Physical instability and geometric effects in frames

Engineering Structures 21 (1999) 557–567

Physical instability and geometric effects in frames

Gabriella Bolzona, Francis Tin-Loib,*

a Dipartimento di Ingegneria Strutturale, Politecnico di Milano, 20133 Milano, Italyb School of Civil and Environmental Engineering, University of New South Wales, Sydney 2052, Australia

Received 27 August 1997; accepted 29 December 1997

Abstract

The interaction between physical instability and geometric (2nd-order) effects in plane frames is investigated herein within afinite element based mathematical programming framework, under the assumptions of quasistatic loading and lumped plasticity.The sources of physical instability consist of inelastic bending hinges characterized by softening nonholonomic (irreversible) momentversus rotation constitutive laws. The complex and insightful response spectrum which can arise is illustrated with the aid of simple,albeit instructive, reference examples. Stability domains are obtained for these examples in the space of the parameters characterizingthe level of member axial force and the amount of hinge softening. Such domains highlight various key features, which are brieflycommented upon, of the interplay between physical and geometric effects. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:Frames; Lumped plasticity; Geometric nonlinearity; Softening

1. Introduction

The stability of structural systems is an important areaof engineering mechanics (see e.g. [1]) and it hasbecome almost mandatory in professional practice tocarry out such an evaluation as part of the assessment ofstructural safety. However, this is often a difficult task,especially when interacting influences as those due tomaterial, geometric and inertia effects are present.

The primary aim of this paper is to investigate, underthe assumptions of quasistatic loading and inviscidbehaviour, the stabilizing or destabilizing contributionsof combined 2nd-order geometric effects and materialproperties (hardening or softening). This is carried outusing, as reference examples, frames of very simplegeometry and loading situations but which, nevertheless,are able to reproduce a wide and complete spectrum ofresponses expected of instability phenomena, such asbifurcation and limit points.

The framework we adopt for carrying out this workhas the following key features: piecewise linear(hardening or softening) constitution of the assumedlumped deformability bending hinges; a description of

* Corresponding author. Tel:+ 61-2-9385-5062; Fax:+ 61-2-9385-6139; E-mail [email protected]

0141-0296/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (98)00018-2

these nonholonomic constitutive laws in complementar-ity format, involving the orthogonality of two sign-con-strained vectors; and finally the obtention of both theor-etical and numerical results from the analysis of a specialand important class of mathematical programming prob-lems, called the linear complementarity problem (LCP)[2]. This powerful and elegant approach is due to Maier[3–7] who, however, has applied it to a far wider rangeof engineering plasticity problems than the one con-sidered herein. Some key papers directly related to ourwork are those by Maier [8] and by Maier and Drucker[9], where general results concerning the interaction oflarge displacements and inelastic behaviour are given.Maier and Perego [10] and, more recently, Bazant andJirasek [11] considered the problem of stability andbifurcations in dynamics with softening but in theabsence of geometric effects. The same approach as weadopt herein has been used, within the small displace-ment context, for limit and elastoplastic analysis underductility constraints [12], and for discussing the equilib-rium branching due to flexural softening [13] of frames.Operative procedures have also been proposed [14,15]for inelastic evolutive analysis and for the determinationof the collapse load of elastic perfectly-plastic or harden-ing frames in the presence of 2nd-order geometriceffects. Numerous other references on mathematical pro-gramming in structural plasticity can be found, for

558 G. Bolzon, F. Tin-Loi /Engineering Structures 21 (1999) 557–567

instance, in the volumes [16] and [17], and in the surveypapers [18] and [19].

The organization of this paper is as follows. In thenext section, we introduce the formulation of our prob-lem in generic rate form. This is carried out for a suitablydiscretized structure and is shown to furnish an LCPwhich makes it possible to obtain not only useful theor-etical results, but also numerical response curves for thestructure. Two simple beams and one portal frame arethen used as reference examples to illustrate the complexand insightful behaviour which can arise under the com-bined influence of physical and geometric effects. Sig-nificant features of various instability phenomena arehighlighted, with theoretical predictions verified numeri-cally.

2. Formulation

The equilibrium configurations of suitably discretizedplane frame structures are sought under the assumptionsof both geometric (2nd-order) and material nonlin-earities. The configuration change of each element canbe described by the six independent degrees of freedomqi, i = 1,..., 6, shown in Fig. 1 and collected in vectorq. Within this framework, the equilibrium of each linearelastic member in its deformed configuration results, invector form, in the following equation relating forcesQto displacementsq:

Q = [kE + kG]q (1)

where kE and kG (see Appendix A) respectively rep-resent the usual linear elastic stiffness matrix, and thegeometric stiffness matrix which accounts for the changeof configuration with loading.

For any frame, the assembly of member equilibriumequations such as Eq. (1) provides the elastic responseto a given loading condition, and allows us to evaluatebending moments at the beam ends as functions of theload parameters, in the absence of inelastic deformatio-ns.

The positive sign-convention we will use in the fol-lowing for bending moments and correspondingrotations is shown in Fig. 2.

For the simply supported beam (SSB) of Fig. 3(a),discretized by two elements and subjected to the axialforce N = aEI/l2 and to the transverse forceF = bEI/l2,

Fig. 1. Beam element.

Fig. 2. Constitutive law characterizing the inelastic hinges.

Fig. 3. (a) Simply supported beam (SSB) and (b) clamped–clampedbeam (CCB).

the global equilibrium equations give, after somestraightforward manipulations:

M+B =

EI6l

(48 + a)(240 +a)3a2 + 416a + 3840

b (2)

For the clamped–clamped beam (CCB) of Fig. 3(b),the moment expressions are:

M+B = M−

A = M−A9 =

EI24l

240 +a

40 + ab (3)

Eq. (2) and Eq. (3), which are linear functions ofbin the present context, furnish well-known basic resultswhen a = 0 (or when no geometric effects are takeninto account). In the presence of axial forces, bendingmoments are decreasing functions ofa, defined fora >− 9.94 (SSB) anda > − 40 (CCB), values which rep-resent the fairly good approximations, for the chosendiscretizations, of Euler’s buckling loads (whose exactvalues are known to be− p2 > − 9.87 for SSB and−4p2 > − 39.48 for CCB).

Equilibrium can be alternatively expressed directly interms of the above bending moments which representthe so-called generalized (in Prager’s sense) stresses ofthe beam, energetically conjugate to the natural

559G. Bolzon, F. Tin-Loi /Engineering Structures 21 (1999) 557–567

(intrinsic) strain parameters represented by the beam endrotations which, in turn, completely define the beamstrain state when axial and shear deformations can beneglected, as assumed herein [15]. These intrinsic gen-eralized stresses and strains, unaffected by rigid bodymotions, are used as natural variables for describing thematerial nonlinearities which are due to the possibleappearance of irreversible rotationsu = u± (i.e. eitheru+or u−) at the ends of each beam element (see Fig. 2).

Inelastic rotations are related to the correspondingbending moments by constitutive laws characterized bya nonholonomic softening (or hardening) irreversiblebehaviour. In this paper, the piecewise linear constitutivelaw represented in Fig. 2 is used.

In the range 0# u # uu, the above referenced lawcan be given the following analytical expressions:

w = M − Mu + hu # 0, u $ 0, wu = 0 (4)

where w, M and Mu stand for w±, M± andM±u respect-

ively, as above introduced foru. The conditionw = 0indicates that the activation limit of the rigid-softeninghinge, or its re-activation limit when the hinge hasexperienced some inelastic rotationu > 0 beforeunloading, has been reached.

For u > uu, a reversible behaviour andM = 0 isassumed.

An alternative rate form of Eq. (4) for hinges whichare at their activation limit (w = 0, 0# u # uu) is the fol-lowing:

w = M + hu # 0, u $ 0, wu = 0 (5)

The above relations (5) apply as well to any nonlinearbending–rotation laws,h being in this case the tangentslope at the consideredM–u point.

By following an approach which dates back to Maier’swork in the early 70s (see e.g. [5]), the inelastic rotationscan be considered as ‘distortions’ (imposed rotations)induced in an assumed wholly elastic structure. In thepresence of geometric effects, these distortions alsoaffect the distribution of bending moments in staticallydeterminate structures. These bending moments can berelated to the imposed rotations by influence functionsZ(a) which can also be obtained from the solution ofthe elastic problem (1).

The total bending moments at the end sections of theelastic beams, due to both the external loads and theinelastic distortions, can then be expressed in compactform as:

M = bM1(a) + Z(a)Q (6)

where vectorM1(a) collects those functions which aregiven for instance by Eq. (2) and Eq. (3) for the SSB

and CCB reference examples, and matrixZ(a) collectsall influence functions (see Appendix A).

The problem formulation is obtained by coupling Eq.(6) with relations (4) or (5).

3. Equilibrium paths

All sections of the virgin structure at the beginning ofthe first loading process behave in an elastic manner,with the bending moments at each member end givenby bM1 (a). By increasing the load multiplier, some sec-tions (one or more) will reach their elastic limitsMu atsome stage during this load application, and inelasticrotations will usually start to develop. However, thesesections could also experience unloading from their elas-tic limit during a continual increasing loading process,as will be shown in our examples.

In the presence of softening, the elastic limit couldrepresent the load carrying capacity of the structure. Tocheck when this is the case, the path of reasoning isas follows.

For the sake of clarity, let us consider first the caseof constant (or almost constant, as is sometimes assumedin simplified frame analysis) axial force.

The evolution of the inelastic rotations of the set ofpotentially active hinges (i.e. those which have reachedtheir activation or re-activation limits) is governed byEq. (5) and Eq. (6) which can be combined to give, underthe above hypothesis:

F = bM1(a) + [Z(a) + H]Q # 0, Q $ 0, (7)

FTQ = 0

Vector Q, in relations (7), collects the inelastic rotationrates of the set of potentially active hinges, and diagonalmatrix H collects the slopesh.

It is well known that the rate problem (7), which hasthe format of a standard LCP once the value ofa isfixed, may admit zero, one, or multiple solutionsdepending on the characteristics (in our case on thedefiniteness) of the matrixK (a,h) = − [Z(a) + H] andof all its principal minors [4,6,8].

When a hardening hinge behaviour is assumed andgeometric effects are neglected, matrixK is positivedefinite,Z andH being both negative definite accordingto our assumed sign-convention. Uniqueness of thestructural response is then ensured.

Within a nonlinear geometric framework, matrixZ(a)is negative definite fora $ 0, but can change its defi-niteness for some value ofa , 0 (members undergoingcompression). In the presence of softening, matrixH ispositive definite according to our sign-convention.Hence in tension, and within some range of compression,Z andH provide opposite contributions toK , allowing


the problem to admit multiple solutions (i.e. bifurcationof the equilibrium branches, in the present context) orno solution at all.

Matrix K characterizes also the stability of the struc-ture under load control, and againZ and H produceopposite stabilizing and destabilizing effects respect-ively, in the presence of tensile forces and softening. Inparticular, stability under load control means that prob-lem (7) admits at least one solution forb > 0, i.e. thatirreversible rotations will eventually occur only byincreasing the loading. A limit point (the load carryingcapacity of the structure, or the threshold to snap-through instability) has been reached if a solution existsonly for b # 0.

To determine the sign ofb , we can restrict our atten-tion to the subset of activated hinges, for whichQ > 0and hence for which relation (7) holds as an equality, i.e.

bM1(a) = − [Z(a) + H]Q = K (a,h)Q (8)

From (8) we can obtain

b =QTKQ

QTM1(9)

and easily see that the sign ofb depends on the sign-definiteness of matrixK only since, for our adopted sign-convention, all the components of vectorM1 are posi-tive.

Let us suppose now thata and b vary together,according to some given lawb = b(a). In this case, theanalytical relations which govern the evolution of theproblem read:

F = (b9M1 + bM 91 + Z9Q)a + [Z + H]Q $ 0, (10)

Q $ 0, FTQ = 0

where a prime denotes differentiation with respect to theload parametera.

The same arguments as before can be applied to (10)in the discussion about stability and bifurcation of theequilibrium paths originating from the current state ofthe structure although the vector (b9M1 + bM 9

1 + Z9Q)and the matrix [Z + H], respectively multiplyinga andQ, no longer remain constant but have to be updatedwith a.

4. Reference examples

We use two simple beams and a frame as referenceexamples to clarify the previous theoretical discussionson the interaction between physical and geometriceffects.

In the first instance, the equilibrium path(s) of the two

simple, yet instructive, reference cases represented inFig. 3 are identified. We also determine the relevantstability domains in the space of the dimensionless para-metersa and h* = hl/EI, which represent respectivelythe member axial force and the slope of the momentversus inelastic rotation relation characterizing thesoftening hinges. For the sake of generality, all theresults are given in dimensionless format, and loads anddisplacements are also scaled by the correspondingvalues at the elastic limit for zero axial force (or no geo-metric effects).

4.1. Simply supported beam (SSB)

Two beam elements,AB and BA9 [see Fig. 3(a)] areused to model the behaviour of this structure. Accordingto this model, inelastic rotations can occur at most inone section, on the symmetry axis, where the linear dis-tribution of bending moment attains its maximum value.The case of constant axial force is considered first.

Let b0 and d0 respectively denote the dimensionlesselastic limit for zero axial force of this structure and thecorresponding deflection scaled by the beam length.Then b0 = 2Mul/EI and d0 = b0/24.

In the presence of axial force, the elastic limitb is anincreasing function of the axial load parametera:

b

b0

= 33840 + 416a + 3a2

(240 +a)(48 + a)(11)

as shown by the dash–dot curve in Fig. 4. The elasticlimit and the load carrying capacity of the structure areof course reduced to zero at the Euler buckling load (aE

> − 9.94).After hinge activation, the behaviour of the structure

depends on both the values ofh* anda, as shown by Fig.

Fig. 4. Dimensionless load versus dimensionless displacement of theSSB forh* = 3 and different axial forces.


Fig. 5. Stability domains of the SSB.

5; Fig. 4 details this behaviour forh* = 3. In particular,comparison with (9) shows that the response under loadcontrol is stable (i.e. the load can be further increased)when:

h* #a(a + 48)(a + 240)

12(3840 + 416a + 3a2)(12)

with the stable and unstable regions being delimited bythe thick line in Fig. 5.

For h* greater than the right hand side of (12), theload carrying capacity of the structure coincides with itselastic limit.

For a = 0, the behaviour is stable for anyh* , 0(hardening hinge); this would be the result, even in thepresence of compression, when no geometric effects aretaken into account.

For h*→ − `, the model simulates an intact elasticstructure and, as expected, Euler’s stability limit is reco-vered.

When EI/lÀh, h*→0, the critical load of Shanley’scolumn [20,21] is obtained in the limit. In this classicalexample, the beam reduces to a rigid bar with a veryshort (b¿l) elastoplastic link of ideal (sandwich) I crosssection (see Fig. 6). According to this model:

Fig. 6. Shanley column.

u =e1 − e2

cb =

Mh

, h =2Ac2

bE1E2

E1 + E2

(13)

and the critical load is obtained whenMcr = Ncrlu/4 =hu. Linearization of Eq. (12) ash*→0 and a→0 alsogives acr = 4h*, i.e. Ncr = 4h/l.

For high values ofh*, the response of the structurecan also be unstable under displacement control (snap-back instability). This occurs when:

h* $(a + 48)(a + 240)

32(120 +a)(14)

The limit for snap-back instability is represented in Fig.5 by the dash–dot line.

For a = 0, or when no geometric effects are taken intoaccount, the limiting slope for avoiding the severe snap-back instability ish* = 3, as also detailed in Fig. 4.

The above limits, given by (12) and (14), apply alsoto any inelastic hinge constitutive law, withh* corre-sponding to the initial tangent modulus of theM–urelationship.

4.2. Clamped–clamped beam (CCB)

According to the assumed CCB beam model [see Fig.3(b)], potential rigid-softening hinges are placed at mid-spanB and end sectionsA andA9.

The dimensionless elastic limit for this structure andits corresponding deflection for zero axial force arerespectivelyb0 = 4Mul/EI and d0 = b0/96.

For different values of axial force, the elastic limitb is an increasing function of the axial load parametera according to the relationship:

b

b0= 6

40 + a

240 +a(15)

Its value is of course reduced to zero at the Euler buck-ling load (aE = − 40).

For any value ofa, the elastic limit is simultaneouslyattained at the three potential hinge locations, but thisdoes not necessarily imply activation of the correspond-ing irreversible rotations. The actual inelasticmechanism(s) and the deformation history of the struc-ture depend on botha and h* as shown by Fig. 7. Inparticular, when

h* $a + 48

12(16)

the elastic limit corresponds to a bifurcation point; thestructure then admits multiple symmetric equilibriumpaths. For higher values ofh*, in particular for


Fig. 7. Inelastic deformation patterns of the CCB depending onsoftening (h*) and axial force (a).

h* $(a + 48)(a + 240)

16(120 +a)(17)

unsymmetric paths may be followed as well as the sym-metric ones. The set of these equilibrium paths corre-sponds to the solutions of LCP (8), whose characteristicsdepend solely on matrixH (i.e. on h*) since we haveassumeda (and henceZ) to be constant.

Most of these mechanisms show instability under loadcontrol (i.e. no solution exists to the rate Eq. (8) forb > 0), independently of the value of the axial force.This kind of instability can be ruled out only for those(a, h*) pairs which satisfy the relationship:

a −40 + a

240 +a24h* +

a + 48 − 12ha + 48 − 24h*

(18)

Sa −40 + a

240 +a48h*D $ 0

This is a subset of the pairs which allow for a type ‘a’mechanism (Fig. 7); this limit of stability is representedby a thick line in Fig. 8.

The complete load–deflection curves fora = 0 (orwhen no geometrical effects are taken into account) andfor some meaningful values ofh* are drawn in Fig. 9.Whena = 0 the behaviour is unstable under load controlfor any h* > 0, and bifurcation of the equilibrium pathinto symmetric, and both symmetric and unsymmetricbranches occurs when 4# h* , 6 andh* $ 6 respect-ively. It is interesting to note that, even for this simplestructure, up to seven different equilibrium branches canemanate from the bifurcation point.

Again, the above stability and bifurcation conditions,

Fig. 8. Stability domains of the CCB.

relations (18) and (16)–(17) respectively, apply to anynonlinearM–u law whereh* is the initial tangent modu-lus.

4.3. CCB with proportional axial and bending loads

Let us consider now the situation of Fig. 10, namelya clamped–clamped beam subjected to axial and trans-verse loads of constant ratio. The elastic solution canstill be represented by Eq. (3) if we setb = ruau. Theelastic limit is attained when:

m(a) =240 +a

40 + auau = r (19)

wherer = 24Mul/rEI = 6b0/r, with b0 = 4Mul/EI beingthe same reference parameter introduced in the pre-vious section.

The meaningful solutionsa of the 2nd-order Eq. (19)are graphically represented in Fig. 11. Note that whenr→`, the axial load dominates (r→0) and the elasticlimit coincides with Euler’s buckling load.

Stability and bifurcation of the equilibrium pathsemanating from the elastic limit depend on the character-istics of the matrix governing the rate problem (10). Inthe present case,bM 1(a) = m(a)I in dimensionless for-mat, whereI represents the 3× 3 identity matrix. Then,since Q = 0 at the beginning of the inelastic process,relations (10) read:

F = rm9(a)aI − K (a,h*)Q $ 0, Q $ 0, FTQ = 0

(20)

where m9(a) obviously represents the derivative, withrespect toa, of the function introduced in (19), evaluatedat the elastic limita = a.

Since the sign of functionm9(a) does not change


Fig. 9. Dimensionless load versus dimensionless displacement of the CCB fora = 0 and different softening hinge characteristics. Letters inparentheses indicate actual deformation patterns, cf. Fig. 7. Continuous lines refer to symmetric solutions; dashed lines to the unsymmetric ones;dash–dot lines to the envelope of the indicated transition configurations, depending onh*. The symmetric solution sequence forh* $ 6 is thesame as for 4# h* # 6.

Fig. 10. CCB under proportional loading.

either in tension or in compression (m9(a) > 0,∀a > 0; m9(a) , 0, ∀a , 0), stability depends on thecharacteristics of matrixK only, as it did in the previouscase of constant axial load. The conditions for stabilityare derived again from Eq. (18), after replacing the gen-eric a with its value a at the elastic limit, which is afunction of r. The stability limit in tension and com-pression, as a function ofr andh*, is represented in Fig.12 by a thick line. The inelastic deformation patterns thestructure follows soon after the elastic limit has beenreached are also shown in the same drawing; dash–dot

Fig. 11. Elastic limit of the CCB under tension and compression.


Fig. 12. Stability domains of the CCB under proportional loading.

lines represent the threshold values ofr andh* for eachindicated pattern. These results are also obtained, as forthe stability limit, by replacinga with a in Eq. (16) andEq. (17).

Fig. 13 shows the overall load–displacement behav-iour of our structure under tensile load forr = 100, i.e.for a > 24.34. The corresponding limit of stability ish* > 2.39 (see Fig. 12), and inelastic deformations startto develop in all possible locations (i.e. at the mid-spansectionB and at the end sectionsA andA9, see Fig. 10)up to h* > 3.01.

Within the stability range, the inelastic deformationrate decreases with the continuously increasing stabiliz-ing axial force and the inelastic rotations themselves tendto asymptotic values which are lower thanuu (see Fig.14).

When the post-elastic behaviour is unstable, a dis-placement controlled process will show the inelasticrotation of the mid-span softening hinge growing fast

Fig. 13. Load multiplier versus dimensionless displacement of theCCB under proportional loading in tension, forr = 100.

Fig. 14. Load multiplier versus inelastic rotations of the CCB underproportional loading in tension, forr = 100. Continuous lines refer tobeam end rotations (u−

A − A9); dashed lines to mid-span rotation (u+B).

initially, up to the critical valueuu, before a stiffeningoverall behaviour is observed again. The elastic limit ofthis structure hence represents the threshold of a possiblesnap-through behaviour with dynamic effects playing asignificant role. Such effects and their interaction withdissipative hinges deserve proper investigation which,however, is beyond the scope of the present work.

Finally, when h* is close enough to the thresholdvalue which corresponds to the next deformation pattern,which involves inelastic rotations developing at the mid-span sectionB only (see Fig. 12), it will happen that,along a post-elastic displacement-controlled process, theinelastic rotations at sectionsA andA9 will stop (and thecorresponding bending moments will decrease) at somestage. The structure will then follow the next defor-mation pattern but with fixed inelastic distortions of thebeam ends, up to the critical valueuu of the inelasticrotation of the softening hinge atB. The load can thenincrease again, first to bring the end sectionsA and A9back to their activation limits and then further on alongan overall stiffening inelastic deformation process.

4.4. Portal frame (PF)

The simple PF of Fig. 15, discretized by beamelements with softening hinges, has been the subject ofrecent investigations [11,22] which showed that a bifur-cation of the structural response occurs ifh* $ 6k/(3 +2k), wherek = (E1I1/l1)/(E2I2/l2) andh* = hl2/E2I2, whengeometrical effects are neglected. In this case, in fact,the same bending moment results in sectionsB and B9at the elastic limit,b = b0 = Mul2/E2I2, and two softeninghinges can simultaneously develop (anti-symmetricsolution) or one section can undergo inelastic rotationwhile the other unloads elastically (unsymmetricsolution).

The possibility of multiple equilibrium paths, how-


Fig. 15. Portal frame (PF).

ever, disappears for this structure (different from thebehaviours presented in the previous sections) when theanalysis is performed within a geometrically nonlinearframework. This is due to the opposite effect on stiffnessof the axial forces in the columnsAB andA9B9, one ofwhich is tensile while the other is compressive although,for equilibrium, their magnitudes are the same. Hence,unequal bending moments result at sectionsB and B9,and inelastic rotations localize in one section only. As anexample, assume beam elongations are negligible. Themagnitude of the axial forces in both columns is equalto aE2I2/l22, wherea = b/r and r = l1/2l2, and if k→`the following bending moments result:

1r

M+Bl2

E2I2

= mB(a) = aS1 +a

61440− a2

3600− 7a2D (21)

1r

M−B9l2

E2I2

= mB9(a) = aS1 +a

61440− a2

3600− 7a2D (22)

It can be clearly seen thatM+B is greater thanM−

B9 forany value of 0, a , aE > 22.678. The elastic limit isthen reached first at sectionB, where inelastic rotationlocalizes. The axial load parameter at the elastic limita, solution of the equationmB(a) = r = b0/r, is graphi-cally represented as a function ofr in Fig. 16.

The stability of the only equilibrium path emanatingfrom the elastic limit depends on the characteristics ofthe equation which governs the present rate problem,i.e.:

wB = rmB9(a)a − [z(a) − h*] uB $ 0, u $ 0, wBuB = 0

(23)

where the expression forz(a) is given in the AppendixA. Relation (23) follows directly from (10) after appro-priate substitutions, as already shown in deriving (20).

It can be easily verified thatmB(a) is a monotonicallyincreasing function ofa, while mB9(a) (i.e. the bendingmoment at the end of the compressed columnA9B9) first

Fig. 16. Elastic limit of the PF.

increases, then decreases, and changes its sign whena= 12. Since the sign of functionm9

B(a) is always positive,stability of the equilibrium path emanating from the elas-tic limit depends only on the sign of the functionz(a) − h*. The stability domain is represented in Fig. 17in the space of the parametersr andh*.

Finally, the frame response forr = 2 (a > 1.786, seeFig. 16), h* = 1 andh* = 3 is represented in Fig. 18,where the load multipliera = b/r is plotted versus thedimensionless displacementd/r = u/rl2 = 2u/l1. Note thatthe structure is quite flexible and hence higher-ordertheory would be required to obtain accurate results.However, the essential role of geometry changes on stab-ility and bifurcation of the equilibrium paths of thepresent structure can be already captured within thepresent approach. In particular, observe that the limit ofstability asa→0 (when 2nd-order theory is appropriate)is h* = 1.5 (h* > 1.475 whenr = 2), while the anti-symmetric solution one would find disregarding geo-metric effects is unstable for anyh* > 0.

Fig. 17. Stability domain of the PF.


Fig. 18. Load multiplier versus dimensionless displacement of the PFfor r = 2 and different softening hinge characteristics.

5. Conclusions

The focus and novelty of this paper is on the interac-tion between material (hardening or softening) and 2nd-order geometric effects in framed structures, illustratedby simple models. Our interest is centred primarily onthe complex manifestations of these two influences atthe global (structural) level.

We present the governing relations for the generic sys-tem in a compact, so called linear complementarity prob-lem (LCP), format. This approach has two main advan-tages. First, it enables a computation-oriented frameworkto be used. Second, and more importantly in our view,it provides a theoretical tool for the qualification of thesolution response domain. In particular, the governingmatrix of the LCP, as is well known in mathematicalprogramming literature, plays a central role (throughconsiderations of its sign-definiteness) in obtainingmechanically meaningful results on the stability of thestructure.

Application to two simple beams and to a portal frameis used to illustrate key features (e.g. bifurcation, limitpoints, snap-back and snap-through instabilities) of theinteracting effects of material and geometrical influ-ences. Complete responses are given to corroborate thetheoretically obtained predictions for a wide range ofparameters. The behaviours of classical Euler and Shan-ley struts can be recovered for the appropriate situations.

Extension to large-scale realistic structures appearspossible and conceptually straightforward. However, itis not easy for problems of large size to capture the mul-tiplicity of solutions (or even to show that no solutionexists) because of the intrinsic combinatorial nature ofthe problem. Computational work in this direction wouldbe a worthwhile research area. A further, possibly moreimportant issue to be addressed in the presence of bifur-cations, is the determination of the most likely equilib-

rium path under load, displacement or a combined load-displacement control.

Acknowledgements

The authors wish to thank Professor Maier for contin-ual encouragement and friendly advice. Dr Bolzon’sappointment as Visiting Fellow in the School of Civiland Environmental Engineering at the University of NewSouth Wales, and support from the Australian ResearchCouncil are also gratefully acknowledged.

Appendix A

Elastic stiffness matrix

kE

EIl3

Al2

I

0 12 Symmetric

0 6l 4l2

−Al2

I0 0

Al2

I

0 − 12 − 6l 0 12

0 6l 2l2 0 − 6l 4l2

(A1)

Geometric stiffness matrix

kG =Nl

0

065

Symmetric

0110

l215

l2

0 0 0 0

0 −65

−110

l 065

0110

l −130

l2 0 −110

l215

l2

(A2)


Influence functions

SSB

M±B = Z(a)u±

B; Z(a) = −EI12l Fa

(48 + a)(240 +a)3840 + 416a + 3a2G

(A3)

CCB

5M7

A

M 7A9

M±B

6 = Z(a)5u7

A

u 7A9

u±B

6; Z(a) = −EI24l 3a b c

b a c

c c d4 (A4)

a =921600 + 61440a + 880a2 + 3a3

2(40 +a)(120 +a);

b = − 457600 + 960a + 13a2

(40 + a)(120 +a)c = −

3840− 64a + a2

4(40 +a);

d =3840 + 416a + a2

4(40 +a)

PF

H M±B

M 7B9

J = Z(a)H u±B

u 7B9

J; Z(a) = −E2I2

96l2Fz − z

− z zG (A5)

z = z(a) =(144 − a2)(3600− a2)

(3600− 7a2)

References

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Physical instability and geometric effects in frames

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