξ)x B0. (2.38)The shape functions are defined by

h A (ξ) = 12(1− ξ), −1 < ξ < 1,

h B (ξ) = 12(1 + ξ), −1 < ξ < 1. (2.39)

The basic coordinate ξ is traditionally normalized to the interval ( −1, 1).

The shape function representation (2.38) can also be written in matrix mat:

for-r0(ξ) = [ hA (ξ)I , h B (ξ)I ] ˜x0. (2.40)

In the displaced configuration the same relation holds, but with ˜ x0 replaced

by ˜ x = ˜ x0+ ˜ u.

2.4 Use of shape functions 27

The Green strain at a point defined by ξ is determined by using the

definition (2.6) on an infinitesimal vector with initial value dr0(ξ) and final

The infinitesimal vector dr in the displaced configuration follows from a

similar formula with ˜ x0 replaced by ˜ x = ˜ x0 + ˜ u. From this the scalarproducts in the strain definition (2.41) are found as

δε G N ds − δ˜u T˜f = 0. (2.47)

This identity is expressed in matrix form by substitution of the virtual strainfrom (2.46) In the present case the integration is trivial For more com-plicated elements the volume integral must be evaluated by approximate

numerical integration in terms of the basic variables (ξ, ).

The variation δ˜u of the displacement vector is arbitrary, and thus the scalar

identity (2.48) gives the vector equilibrium equation

equi-Example 2.1 Equilibrium and stiffness of two-element truss.The equilibrium equations and the tangent stiffness of the two-element trussshown in Fig 2.6 are derived and used to describe a lateral bifurcation

instability and the associated displacement pattern (Pecknold et al., 1985).

The equations are used to illustrate special numerical solution techniques inChapter 8

Fig 2.6 Two-element truss supported by a lateral spring.

Figure 2.6 shows a truss consisting of two identical bar elements with

stiff-ness AE The initial geometry is specified by the lengths a, b and c, where c

indicates the deviation from vertical, e.g due to a geometric imperfection

The initial length of the elements is denoted l0 Lateral support is provided

by a linear spring with stiffness k This spring remains horizontal during deformation The load consists of a vertical force f1 and a horizontal force

f2 acting at the top of the truss In the present example the general librium equations and tangent stiffness are derived The specific results are

equi-here limited to the ‘perfect’ structure, c = 0 and f2 = 0