2.02 stabilityofelastic,anelastic,and …cee.northwestern.edu/people/bazant/pdfs/papers/s49.pdf ·...

2.02Stabilityof Elastic,Anelastic, andDisintegrating Structures, and FiniteStrain E¡ects: anOverviewZ.P. BAZ" ANTNorthwesternUniversity, Evanston, IL, USA

2.02.1 INTRODUCTION 47

2.02.2 ELASTIC STRUCTURES 48

2.02.2.1 Buckling of Columns, Frames, and Arches 482.02.2.2 Dynamic Stability Analysis and Chaos 502.02.2.3 Energy Analysis of Stability of Elastic Structures, Postcritical Behavior, and Catastrophe Theory 522.02.2.4 Shells, Plates, Thin-wall Beams, and Sandwiches 55

2.02.3 ANELASTIC AND DISINTEGRATING STRUCTURES 58

2.02.3.1 Elastoplastic Buckling and Shanley’s Bifurcation 582.02.3.2 Stability Implications of Normality Rule and Vertex Effect 602.02.3.3 Viscoelastic and Viscoplastic Buckling 612.02.3.4 Thermodynamics of Structures, Inelastic Stability, Bifurcation and Friction 622.02.3.5 Stability Problems of Fracture Mechanics 652.02.3.6 Damage Localization Instabilities and Size Effect 67

2.02.4 NONLINEAR 3D FINITE-STRAIN EFFECTS ON STABILITY 71

2.02.4.1 Finite-strain Effects in Bulky or Massive Bodies 712.02.4.2 Nonlinear Finite-strain Effects in Columns, Plates and Shells Soft in Shear 722.02.4.3 Discrepancy between Engesser’s and Haringx’s Formulas: New Paradox and its Resolution 732.02.4.4 Shear Stiffness Associated with Second-order Strain 742.02.4.5 Differential Equations of Equilibrium Associated with Different Finite-strain Measures 752.02.4.6 Ramifications 77

2.02.5 CLOSING REMARKS 77

2.02.5.1 Implications for Large-strain FE Analysis 772.02.5.2 Looking Back and Forward 77

2.02.6 REFERENCES 77

2.02.1 INTRODUCTION

Stability is a fundamental problem of solidmechanics whose solution determines theultimate loads at which structures fail or deflectexcessively. It is an old problem which has beenstudied for two and half centuries. The evolu-tion of the theory was for a long time focusedon elastic structures for which the loss of

stability is the primary cause of failure, thematerial strength having little or no relevance.Throughout the twentieth century, interestgradually expanded to anelastic structureswhere stability loss and material failure areintertwined due to plastic behavior and creep.Since the mid-1970s, the destabilizing effects ofmaterial disintegration caused by damage

47

localization and fracture propagation, as wellas the finite strain effects on stability of three-dimensional (3D) bodies, have received en-ormous attention. Thus, as of early 2000s, thetheory of structural stability may be seen as avery broad field which encompasses most ofsolid mechanics and intersects with most if itsdomains, while resting on the same basicconcepts and utilizing the same mathematicalapproaches.The present chapter, taking this very broad

viewpoint, will attempt a synthesizing review ofthe main results in the entire field of stabilitytheory as it exists today. The rich spectrum ofresults in the classical theory of elastic stability,including the difficult but beautiful subject ofshell bucking, can be covered only succinctly inthis, admittedly overambitious, review. Anenhanced emphasis, out of proportion to thescope of existing results, will be placed onanelastic structures, especially on the challen-ging modern problems of stability of structureslosing their integrity because of distributeddamage or localized fracture, and on the finitestrain effects in 3D bodies, an intriguing topicthat has been generating controversies andlively polemics for almost a century. Thediscussion will focus on these ‘‘hot’’ topics inmuch more detail than other stability studies.While the first two sections are of a reviewcharacter, the last section, devoted to 3D finitestrain effects on stability, will present inconsiderable detail a new explanation of thedifferences between the Engesser- and Haringx-type theories of sandwich buckling, widelyregarded as paradoxical.To avoid excessive length and to make this

review easily accessible to engineers andscientists from other fields desiring to acquaintthemselves with structural stability theory,mathematics will be kept to the bare minimum.The differential equations as well as thederivations will be omitted. Nevertheless, toenhance understanding, the physical causesand mechanisms of various types of instabil-ities will be discussed, albeit concisely.Apart from some selected contributions, only

the main literature sources to the vast field ofstability can be cited here. Extensive literaturereferences and a detailed exposition of most ofthe material covered here can be found in thebook by Bazant and Cedolin (1991). Perspica-cious reviews of the more classical material onelastic and plastic stability, considerably moredetailed and much more mathematical, weregiven by Hutchinson (1974) and Budianski andHutchinson (1972), and a valuable review ofexperimental evidence is found in Singer et al.(1998). The earlier works are covered inTimoshenko and Gere (1961).

2.02.2 ELASTIC STRUCTURES

2.02.2.1 Buckling of Columns, Frames, andArches

The primary cause of failure of slenderelastic beams or frames carrying large com-pressive axial forces due to gravity loads is notfailure of the material but attainment of Euler’s(1744) first critical load,

Pcr1 ¼p2

L2

� �EI ð1Þ

where E¼Young’s modulus, I¼ centroidalmoment of inertia of the cross-section, andL¼ effective length representing the half-wavelength of deflection curve, which dependson the boundary conditions. For the idealizedcase of a perfect structure, Pcr represents astate of neutral equilibrium (i.e., a state atwhich the deflections can increase at constantload), and a state of symmetry-breakingbifurcation of the equilibrium path in theload-deflection space. The critical stress

scr ¼Pcr

A¼ p2E

L=rð Þ2ð2Þ

where A¼ cross-section area and r2 ¼ I=A;depends only on E and the slenderness L/r,and exhibits no size effect.The analysis of beam or column buckling

requires that the deflections of the structure betaken into account in writing the differentialequations of the beam or frame based on thetheory of bending. For reasons of geometry,the problem is nonlinear at large deflections,but linearization for small deflections leads to alinear fourth-order ordinary differential equa-tion for the deflections. The critical load isfound from a linear eigenvalue problem.Real columns inevitably have imperfections

such as an initial curvature, axial load eccen-tricity, or small lateral loads, and often aresubjected to large initial bending moments M0:Because of the imperfections, the deflectionincreases rapidly when Pcr1 is approached.Small deflections w near Pcr are given byYoung’s (1807) formula

w ¼ w0

1� P=Pcr1ð Þ ð3Þ

where constant w0 characterizes the initialimperfection or initial bending moments.While this formula is exact only for initialsinusoidal curvature, it represents an asympto-tic approximation near Pcr1 for all kinds ofinitial imperfections. Therefore it has beenadopted as a universal basis of design codes(a less simple formula, which was used for steel

48 Stability of Elastic, Anelastic, and Disintegrating Structures, and Finite Strain Effects

structures, is obtained for an eccentric load butit is asymptotically equivalent near the criticalload). The deflection given by Young’s formulais used to calculate the initial moment magni-fication by P and w. The magnification has afactor depending on the initial moment dis-tribution along the column, which is given indesign codes by approximate formulas.Although the basic cases of critical loads of

columns with various end conditions weresolved by Euler as early as 1744 (long beforethe theory of bending was completed byNavier), developments in column theory havenevertheless proceeded until modern times.Simple but realistic design formulas takinginto account geometrical imperfections andinitial bending moments in metallic and con-crete columns or frames were incorporated intodesign codes by the middle of the twentiethcentury.The curve of finite deflections of very slender

elastic columns of constant cross-section (acurve called the ‘‘elastica’’) was solved exactly(Kirchhoff, 1859) in terms of elliptic integrals.Very large deflections of arbitrary elasticcolumns and frames can today be easilycalculated by nonlinear finite element (FE)codes.Many extensions of the column buckling

theory have been studied. The basic cases ofspatial buckling of beams subjected to axialforce and various kinds of torque are amenableto simple analytical solutions. If the column isnot slender, or if it is orthotropic, with a lowshear modulus (as in fiber composites), thebending theory must be enhanced by takinginto account shear deformations, which cansubstantially decrease the value of critical load(note a later comment on the formal equiva-lence but different applicability of the Engesserand Haringx formulas). Consideration of theeffect of pressure in compressed fluid-filledpipes, or of axial pre-stress introduced byembedded tendons, is a problem appearing atfirst tricky but in fact easy to handle.The buckling of elastic frames, i.e., assem-

blages of rigidly or flexibly connected beams, isnormally analyzed by the stiffness method, onthe basis of the stiffness matrices of beamelements. The stiffness matrix relates theincrements of the bending moments and shearand axial forces at the ends of the element tothe increments of the associated displacementsand rotations. Because of the second-ordermoments of the initial axial force on thedeflections, the stiffness coefficients dependon the axial force. If the beam has a uniformcross section, this dependence can be describedby simple trigonometric, hyperbolic and poly-nomial functions of the initial axial load,

derived by James (1935) and Livesley andChandler (1956) and others. Trigonometricexpressions for the inverse flexibility matrixcoefficients were derived earlier by von Misesand Ratzensdorfer (1926) and Chwalla (1928).There are two possible approaches to calcu-

lating the critical loads of frames: (i) the beamis subdivided into many sufficiently short beamelements, in which case the dependence of thestiffness matrix on the initial axial force getsautomatically linearized and the critical load(or parameter of the load system) can then beobtained from a linear matrix eigenvalueproblem; or (ii) the stiffness matrix of theentire beam is used. In the latter approach, thenumber of unknowns is one or two orders ofmagnitude less but the matrix eigenvalueproblem is nonlinear because the stiffnesscoefficients are highly nonlinear functions ofthe load. The latter approach is computation-ally far more efficient but requires a morecomplicated iterative solution based on succes-sive tangential linearizations of the stiffnessmatrix.Designers sometimes simplify the frame

buckling problem by considering a columnwithin the frame as a separate elastic columnwith flexible end restraints, isolated from theframe. However, such a trivialization ofteninvolves large errors on the unsafe side becausethe dependence of the effective tangentialstiffness of the end restraints on the unknowncritical load is neglected.The redundant internal forces in frames can

vary at the critical state while the load isconstant. This means that the flexibility matrixof the primary statically determinate structurebecomes singular at the critical load. Typically,however, the coefficients of this matrix becomeinfinite and the matrix loses positive definite-ness before the critical load is reached. Thismakes the flexibility matrix unsuitable forchecking stability and the flexibility methodcomputationally unsuitable for structures withmore than a few redundant internal forces.Very large regular rectangular frames or

lattices lead to linear difference equations. Thebasic cases can then be solved exactly by themethods of difference calculus (Bazant andCedolin, 1991, Section 2.02.3). The simplestmodes, easily solved by one-line formula, arethe internal buckling of either sway or nonswaytype, and the boundary buckling. Approximat-ing the difference equations as partial differ-ential equations (Bazant, 1971b), one can solvethe long-wave (global) buckling of the frame asa continuum. Because the rotations f of thejoints are independent of the rotations y of thechords of the beams, the proper approximation(Bazant, 1971b, Bazant and Cedolin, 1991,

Elastic Structures 49

section 2.10), which must be exact up to thesecond-order terms, is the micropolar conti-nuum. (Be warned that several micropolarapproximations found in the literature areincorrect because cross terms such as ff,xx

are missing yet contribute to the quadraticterms of energy density (x¼ spatial coordi-nate).) The critical loads of large regularframes or lattices with rectangular boundarieswere thus solved in 1973 analytically (Bazantand Cedolin, 1991). Various types of built-upor latticed columns can be continuouslyapproximated as a continuous column but theshear deformation must be taken into account(see later comments on the Engesser andHaringx formulas).An intricate case for which a sophisticated

special theory has evolved (cf. Bazant andCedolin, 1991, section 2.8) is the buckling ofhigh arches and slender rings (or cylindricalshells in the transverse plane). Their buckling isdescribed by a fourth-order linear differentialequation for arch deflections. Its coefficientsare not constant but vary along the arch withits initial curvature. They also depend on theload in a more complicated way than in thecase of columns. The axial inextensibility of thearch, which is a justifiable simplifying assump-tion for high (or deep) arches, presents furtherrestrictions. In the case of hinged arches, theinextensibility constraint excludes the odd-numbered critical loads, which correspond tobuckling modes with an odd number of waves.(Noting this restriction, Hurlbrink (1908)corrected the previously accepted Boussinesqsolution of the lowest critical value of uniformload.)

2.02.2.2 Dynamic Stability Analysis andChaos

Up to now, we tacitly implied the loads to beconservative. Such problems can be solvedstatically and do not necessitate the use of thegeneral criterion of stability, which is dynamic.However, for structures subjected to noncon-servative loads, stability may, and often is, lostin a dynamic manner. Important examples ofnonconservative loads are: (i) loads withprescribed time variation (e.g., pulsatingloads), (ii) loads generated by the flow of gases(wind loads) or liquids, and (iii) reactions fromjet or rocket propulsion. An idealization of (ii)and (iii) are the follower loads, whose orienta-tion follows the rotation of the structure.The fundamental definition of stability that

is generally accepted in all fields of science isdue to Liapunov (1893) and may be simplystated as follows. A given solution (or, as a

special case, a given equilibrium state) of adynamic system is stable if all the possiblesmall perturbations of the initial conditions canlead only to small changes of the solution (orresponse). (In the case of continuous struc-tures, it makes sense to measure the changes interms of an overall norm of the deflectiondistribution rather than locally.)Based on Liapunov’s definition, it can be

shown that, under certain mild restrictions,stability can be decided by analyzing alinearized system. This, for instance, justifiesconsidering only linearized small-strain expres-sions in columns, plates, and shells. Stablesystems, when perturbed, develop vibrations,either undamped or damped. The vibrationfrequency depends on the applied load.It is further easily proved that stability is lost

when the frequency of natural vibrationsbecomes complex (Borchhardt’s criterion). Inconservative systems, the frequency diminisheswith increasing static load and becomes zero atthe limit of stability, which represents thecritical state of neutral equilibrium (Figure1(a)). Stability is then lost through nonaccel-erated (static) motion away from the initialequilibrium state, called the divergence (foraircraft wings) or (more generally) staticbuckling.An important property of nonconservative

systems is that the frequency at the loss ofstability can be nonzero (Figure 1(b)), in whichcase a static stability analysis is inapplicable.Typically, dynamic instability is caused byvibrations of ever increasing amplitude, duringwhich the structure moves in such a mannerthat it absorbs an unbounded amount ofenergy from the nonconservative load (suchas wind or pulsating load). The dynamicinstability, also called flutter, is an importantconsideration for aircraft wings, as well as tallguyed masts, chimneys and suspension bridges(Simiu and Scanlan, 1986). A famous exampleis the aeroelastic instability that destroyed theTacoma Narrows Bridge in 1940.Pulsating loads, produced, e.g., by unbal-

anced rotating machinery or traveling vehicles,can lead to dynamic instability in the form ofparametric resonance (Rayleigh, 1894). Thestructure moves in such a way that the axialmode absorbs an unbounded amount of energyfrom the load. This kind of resonance occurs atdouble the natural frequency of lateral vibra-tions corresponding to the static (average)value of the load (Figure 1(c)). The doublingof frequency is explained by the fact that thesecond-order axial (or in-plane) strains due tolateral deflections of columns (or plates), onwhich the axial (or in-plane) forces work, havedouble the frequency of the lateral deflections.


Due to higher-order Fourier components ofaxial strain history, a milder parametricresonance may also occur at other integermultiples of the natural frequency. An impor-tant aspect of parametric resonance is that thestructure is stabilized by sufficient damping. Inconservative systems, by contrast, the dampinghas no stabilizing influence.In a rotating coordinate system, forces

resulting from apparent accelerations, such asthe Coriolis force and gyroscopic moments,can stabilize the structure even though they dono work on the motion of the structure. TheCoriolis force, e.g., causes an unbalancedrotating shaft to regain stability at supercriticalrotation velocities.An important consequence of Liapunov’s

definition of stability is the Lagrange–Dirichlettheorem (Lagrange, 1788). When the totalenergy is continuous and all the forces areconservative or dissipative, the equilibrium isstable if the potential energy of the structure asa function of all the generalized displacementsis positive definite (i.e., has a strict minimum).This theorem greatly simplifies the analysis

of conservative systems, making it possible touse static analysis, avoiding the complicationsof dynamic analysis of a structure withsymmetry-breaking imperfections. To simplifythe analysis of nonconservative systems, it is of

great interest to find test functions analogousto potential energy, called Liapunov functions,which make it possible to decide stabilitywithout dynamic analysis. Such functions,however, have been discovered only for somespecial situations with nonconservative loads.Nonlinear dynamic systems that cannot be

linearized can exhibit a complex dynamicresponse that is nonperiodic and appears tobe random (Figure 2). Such a response, calledchaos, shows nevertheless a certain degree oforder and cannot be described by methods ofrandom dynamics. Great attention has beendevoted to the trajectories of response of suchsystems in the phase space (a space whosecoordinates are the generalized displacementsand their velocities). A typical property ofdamped stable linear oscillators is that thetrajectory is attracted, in several characteristicways, to a single point, called the attractor. Fora nonlinear oscillator, the trajectory appears aschaotic but, on closer scrutiny, is often foundto be attracted to something called the ‘strangeattractor’, which describes a hidden order inthe response and normally has a fractalstructure. Chaotic systems are inherently un-stable—very small perturbations produce tra-jectories that exponentially diverge from theoriginal trajectory (Figure 2). This makes theresponse over longer periods of time unpre-

Figure 1 (a) Dependence of free vibration frequency o on static load P for conservative systems. (b) Typicalexample for nonconservative systems. (c) Strutt diagram for parametric resonance (p¼ amplitude parameterof pulsating load; O¼ forcing frequency; b¼ damping parameter).


dictable (Thompson, 1982, 1989; Thompsonand Stewart, 1986; Moon, 1986).

2.02.2.3 Energy Analysis of Stability ofElastic Structures, PostcriticalBehavior, and Catastrophe Theory

By virtue of the Lagrange–Dirichlet theo-rem, stability of equilibrium of elastic struc-tures under conservative loads can be decidedby checking the positive definiteness (existenceof a strict minimum) of the potential energy Pas a function of displacement vector q near theminimum point of P. Aside from q, P alsodepends on various control parameters includ-ing load P and, for imperfect structures, alsoon the imperfection magnitude a characterizingthe deviation from symmetry.The Taylor series expansion of the potential

energy in terms of displacement vector q aboutthe equilibrium state (defined as q¼ 0) beginswith the quadratic term called the secondvariation,

d2P ¼ qTKq

2ð4Þ

where

K ¼ @2P@qT@q

ð5Þ

which represents the tangential stiffness matrix.The structure is stable if d2P (or K) is positivedefinite. If the potential exists, K is symmetric(and real), and so the structure is stable if andonly if all the eigenvalues of K are positive; orequivalently, if and only if all the principalminors of K are positive (Sylvester’s criterion).Of main interest is the lowest critical load,

P¼Pcr1, for which K becomes singular and thequadratic form d2P positive semidefinite. Forcolumns and frames (but not necessarily plates

and shells), the lowest critical load is the first,for which the buckling wavelength is thelongest. In conservative systems, the lowestcritical load represents the limit of stability. Forhigher loads P, the quadratic form is indefiniteor negative definite, which implies that con-servative systems are unstable for P4Pcr1.The behavior after the critical load is

reached, called the postcritical behavior (Fig-ure 3), is determined by the term of the Taylorseries expansion of P that comes next after thequadratic term. An important aspect is thepostcritical imperfection sensitivity, which de-scribes how the maximum load, Pmax, isaffected by the magnitude of small imperfec-tion a. Despite tremendous variety of structur-al forms, all the initial postcritical behavior cantake only a few typical forms.

Figure 3 Responses of perfect and imperfectstructures in bifurcation buckling: (a) stable sym-metric, (b) unstable symmetric, (c) asymmetric, and(right) Koiter’s power laws (l¼ load parameter,q¼ lateral deflection, q0¼ q at lmax).

Figure 2 Chaotic vibration of nonlinear system (divergent response change caused by a very small change ininitial conditions).


For many structures, P can be considered afunction of only one deflection parameter q.Typical are symmetric structures which, in theabsence of imperfections, remain symmetric upto the lowest critical load Pcr: They possess apotential energy function P(q) that containsthe quadratic and quartic terms but misses thecubic terms in the Taylor series expansion.Such structures exhibit postcritical behaviorthat is symmetric with respect to q, termedsymmetric bifurcation. Depending on the signof the quartic term, the critical state may bestable or unstable, which is called the stable orunstable symmetric bifurcation (Figures 3(a)and (b)). The former (which is typical ofcolumns, symmetric frames, and plates) isimperfection insensitive. The latter is imperfec-tion sensitive (a behavior found is some typesof frames; Bazant and Cedolin, 1991; Bazantand Xiang, 1997a); this means that theequilibrium load value of perfect structure(a¼ 0) decreases with |q|, and that Pmaxopcrfor aa0:Imperfection sensitivity, i.e. the reduction of

Pmax caused by imperfection, is stronger forelastic structures that exhibit bifurcation andpossess the cubic terms in P(q). In that case,which is called asymmetric bifurcation, thecritical state is always unstable and thestructure is always imperfection sensitive(Figures 3(c), 4). Such behavior is typicallyexhibited by asymmetric frames (the classicalexample being the G-shaped frame; Figure 4),

and in an extreme manner by many shells (e.g.,spherical shells, or cylindrical shells under axialcompression or bending (but not under radialpressure or torsion)).A famous result of elastic stability theory is

that, for all types of imperfection sensitiveelastic structures, the postcritical imperfectionsensitivity of bifurcation buckling can be onlyof two types, characterized by Koiter’s (1945)power laws:

DPmax

Pcrp a2=3 or a1=2 ð6Þ

where DPmax ¼ Pcr � Pmax: The former appliesto unstable symmetric bifurcation, and thelatter to asymmetric bifurcation. The latter isgenerally more dangerous because DPmax ismuch larger when a is sufficiently small.Imperfection sensitivity is also important fordynamic buckling (Budiansky and Hutchinson,1964).Consider now elastic structures that possess

no cubic term in P(q) and exhibit no bifurca-tion, due to nonexistence of symmetric deflec-tions. In this case, called snapthrough, thedeflection curve P(q) has a limit point (peak,maximum point, Pmax), which represents thestability limit (critical state) if the load is agravity load (dead load). After the limit point,the response under gravity load becomesdynamic; the structure ‘snaps through’, inaccelerated motion. Such behavior is typicalof flat arches or shallow (cylindrical orspherical) shells, in which the failure is causedby shortening of the arch or in-plane normalstrains of the shell. The postpeak equilibriumload-deflection curve exhibits postpeak soft-ening, which is unstable for load control(gravity load). For displacement control, thepostpeak softening is stable, but only as long asthe slope of the curve is negative. If the slopebecomes vertical, stability is lost despitedisplacement control, which is called snap-down. After snapdown, the slope of the curvemay become positive again but the states areunstable. Snapdown is exhibited by elastic flatarches or shallow shells loaded through asufficiently soft spring.In bifurcation buckling, the deflection curves

of imperfect structures exhibit a limit pointwith snapthrough rather than bifurcation.Since inevitably some imperfections are alwayspresent, bifurcation buckling is merely anabstraction, albeit a very useful one.When a structure has two equal or nearly

equal critical loads corresponding to differentbuckling modes, the modes usually interact in away to cause imperfection sensitivity. Thisoccurs in built-up (latticed) columns if the

Figure 4 Roorda’s (1971) experimental verificationof calculated postcritical response in asymmetricbifurcation of a G-frame.


flanges or lattice members buckle locally at thesame load as the column as a whole, instiffened plate girders if the stiffening ribsbuckle locally at the same load as the web, inbox girders if the stiffeners buckle locally at thesame load as the plates, or if the plates buckleat the same load as the box, etc. Disregardingpostcritical behavior, designs with coincidentcritical modes would seem to yield optimumweight, but in fact they represent what is calledthe ‘‘naive’’ optimal design, which must beavoided because it produces high imperfectionsensitivity.The type of buckling of a system governed

by a potential is qualitatively fully determinedby the topological characteristics of the poten-tial surface. The problem is analogous toinstabilities encountered in various fields ofphysics and other sciences and is generallydescribed by the catastrophe theory. Considerthe potential P as a general function of freevariables q1;y; qn; and of control parametersl1;y; lm corresponding to the load and theimperfections. If n¼m¼ 1, there exists onlyone type of catastrophe called the fold,equivalent to snapthrough or to asymmetricbifurcation. If n¼ 1 and mr2, there exists asecond catastrophe called the cusp, equivalentto symmetric bifurcation (Figure 5). If nr2and mr4, there exist seven catastrophes, thefive additional ones being called the swallow-tail, butterfly, hyperbolic umbilic, elliptic um-bilic, parabolic umbilic, and double cusp (thisremarkable result has been rigorously provedby Thom, 1975). Examples of elastic structuresthat exhibit all these seven catastrophes havebeen given, although some have an air ofartificiality. Completely general though thecatastrophe theory is usually perceived to be,its present form is nevertheless inapplicable toelastoplastic, damaging and fracturing struc-tures—a very important class.The potential energy concept is useful as the

basis of direct variational methods of calculat-ing the critical loads of continuous elasticstructures. The deflection field is described as

w xð Þ ¼XNi¼n

qifi xð Þ ð7Þ

where fiðxÞ are chosen as linearly independent(preferably orthogonal) functions of the co-ordinate vector x (1D, 2D, or 3D). In the Ritz(or Rayleigh–Ritz) variational method, thepotential energy P is minimized with respectto deflection parameters qi. This yields thenecessary conditions

@P@qi

¼ 0 ð8Þ

for all qi, which represent equations for qi. If Pis quadratic, the equations are linear, and ifthey are homogeneous one has a matrixeigenvalue problem for the critical load. Thesolution converges for n-N if functions fi(x)form a complete system. For a finite n, one hasan upper bound approximation on Pcr1, whichis close enough if n is large enough or if theselected functions fi(x) can closely approxi-mate the deflection shape. The FE method forconservative problems can be regarded as aspecial case of the Ritz variational method.Often it suffices to use only one judiciously

chosen function f1(x); the critical load is thengiven by the Rayleigh (1894) quotient

PR ¼ U

%Wð9Þ

where U¼ strain-energy expression (quadratic)in terms of displacement distribution, w, and%W ¼ expression for work per unit load, P¼ 1(also quadratic). Analyzing d2PR; one can

Figure 5 Surfaces of potential P for three bucklingtypes. (a) Snapthrough or limit point (fold cata-strophe). (b) Unstable symmetric bifurcation (cuspcatastrophe). (c) Asymmetric bifurcation (also foldcatastrophe); q¼ deflection, P¼ l¼ load parameter.


prove that PR represents an upper bound onPcr1. The Ritz method is equivalent to mini-mization of PR with respect to f1 xð Þ (con-sidered as unknown). In mathematics, theRayleigh quotient PR is equivalently expressedin terms of the differential operators of theboundary value problem (provided the pro-blem is self-adjoint, which is automatically thecase if P exists).For statically determinate columns, an upper

bound that is always closer to Pcr1 than PR isgiven by the Timoshenko quotient

PT ¼%W

U1ð10Þ

where U1 is the complementary strain energycalculated from the second-order bendingmoments M caused by a unit load (P¼ 1)acting on the deflections w, expressed in termsof w. PT can be shown to represent nothing elsebut the Rayleigh quotient as defined in mathe-matics on the basis of the differential operatorsof the second-order differential equation forbuckling of statically determinate columns.From the viewpoint of design safety, it

would be preferable to have lower boundsrather than upper bounds on Pcr1; they existbut, unfortunately, in most cases are not closeenough to be useful (Bazant and Cedolin, 1991,section 5.8).Based on the calculus of variations, from P

as a functional of deflection w(x), one canderive the differential equation of the problem.This approach, which is useful for morecomplicated problems such as thin-wall bars,also yields the boundary conditions that arecompatible with the existence of a potential.They are of two kinds: kinematic (essential) orstatic (natural).

2.02.2.4 Shells, Plates, Thin-wall Beams, andSandwiches

The design of shells and plates is dominatedby stability. Shell buckling is a problem with afascinating history. After the critical loads ofexternally pressurized spherical shells andaxially compressed cylindrical shells werecalculated at the dawn of the twentieth century(Lorenz, 1908; Timoshenko, 1910; Southwell,1914), they were found to be 3–8 times largerthan the experimental failure loads (Figure6(b)). Despite persistent efforts, the discre-pancy remained unexplained until von Karmanand Tsien (1941) (Figure 6(a)) discovered theanswer to lie in the extreme imperfectionsensitivity of nonlinear postcritical behaviorwhich causes the bifurcation to be asymmetric(their celebrated result was later found to fit the

broader context of Koiter’s (1945) postcriticaltheory, already discussed). The asymmetry iscaused by the fact that the radial resultant ofthe in-plane stresses in a curved shell has adeflection-dependent quadratic term whichresists an inward buckle and assists an outwardbuckle (this resultant is analogous to a non-linear quadratic lateral spring support of acolumn; von Karman and Tsien, 1941).The buckling of some shells is imperfection

insensitive, permitting the design to be basedon critical loads (e.g., a cylindrical shellsubjected to lateral pressure or torsion; Figure6(c)). The archetypical cases of imperfection-sensitive shells are the externally pressurizedspherical shell and the cylindrical shell sub-jected to axial compression or bending. Intheory, a perfect shell of the imperfection-sensitive type exhibits, after bifurcation, adynamic snapdown to a postcritical load valuetypically equal to only 15–35% of Pcr1, but inpractice the imperfections are always so largethat the load cannot exceed these values atreasonable deflections (see the curves forimperfection parameter 0.1 or 0.2 in Figure6(a)). That the critical loads can indeed beclosely approached was experimentally demon-strated only when Almroth et al. (1964) andTennyson (1969) succeeded in fabricatingcylindrical shells with extraordinarily smallimperfections.The critical loads for axisymmetric buckling

(Figure 6(d), bottom) represent a 1D problemwhich can be solved easily. The 2D non-axisymmetric buckling modes (Figure 6(d),top) are harder to solve. Often the problemcan be simplified by considering the shell asshallow (which means that the rise of everypossible buckle is small compared to the chordof the buckle arc). A famous result for suchshells was Donnell’s (1934) reduction of thecritical load problem to one linearized eighth-order partial differential equation for shelldeflection. A system of eight linearized partialdifferential equations, known as the Donnell–Mushtari–Vlasov theory, was obtained forgeneral shallow shells. Based on the shallowshell approximation, critical loads for manymodes of various shells have been solvedanalytically, before the FE era.Calculation of the postcritical behavior and

failure loads of shells is a difficult problem,sometimes even with FE programs. Althoughverification of an important design by FEs isimperative, the design is generally based oncritical load solutions reduced by an empirical‘‘knock-down’’ factor f which accounts forimperfection sensitivity and has been tabulatedfor various practical cases (e.g., Kollar andDulacska 1984). For instance, for a cylindrical


shell of radius R and thickness h, the axialinternal force at failure is expressed as

N�xx ¼ fEh2

Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1� n2ð Þ

p ð11Þ

where n¼Poisson’s ratio, and f depends onR/h (e.g., Budiansky and Hutchinson, 1972;Budiansky, 1974). For combined loadings, alinear interaction diagram may be safelyassumed in design (as, e.g., between themaximum Nxx values due to bending and toaxial loading of a cylindrical shell). Because theimperfections are highly random and have alarge influence, a probabilistic estimation of thefailure load is appropriate (e.g. Bolotin, 1969).The reason for the extreme imperfection

sensitivity of shells may be seen in the existenceof many buckling modes with nearly equalcritical stresses. For example, the criticalstresses for many nonaxisymmetric bucklingmodes of an axially compressed cylindricalshell are only slightly lower than that for the

axisymmetric pattern. In postcritical deflec-tions, the buckle patterns may change and thefinal one is the diamond buckle pattern(Yoshimura pattern, Figure 6(e)). When asignificant lateral pressure is superimposed onaxial compression, the critical loads for differ-ent modes are no longer close to each other,and such a shell ceases to be imperfectionsensitive.Some nonlinear problems of shells were

approximately solved before the advent ofFEs (e.g., by variational series expansions).Many older analytical solutions, however, havenow lost much of their value because of theircomplexity. But those that are simple never-theless remain valuable for the understandingthey convey and for use as checks on FE codes,and are invaluable for design optimization andprobabilistic modeling.Let us now discuss the simpler problem of

plates, which became well understood abouttwo decades earlier than shells. Fourier serieswith Ritz variational methods were used before

Figure 6 (a) Von Karman and Tsien’s (1941) postcritical diagrams of axial compression force versusshortening of cylindrical shell. (b) and (c) Comparison of critical loads of perfect shells with measured failureloads for cylindrical shells under axial compression (b) or radial pressure (c) (R, h, l¼ radius, thickness andlength of shell). (d) Critical modes. (e) Yoshimura’s final buckle pattern.


the middle of the twentieth century to solve thecritical loads of rectangular or circular plateswith various edge conditions and diversecombinations of in-plane normal and shearforces (e.g. Timoshenko and Gere, 1961). Likeshells, plates exhibit many buckling modes.Unlike shells, the critical loads are far apart(and thus they cannot interact to causeimperfection sensitivity). In contrast to col-umns and in similarity to shells, the criticalmode with the longest wavelength in onedirection is not necessarily the lowest criticalload for plates.In contrast to shells, the buckling of plates is

not imperfection sensitive. This was establishedearly in the nineteenth century on the basis ofthe approximate solutions of the famous vonKarman (1910)–Foppl (1907) nonlinear equa-tions for initial postcritical behavior (twocoupled fourth-order partial differential equa-tions for the deflection and the Airy stressfunction of in-plane stresses). If large deflec-tions take place, plates supported along theirentire boundary show a substantial postcriticalreserve. Its source is the ability of a plate toredistribute the in-plane compressive forcesinto compressed cylindrically buckled stripsalong the boundaries (for compression parallelto supported edges) and into diagonal ten-sioned strips (for shear loading). When largebuckles develop and the cylindrically buckledstrips carry most of the in-plane forces, theplate acts essentially as a truss. Ultimately, thebuckled strips yield, which means that simplelimit analysis can be applied to the truss. Sucha truss analogy was exploited by von Karmanet al. (1932) to deduce stunningly simpleapproximate formulas for the maximum post-critical load Pmax of plates subjected tocompression and shear; for simply supportedrectangular plates compressed in the directionof one side,

Pmax E kh2ffiffiffiffiffiffiffiffiffiEsY

pð12Þ

which, remarkably, does not depend on theplate dimensions (h¼ plate thickness,sY¼ yield stress, and kEconstant).Transverse shear deformations are unimpor-

tant in thin plates and shells. But they may besignificant in composite shells, and dominatethe buckling of sandwich plates and shells (e.g.,Plantema, 1966), i.e., plates that consist of stiffbut thin face sheets (skins) bonded to a softcore (a foam or honeycomb). Local buckling ofthe skin, which provokes delamination frac-ture, is an additional very important mode ofinstability of sandwiches. In sandwich shells,the critical loads for axisymmetric bucklingmodes and modes with periodic buckles along

the circumference are distinctly separated,which suppresses postcritical imperfection sen-sitivity (Tennyson and Chan, 1990). Micro-buckling of fibers due to shear causescompressive kink-band failure in composites(Rosen, 1965; Budiansky, 1983; Fleck, 1997;Bazant et al., 1999).Long thin-wall girders (e.g., metallic cold-

formed profiles, concrete or welded steelgirders for large bridges and buildings) repre-sent long shells that can be approximatelytreated according to the theory of thin-wallbeams. The first important result in this broaddomain was contributed by Prandtl (1899) inhis dissertation on lateral buckling of beams ofrectangular cross section subjected to bending,which he obtained in terms of Bessel functionsand verified by his experiments. His pioneeringwork, as well as the simultaneous solution of aspecial case of lateral buckling by Michell(1899), stimulated rapid further progress.The theory of thin-wall beams can in general

be regarded as a semi-variational approach(Kantorovich variational method) in which thebasic modes of deformation in the transversedirections are judiciously assumed and energyminimization then yields ordinary differentialequations for the deflections and torsionalrotations as well as the parameters of thesemodes as functions of the longitudinal coordi-nate. The reduction of the problem to ordinarydifferential equations greatly simplifies stabilityanalysis. Beside the deformation modes de-scribed by Saint-Venant torsion theory and bythe theory of bending with plane cross sections,one must consider for beams of open profile amode in which the cross section warps out ofplane. The bimoment produced by the warpingmode is an important mechanism in resistingtorsion (the resistance being provided by axialnormal stresses characterized by the bimo-ment). In box girders, one must also includemodes describing the bending deformation ofthe cross section within its own plane. 1D FEsof thin-wall beams of open or closed crosssection, incorporating the out-of-plane warp-ing mode and the in-plane cross-section defor-mation mode, have been formulated and usedto analyze critical loads and postcriticalbehavior.Solutions based on the warping torsion

theory describe the important cases of lateralbuckling of thin-wall beams, in which ahorizontal beam subjected to bending in thevertical plane twists and bends laterally, and ofaxial-torsional buckling of an axially com-pressed column.Limited though the studies of postcri-

tical behavior have been, some FE studiesindicate that lateral buckling of beams is not


imperfection sensitive and has a high post-critical reserve.

2.02.3 ANELASTIC ANDDISINTEGRATING STRUCTURES

2.02.3.1 Elastoplastic Buckling and Shanley’sBifurcation

Elastic structures fail either by exhaustingmaterial strength or by instability. Structuresthat are not elastic may, and usually do,fail by a combination of both. The evolutionof material failure becomes a part of process ofstability loss of the structure. Consideration ofinelastic behavior and material failure is todayan essential ingredient of a sound assessment ofstability of structures.A salient property of elastoplastic behavior

is the irreversibility at unloading, manifestedby the fact that the unloading modulus Eu islarger that the tangent modulus Et for furtherloading (in absence of damage, Eu¼E). Thisproperty causes a type of behavior not seen inelastic structures: the bifurcation of equili-brium path in the load-deflection space can,and in fact does, occur at increasing, ratherthan constant, load P. This phenomenon wasdiscovered by Shanley (1947) in a revolutioniz-ing paper which corrected a previous erroneousconcept that had lasted over half a century.The problem has a complicated history. In

two subsequent studies, Engesser (1889, 1895)proposed two different formulas for the load atwhich a perfect elastoplastic column begins tobuckle:

Pt ¼p2

L2EtI or Pr ¼

p2

L2ErI ð13Þ

where Pt is called the tangent modulus load,and Pr the reduced modulus load (Figure 7(a));Et is the tangent modulus of material forfurther loading from the stress level at initialunbuckled state, and the reduced modulus Er

(dependent on cross section geometry; Enges-ser, 1895, 1898) is calculated as the effectivemodulus for buckling at constant load (forwhich Et applies at the side of neutral axis thatundergoes further shortening, while Eu appliesat the side that undergoes extension (unload-ing) during deflection). The reduced modulustheory, yielding Pr, was supported and refinedby von Karman (1910). It had been accepted asvalid for five decades, until tests on aluminumalloys and high strength steels, exhibiting aslowly decreasing Et, revealed the buckling tobegin at Pt, which can be much smaller than Pr.Based on Shanley’s (1947) epoch-making

discovery, which was immediately accepted by

von Karman and was later generalized to 3Dsolids by Hill (1962) and others, the firstbifurcation of the equilibrium load-deflectionpath (Figure 7(a)) occurs when the tangentialstiffness matrix Kt of the structure becomessingular or, in the case of a discontinuousevolution of Et, when the smallest eigenvalue ofKt jumps from positive to negative (the reasonwill be mentioned later, in Equation (21)). Nounloading occurs anywhere in the structure,and the analysis based on Kt is often calledHill’s (1962) method of linear comparisonsolid. This is a valuable simplification. How-ever, the problem of bifurcation load is stillnonlinear because Et depends on Pt as de-scribed by the elastoplastic constitutive law.In contrast to elastic bifurcation, the Shan-

ley first bifurcation state does not representneutral equilibrium; the initial postbifurcationstates are always stable. The symmetry-break-ing secondary path (i.e., lateral deflection) isnevertheless the path that must occur inreality. These facts can be proved either byanalysis of imperfections or, less tediously andmore generally, by calculating the secondvariation d2S of the entropy increment (orHelmholtz free energy increment) of a perfectstructure and the differences in entropy incre-ment between the primary and secondarypaths (Bazant and Cedolin, 1991, section10.2; Figure 7(b)).In an elastoplastic column, all the (infinitely

many) undeflected states between Pt and Pr

represent bifurcation states (Figure 7(a)). Theyare stable if the load is controlled, and Pr is thestability limit. If the load-point displacement iscontrolled, the stability limit is higher than Pr.The maximum load, Pmax, is in practice usuallymuch closer to Pt than to Pr (Figure 7(a)). It isfor this reason that Pt is so important. Thedesign of metallic columns, frames, thin-wallbeams and shells is now generally based on thetangential modulus, and the concept is alsovalid for concrete (Bazant and Cedolin, 1991;Bazant and Xiang, 1997b).The initial postbifurcation behavior of per-

fect and imperfect structures near Pt is stable.In contrast to elastic structures, the deflectionsto the right and left are not symmetric if thecross section is nonsymmetric, and the Pr andPmax values for buckling to opposite sides aredifferent.When Et as a function of stress drops

suddenly, which occurs for mild steel, Pr can,and normally does, coincide with Pt. In thislight, it might seem strange that tests of hot-rolled steel profiles made of mild steel exhibit alarge difference between Pr and Pt. The reasonis that very large residual thermal stresses getlocked in after initial cooling in the steel mill


(Osgood 1951; Yang et al. 1952). They causethe cross-section parts with different residualstresses to begin yielding at very different loadvalues, thus rendering the overall force–defor-mation diagrams of the cross section smoothlycurved. Discovery of this phenomenon re-solved several previous decades of groping inefforts to explain the experiments in terms ofimperfections alone. Thus Shanley’s theory isimportant not only for alloy steels but also forhot-rolled profiles made of low-carbon mildsteel which passes from elastic behavior toperfectly plastic yielding almost instantly.The buckling behavior of reinforced concrete

columns is complicated by tensile cracking.

Young’s formula for the effect of imperfectionson the deflection and moment magnification isused in design, complemented by an empiricalformula for the effective modulus. The magni-fied moment M and axial force P are thencompared to a semi-empirical strength envel-ope in the (P, M) plane. Consideration ofperfect columns is avoided by prescribing forconsideration in design a certain minimumload eccentricity (Figure 8).Our discussion so far pertains to small

deflection behavior. Large plastic deflectionsof columns are of interest mainly for predictingthe energy absorption capability under blast,impact, and earthquake. If the yield plateau of

Figure 7 Inelastic buckling. (a) Shanley bifurcation in plastic structures with a smooth or bilinear stress–strain diagram. (b) Corresponding surfaces of the second variation of entropy d2S (the negative of potentiald2F) at Shanley bifurcation (bottom) and at reduced modulus load (top). (c) Reduction of tangential shearmodulus derived from torsional critical loads of axially compressed elastic-plastic cruciform columnsmeasured by Gerard and Becker (1957), and comparison with (smoothed) results of microplane model forsteel (solid curve, Brocca and Bazant, 2000).

Anelastic and Disintegrating Structures 59

the material is long enough, the fully plasticsoftening postpeak response of columns andframes can be determined relatively easily byanalyzing a plastic hinge mechanism subjectedto axial forces. When the structure is slenderand has high initial bending moments or largeimperfections, which is often the case, thetransition from elastic behavior to the hingemechanism is so rapid that it may be con-sidered as sudden. A close upper bound for theenergy absorption capability of the structure isthen provided by the area under the load-deflection curves for elastic behavior and forthe plastic hinge mechanism. The inter-section of these curves is an upper boundon the maximum load (provided the elasticload-deflection curve is rising up to theintersection). In the case of buckling of thinplastic plates under in-plane forces, one maysimilarly base a simple calculation on a yieldline mechanism.Very difficult 3D problems are the plastic

localization instabilities caused by the geome-trically nonlinear effects of finite strain (Rud-nicki and Rice 1975; Bazant and Cedolin, 1991,chap. 13). The best known example is necking(e.g., Hutchinson and Neale, 1978)—a narrow-ing of the cross section of a bar that is initiallyunder uniform tensile strain. The narrowingcauses the load to peak and then decrease atincreasing displacement. Finite-strain FE solu-tions (e.g., Needleman, 1982; Tvergaard, 1982)are very sensitive to the precise form of theplastic constitutive law. Geometric nonlinear-ity, however, is not the only cause of necking.A complete analysis needs to take into accountalso the damage due to the growth of voids ormicrocracks (which is discussed in a latersection).

Analysis of plastic localization instabilities isrequired to understand the bursting of pipesand other shells caused by internal pressure,bending failure of tubes due to ovalization ofcross-section, postcritical reserve in plates andthin-wall beams, etc.

2.02.3.2 Stability Implications of NormalityRule and Vertex Effect

When multiaxial stresses and strains arise inbuckling, a strong sensitivity to the multiaxialnature of the elastoplastic constitutive law isencountered. A central concept in the formula-tion of these laws is the normality rule. Thisrule requires that, in the 9D spaces of thecomponents of the stress tensor and the strainrate tensor, the strain rate (plastic flow) vectormust be normal to the current yield surface. Aspecial form of the normality rule, in thecontext of the J2 flow theory, was proposed byPrandtl (1924). Later, the normality rule wasgeneralized for any type of yield surface andwas derived from certain plausible workinequalities (Hill, 1950; Drucker, 1951). Itwas recognized that adherence to this rule,originated by Prandtl, is essential (Drucker,1951; Rice, 1971) for ensuring stability of thematerial, particularly for preventing spuriousstrain localization instabilities (to be discussedlater).Various experimentally observed phenomena

such as the dilatancy of frictional slip were laterfound to be in apparent violation of thenormality rule. One such phenomenon, theso-called vertex effect, was brought to light byGerard and Becker’s (1957) tests of torsionalbuckling of short axially compressed thin-wall

Figure 8 Left: typical failure envelope (interaction diagram) and curves of axial load P versus maximumbending moment M for a reinforced concrete column of slenderness L/r¼ 70, loaded at various constantrelative load eccentricities e/h (P¼ load in 1000 lb., M¼maximum bending moment in 1000 lb� in.). Right:corresponding diagrams of P versus axial displacement u.


cruciform columns (Figure 7(c)). The bifurca-tion load of these columns is proportional tothe incremental material stiffness ‘‘to-the-side,’’i.e., the stiffness (in this case the shear modulus)for a stress increment vector that is tangentialto the yield surface in the 9D space of stresscomponents (top right in Figure 7(c)). Accord-ing to the classical J2-plasticity theory based onthe normality rule, as well as all the plasticitytheories with a single loading potential surfacein the deviatoric stress space, the stiffness ‘‘to-the-side’’ is equal to the initial elastic stiffness.Thus the critical load for torsional buckling ofa cruciform column is predicted by thesetheories to be equal to the elastic critical load,regardless of the axial compressive stress. Thisis grossly incorrect and far higher than themeasured values (Figure 7(c) left).This inadequate performance of the plasti-

city models with a single loading potential inthe deviatoric stress space, which, as of early2000s, dominate the computational practice,has not been taken seriously for a long time,and has almost been forgotten in researchwithout ever being satisfactorily resolved. Fourtheories that have been used to deal with thisparamount problem of plasticity may bementioned.(i) Hencky’s deformation theory of plasti-

city. It predicts the stiffness to-the-side to besecant stiffness, which can be much smallerthan the initial elastic stiffness and gives arelatively good agreement with Gerard andBecker’s buckling experiments. However,Hencky’s is not a fully consistent theorysuitable for general FE programs.(ii) Enhancement of the plastic potential

surface with a vertex traveling with the currentstress point (Rice, 1975; Rudnicki and Rice,1975; Hutchinson and Neale, 1978; Hutchin-son and Tvergaard, 1980; Bazant, 1980; Bazantand Cedolin, 1991; section 10.7). This ap-proach, however, does not seem effective forgeneral FE programs.(iii) Multisurface plasticity, proposed by

Koiter (1953). This theory reflects the actualsource of the vertex effect, which is due to anintersection of several loading potential sur-faces at the current stress point. An importantadvantage of introducing multiple loadingsurfaces is that the vertex effect can be modeledeven while the normality rule, crucially im-portant for the soundness of constitutive laws(Rice, 1971), can be satisfied separately for theindividual surfaces. The multisurface constitu-tive model, however, is hard to identify fromtest data and has not been developed for large-scale computations, partly because very manysimultaneous loading surfaces would be re-quired for a general model.

(iv) Taylor-type crystallographic models,initiated by Taylor (1938), worked out in detailfirst by Batdorf and Budianski (1949) and laterrefined by others, (e.g., Rice, 1971; Bronkhorstet al., 1992; Hutchinson, 1970; Butler andMcDowell, 1998), and the related microplanemodel (Bazant, 1984; Bazant and Planas, 1998,chap. 14; Carol and Bazant 1997; Brocca andBazant, 2000; Caner and Bazant, 2000; Caneret al., 2002). Both types of constitutive modelsare defined not in terms of tensors and theirinvariants but in terms of the stress and straincomponents on surfaces of various orientationsin the material (called the microplanes for thelatter type). Either the stress components (forTaylor models) or the strain components (formicroplane models) are assumed to be theprojections of the stress or strain tensor. Theseare essentially multisurface plasticity models,though not expressed in terms of tensors andtheir invariants. They can separately satisfy thenormality rules, either on each crystallographicplane or on each microplane, yet exhibit thevertex behavior automatically, by describing itsphysical source directly. The microplane mod-el, with efficient (optimal Gaussian) numericalintegration over all spatial orientations ofmicroplanes (Bazant and Oh, 1986), has beendeveloped for large-scale computation of con-crete and metals and has been shown (Broccaand Bazant, 2000) to automatically reproducethe inelastic buckling data for metals (Gerardand Becker, 1957) (see Figure 7(d)) as well asconcrete (Caner et al., 2002).

2.02.3.3 Viscoelastic and ViscoplasticBuckling

Viscoelastic or viscoplastic behavior dissi-pates energy, which does not destabilize astructure. Thus, according to the Lagrange–Dirichlet theorem, the stability limit of aviscoelastic structure must still be decided bythe loss of positive definiteness of the potentialfor the elastic part of response. However, thestability limit is often not the issue of mainpractical interest. Rather, it is the maximumload for which the deflections and stressesduring the desired lifetime remain tolerable, orthe critical time for which the deflections andstresses under the given design load remaintolerable.The differential or integral equations that

govern viscoelastic buckling can be obtainedfrom those that govern elastic buckling byreplacing the elastic constants with the corre-sponding viscoelastic operator. This operatorcan be of a differential type, based on theMaxwell or Kelvin chain rheological model, or


of an integral type, corresponding to acontinuous relaxation or retardation spectrum.If there is no aging, a Laplace transform can beused to reduce the problem to elastic buckling,and inversion of the Laplace transform thenyields the time evolution of buckling deflec-tions. For viscoelastic materials that are solids(i.e., do not possess a purely viscous response),a structure loaded to the stability limit takes aninfinite time to develop a finite deflection as aresult of an infinitesimal disturbance or im-perfection (e.g., Freudenthal, 1950; Hilton,1952). Therefore the long-time critical load,PcrN, which is obtained by replacing theinstantaneous elastic modulus E in the criticalload solution with the long-time elastic mod-ulus EN, is normally unimportant. Importantfor design is the time to reach, for givenimperfections, the maximum tolerable deflec-tion or the maximum (second-order) stress dueto buckling. This time must not be less than therequired design lifetime.Viscoplastic buckling is different. In contrast

to viscoelastic structures, there exists a finitecritical timed t* at which the deflectiontriggered by an infinitely small imperfectionbecomes finite (or the deflection triggered by afinite imperfection becomes infinite accordingto the geometrically linearized theory); e.g.,Hoff (1958). The basis of design is to ensurethat t* exceeds the required lifetime. The higherthe load, the smaller is t*. If the viscoplasticmaterial does not have a finite elastic limit(e.g., if the viscoplastic strain rate is a powerfunction of stress magnitude), the critical loadaccording to Liapunov’s stability definition(implying an infinite load duration) is zero.Long-time buckling of concrete structures is

a very complicated but important phenomenonwhich has caused some slender columns andshells to collapse after many years of service. Inthe low (service) stress range, concrete isviscoelastic but exhibits aging, caused bychemical processes of hydration and by relaxa-tion of micro-pre-stress induced by drying andchemical changes—processes going on formany years. The consequence is that theVolterra integral equation for strain historyin terms of stress history has a nonconvolutionkernel, and the discrete Kelvin or Maxwellchain model has age-dependent viscosities andelastic moduli. The aging makes the Laplacetransform methods ineffective even in theviscoelastic range, but a numerical integrationof the evolution of buckling response in time iseasy. Simultaneous drying intensifies creep,and so the problem is coupled with diffusionprocesses. Furthermore, concrete undergoescracking during buckling. This and the creepcause a gradual stress transfer from concrete to

steel reinforcement. No wonder that crudeempirical design procedures are still in use.Detailed FE solutions that agree reasonably

well with tests have nevertheless been achieved.The design of concrete columns relies on anempirical overconservative reduction of theeffective elastic modulus, reflecting creep, agingand cracking (as well as the Shanley effect).For creep with aging, most effective is the useof the age-adjusted effective modulus method,which is based on a theorem stating that if thestrain history is linearly dependent on thecompliance function, then the stress history islinearly dependent on the corresponding re-laxation function (Bazant and Cedolin, 1991).Since the prediction of lifetime is rather

uncertain, a statistical approach is appropriate.

2.02.3.4 Thermodynamics of Structures,Inelastic Stability, Bifurcation andFriction

So far our review of inelastic structures hasnot dealt with the problem of stability.Stability cannot, in principle, be based onelastic potential energy. This does not mean,however, that stability would have to beanalyzed dynamically, according to Liapunov’sstability criterion. An energy approach tostability, which is much simpler, can be basedon thermodynamics of structure (Bazant andCedolin, 1991) (a subject distinct from thethermodynamics of constitutive equations ofmaterials).According to Gibbs’s form of the second law

of thermodynamics, a thermodynamic systemis stable if the entropy increment calculated forevery admissible infinitesimal change of systemstate is negative. If this increment is zero forsome change, the system is critical, and if it ispositive for some change, the change musthappen, which means that the initial equili-brium state of the system is unstable. This is ageneral thermodynamic definition of stabilityof equilibrium, which is equivalent to Liapu-nov’s definition (except when dealing withstability of motion). Its beauty is that theanalysis of the motion caused by initialimperfections, which can be rather laborious,may be skipped. It suffices to analyze theperfect structure, which is much simpler,especially if the structure is inelastic.An inelastic structure is normally far from a

state of thermodynamic equilibrium, at whichall the dissipative processes would come to astandstill. In principle, this prohibits usingclassical thermodynamics, which deals onlywith states infinitely close to thermodynamicequilibrium. However, the use of irreversible


thermodynamics would cause great complica-tion. It can fortunately be circumvented by thehypothesis of a tangentially equivalent inelasticstructure (Bazant and Cedolin, 1991). This is astructure characterized by the tangential mod-uli, assumed to behave equivalently to theactual inelastic structure for small changes ofstate. The existence of such a structure is ofcourse tacitly implied whenever the loadincrement in a computer program is analyzedon the basis of the tangent elastic modulus. Ateach stage of loading, there are many tangen-tially equivalent elastic structures correspond-ing to various possible combinations oftangential moduli for loading and unloadingin various parts of the structures. In theory,they all need to be considered.The deformations of elastic structures are

reversible changes of state, and thus they donot change the entropy of the structure.However, the thermodynamic system mustinclude the load, whose changes are definedindependently of the equilibrium of the struc-ture. For example, if the structure is inequilibrium under gravity loads P at initialdeflections q0, and its deflections q are thenchanged by dq away from equilibrium, theequilibrium values of the reactions f(q) changebut P does not. The disequilibrium createsentropy change

DS ¼Z

P � f ðqÞ½ �Tdq ð14Þ

Substituting

f qð ÞEP þ K t q � q0ð Þ ð15Þ

and integrating from q0 to q0þ dq, one has

�TDS ¼ d2W ¼ 12dqTK tdq ð16Þ

up to the second-order terms; here p, q, f arecolumn matrices, Kt a square matrix; T is theabsolute temperature; d2W is the second-orderwork of the reactions, i.e. the equilibrium loadvalues (in the rare case that d2W¼ 0, the fourthvariation needs to be analyzed similarly, whichwe do not discuss here). If the loads vary, thesecond-order work of loads must be added tothe right-hand side of Equation (15). It can beshown that under isothermal conditions,d2W ¼ d2F ¼ second variation of Helmholtzfree energy (or total energy) of the structure-load system under isothermal (or adiabatic)conditions. These energies represent the poten-tial energies of the tangentially equivalentelastic structure expressed in terms of theisothermal (or isentropic) tangential moduliof the material.

The conceptual difference from elastic sta-bility is that there are many potential energyexpressions to consider, each of them valid in adifferent sector of the space of q, correspond-ing to different possible combinations ofloading and unloading in various parts of thestructure. The surfaces of second variation ofentropy S (the negative of potential surfaces) indifferent sectors of this space are quadraticsurfaces, which are joined continuously andwith a continuous slope across the sectorboundaries (Figure 7(b)). The eigenvector fora sectorial quadratic surface may lie inside oroutside the corresponding sector. Only if it liesinside, the loss of positive definiteness of thepotential surface (or negative definiteness ofthe entropy surface) in that sector is real andrepresents stability loss (e.g., in an elastoplasticcolumn under gravity load, such a situation isfirst encountered when P reaches Pr, not Pt).For the basic case of structures with a single

load (or load parameter) P and the associateddisplacement q, the thermodynamic analysisshows that elastic as well as inelastic structuresare stable as long as

dP=dq40 ð17Þ

The stability limit is reached at the maximum(peak) load, and the postpeak states for whichdP/dqo0 are unstable. Under displacementcontrol, the postpeak states are stable unlessthe softening slope (slope of the descendingP(q)) curve becomes vertical; if the slopereverses from negative to positive, which iscalled the snapback, the structure becomesunstable (a stable response can nevertheless beobtained by controlling some internal displace-ments (e.g., the relative displacement across thedamage band or crack)). A structure thatexhibits postpeak softening (dP/dqo0) willexhibit a snapback when it is loaded througha sufficiently soft spring. The snapback in-stability is reached when

dP=dq ¼ �C ð18Þ

where C¼ stiffness of the loading spring. Forelastic structures, postpeak softening can becaused only by buckling. Otherwise, it can alsobe caused by fracture or damage.Thermodynamic analysis of bifurcation re-

quires decomposing the infinitesimal loadingstep into two substeps. The first is a changeaway from equilibrium, in which the controlledloads or controlled displacements are changedwithout any change in the stress and deforma-tion of the structure. The second substep is arestoration of equilibrium at constant controls.The equilibrium state approached in the


second substep may lie either on the primary(symmetry preserving) path or on the second-ary (symmetry breaking) path, depending onwhich approach to equilibrium produces alarger entropy increment. In this manner onecan prove in general, without consideringimperfections, that the secondary path mustoccur. It is called the stable path. This is adifferent concept than that of stability ofequilibrium (for inelastic structures, the post-bifurcation equilibrium states on both theprimary and the secondary paths may bestable).At small enough loads, the eigenvector for

the quadratic entropy surface corresponding tothe secondary path lies outside the sector of thedisplacement space in which this surface isvalid. As the load increases, the eigenvector isturning, and at the first bifurcation load thissector first moves into the sector of validity ofthis surface. At that moment, the incrementalequilibrium equations

K tdq 1ð Þ ¼ df ð19Þ

and

K tdq 2ð Þ ¼ df ð20Þ

for paths (1) and (2), respectively, must be validsimultaneously for the same load increment df.So, by subtraction, one gets a homogeneouslinear matrix equation for the difference ofdisplacement increments:

K t dq 2ð Þ � dq 1ð Þ� �

¼ 0 ð21Þ

Hence, the first bifurcation is indicated bysingularity of the tangential-stiffness matrix Kt

(Bazant and Cedolin, 1991), chap. 10; Hill,1958; Maier et al., 1973; Petryk, 1985; Nguyen,1987). This matrix corresponds to loading (asopposed to unloading) at all points of thestructures. Before the first bifurcation, theeigenvector of Kt is inadmissible because, inthe displacement space, it lies outside the sectorfor which no point of the structure undergoesunloading. The orientation of this eigenvectorrotates during loading and, at the first bifurca-tion, it lies at the boundary of this sector, thusmaking bifurcated path (2) admissible, for thefirst time during the loading process. After thefirst bifurcation, the eigenvector for the stateson path (1) rotates inside this sector, and so acontinuous series of bifurcation states lies onpath (1).The first bifurcation for a structure in which

the tangential moduli are changing continu-ously can be determined by linear analysis of asolid without unloading (known as Hill’s

(1962) linear comparison solid), in which thetangent modulus for loading applies every-where. If the tangential moduli change by ajump, then the first bifurcation occurs when thesmallest eigenvalue of Kt jumps from a positiveto a negative value.The use of the initial elastic stiffness matrix

in load step iterations in FE programs may bedeceptive. Convergence is lost if stability is lost,but not necessarily if the first bifurcation stateon the primary loading path has been passed.Thus the iterations based on the initial elasticstiffness may, deceptively, converge for stablestates lying on the primary path even if the firstbifurcation state has been missed. Thus, ifthere is any danger of bifurcation, it isimportant to calculate the tangential stiffnessmatrix and check its positive definiteness.Imperfections in the sense of the eigenvectorof Kt must be introduced to ensure that thebifurcation response be triggered in the com-putations (e.g., de Borst, 1987, 1989).Phenomena such as friction or damage (or

violation of the normality rule of plasticity)may cause the tangential stiffness matrix K tobe nonsymmetric. If K is symmetric, positivedefiniteness is lost when det K¼ 0, whichmeans that the stability loss under load control(gravity load) is characterized by neutralequilibrium. However, if K is nonsymmetric,the thermodynamic condition of stability limit(critical state), which reads

dqTKdq ¼ 0 ð22Þ

can be satisfied not only when

det K ¼ 0 ð23Þ

i.e., when there is neutral equilibrium,

Kdq ¼ 0 ð24Þ

but also when the vector

Kdq ¼ df ð25Þ

is orthogonal to df, i.e., when

dqTdf ¼ 0 ð26Þ

This means that, for nonsymmetric K; dis-placements with nonzero load incrementsmay occur at zero work. Stability is decidedby the symmetric part K of matrix K, whereasbifurcation or neutral equilibrium is decided bythe singularity of the nonsymmetricmatrix K.According to the Bromwich theorem, the

smallest eigenvalue of the symmetric K is lessthan or equal to the smallest real part of theeigenvalues of the nonsymmetric K. This


means that, in the case of friction or symmetry-breaking damage, positiveness of the real partsof the eigenvalues of the tangential stiffnessmatrix does not guarantee stability. Stabilitymay be lost before the loading process leads toa bifurcation or a neutral equilibrium state,

df ¼ 0 ð27Þ

at nonzero dq.An inelastic structure may also be destabi-

lized by load cycles. From the energetic view-point, stability in the standard sense (i.e., forone-way deviations from the equilibrium state)does not imply stability for infinitesimal devia-tion cycles and, vice versa, the latter does notimply the former. If infinitesimal strain cyclesof a small material element increase entropy(i.e., produce negative second-order work), thematerial is locally unstable, which may(although need not) destabilize the structure.Drucker’s (1951) postulate (equivalent to Hill’s(1950) principle of maximum plastic work)prohibits such constitutive laws. This providesuseful restrictions such as convexity of the yieldsurface and the normality rule for the strainrate vector in the 9D strain space. It alsoprevents the constitutive law from causinginstabilities such as strain localization.In the case of internal friction, however, the

normality rule needs to be in some way relaxed.But then the flow rule, called nonassociated,typically produces localization instabilities(such as shear bands, cracking bands) whichmay or may not be real. It appears that theneed for a nonassociated flow rule may oftenarise from adopting a single yield surface wherein reality multiple yield surfaces intersecting atthe current state point of the 9D strain spaceshould be considered. The normal strain-ratevectors for all these surfaces may get super-posed in a way that makes the resultant strain-rate vector appear not to be normal to thesingle yield surface defined by the yield limitsfor radial loadings. Such behavior is capturedby the slip theory of plasticity of Taylor (1938)and Batdorf and Budiansky (1949), as well asby later generalization and modification in theform of the microplane constitutive model,which implies the existence of as many loadingsurfaces as there are mircoplanes (Bazant andCedolin, 1991; Carol and Bazant, 1997).By extensions of Mandel’s (1964) stability

analysis of an elastically restrained frictionallysliding block, it has been proved that, infrictional materials, a nonassociated flow ruledoes not cause instability if the strain-ratevector lies within a certain fan of directions.The fan is bounded by the normal to the yieldsurface and the normal to the surface of the so-

called frictionally-blocked second-order elasticenergy density. Simple formulations are avail-able for two cases: (i) direct internal friction,representing the effect of hydrostatic pressureon deviatoric yielding; and (ii) so-called inversefriction, representing the effect of stress invar-iant J2 on the inelastic volume change (Bazantand Cedolin, 1991, section 10.7).

2.02.3.5 Stability Problems of FractureMechanics

Plasticity and hardening damage of thematerial profoundly affects the stability ofstructures but this does not cause instability.The cause lies solely in the nonlinear geometriceffect of deformations. Alternatively, fractureand softening damage (such as microcrackingor plastic micro-void growth) can destabilize astructure, even in absence of the nonlineargeometric effect.The energy balance condition of crack

propagation is g¼R(c), where g¼ energyrelease rate, and R(c)¼R-curve¼ critical valueof g depending in general on the crackextension c. This condition plays the role ofan equilibrium condition—the crack can growstatically. For g4R(c) the growth is dynamic,and for goR(c) no growth is possible. Underload control, the limit of stability of a structurewith a statically growing crack is reached atmaximum load, which occurs when

@g=@c½ �P¼ @R cð Þ=@c ð28Þ

Replacing¼with4yields the stability condi-tion. The stability limit for loading through aspring, which characterizes the ductility of astructure, can be determined by calculating theload-deflection curve, which can be done onthe basis of a solution of the stress intensityfactor. Under displacement control, fracturegrowth becomes unstable when the postpeakdescending load-deflection curve reaches asnapback (Figure 9(a)). Some structure geo-metries never exhibit a snapback instability(unless loaded through a soft enough spring),while others do. The former is the case, e.g., fornotched three-point bend beams. The latteris the case whenever the ligament is subjectedto a normal force (which is called ligamenttearing, occurring, e.g., in notched tensile stripspecimens).The R-curve is the simplest (and crudest)

way to take into account the finiteness of thefracture process zone at the crack tip. Thisproperty, which is more accurately describedby the cohesive crack model or the crack bandmodel, gives rise to a deterministic size effecton the nominal strength of geometrically


similar structures with geometrically similarfailure modes. By measuring the size effect innotched specimens, one can deduce the R-curveand the main parameters of the cohesive crackmodel.Simultaneous growth of many cracks (of

length ai, 1¼ 1, 2,y) typically produces bifur-cation as well as stability loss (Figures 10 and11(a)). The incremental potential d2F (equal to

the negative of the second-order entropy, orfree energy) is characterized by a matrix thatinvolves the partial derivatives @Ki=@aj whereKi is the stress intensity factor of crack i. Thepotential surface is a patch-up of manyquadratic sectors joined continuously andsmoothly (Figures 11(b) and (d)). In multicracksystems, the advance of one crack tip oftencauses unloading of the nearby crack tip, withthe result that only one of the cracks can grow.This for instance happens for symmetricallyedge-cracked and center-cracked tensionedstrip specimens, in which there are two inter-acting crack tips.Another example is the system of parallel

equidistant cracks in an elastic halfplane,driven by cooling or drying shrinkage (Figure10(a); Bazant and Cedolin, 1991, section 11.2).As the cooling front advances into the half-plane, the cracks first grow while keeping equallength (primary path). At a certain moment(Point A in Figure 10(b)), a stable bifurcationof the equilibrium path is reached. For thebifurcated (secondary) path, which is the stablepath (i.e., must occur), every other crack stopsgrowing (Figure 10(a)) and gradually closes,while the growing cracks gradually open wider.If the cracks are somehow kept equally longafter the bifurcation, a stability limit (B inFigure 10(b)) is eventually reached. Thebehavior is analogous to elastoplastic columns.The parallel crack system, however, may bestable, with no crack arrest, if the coolingtemperature profile is rather flat and has asufficiently steep front, or if the halfplane isreinforced by bars near the surface.Similar bifurcations and instabilities are

exhibited by parallel cracks in the tensile zoneof beams subjected to bending. Such bifurca-

Figure 9 (a) Typical diagrams of nominal stressversus relative deflection for structures of differentsizes, exhibiting strain-softening damage of fracture(with stability limits for loading device of stiffnessCs). (b) Effect of localization in compressed concreteprisms of various lengths calculated by crack bandmodel, and test data of van Mier (1986).

Figure 10 Evolution of a system of parallel cooling cracks and depth of cooling penetration l versus lengtha2 of leading cracks.


tions determine the spacing of deep cracks andductility of the structure. The initial crackspacing can be determined (Li et al., 1995) byconsidering a sudden formation of cracks oflength a0 and stipulating three conditions: (i)the material tensile strength is reached beforethe cracks appear, (ii) the energy released fromthe structure caused by a finite crack jump oflength a0 is equal to the total energy dissipatedby the creation of the initial cracks of length a0,and (iii) the energy release rate for furthergrowth of cracks of length a0 is critical. Thebifurcations in a system of parallel cracks areimportant because they decide the width ofcracks, which controls the overall effectivepermeability and the rate of ingress of corro-sive agents into structures.

In 3D, an approximate analysis of theentropy increment confirms that a hexagonalpattern of cooling or drying cracks on ahalfspace surface (as seen, e.g., on a dryinglake bed or a cooling lava flow) should befavored over triangular and square patterns aswell as parallel planar cracks. However,accurate solutions of the bifurcation andstability of 3D cracks seem unavailable.

2.02.3.6 Damage Localization Instabilities andSize Effect

Continuum damage mechanics provides asmeared description of the growth of micro-cracks in brittle materials or voids in plastic

Figure 11 (a) Double-edge cracked specimen—diagram of average stress versus displacement, withbifurcations (solid curve: one crack growing; dashed: both growing). (c) Longitudinal localization of strainsoftening in a tensioned bar. (b and d) Corresponding surfaces of incremental potential d2F (¼minus second-order entropy increment) showing bifurcations and instabilities (02 or 05) under controlled axial displacement.


materials. Of main interest is the damage thatcauses strain softening—a phenomenon mani-fested by a negative slope of the stress–straincurve and generally by a loss of positivedefiniteness of the matrix of tangential moduli.Such a material behavior violates Drucker’spostulate (or postulates of maximum plasticwork, Jirasek and Bazant, 2002) and leads tolocalization instabilities and bifurcations, andto size effect (Bazant and Chen, 1997; Bazant,1999). Formation of softening hinges in beamsand frames also causes bifurcations (Maieret al., 1973) and, inevitably, localizationinstabilities and size effect (Bazant, 1986,2001, 2002a, 2002b; Jirasek and Bazant, 2002;Bazant and Jirasek, 2002), in not only staticbut also in dynamic (seismic) loading (Bazantand Jirasek, 1996), and so does softening infriction (particularly the frictional stress dropfrom static friction to dynamic friction); seealso Rice and Ruina (1982).Although strain-softening stress–strain rela-

tions have been used in concrete engineeringsince the 1950s and in FE analysis since 1968(Bazant, 1986), until about 1985 most mechan-icians regarded all strain-softening studies withcontempt (research into this ‘‘dubious’’ con-cept was even banned in some communistcountries), although, curiously, the continuumdamage mechanics, which exhibits (withoutregularization based on some characteristiclength) the same localization instabilitiesas any strain softening model, neverthelessescaped condemnation, perhaps because thestrain softening was disguised as a separatedamage variable while the ‘‘true’’ stress ex-hibited only hardening (a regrettable conse-quence has been that efforts to regularizecontinuum damage mechanics formulationshave been lagging).Today, however, the strain-softening dam-

age is a well-established and indispensableconcept—but of course only within the contextof some nonlocal material model possessing acharacteristic length.In absence of a characteristic length, one

objectionable property of strain softening isthat if the elastic moduli matrix ceases beingpositive definite, the material cannot propagatewaves and the dynamic initial-boundary valueproblem changes its type from hyperbolic toelliptic (Hadamard, 1903). Since the unloadingmodulus is always positive (as experimentallydiscovered for concrete in the 1960s), thematerial can nevertheless propagate unloadingwaves. A strain-softening material can alsopropagate loading waves of a sufficiently steepwave front (Bazant and Li, 1997) because, astranspired from experiments (Bazant et al.,1995), a sudden increase of the loading rate

temporarily reverses strain softening to strainhardening (which explains why ultrasoundcan pass through strain-softening concretespecimens).That the concept of strain-softening is not

mathematically meaningless was demon-strated by analyzing converging step wavesproduced in a rod when the ends are suddenlyforced to move inward at constant velocity.After the waves meet, two kinds of responsescan occur: (i) if the strain front magnitude isless than one half of the strength limit (onsetof strain softening), the waves get super-imposed and the stress front doubles; butotherwise (ii) the stress drops instantly to zero,the bar splits, and unloading waves emanatefrom the split. The dynamic problem is ill-posed (and the solution is unstable) becausean infinitely small change in the velocityimposed at the ends of the rod can cause afinite change in the response. The solutionnevertheless exists. But there is a problem: thesplitting of the bar occurs with zero energydissipation, which makes a strain-softeningconcept with no characteristic length physi-cally unacceptable (Bazant and Cedolin,1991). Similar unstable and physically unrea-listic solutions have been mathematicallydemonstrated for waves in a rod in whichthe strain softening is followed by reharden-ing, and also for radially converging waves ina strain-softening sphere.Rudnicki and Rice (1975) and Rice (1975,

1976) pioneered analysis of static localizationof plastic (nonsoftening) strain into a planarband (e.g., a shear band) in an infinite body.Such localization is caused by the nonlineargeometric effect of finite strain and occurs verynear the yield stress value. If strain softeningtakes place, its destabilizing effect is normallymuch stronger than that of geometric non-linearity. Bifurcation of the localization type isdecided merely by the constitutive law; itoccurs when the so-called acoustic tensor

A ¼ ndC tdn ð29Þ

becomes singular (where C t¼ fourth-ordertangential moduli tensor, and n¼ unit normalof the localization band). Various damageconstitutive laws have been analyzed withregard to the singularity of A.Localization can also be triggered by a lack

of normality of plastic flow (de Borst, 1988;Leroy and Ortiz, 1989). In the case of infinitespace, this condition represents the stabilitylimit as well. But for an infinite planarlocalization band of finite thickness h locatedwithin an infinite layer of finite thickness L4h,the stability limit occurs later and is character-


ized by singularity of the tensor

Z ¼ nd C t þ Cuh= L � hð Þ½ �dn ð30Þ

where C u¼material stiffness tensor for un-loading. For L=h-N; the condition for theacoustic tensor is recovered.When a localization band much longer than

its thickness cannot develop, because of a finitesize of the body, a more realistic, yet analyti-cally solvable, model is the localization ofsoftening damage into an ellipsoidal region.This has been solved with the help of theEshelby theorem giving the strain of anellipsoidal plug that has been fitted into adifferent ellipsoidal hole in an infinite elasticbody (Bazant and Cedolin, 1991, section 13.4).The bifurcations and instabilities are found todepend on the aspect ratios of the ellipsoid andoccur later in the loading process than for aninfinite band.The usefulness of the bifurcation studies of

constitutive laws, however, is rather limited—they reveal only the onset of localization, whileoften it is much more important to determinethe postbifurcation behavior of a body con-taining a damage localization region. In thisregard, the concept of strain softening (orcontinuum damage) requires a characteristiclength (or material length) in order to over-come certain fundamental difficulties.In a strain-softening bar, a uniform axial

strain cannot occur. The strain inevitablylocalizes into a band (or fracture process zone)of the smallest possible length. Consequently,the longer the bar, the steeper is the postpeakcurve of stress vs. average strain (Figure 11(c)).A nonlocalized strain distribution loses stabi-lity when the softening load-deflection slopereaches a certain magnitude (Figure 11(a))depending on the structure size and thestiffness of the loading device (the stabilitylimit is the only objective and sound approachto defining the ductility of a structure; Bazant,1976).An analytical solution of a tensioned bar

subdivided into FEs exhibits bifurcations.Their consequence is that the strain softeningmust usually localize into a single FE, nomatter how small the element may be. Numer-ical FE computations of strain-softening da-mage under static loading may converge, but toa wrong solution. They reveal spurious meshsensitivity—the postpeak load-deflectioncurves of structures with strain softening areunobjective because very different peak loadsand postpeak responses are obtained fordifferent element sizes. The energy dissipatedby breaking the structure converges to zero asthe FE size is reduced to zero. These are

physically unrealistic features. The remedyconsists in introducing some form of anonlocal continuum model possessing a char-acteristic length (Bazant and Jirasek, 2002;Jirasek and Bazant, 2002), of which there arebasically three types:(i) the crack band model,(ii) the integral-type nonlocal model, and(iii) the second-gradient model;

(the first-gradient or Cosserat-type models didnot turn out to be effective for localization oftensile damage). The first and third type can beregarded as approximations to the second.The crack band model (Bazant, 1976, 1982;

Bazant and Oh, 1983; Bazant and Cedolin,1991, ch. 13; Bazant and Planas, 1998), is thecrudest but simplest (and thus preferred inconcrete design firms and commercial pro-grams). Its salient characteristic is to impose acertain characteristic size h of the FE, whichmust be regarded as a material property,representing a characteristic length l of thematerial; l corresponds to the width of thecrack band (or strain-softening damage band).In the case that localization of strain-softeningdamage is not prevented by some restraintssuch as reinforcing bars, the crack band modelis essentially equivalent to the cohesive (orfictitious) crack model, in which the crackopening is equal to the cracking strain accu-mulated over the width of the crack band.Various semi-empirical energy-based numeri-cal corrections need to be used when the banddoes not propagate along the mesh lines(Cervenka and Cervenka, 1998).In the nonlocal continuum concept, which

was introduced for elasticity by Eringen (1965),the stress at a given point depends not only onthe strain at that point but also on a certainaverage of the strain field in a neighborhood ofthe point, whose effective size corresponds tothe characteristic 1ength l of the continuum.The average is weighted by a bell-shapedweight function. When the nonlocal conceptwas introduced in 1984 for the purpose oflimiting the localization of strain softening, thetotal strain was averaged over the neighbor-hood. However, this caused two problems: thenumerical implementation was cumbersome(requiring imbrication of FEs), and zero-energy periodic modes of instability developedand had to be suppressed by an artificialparallel overlay.The problems of the total strain averaging

were overcome in 1987 by the nonlocal damagemodel, in which the spatial averaging is doneon the damage variable or the inelastic part ofstrain. With this concept, localization of strainto form a displacement discontinuity line isimpossible (the reason for this is that if the


strain profile became a function with C0

continuity (Dirac delta function), the averagingintegral would convert the profile in the nextiteration to a function with C1 continuity(Heaviside step function)). The nonlocality ofdamage can be physically justified (Bazant andJirasek, 2002) by the smoothing of an array ofdiscrete cracks and by crack interactions (thelatter, however, also points to a more compli-cated nonlocal model which distinguishes thedirections of amplifying and shielding crackinteractions; Bazant, 1994; Bazant and Planas,1998). FE studies demonstrate that the non-local damage model is free of zero-energyperiodic modes of instability, eliminates spur-ious sensitivity, effectively limits excessivedamage localization, and correctly simulatesthe size effect on the nominal strength as wellpostpeak behavior. However, studies (Jirasek,1998b) revealed that the nonlocal processing ofinelastic strain does not suffice for simulatingcomplete fracture with a zero stress across thesoftening band. Rather, an ‘‘over-nonlocal’’inelastic strain, proposed simultaneously andindependently in 1996 by Planas and colleagues(cf. Bazant and Planas, 1998) and by Strom-berg and Ristinmaa (1996) is needed for that.The nonlocal concept makes sense only if the

FEs are at least three-times smaller than thecharacteristic length l, and this is also requiredto make the propagation direction of a damageband independent of the mesh layout. If thenumber of elements of this size would beexcessive, one may use the crack band model,for which the FE size should optimally be l butcan be much larger. If the number remainsexcessive, one may use a variant of the crackband or cohesive crack model, the idea ofwhich is to embed in the FE either a crackband (a strain-softening strip limited by straindiscontinuity lines) or a cohesive crack (a lineof displacement discontinuity), (Ortiz et al.,1987, Belytschko et al., 1988). There are twobasic simple forms of such models—kinemati-cally consistent and statically consistent. But itis their combination that is found to be optimal(Jirasek, 1998a, 1998b). The embedded dis-continuity models are best to model the finalstage of damage in which the distributedcracking coalesces into a distinct fracture. Butthere is a mesh bias for the propagationdirection—when the mesh lines are not laid inthe correct direction of fracture propagation,the correct orientation of the crack band ordiscontinuity is not obtained, and the error inorientation can be very large. As shown byJirasek (1998a), it is best to use the nonlocalmodel at the beginning, in order to capture thecorrect discontinuity orientation, and laterswitch to an embedded discontinuity model.

In the nonlocal damage model, the need tocalculate averages over a group of FEsincreases the bandwidth, and the use ofasymmetric averaging near the boundariesmakes the stiffness matrix nonsymmetric.Although the asymmetry resulting from non-locality causes does not destabilize computa-tional simulations, it is inconvenient for sometypes of numerical algorithm. This drawbackcan be avoided by second-gradient damagemodel—a weakly nonlocal model in which thenonlocal variable is obtained from a linearcombination of the local variable and itsLaplacian, as proposed in 1984 (Bazant andCedolin, 1991, Equation 13.10.25) for a modelderived (and physically justified) by Taylorseries expansion of the weight kernel in thenonlocal averaging integral and by truncationafter the quadratic term. The truncation,however, is physically questionable becausecrack interactions, from which a nonlocalmodel can be derived (Bazant, 1994), have arather slow decay with distance.A drawback of the second-gradient approx-

imation of the nonlocal model is the C1

continuity required for the strain field in theFE. This drawback can be circumvented by astrongly nonlocal formulation of Peerlings et al.(1996) obtained by expressing the local variableas a linear combination of the nonlocalvariable and its Laplacian. As discovered byPeerlings et al. (1996), this drawback can becircumvented by inverting the nonlocal opera-tor—the nonlocal variable is expressed as alinear combination of the nonlocal variableand its Laplacian, which provides an addi-tional separate partial differential equation(Helmholtz equation) from which the nonlocalvariable is solved. De Borst and co-workershave shown that this approach yields verygood results.The main practical consequence of the

damage localization instabilities and of theexistence of a characteristic length ofthe continuum is the size effect, both on thenominal strength of structures and on thepostpeak response. Numerical studies of non-local and gradient models of geometricallysimilar structures of different sizes, exhibitingsimilar damage band paths, show that themaximum loads approximately follow thesimple size effect law proposed in 1984 byBazant (e.g., Bazant and Cedolin, 1991, chaps.12 and 13; Bazant and Planas, 1998; Bazantand Chen, 1997). This law has been amplyverified experimentally, and has been derived(Bazant 1997, 1999, 2002a; Bazant and Chen,1997; Bazant et al. 1999) from fracturemechanics and limiting plastic solutions byapplying the technique of asymptotic matching.


The size effect has been shown important forthe design of large concrete structures, forassessments of failure and seismic response oflarge bodies in geotechnical engineering andgeophysics, for Arctic ice studies, etc. The sizeeffect law provides an important check on thevalidity of a numerical model, and the sizeeffect measurements can be exploited tocalibrate the main model parameters. Largesize effects due to damage localization andstable growth of large fractures are observednot only in concrete (and mortar)—the pro-blem so far studied most, but also in tensile andcompression failures of fiber-polymer compo-sites (Bazant et al., 1999), sea ice (Bazant andKim, 1998), rocks and ceramics (Bazant andPlanas, 1998), wood, and probably in all theother quasibrittle material including toughenedceramics, polymer and asphalt concretes, par-ticulate composites, wood particle board, bone,biological shells, stiff clays, cemented sands,grouted soils, coal, paper, various refractories,some special tough alloys, rigid foams andfilled elastomers (Bazant, 2002a).

2.02.4 NONLINEAR 3D FINITE-STRAINEFFECTS ON STABILITY

2.02.4.1 Finite-strain Effects in Bulky orMassive Bodies

To determine the critical state, the potentialenergy must be expressed accurately up to thequadratic terms. For the strain energy, thiscondition is satisfied if the small (linearized)strain expressions are used. But the potentialenergy also includes the work of the initialstresses Sij which are finite. Therefore, thestrains on which Sij work must be expressedaccurately up to their second-order terms (thesubscripts label the Cartesian coordinatesxi; i ¼ 1; 2; 3). Therefore, a finite strain expres-sion that is correct up to the second-orderterms in the displacement gradient must beused.Any tensor e satisfying the following three

conditions can be used as a measure of finitestrain: (i) e must vanish for all rigid-bodymotions; (ii) e must be a symmetric tensor(because the stress tensor, which is symmetric,would do zero work on an antisymmetric part);and (iii) e must be continuous and continuouslydifferentiable, and its matrix must be invertible(Bazant and Cedolin, 1991; Rice 1993). Forconvenience, a fourth condition is imposed; emust, for small deformations, reduce to thestandard small-strain tensor. There are manyfinite strain tensors satisfying these conditions(Bazant and Cedolin, 1991; Rice, 1993; Ogden,1984). A very general class of finite strain

tensors consists of the Doyle–Ericksen (1956)strain tensors,

e ¼ 1

mUm � 1ð Þ for ma0; e ¼ lnU form ¼ 0

ð31Þ

where U¼ right-stretch tensor and m is anarbitrary real number. These tensors have thesecond order approximation:

emð Þ

ij ¼ eij � 12m eikekj ð32Þ

where eik is the small (linearized) strain tensor,m ¼ 2 gives Green’s Lagrangian strain tensor,while m ¼ 1 and m-0 provide the second-order approximations to Biot’s (1965) straintensor and to Hencky’s (logarithmic) straintensor, respectively.For beams, plates and shells with negligible

transverse shear deformations, the choice of mmakes no difference because the second-orderwork of the initial axial or in-plane stressesdepends only on the rotations of the crosssections or the normals. However, for massivebodies, and for slender beams or shells soft inshear (such as sandwich structures), the choiceof m does make a difference.Historically, beginning with Southwell

(1914) and Biezeno and Hencky (1928), variousstability formulations associated by work withm ¼ 2; 1; 0;�1;�2 have been proposed. Thedifferential equations of equilibrium for devia-tions from the initial state, the quadraticvariational principles and the critical loadsolutions for these formulations seemed todiffer and be in conflict. This generatedunnecessary long-running polemics and con-troversies (see, e.g., the preface of Biot’s (1965)book, or various comments in works ofTruesdell). As it turned out, however, all theseformulations are equivalent (Bazant, 1971a;Bazant and Cedolin, 1991, chapter 11) becausethe tangential moduli tensors C

mð Þijkl for different

m and the same material are not the same;according to the requirement of equality ofsecond variation of work, they are related as

Cmð Þ

ijkl ¼Cijkl þð2� mÞ

4

� ðSikdjm þ Sjkdim þ Simdjk þ SimdikÞ ð33Þ

(Bazant, 1971a; Bazant and Cedolin, 1991,section 11.4) where Cijkl are the tangentialmoduli associated with Green’s Lagrangianstrain tensor m ¼ 2ð Þ; and dij is Kronecker’sdelta (note that in the literature there also existformulations in which the objective stressincrements do not correspond to the same mas the moduli Cijkl ; but this is incorrect).Obviously, the differences among the C

mð Þijkm

Nonlinear 3D Finite-strain Effects on Stability 71

values for different m are insignificant if the(suitable) norms

Sij xð Þ { C

mð Þijkm xð Þ

ðfor every xÞ ð34Þ

where x are coordinate vectors of points in thestructure.The 3D buckling instabilities and the differ-

ences in tangential moduli corresponding todifferent choices of the finite strain tensor areimportant if and only if some of the principalvalues of the tangential elastic moduli tensor isof the same order of magnitude as the initialstresses at the same point or, in the case ofslender structures, at the same cross section.This can for instance occur for materials thatundergo a drastic reduction of tangentialstiffness due to plasticity or damage anddevelop shear bands, cracking bands or crush-ing bands. Simple analytical solutions havebeen obtained for a number of importantproblems, e.g., periodic internal buckling of acompressed massive orthotropic body, periodicsurface buckling of a compressed othotropichalfspace or layered halfspace (folding ofstrata, in geology), and bulging or bucklingof a compressed thick orthotropic rectangularspecimen (Figure 12; see Biot (1965) and ageneralization in Bazant (1971a) and in Bazantand Cedolin, 1991, chap. 11).

2.02.4.2 Nonlinear Finite-strain Effects inColumns, Plates and Shells Soft inShear

Another important class of stability pro-blems where the differences in tangentialmoduli corresponding to different choices ofthe finite strain tensor are important consists ofcolumns, plates and shells soft in shear. Thisimportant class includes:(i) highly anisotropic materials such as

composites with a soft (polymeric) matrix,reinforced by stiff (carbon, glass) fibers in onlyone or two directions;(ii) composite slender structures with a core

soft in shear, such as sandwich columns, platesand shells;

(iii) continuum approximations of built-upcolumns, either battened columns or latticecolumns, the cells of which are normally soft inshear;(iv) continuum approximations to layered

elastomeric bearings of the type used forbridges and for seismic isolation of buildings;and(v) helical springs, the shear stiffness of

which is typically quite low.There has been a long-lasting (but unneces-

sary) controversy between two different for-mulas for critical stress, one derived byEngesser (1889), with applications for built-up columns in mind, and another derived byHaringx (1942) for helical springs;

Pcr ¼PE

1þ PE=GAð Þ ðEngesserÞ ð35Þ

Pcr ¼GA

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4PE

GA

r� 1

!ðHaringxÞ ð36Þ

(see also Timoshenko and Gere, 1961); E, G areelastic Young’s and shear moduli respectively,PE ¼ p2=l2

�EI¼Euler’s critical load (for a

column without shear deformations), l is theeffective buckling length, and EI, GA arebending stiffness and shear stiffness of thecross-section, respectively (note that, in gen-eral, A ¼ kA0 where A0¼ actual cross-sectionarea and k¼Timoshenko’s shear correctionfactor, which is greater than, but close to, 1; fora sandwich kE1). Each of these two formulaecan be regarded as a different and equallyplausible generalization of the Timoshenkobeam theory (Timoshenko, 1921), which doesnot deal with finite-strain effects and appliesonly to beams carrying negligible axial force.The discrepancy between these two formulae

used to be, before 1971, regarded as acontroversial paradox. Then it was shown(Bazant, 1971a; Bazant and Cedolin, 1991,chap. 11) that this classical paradox is causedby a dependence of the tangential shearmodulus C1212 ¼ G on the axial stress S11 ¼�P=A; which inevitably is different for differ-ent choices of the finite-strain measure, i.e., fordifferent m. It turns out that Engesser’sformula corresponds to Green’s Lagrangianstrain tensor (m¼ 2), and Haringx’s formula tothe Lagrangian Almansi strain tensor (m¼ –2).Therefore, the shear moduli in Engesser’s andHaringx’s formulas (35) and (36) should havedifferent notations; we denote them from nowon as G 2ð Þ and G �2ð Þ; respectively. From (33), itfollows that

G 2ð Þ ¼ G �2ð Þ þ P=A ð37Þ

Figure 12 Basic types of 3D buckling.


The difference in the E-values indicated by (33)for m¼ 2 and m¼ –2 can be neglected becausethe axial stress is always negligible compared tothe E value for the skins.Replacing G in Engesser’s formula with G þ

Pcr=A and solving Pcr from the resultingequation (Bazant and Cedolin, 1991, p. 738),one obtains Haringx’s formula, which makesthe equivalence blatant and resolves the oldparadox. The difference in shear moduli in (37)is important only if the axial stress S11 ¼�P=A is not negligible compared to G.Equation (37) makes it also clear that if theshear modulus is constant (independent ofstress �P/A) for one formula, it cannot beconsidered constant for the other formula.

2.02.4.3 Discrepancy between Engesser’s andHaringx’s Formulas: New Paradoxand its Resolution

Though a new kind of paradox has arisenfrom some experimental studies and 3D FEsimulations of sandwich columns (Kardoma-teas, et al., 2002; cf. Bazant, 2002b). Let l bethe length of a pin-ended sandwich column; hthe thickness and G the elastic shear modulusof the core. The skins have thickness t andaxial elastic modulus E (Figure 13, left).Young’s modulus of the core is negligible, butsince t{h the entire shear force is carried bythe core. Therefore, one may substitute

EI ¼ R ¼ Ebt h þ tð Þ2=2þ Ebt3=6EEbth2=2 ð38Þ

which represents the bending stiffness of thesandwich t{hð Þ; and

GA ¼ H ¼ Gbh ð39Þ

which represents the shear stiffness of thesandwich, b being the cross-section width.With these substitutions, the Engesser andHaringx formulas become

Pcr ¼PE

1þ PE=Gbhð Þ ðEngesser typeÞ ð40Þ

Pcr ¼Gbh

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4PE

Gbh

r� 1

" #ðHaringx typeÞ ð41Þ

where PE is the Euler load,

PE ¼ p2

l2EI E

p2

l2Ebth2

2ð42Þ

Similar to (37), one may check that, by makingthe replacement

Gcore’Gcore �2t

hsskins ð43Þ

where sskins ¼ �Pcr=2bt and G¼Gcore¼ shearmodulus of the core, the Engesser-type formula(40) gets transformed into the Haringx-typeformula (41).Although the foregoing replacement works,

it is, however, purely formalistic, with nophysical basis. It is certainly paradoxical thatthe shear modulus in the core should dependon the axial stress in the skins. Therefore, thereason for the discrepancy between these twoformulas cannot be caused by the differences inthe shear modulus G of the core material, asgiven by Equation (37). Besides, there is noreason for the G-moduli associated withdiverse strain measures to differ because theaxial stress in the core of sandwich columns isalways negligible compared to the shearmodulus of core.

Figure 13 Top left: buckling of hinged sandwich column. Bottom left: rotation angles and internal forces insandwich column. Right: second-order shortening due to shear deformations.


Realizing these points created a new kindof paradox and generated polemics at recentconferences. To explain it, Bazant (2002b)noticed that one must not limit considera-tion to the material level, as in Equation (43).Rather, one must consider from the outset asandwich column constrained by the hypoth-esis that (in slender enough columns) the crosssections of the core must remain plane. It maybe shown (Bazant, 2002b) that the secondvariation of the potential energy P of asandwich column expressed on the basis ofDoyle–Ericksen finite-strain tensors with arbi-trary m can be expressed as

d2P ¼ 1

2

Z L

0

�RðmÞc02

þ H mð Þ þ 1

42� mð ÞP

� �w0 � cð Þ2�Pw02

�dx

ð44Þ

where w¼w(x)¼ lateral deflection as a func-tion of axial coordinate x, H(m) and R(m) arethe shear and bending stiffness associated withm, and the primes denote derivatives withrespect to x. Proceeding in the standard way(Bazant and Cedolin, 1991) according to thecalculus of variations, one obtains for m¼ 2and m¼ –2 the linear differential equation

w00 þ k2w ¼ 0 ð45Þ

for column deflections w(x), and the boundaryconditions w¼ 0 at x¼ 0 or l. The expressionsfor parameter k2 as a function of load P aredifferent for different m, and this leads to twodifferent critical load formulas analogous tothose of Engesser and Haringx:

for m ¼ 2 :

Pcr ¼Pð2ÞE

1þ Pð2ÞE =H 2ð Þ

� �with P

ð2ÞE ¼ p2

l2R 2ð Þ ð46Þ

for m ¼ �2 :

Pcr ¼H �2ð Þ

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4P

1�2ð ÞE

H �2ð Þ

s� 1

24

35

with P�2ð ÞE ¼ p2

L2R �2ð Þ ð47Þ

It has been shown (Bazant, 1971a; Bazantand Cedolin, 1991) that the case m¼ 2 isassociated by work with Truesdell’s (1955)(objective) stress rate, and the case m¼ –2with Cotter and Rivlin (1955) convected(objective) stress rate or with Lie derivatives

of Kirchhoff stress. Using other m values, onecould further obtain from Equation (44) aninfinite number of sandwich buckling formu-las. Curiously, however, no investigatorsproposed critical load formulas associatedwith m other than 2 and –2, although manyinvestigators (e.g., Biot, 1985; Biezeno andHencky, 1928; Neuber, 1965; Jaumann, 1911;Southwell, 1914; Oldroyd, 1950; Truesdell,1955; Cotter and Rivlin, 1955—see Bazantand Cedolin, 1991, chap. 11) introducedformulations for objective stress rates, 3Dstability criteria, surface buckling, internalbuckling, and incremental differential equa-tions of equilibrium associated with m¼ 1, 0,and –1.In analogy to Equation (37) and similarity to

Equation (43), one may expect the shearstiffnesses for the Engesser’s and Haringx’sformulas to be related as

H 2ð Þ ¼ H �2ð Þ þ Ph=2t ð48Þ

Indeed, when this relation is substituted intoEngesser’s formula (46) and the resultingequation is solved for P ¼ Pcr; Haringx’sformula (47) ensues. However, unlike homo-geneous columns soft in shear, the foregoingtransformation cannot be physically justified inthe sense of Equation (43), i.e., on the basis ofthe general transformation of tangential mod-uli in Equation (33) nor its special case inEquation (37). The reason is that the axialstress S0 in the core is much smaller than theaxial elastic modulus E for the core, and in factnegligible.From this viewpoint, the transformation in

Equation (43) appears illogical: why should theshear modulus of the core be adjusted accord-ing to the axial stress in the skins? This hasbecome a new apparent paradox. To resolve it,we need to take a closer look at the definitionof the shear stiffness H of a sandwich, whichwe do next.

2.02.4.4 Shear Stiffness Associated withSecond-order Strain

Let us imagine a homogeneous pure sheardeformation of an element Dx of the sandwichcolumn:

u1 ¼ u1; 1 ¼ u1; 3 ¼ e11 ¼ 0;

u3; 1 ¼ g; e13 ¼ e31 ¼ g=2 ð49Þ

With these expressions, the second-order smallincremental potential energy of the element,corresponding to (first-order) small strains, is


obtained as

d2W ¼Dx

ZA

� P

2bt

1

2uk;1uk;1 � aek1ek1

� ��

þ12Gmg2

�dA ð50Þ

Upon rearrangements, the incremental poten-tial energy (per height of column, Dx ¼ 1) is

d2W ¼ bhDx G mð Þ � 2þ m

4

P

bh

� �g2

2ð51Þ

In particular, for m¼ 2 (Engesser type) andm¼ –2 (Haringx type),

d2W ¼

bh Dx G 2ð Þ � P=bhð Þ� �

g2=2

ðEngesser type GÞbh Dx G �2ð Þg2=2

ðHaringx type GÞ

8>>>><>>>>:

ð52Þ

Since the foam core in an axially loadedsandwich column carries no appreciable axialstresses, we should use that definition of G (m)

for which the shear stiffness of the core requiresno correction for the effect of the axial force Pcarried by the skins. As we see, that is thelatter, Haringx-type, expression (for m¼ –2).In that case, the shear modulus G (–2) is thesame as that obtained in a pure shear testwithout normal stress, e.g., in the torsion testof a thin-wall tube made of the rigid foam.The solutions for ma� 2; including the

Engesser-type formula, are, of course, equiva-lent. But if they are used, the shear modulus ofthe core must be corrected for the effect of theaxial forces F ¼ P=2 carried by the skins. Itwould be wrong to use in them the G valuemeasured in a pure shear test of the foam, inwhich no normal force acts on the shear plane.Intuitive understanding can be gained from

Figure 13 (right), which shows two kinds ofshear deformation of an element (of heightDx ¼ 1) of a sandwich column. In the firstkind (Figure 13, top right), corresponding tothe deformation described by Equation (49),the shearing of the element is accompaniedby second-order axial extension of the skins,equal to

d2u ¼ 1� cos gEg2=2 ð53Þ

(per unit height). If the initial forces F werenegligible, this second-order small extensionwould make no difference but since they arenot, one must take into account the work ofthe initial forces of F on this extension, which is

d2W ¼ 2Fg2=2 �

bh or � bhS0 �g2=2 �

ð54Þ

(per unit height, Dx ¼ 1). This work must beadded to the work of the shear stresses,

d2W ¼ Gg2=2 �

bh ð55Þ

in order to obtain the complete second-orderwork expression. In the second kind of sheardeformation (Figure 13, bottom right), theinitial forces F do no work. So, the incrementalsecond-order work expressions for these twokinds of shear deformation, respectively, are

d2W ¼bh G 2ð Þ þ S0 �

g2=2 ðcase aÞG �2ð Þg2=2 ðcase bÞ

(ð56Þ

These two cases (Figure 13 right) give the sameincremental second-order work if G 2ð Þ ¼G �2ð Þ � S0 or G 2ð Þ ¼ G �2ð Þ þ 2F=bh: We seethat these relations coincide with (37).From the foregoing comparisons and the

discussion of Figure 13 (right), it is nowobvious that a constant shear modulus G,equal to the shear modulus obtained in a sheartest of the foam (e.g., a torsional test of ahollow tube), can be used only in the Haringx-type formula m ¼ �2ð Þ:Kardomateas et al. (2002) and others studied

the differences between the Engesser-type andHaringx-type formulas experimentally and byFE analysis. They concluded that the Haringx-type formula gives better predictions. Sincethey tacitly adopted a constant value ofincremental modulus G, this is indeed theconclusion that they should have obtained.The present theoretical analysis explains why.When can the differences between the

column solutions for different m, and particu-larly between the formulas of Engesser andHaringx, be ignored? When P{H mð Þ ¼ G mð Þbh:

2.02.4.5 Differential Equations of EquilibriumAssociated with DifferentFinite-strain Measures

The Engesser and Haringx formulas can alsobe derived from the differential equations ofequilibrium. This is discussed for a homoge-neous column weak is shear on p. 738 inBazant and Cedolin (1991), and we will nowindicate it for a sandwich. Figure 13 (right)shows two kinds of cross sections of asandwich column in a deflected position: (a)the cross section that is normal to the deflectedcolumn axis, on which the shear force due toaxial load is

Q ¼ Pw0 ð57Þ

and (b) the cross section that was normal to thecolumn axis in the initial undeflected state, on


which the shear force due to axial load is

%Q ¼ Pc ð58Þ

From equilibrium, for a simply supported(hinged) column, the bending moment is M ¼�Pw in both cases. The force-deformationrelations are

M ¼ Ebth2c0=2; ð59Þ

and

Q or %Q ¼ Gbhg or Gbh w0 � cð Þ ð60Þ

in case a or b, respectively. Eliminating M; g;cand Q or %Q from the foregoing relations, weget a differential equation of the form w00 þk2w ¼ 0; i.e., the same as Equation (45), where

k2 ¼ GbhP

E bth2=2ð Þ Gbh � Pð Þ ð61Þ

or

k2 ¼ P2 þ GbhP

E bth2=2ð ÞGbhð62Þ

respectively. Setting k ¼ p=l and solving for P,we find the former equation to lead toEngesser’s formula (35) and the latter toHaringx’s formula (36). As we see, Engesser’sformula (m¼ 2) is obtained when the sheardeformation g is assumed to be caused by theshear force acting on the cross-section that isnormal to the deflected axis of column, andHaringx’s formula (m¼ –2) when caused by theshear force acting on the rotated cross sectionthat was normal to the beam axis in the initialstate.The foregoing equilibrium derivation, how-

ever, does not show that the values of shearstiffness in both formulas must be different.

Figure 14 Top left: deformation of a helical spring. Top right: deformation of a layered elastomeric bearing.Bottom left: built-up columns (battened, and pin-jointed lattice). Bottom right: deformations of cells of built-up columns showing second-order axial strain due to shear (top middle and right).


Especially, it does not show that the shearstiffness in the direction of the rotated crosssection can be kept constant, while the shearstiffness in the direction of normal to thedeflected axis must be considered to depend onthe axial force. This has been a perennialsource of confusion. To dispel it, the work ofthe shear forces must be considered, and so, anenergy approach is appropriate.

2.02.4.6 Ramifications

The present analysis can be easily adapted tocomposites with stiff axial fibers and a matrixvery soft in shear, as well as to built-up(battened or lattice) columns, helical springs,and layered elastomeric bearings (Figure 14).The conclusions are identical (Bazant, 2002a).For general structures, Equation (34) is only

a necessary condition for making the differ-ences among the finite-strain measures e mð Þ fordifferent m irrelevant. A condition that issufficient is

maxxð Þ

Sij xð Þ {min

xð ÞC

mð Þijkm xð Þ

ð63Þ

2.02.5 CLOSING REMARKS

2.02.5.1 Implications for Large-strain FEAnalysis

The present conclusions about sandwichbuckling have profound implications forlarge-strain FE analysis in general. The use ofthe standard updated Lagrangian formulation,in which the constitutive law is defined in termsof the strain rate and the Jaumann (co-rotational) rate of Kirchhoff stress, could notgive the correct result for a linearly elasticsandwich column, in which the stress at eachpoint is negligible compared to the elasticmoduli at that point. Since the Jaumann ratecorresponds to m¼ 0, the result would have tolie in the middle between Haringx’s formula(m¼�2) and Engesser’s formula (m¼ 2). Theobjective stress rate energetically associatedwith m¼�2 is the Lie derivative of Kirchhoffstress (Bazant and Cedolin, 1991, section 11.4).Therefore, only this rate can yield the correctFE result.

2.02.5.2 Looking Back and Forward

Historically, a neglect of the shear deforma-tions in built-up columns (which was inconformity to the building code of the time)had one particularly tragic consequence. Asshown by Prandtl (1907) and others, disregard

of Engesser’s (1889) formula by the design codeof the time was the main culprit in the collapsein 1907 of the record-span Quebec bridge overSt. Lawrence, with a great loss of life.While there are many examples of theories

that have become complete or almost com-plete, the theory of stability of structures, at thedawn of the third millenium, has not yetreached that point. As far as elastic stabilityis concerned, its understanding is now verygood, although even here a rapid progress hasbeen happening in certain particular directions,for instance those of chaos or general cata-strophes. Almost the same could be said ofstability of anelastic structures exhibitingplasticity, viscoelasticity, and viscoplasticity.However, as far as stability of structures

disintegrating because of damage and fractureis concerned, this is a challenge for the future.Major advances are still to be expected.

ACKNOWLEDGMENTS

Partial financial support under grants fromONR (number N00014-91-J-1109) and NSF(number CMS-9713944) to Northwestern Uni-versity is gratefully appreciated.With the kind permission of Wiley-VCH

GmbH, Berlin, and of ZAMM editor, a majorportion of Sections 2.02.2 and 2.02.3, dealingwith elastic structures and with inelastic anddisintegrating structures, is reprinted fromBazant’s (2000) article with only minor up-dates.

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Copyright 2000, Wiley–VCH. Published by Elsevier Ltd. All rights reserved. Comprehensive Structural Integrity

First published as Z. P. Bazant, 2000, Zeitschrift fur Angew. Math. und Mech.,80, 709–732.No part of this publication may be reproduced, stored in any retrieval system ortransmitted in any form or by any means: electronic, electrostatic, magnetic tape,mechanical, photocopying, recording or otherwise, without permission in writingfrom the publishers.

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