Engineering Structures 30 (2008) 1895–1904www.elsevier.com/locate/engstruct

Limit and shakedown analysis of 3-D steel frames

Hoang Van Long∗, Nguyen Dang Hung

LTAS- Fracture Mechanics, University of Liege, Chemin des Chevreuils, 1, B 52/3, 4000 Liege, Belgium

Received 13 July 2007; received in revised form 6 December 2007; accepted 6 December 2007Available online 22 January 2008

Abstract

This paper presents an efficient algorithm for both limit and shakedown analysis of 3-D steel frames by the kinematic method using linearprogramming technique. Several features in the application of linear programming for rigid-plastic analysis of three-dimensional steel framesare discussed, as: change of the variables, automatic choice of the initial basic matrix for the simplex algorithm, direct calculation of the dualvariables by primal–dual technique. Some numerical examples are presented to demonstrate the robustness, efficiency of the proposed techniqueand computer program.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Limit analysis; Shakedown analysis; Plastic hinges; Space frames; Linear programming

1. Introduction

The fundamental theory of plastic analysis of the framestructures was pointed out fifty years ago. This techniqueconsisting of an application of the mathematical programmingis widely exposed in the literature [1–6]. When the linearizedcondition of plasticity admissibility is adopted, the plasticanalysis problem can be reduced to a linear programming (LP)problem where simplex technique is largely used. This directionhas been deeply exploited in the years 1970–1990, and someinteresting computer programs have been developed [7–11].Unfortunately, for the last two decades, the research in this areahas been sporadic and limited; practical engineering has not yetresponded.

The major advantage of the shakedown analysis issolely applicable for the arbitrary loading histories (oftenin the practices). However, under geometrical nonlinearityconditions, usually considered for steel structures, difficultieshave generally appeared in this kind of problem (shakedownanalysis).

In recent years, numerous authors have concentrated theirefforts on the advanced nonlinear analysis of 3-D steelframes [12–17]. Modern analysis must take into account the

∗ Corresponding author.E-mail address: VanLong.Hoang@student.ulg.ac.be (H. Van Long).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.12.009

interrelated effects of material inelasticity and geometricalnonlinearity in furnishing an adequate response to the problemsof structural systems and their components. The complicated3-D steel structures may be solved in this direction. However,this development is based on the step by step methods that maycontain a lot of difficulties in considering the cases of arbitraryloading histories.

Consequently, the parallel development of both step bystep and direct methods (by mathematical programming) isnecessary. They give a better view of the behaviour of thereal structure and also they may mutually make up fortheir deficiencies. It is the fundamental motivation of ourwork: the development of the theoretical foundations and thepractical software useful for inelastic structures analysis andoptimization. In the following, one can find a brief presentationof a computer program, namely CEPAO:

This package had been developed in the Department ofStructural Mechanics and Stability of Constructions of theUniversity of Liege by Nguyen-Dang Hung et al. in the1980s [9–11]. Indeed, CEPAO was a unified package devoted toautomatically solving the following problems for 2-D frames:Elastic analysis, limit rigid-plastic analysis with proportionalloadings; step by step elastic–plastic analysis; shakedownanalysis with variable repeated loadings; optimal plastic designwith fixed loading; optimal plastic design with choice ofdiscrete profiles and stability checks; shakedown plastic design

1896 H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904

with variable repeated loadings; shakedown plastic design withupdating of elastic response in terms of the plastic capacity.

With the CEPAO, efficient choice between statical andkinematic formulations is realized leading to a minimumnumber of variables; also there is a considerable reductionof the dimension of every procedure. The basic matrixof LP algorithm is implemented under the form of areduced sequential vector which is modified during eachiteration. An automatic procedure is proposed to build upthe common characteristic matrices of elastic–plastic or rigid-plastic calculation, particularly the matrix of the independentequilibrium equations. Application of duality aspects in the LPtechnique allows direct calculation of dual variables and avoidsexpensive reanalysis of every problem.

At this time, the CEPAO is extended to the case of spacesteel frames with the following problems: rigid-plastic analysisand design by LP; elastic–plastic analysis by the step-by-stepmethod in both first and second order (P-delta effects). The limitand shakedown analysis is based on the upper bound theoremwhile the rigid-plastic design is supported by the lower boundtheorem. It is important to indicate here that in the case offrame structures both kinematic and statical methods lead tothe same one and only solution. The geometrical nonlinearity iscompletely ignored in this direct method (by LP).

Because of the limitation of an article, the present workdescribes only details of the module of limit and shakedownanalysis where some original contributions are indicated in theSection 3. We hope that other contents will be presented in ournext papers.

2. Assumptions and modelling plastic hinges

The following assumptions have been made:

– Loading is quasi-static and service load domain is specifiedby linear constraints;

– The torsional stiffness and the effect of the shear forces arenegligible;

– The rigid — perfectly plastic material is used in the limitanalysis; the elastic — perfectly plastic material is appliedin the other problems.

– Plastic hinges are located at critical sections.

Modelling plastic hingesSince the effect of both shear forces and torsional moments

are ignored, the condition of plastic admissibility at the criticalsections becomes Φ(N , My, Mz) ≤ 0, with N is the normalforce and My , Mz are respectively bending moments about toy and z axes. The plastic hinge modelling is described by thechoice of net displacement (relative) - force relationship at thecritical sections. In present work, the normality rule is adopted.∆

θyθz

= λ

∂Φ/∂ N∂Φ/∂ My∂Φ/∂ Mz

,

or, symbolically:

ei= λiNi

C , (1)

where λi is the plastic deformation magnitude; ei is the vectorof longitudinal displacement and two rotations of i th section;Ni

C is a gradient vector of the yield surface Φ.The application of the LP techniques requires that the

nonlinear yield surfaces must be linearized. In civil engineeringpractices, for bisymmetrical wide-flange shapes, severalStandards replace the curvilinear yield surface by a polyhedronsixteen-facet:

α1|N |

NP+ α2

∣∣My∣∣

MPy+ α3

|Mz |

MPz= 1 for

|N |

NP≥ α0; (2a)

α4|N |

NP+ α5

∣∣My∣∣

MPy+ α6

|Mz |

MPz= 1 for

|N |

NP< α0; (2b)

where: MPy,MPz are the plastic moment capacity with respectto y and z axis, NP is the squash load, 0 ≤ α0 < 1, α1, . . . α6are the dimensionless coefficients. The Eqs. (2a) and (2b) mayalso be written

a1 |N | + a2∣∣My

∣∣ + a3 |Mz | = S0 for|N |

NP≥ α0; (3a)

a4 |N | + a5∣∣My

∣∣ + a6 |Mz | = S0 for|N |

NP< α0; (3b)

with S0 is a referential value, and a1, . . . a6 are the nonzerocoefficients.

At the i th critical section, the plastic admissibility definedby Eqs. (3a) and (3b) has the following form:

Yisi≤ si

0, (4)

where matrix Yi contains the coefficients a1, . . . a6; si collectsthe vector of internal forces; the column matrix si

0 contains thecorresponding terms S0.

According to the definitions of the matrices NC , Y, we cansee that

NiC = YiT. (5)

The detailed form of matrix NC is below-mentioned by Eq.(13).

3. Application of LP

A systematic treatment of the application of LP in plasticanalysis can be found in [4,5]. In the present work, we restrictourselves to describing some practical aspects of the CEPAOpackage applied to the case of 3-D steel frames. They are:the further reduction of the kinematic approach (Sections 3.2.2and 3.3.2), and the direct calculation of the internal force (orresidual internal force) distribution (Sections 3.2.3 and 3.3.3).

3.1. General formulation

In the CEPAO, the canonical formulation of the LP isconsidered:

Min π = cTx|Wx = b (6)

where π is the objective function; x, c, b are respectivelythe vector of variables, of costs and of second member. W

H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904 1897

is called the matrix of constraint. For the sake of simplicity,the objective function has a state variable, and the matrixformulation is arranged such that the basic matrix of the initialsolution appears clearly as follows:

[−cT

1 1 −cT2

W1 0 W2

] x1π

x2

=

[0b

]. (7)

The basic matrix of the initial solution is

X0 =

[1 −cT

20 W2

].

Eq. (7) can then be written under a general form

W∗x∗= b∗. (8)

The matrices W∗, x∗, b∗ and X0 for both limit and shakedownanalysis problems will be accurately calculated in the followingsections:

3.2. Limit analysis by kinematic method

3.2.1. Kinematic approachA kinematically admissible state is defined by a collapse

mechanism that satisfies the condition of compatibility. It leadsto a positive external power supplied by the reference loading.Based on the upper bound theorem of limit analysis, thekinematic formulation of limit analysis can be stated as a LPproblem.

Min φ = sT0 λ

∣∣∣∣∣∣NCλ − Bd = 0

fTd = ξ

λ ≥ 0.

(9)

The safety factor will be obtained by

µ+ = φ/ξ.

In Eq. (9), λ is the vector of the plastic deformation magnitude;B is the kinematic matrix defined in Appendix A; d, f arerespectively the vector of independent displacements and thevector of external load; ξ is a constant (generally, one takesξ = 1).

3.2.2. Further reduction of the kinematic approachIn the kinematic method, the unknowns are the plastic

deformation magnitude, λ, and the independent displacement,d (negative or positive). In LP procedure we need nonnegativevariables so that we adopt the change of the variables as in thefollowing:

d′= d + d0 so that d′

≥ 0.

The way to fix the value of d0, which depends on the realstructure, such that d′ are always nonnegative is explained inthe Appendix C. Now, the problem of Eq. (9) becomes

Min φ = sT0 λ

∣∣∣∣∣∣NCλ − Bd′

= −Bd0

fTd′= ξ + fTd0

λ, d′≥ 0.

(10)

Therefore, the vector of variables, matrix of constraint, vectorof second member corresponding to the problem of Eq. (8) forlimit analysis are given below.

x∗T=

[π xT η

]=

[π d′T λT η

]b∗T

=[0 bT]

=[0 −Bd0 ξ + fTd0

]W∗

=

1 0T−sT

0 00 −B NC 00 fT 0T 1

where η is an artificial variable which must be taken out of thebasic vector in the simplex process.

Because of the use of the simplex technique, finding aninitial admissible solution such that the initial value of anyvariable (except the objective function) must be nonnegativeis needed. To satisfy this requirement, it appears that thefollowing arrangement leads to good behaviour of the automaticcalculation.

The linearized condition of plastic admissibility for the i thsection (Eq. (4)) may be expanded as follows:

〈1〉

〈2〉

〈3〉

〈4〉

〈5〉

〈6〉

〈7〉

〈8〉

〈9〉

〈10〉

〈11〉

〈12〉

〈13〉

〈14〉

〈15〉

〈16〉

ai1 −ai

2 −ai3

ai1 ai

2 −ai3

ai1 ai

2 ai3

−ai1 ai

2 ai3

−ai1 −ai

2 ai3

−ai1 −ai

2 −ai3

ai1 −ai

2 ai3

−ai1 ai

2 −ai3

ai4 −ai

5 −ai6

ai4 ai

5 −ai6

ai4 ai

5 ai6

−ai4 ai

5 ai6

−ai4 −ai

5 ai6

−ai4 −ai

5 −ai6

ai4 −ai

5 ai6

−ai4 ai

5 −ai6

N i

M iy

M iz

≤

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

Si0

. (11)

The Fig. 1 describes the projection of 16 planar facets of thepolyhedral stress-resultant yield surface corresponding to the16 inequalities numbered on the Eq. (11).

According to Eqs. (4) and (5), we see that Eq. (11) can bewritten under symbolic formulation

NiTC si

≤ si0. (12)

Put

NiC =

ai1 ai

1 ai1

−ai2 ai

2 ai2

−ai3 −ai

3 ai3

.

Let us note that: NiC is always nonsingular because ai

1, ai2, ai

3are certainly positive.

1898 H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904

Fig. 1. Projection of the yield surface on the plan My O Mz .

Then, matrix NiC in Eq. (12) may be decomposed into three

submatrices:

NiC =

[N

iC −N

iC N

iC

], (13)

with NiC is the rest of Ni

C after deducting NiC and −N

iC.

The decomposition of matrix NiC leads then to the following

form:

siT0 =

[siT

0 siT0 siT

0

]=

[Si

0 · · · Si0

];

λiT=

[λ

iTλ

iT+3 λ

iT],

where

λiT

=[λi

1 λi2 λi

3

];

λiT+3 =

[λi

4 λi5 λi

6

];

λi=

[λi

7 . . . λi16

].

Let now Si be a diagonal matrix, such that

Si= diag [1 x sign of ((N

iC)−1bi)],

with

biT=

[b3(i−1)+1 b3(i−1)+2 b3(i−1)+3

].

Let E be a unity matrix of dimension 3 × 3.And consider now the new plastic deformation magnitude

distribution:

λi′T=

[λ

i′Tλ

i′T+3 λ

iT],

in which

λi′

= 0.5(E + Si)λi+ 0.5(E − Si)λ

i+3;

λi′

+3 = 0.5(E − Si)λi+ 0.5(E + Si)λ

i+3.

With the mentioned arrangement, and if the case of initial basis

of variables is[λ

1′Tλ

2′T. . . λ

n′

s T], the initial basic matrix

may be determined as follows:

X0 =

1 −s1T0 −s2T

0 . . . −snsT0 0

0 N1CS1 0 . . . 0 0

0 0 N2CS2 . . . 0 0

......

... . . ....

...

0 0 0 . . . NnsC Sns 0

0 0T 0T . . . 0T 1

,

in which, ns is the number of critical sections.Easily, we may demonstrate that the initial solution

x0 = X−10 b

is certainly non-negative.

3.2.3. Direct calculation of the internal force distributionThe strain rate at critical sections is chosen as variables

in kinematical approach. The collapse factor and mechanismare given as output. To obtain the internal force distributionwhile avoiding the static approach, the dual properties of LPare used. The physical significance of the dual variables may beestablished as follows:

The canonical dual form of the LP problem of Eq. (6) is

Max (bTy + 0Th)

∣∣∣∣WTy + h = ch ≥ 0,

(14)

in Eq. (14), yT=

[sT µ−

],

and h are the nonnegative slack variables:

hT=

[0T h1T h2T . . . hns T]

, with:

hiT=

[h

iTh

iT+3 h

i].

It may be seen from the equality (14) that the internal forces arerelated to the slack variables h:

si= (N

iTC )−1(si

0 − hi). (15)

H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904 1899

It can be shown that the slack variables h are identified exactlyas the reduced costs c of the primal problem (8):

h = c = (X - 1op (1, :))W∗

where X−1op (1, :) is the first row of the inverses basic matrix at

optimal solution.The reduced costs c necessary for the convergence test of

the simplex algorithm are variable in the output of the primalcalculation. The automatic computation by (15) of the internalforces distribution is independent of the type of collapse:partial, complete or over-complete.

3.3. Shakedown analysis by kinematic method

3.3.1. Kinematic approachBased on the upper bound theorem of shakedown analysis,

the safety factor can be determined by minimizing thekinematically admissible multiplier. Since the service loaddomain is specified by linear constraints, the kinematicapproach leads to a LP problem:

Min φ = sT0 λ

∣∣∣∣∣∣NCλ − Bd = 0

sTENCλ = ξ

λ ≥ 0,

(16)

where sE is the envelope of the elastic responses of theconsidered loading domain.

The safety factor will be obtained by

µs+ = φ/ξ.

3.3.2. Further reduction of the kinematic approachAs in the limit analysis, by an appropriate choice of d0 such

that:

d′= d + d0 ≥ 0,

and by using the new plastic deformation magnitudedistribution, the vector of variables, matrix of constraints andvector of second member corresponding to the problem ofEq. (8) for shakedown analysis have the following form:

x∗T=

[π d′ λ η

];

b∗T=

[0 −Bd0 ξ

];

W∗=

1 0T−sT

0 00 −B NC 00 0T sT

ENC 1

.

With initial basic matrix:

X0 =

1 −s1T0 −s2T

0 . . . −snsT0 0

0 N1CS1 0 . . . 0 0

0 0 N2CS2 . . . 0 0

......

... . . ....

...

0 0 0 . . . NnsC Sns 0

0 s1TE

(N

1CS1

)s2T

E

(N

2CS2

). . . snsT

E

(N

nsC Sns

)1

And the initial basic variables:[λ

1′Tλ

2′T. . . λ

ns ′T].

The problems of Eqs. (10) and (16) are similar except for thechoice of the initial admissible point in the permissible domainand the shakedown analysis requires preliminary calculation ofelastic responses.

3.3.3. Direct calculation of the residual internal forcedistribution

Again the dual form of Eq. (16) is written similarly to Eq.(14) with:

yT=

[ρT µs−

]; hT

=[0T h1T h2T . . . hns T]

where ρ is the residual internal force vector, hiT=[

hiT

hiT+3 h

i].

From Eq. (14), the residual internal forces are related to theslack variables h as the following relation:

ρi= (N

iTC )−1(si

0 − µsNiCsE − h

i).

As h is identified to be the reduced costs of the primal problem(16), the distribution of residual internal force is directlyobtained without performing a second static approach.

4. Numerical examples and discussions

The presentation of the two following examples aims at acomparison of the CEPAO results with those of some otherauthors, and the comparison of the ultimate states of the framesfined by different models in CEPAO. Therefore, we presentnot only the results given by limit and shakedown analysisbut also those calculated by the step-by-step method (a briefpresentation is presented in Appendix B).

In those examples, with the elastic–plastic analysis by hinge-by-hinge method (first and second order), the plastic interactionfunction proposed by Orbison [18] for compact wide-flangesections is introduced in the CEPAO.

Φ = 1.15n2+ m2

y + m4z + 3.67n2m2

y + 3n6m2z

+ 4.65m2z m4

y − 1 = 0,

in which, n = N/Np is ratio of the axial force to the squashload, m y = My/Mpy and mz = Mz/Mpz are the ratios ofthe major-axis and minor-axis moments to the correspondingplastic moments. This yield surface is already used in severalreferences [12–14] that we consult to compare with our results.

In the direct analysis by LP, the plastic strength of cross-sections used in the AISC [19] is installed in the CEPAO, withthe value of a1, . . . , a6 and α in the Eqs. (3a) and (3b) are:a1 = S0/Np; a2 = 8S0/9Myp, a3 = 8S0/9Mzp, a4 = S0/2Np,a5 = S0/Myp, a6 = S0/Mzp, α = 0.2.

Example a — Six-storey space frame: Fig. 2 shown Orbison’ssix-story space frame. The yield strength of all members is250 MPa and Young’modulus is 206.850 MPa. Uniform floorpressure of 4.8β1 kN/m2; wind loads are simulated by pointloads of 26.7β2 kN in the Y -direction at every beam–columnjoint. In which, β1, β2 are the factors that define the loadingdomain.

1900 H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904

Fig. 2. Example a— Six-storey space frame ((a) perspective view, (b) planview).

Example b — Twenty-storey space frame: Twenty-storey spaceframe with dimensions and properties shown in Fig. 3. Theyield strength of all members is 344.8 MPa and Young’modulusis 200 MPa. Uniform floor pressure of 4.8β1 kN/m2; windloads = 0.96β2 kN/m2, acting in the Y direction.

Concerning the loading domain (for two examples), twocases are considered for shakedown analysis: (a) 0 ≤ β1 ≤

1, 0 ≤ β2 ≤ 1 and (b) 0 ≤ β1 ≤ 1, −1 ≤ β2 ≤ 1. For fixedor proportional loading, we obviously must have: β1 = β2 = 1.The uniformly distributed loads are lumped at the joints offrames.

Diverse models have been adopted by some researchto capture both the material inelasticity and geometricalnonlinearity [12–17], the corresponding load ratios are well inaccord (see Table 1). In which, the second-order plastic-hingemodel has been used in the CEPAO, the large deflection isignored.

The results analysed by CEPAO with different methodsshown on the Table 2, Figs. 4 and 5 point out:

– An expectable coincidence of results calculated by limitanalysis and elastic–plastic analysis first order, it allows usto deduce the good convergence between the dual methods

Fig. 3. Example b— Twenty-storey space frame ((a) perspective view, (b) planview).

Table 1Comparison of results (elastic–plastic 2nd order)

Author Model Load multiplierExample a Example b

Liew JYR-2000 [12] Plastic hinge 2.010 –Kim SE-2001 [13] Plastic hinge 2.066 –Chiorean CG-2005 [14] Distributed

plasticity2.124(n = 30)

1.062 (n = 30)

Chiorean CG-2005 [14] Distributedplasticity

1.998(n = 300)

1.005 (n = 300)

Cuong NH-2006 [15] Fiber plastichinge

2.040 1.003

Liew JYR-2001 [16] Plastic hinge – 1.031Jiang XM-2002 [17] Fiber

element– 1.000

CEPAO-2007 Plastic hinge 2.033 1.024

H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904 1901

(a) Example a.

(b) Example b.

Fig. 4. Deformation at limit state given by CEPAO (From the left to the right: Elastic–plastic first order, Elastic–plastic second order; Limit analysis; Shakedownanalysis, load domain a; Shakedown analysis, load domain b. The points on Fig. (a) indicate the plastic hinges.).

Fig. 5. Load–deflection results at point A (Figs. 2 and 3) given by CEPAO.

in the CEPAO (kinematic and static methods) and the goodcorrelation between the Orbison’yield surface and this in AISC-LRFD.

– In the case of symmetrical horizontal loading (seismic loador wind load), the load multipliers determined by shakedownanalysis are the smallest (alternating plastic occurs).

1902 H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904

Table 2Ultimate strengths of the frames given by CEPAO with different analysis

Method Load multiplier Limit stateExamplea

Exampleb

Hinge-by-hinge, first order 2.489 1.689 Formation of a mechanismHinge-by-hinge, secondorder

2.033 1.024 Unstableness

Limit analysis 2.412 1.698 Formation of a mechanismShakedown analysis,domain load a

2.311 1.614 Incremental plasticity

Shakedown analysis,domain load b

1.670 0.987 Alternating plasticity

5. Conclusions

From the performed work, we can draw the followingconclusions:

It appears that the canonical formulas in both limit andshakedown analysis using LP for 3-D steel frames may bereduced by a special change of the variables and by a naturalchoice of the initial basic matrix useful for the simplexalgorithm. The distribution of the internal forces may bedirectly calculated by the application of duality aspects inthe LP technique. This allows avoiding expensive reanalysisof the primal problem. The above-mentioned techniques arevery suitable for automatic computation; consequently, theyhave been completely implemented in CEPAO package. By theway, the problem of ultimate strengths of the large-scale 3-D steel frames under fix or repeated loading, in the sense ofrespectively limit and shakedown analysis, can be solved nowby the CEPAO package in an automatic manner look like anyfinite element algorithm devoted to 3D frame structures. Thispaper shows also that the simplex technique still is a necessarytool in the automatic plastic analysis of 3-D steel frameworksafter a less eventful period of the application of LP in theanalysis of frame structures.

Appendix A. Compatibility relation

Let eTk =

[∆A θy A θz A ∆B θy B θzB

]be the vector

of the axial displacement and the net rotation of the memberends (Fig. A.1(a)). Assemble for the frameworks (system of theelements) we have the vector e.

Let dTk = [d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 dek]

be the vector of the member independent displacements inthe global coordinate system OXYZ, as shown in Fig. A.1(b).Assembled for the frameworks we obtain the vector d.

In the sense of limit analysis, we may think that: d1, d2,d3, d7, d8, d9 are the displacements corresponding to thedeflection mechanisms (beam and sideways); d4, d5, d6, d10,d11, d12 are the displacements showing the joints mechanisms;dek displacement in the longitudinal direction of the element,describes the bar mechanisms (the bar translates along thisaxis). Since the torsional stiffness of the elements is negligible,we must eliminate the degree of freedom that only provokespure torsion in the bars.

The compatibility relation is defined as

e = Bd,

where B namely the kinematic matrix that is determined by

B =

∑k

AkTkLk. (A.1)

In Eq. (A.1), Lk is a localization Boolean matrix of member k;and

Ak =

1 0 0 0 0 0 0 0 0 0 −1

0 0 −1lk

1 0 0 01lk

0 0 0

0 −1lk

0 0 −1 01lk

0 0 0 0

0 0 0 0 0 1 0 0 0 0 −1

0 01lk

0 0 0 0 −1lk

−1 0 0

01lk

0 0 0 0 −1lk

0 0 1 0

,

Tk =

Ck

C′

kCk

C′

k1

,

with:lk is the length of element k;

Ck =

[c11 c12 c13c21 c22 c23c31 c32 c33

]is the matrix of direction cosines of

element k;

C′

k =

[c21 c22 c23c31 c32 c33

].

Appendix B. Hinge-by-hinge method

In the CEPAO, the step by step method is used for theincrement nonlinear elastic–plastic analysis. After each step,a new plastic hinge occurs, the elastic–plastic constitutiveequation is then updated and it replaces the elastic constitutivein the elastic analysis. The other procedures are identical tothose of the elastic analysis, even when the P-delta effect istaken into account. Therefore, in a brief presentation, we onlypresent the construction of the elastic–plastic matrix.

Let 1eC and 1eR be the vectors of relative displacementincrements at the yielded sections (plastic hinges), and at theelastic sections. Elastic constitutive equation (the Hooke’s low)for the structure may be written as follows:[

1sR1sC

]=

[DRR DRC

DTRC DCC

] [1eR − 0

1eC − 1epC

], (B.1)

in which, 1epC are the plastic strain increments at the plastic

hinges, 1sR and 1sC are the vectors of internal forceincrements at the elastic sections and the plastic hinges.

Based on the Drucker’s normality rule, plastic strainincrements are normal to the yield surface and orthogonal to

H. Van Long, N. Dang Hung / Engineering Structures 30 (2008) 1895–1904 1903

(a) Relative displacements at critical sections. (b) Member’s independent displacements (globalaxis).

Fig. A.1. Member k.

the internal force increments. Thus, for the sections (elementends) in the plastic state, we have

1epTC 1sC = 0 and, (B.2a)

1epC = NCλ, (B.2b)

(λ and NC are defined in Eq. (1)).From Eqs. (B.2a) and (B.2b) and noting that λ is arbitrary,

we obtain

NTC1sC = 0. (B.3)

Using (B.1) and (B.2b) and (B.3), the plastic deformationmagnitude may be deduced

λ =[(NT

C DCCNC )−1NTC DT

RC (NTC DCCNC )−1NT

C DCC][

1eR1eC

],

or:[0λ

]=

[0 0

R1 R2

] [1eR1eC

]. (B.4)

Eq. (B.2b) may be rewritten in the following form:[0

1epC

]=

[0 00 NC

] [0λ

]. (B.5)

Substituting (B.4) in (B.5), one obtains[0

1epC

]=

[0 0

NC R1 NC R2

] [1eR1eC

]. (B.6)

From (B.1) and (B.6), one finally obtains the elastic–plasticconstitutive relation:[1sR1sC

]=

[DRR − DRCNC R1 DRC − DRCNC R2

DTRC − DCCNC R1 DCC − DCCNC R2

] [1eR1eC

].

Appendix C. Determination of the value of d0

Suppose that d is the real displacement field (the realmechanism), in which, dmax is the largest (absolute value). Incase of limit analysis, the safety factor is determined by theequilibrium between the internal power and the external power.

µ+ = sT0 λ/fTd = φ/ξ (C.1)

where the symbols are defined in Eq. (9).

Based on the upper bound theorem of limit analysis, we have

µ+ ≤ µ∗, (C.2)

with µ∗ is a load factor of any licit mechanism. By giving anylicit displacement field d∗ (for example, only one componentequals unity, and all other components are nil), µ∗ may beeasily obtained.

From the Eqs. (C.1) and (C.2), one has

φ/ξ ≤ µ∗. (C.3)

On the point of view of geometry (kinematic), with the realmechanism, d, there is at least a plastic deformation component,e, such that:

e ≥ dmax/Hmax,

with Hmax is the maximum dimension of the structure.Therefore, a lower bound of the internal power may be

evaluated.

φ ≥ sp mindmax/Hmax, (C.4)

in which, sp min is the smallest among the plastic capacity (NP ,Mpy , Mpz) of all the sections on the structure.

From the Eqs. (C.3) and (C.4), the maximum displacementis constrained by an upper bound.

dmax ≤ ξµ∗ Hmax/sp min.

Then, any value of d0 that satisfies

d0 ≥ ξµ∗ Hmax/sp min ≥ dmax,

will lead: d′= d + d0 is always nonnegative.

With similar argument, the value of d0 for the shakedownanalysis may be obtained.

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