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Engineering Structures 28 (2006) 593–608

www.elsevier.com/locate/engstruct

Inelastic flexural strength of aluminium alloys structures

M. Manganielloa, G. De Matteis b,∗, R. Landolfoa

a Department of Constructions and Mathematical Methods in Architecture, University of Naples “Federico II”, Via Monteoliveto 3, I-80134 Napoli, Italyb University of Chieti-Pescara “G. D’Annunzio”, Faculty of Architecture (PRICOS), Viale Pindaro 42, I-65127 Pescara, Italy

Received 20 January 2005; received in revised form 15 September 2005; accepted 16 September 2005

Available online 7 November 2005

Abstract

In this paper, the results of an extensive numerical study devoted to the evaluation of the inelastic flexural behaviour of aluminium alloy

structures are provided. The main aim of the research is to determine the required ductility for applying simplified methods of plastic analysis

(i.e. plastic hinge method) to structural systems made of materials characterised by a continuous hardening and with limited deformation capacity.

Therefore, the cross-section rotational capacity necessary to attain predefined levels of load bearing capacity is evaluated for different structural

schemes and then compared to the available rotational capacity corresponding to fixed thresholds of ultimate cross-section curvature. The influence

of both geometrical (cross-section shape factor and structural scheme) and mechanical (material hardening and ultimate deformation capacity)

parameters is taken into account. The parametric analysis is performed by using a numerical model implemented in the implicit non-linear FE

code ABAQUS/Standard and calibrated on available experimental tests. On the basis of the above analysis, the limit values for the rotational

capacity of a cross-section in bending necessary to guarantee adequate inelastic redistribution of internal forces for continuous beams and framed

structures are given. Finally, new indications for the application of the modified plastic hinge method included in Eurocode 9 are provided.c 2005 Elsevier Ltd. All rights reserved.

Keywords: Aluminium alloys; Ductility; Inelastic behaviour; Material hardening; Plastic hinge method; Rotational capacity

1. Introduction

Although the first building structures made of aluminium

alloys appeared in Europe in the early Fifties of the past

century, their use in the field of structural engineering is

still very limited [1]. Nevertheless, it has to be recognised

that, thanks to high strength, lightness, corrosion resistance,

formability and recycling process, the use of aluminium alloys

in some structural applications where other metal materials

are not competitive has shown a continuous and consistent

growth. Since for many years aluminium alloys have beennearly exclusively used in aeronautical and marine applications,

where the necessity to avoid failure modes induced by fatigue

led to considering only the elastic behaviour of the material,

the possibility to exploit their inelastic strength has been

constantly ignored for a long time. Nowadays, the optimisation

∗ Corresponding address: Department of Structural Analysis and Design,University of Naples “Federico II”, P. le Tecchio 80, I-80125 Napoli, Italy. Tel.:+39 081 768 2444; fax: +39 081 593 4792.

E-mail address: [email protected] (G. De Matteis).

of structural design and also the increasing use of aluminium

alloys in the field of civil engineering leads us to go deeply

inside the research activity concerning the possibility to fully

exploit the material inelastic capacity [2].

The post-elastic response of aluminium alloy structures

is significantly different from steel. This is due to the

material behaviour, which is characterised by a continuous

and remarkable strain hardening (round-house type material)

and also by a limited ductility. For these reasons, the limit

analysis methods commonly used for steel, namely the plastic

hinge method, which are tightly based on the hypothesis of perfect plasticity and unlimited ductility of the material [3],

are not applicable for aluminium alloys as well. In fact, the

above assumptions ensure that the ultimate condition of a

structure is attained when finite deformations of at least one

part can occur without any change of the external loads and,

therefore, the bending moment distribution or the applied load

multiplier remain constant as the system deforms [4]. On the

contrary, in the case of hardening materials the load multiplier

is always increasing with respect to any displacement parameter

and the internal moment distribution depends upon kinematic

0141-0296/$ - see front matter c 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2005.09.014

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594 M. Manganiello et al. / Engineering Structures 28 (2006) 593–608

Nomenclature

M bending moment;χ cross-section curvature;

n Ramberg–Osgood coefficient;

f 0.2 conventional yield stress (0.2% offset deforma-

tion); f U maximum stress on the σ –ε curve;εU residual deformation corresponding to f U ;

M PL full plastic bending moment;

M PL modified full plastic bending moment;W PL plastic modulus about the neutral axis;

W EL section modulus about the neutral axis;

α cross-section shape factor;η numerical multiplier of the conventional elastic

stress limit f 0.2;

v0.2 elastic displacement in the middle cross-section;vU ultimate displacement in the middle cross-

section;F 0.2 applied limit elastic load;

F U applied ultimate load;

F EPP ultimate load evaluated by means of the plastic

hinge method;χ5 ultimate curvature limit for brittle alloys provided

by EC 9 (= 5χ0.2);

χ10 ultimate curvature limit for ductile alloys

provided by EC 9 (= 10χ0.2);

χ x generic ultimate curvature limit ( x times the

elastic curvature limit χ0.2); Lr length ratio;

M max maximum bending moment on the beam inter-

ested by the plastic mechanism; M min minimum bending moment on the beam inter-

ested by the plastic mechanism;θ OO cross-section rotation on the inflection point;

β rotational capacity;βrequired required rotational capacity;

βavailable available rotational capacity;

β(EPP) rotational capacity required to attain the load

F EPP;

A1, A 2, B1, B2 numerical constants related to the evalua-

tion of rotational capacity;

a, b, c numerical constants related to the evaluation of

the η factor provided by EC 9; A, B, C , D numerical constants related to the evaluation

of proposed η factor;SB1 standard beam subjected to a middle concentrated

load;

SB2 standard beam subjected to either a uniform load

or two concentrated forces;

β(χ5) rotational capacity measured when the curvature

limit χ5 is reached;

β(χ10) rotational capacity measured when the curvature

limit χ10 is reached;

β(χ x ) rotational capacity measured when the curvature

limit χ x is reached;

n∗ values of the n parameter for which the required

and available ductility are coincident;

M 0 lower plastic moment in a plastic hinge with

hardening behaviour;

θ relative rotation at a plastic hinge;

k hardening factor.

conditions of the structural system (Fig. 1). Moreover, the

concept of concentrated plasticity, commonly adopted in the

case of perfectly plastic material schematisation, is not suitable

for round-house type materials, because the extension of the

plastic zone strongly depends on the material hardening level.

For all these reasons, the application of the conventional plastic

hinge method to aluminium structures does not provide the load

corresponding to an effective collapse mechanism [5].

On the other hand, the application of rigorous methods

of analysis for structures loaded beyond the elastic range,

which are based on incremental procedures applied on discrete

structural models, is burdensome and not compatible with

practical applications. Besides, the application of the plastic

hinge method, which is based on the assumption of a

concentrated plastic zone, is very useful also because it allows

the evaluation of a ultimate load independently of the actual

material features. Therefore, the computational advantages

related to a methodology of analysis based on the simplified

hypotheses of concentrated plasticity and perfectly plastic

force–displacement behaviour of plastic zones induce us to

extend this approach, with appropriate modifications, also to

materials which are neither sharp-knee type nor characterised

by unlimited ductility resources.

2. Previous works

In the first half of the Sixties, the basic virtual work solution

for defining the plastic collapse load was extended to cover the

effect of the material strain hardening. A simple approach to

solve the problem related to the effect of strain hardening in

the elasto-plastic solution of structures was proposed by Sawko

[6,7]. Several tests demonstrated that a good agreement

between experimental and predicted behaviour could be

obtained if the bending moment at the plastic hinge was

assumed to change in magnitude with the relevant rotation.

A simplified linear relationship for the plastic moment versus

hinge rotation was considered ( M PL = M 0 + k · θ), accountingfor the lower plastic moment M 0, the strain hardening factor

(k ) and the rotation at the plastic hinge (θ ). An elasto-plastic

analysis programme for grillages was set up. Although a time

consuming computation was required, a notable improvement

in the numerical evaluation of structural deflections was

obtained [6]. It was also emphasized that because the collapse

condition generally does not occur by pure rotation of the

hinges, a collapse criterion based on the maximum reached

deflection could be more reliable than the one related to

the “collapse load” evaluation [7]. Therefore, the collapse

condition was defined assuming a limit displacement equal to

the one related to the design load multiplied by a numerical



M. Manganiello et al. / Engineering Structures 28 (2006) 593–608 595

Fig. 1. Comparison between round-house materials and steel.

factor equal to 4 [8]. However, the application of this approach

was based on the assumption of the deflection correspondingto the carrying capacity of the structure. Obviously, a larger

ultimate deflection would lead to a greater collapse load. The

same is not true for the simple plastic theory, where the collapse

load is independent from the displacement. The possibility to

remove the concentrated plasticity hypothesis was not taken

into account.

First studies concerning the inelastic behaviour of alu-

minium alloy structures were carried out at the end of the

Seventies [9,10]. Simplified hypotheses for characterising the

actual non-linear structural response of the system, both re-

lated to the material behaviour and geometrical configuration,

were done in order to set up a numerical model by means of

carrying out a parametric investigation. The comparison with

the load–displacement behaviour obtained by means of the

common plastic hinge method implemented using the plastic

moment value M PL = f y · W PL ( f y ≡ f 0.2), led to the follow-

ing preliminary conclusions [11]: (a) the plastic hinge method

implemented using M PL = f 0.2 · W PL could be too conserva-

tive for strong hardening alloys and could be unsafe in the case

of weak hardening alloys; (b) for cross-sections characterised

by high values of cross-section shape factor (α = W PL / W EL),

larger plastic deformation capacity is required to allow the ex-

ploitation of the full strength; (c) the need for redistributing

bending moments beyond the elastic limit basically depends

upon the structural scheme.For the sake of clarity, in Fig. 2, a comparison between the

behavioural curves of a typical aluminium alloy continuous

beam corresponding to different inelastic methods of analysis

(namely, plastic hinge method implemented using M PL = f 0.2 ·

W PL and FE analysis based on a Ramberg–Osgood material

model) is shown. In such a diagram, load ( F ) and displacement

(v) are normalised with respect to the values (v0.2 and F 0.2,

respectively) corresponding to the attainment of the material

conventional elastic limit f 0.2. The two round-off curves

emphasize the behaviour of two aluminium alloys characterised

by different material hardening degrees. In particular, n = 10

is representative of a strong hardening alloy, while n = 25

corresponds to a weak hardening alloy. The piecewise curve

represents the load–displacement behaviour obtained by meansof the plastic hinge method and the F U value corresponds to the

load level defined when the ultimate curvature (χU ) is reached

in the most stressed cross-section.

In the above studies, the ductility demand was evaluated in

a conventional way, by considering the deflection for which

the plastic hinge methods and the more accurate elasto-plastic

incremental approach provide the same load bearing capacity

(see points A and B in Fig. 2). According to this assumption,

it was evident that the ductility demand is higher for weak

hardening alloys than for strong hardening alloys.

The remarkable advantages related to the use of simplified

material models induced us to investigate the possibilityto adopt elastic–perfectly plastic schematisation even for

hardening materials. To this purpose, the aforementioned

analyses were interpreted [12] and extended [13] with the

main scope to permit a reliable application of the plastic

hinge method to structures made of strain hardening materials

with limited ductility. In particular, a simplified analysis

methodology, which is presently adopted by Eurocode 9 [14],

was proposed [12]. It was based on the assumption of an

equivalent yield stress ( f y ) defined as the nominal conventional

elastic limit stress ( f 0.2) corrected by a numerical factor

η accounting for the material strain hardening and ductility

( f y = η · f 0.2). Therefore, the proposed method basicallyconsisted in the implementation of the plastic hinge method

by using as plastic moment the modified value M PL = η ·

f 0.2 · W PL. Once defined on the round-off curve the actual

ultimate condition (F U –vU ) corresponding to the attainment

of a fixed curvature limit χU (point C 1 for a strong hardening

alloy), the factor η was obtained by dividing the load F U

for the one corresponding to the same displacement (vU /v0.2)

but evaluated on the piecewise curve (point C 2). Therefore,

for a fixed structural configuration, the factor η was given

as a function of geometrical (cross-section shape factor) and

mechanical (material strain hardening and ultimate curvature

of the cross-section) parameters.




Fig. 2. Basic assumptions.

3. Aim of the study

The European design codes for metal structures, namely

Eurocodes 3 and 9, give the possibility to implement the

plastic global analysis whenever the structural members are

characterised by a large rotation capacity at the actual location

of the plastic hinges. Amongst different types of inelastic

analysis, the plastic hinge method unduly represents the most

simplified approach. However, it can be applied provided that

the structural ductility is sufficient to enable the development

of the relevant plastic mechanisms. Therefore, it should be

based upon a preventive evaluation of both the rotation capacity

actually demanded of plastic hinges in order to develop the

relevant collapse mechanism and then on the comparison with

the actual available cross-section rotational capacity so as

to check that such a rotation demand is actually compatible

with utilised members. According to the Eurocodes, this is

implicitly assumed when member cross-sections are ductile,

i.e. belonging to class 1, even if class limits for cross-

section classification are simply given as a function of the

maximum slenderness parameter of the different constituting

plate elements.

The initial aim of the study is to determine the entity of the

required rotational capacity for commonly adopted structural

schemes (continuous beams and simple portal frames) and then

to check whether the value implicitly assumed in Eurocode 9 to

set up the cross-section classification criterion for aluminiumalloy structures [15] is correct or not. Therefore, the cross-

section rotational capacity demand to attain a pre-defined level

of load bearing capacity of the examined structure is evaluated

and then compared with the one corresponding to some fixed

thresholds of the ultimate cross-sectional curvature, defining

the available rotational capacity of the cross-section based

on material ductility. To this purpose, the actual response

of examined structures has been evaluated by means of

material models with diffuse plasticity. In order to calibrate

the adopted FE model, a preliminary comparison with available

experimental results has been performed. Then, the influence

of different parameters, namely shape factor, hardening degree

and elastic bending moment distribution, on the inelastic

behaviour of simple structural schemes is investigated.A second part of the paper is devoted to investigating the

correctness of the conventional plastic hinge method when

applied to aluminium alloy structures. Therefore, such a

method is calibrated by removing some simplified hypotheses

(i.e. constant length for the plastic hinge, fixed independently

from the material hardening degree and assumed equal to the

half beam height (h/2), when the plastic hinge forms close

to the clamped end, and equal to h in the other cases) on

which previous studies were based. A comparison with current

provisions of Eurocode 9 related to the evaluation of the load

bearing capacity of continuous beams is also provided and

discussed.

4. The numerical model

4.1. Available experimental data

The finite element model used in the current study has been

set up on the basis of existing experimental tests [16] related

to simple supported and continuous beams having different

geometries (number and length of the spans), different profile

cross-sections and different base materials (strength, ductility

and hardening degree) — see Tables 1 and 2. The non-linear

response of the examined structures is due to the material

behaviour only, since for the adopted cross-sections a local

buckling phenomenon occurs for very large displacements.

Also, lateral–torsional buckling is inhibited. In particular, it

is worth noticing that the effect of strain hardening is taken

into account considering two aluminium alloys, namely AA

6082, which represents a heat-treated alloy with limited strain

hardening, and AA 5083, corresponding to a non-heat-treated

alloy and having a significant strain hardening.

4.2. The adopted FEM model

The adopted finite element model has been implemented

into the non-linear finite element code ABAQUS/Standard [17].

Since local instability phenomena have not been considered,




Table 1

Mechanical parameters for the different cross-sections [15]

Beam f 0.2 (N mm−2) f U (N mm−2) εU E (N mm−2) n α I × 10−6 (mm4)

H5 165 295 15.4 69 000 7.4 1.56 2.89

H6 278 309 8.5 69 000 35.0 1.56 2.89

I5 165 295 15.4 69 000 7.4 1.16 8.18

I6 278 309 8.5 69 000 35.0 1.16 8.18

R6 302 317 8.3 70 000 75.7 1.28 5.40

Table 2

Geometrical lengths of beam segments [15]

Beam L AB = L DE (mm) LBC (m) LCD (m) Profile Alloy v0.2 (mm) F 0.2 (kN)

H5 0 1200 1200H

5083 - O 22.05 15.87

H6 0 1200 1200 6082 - T6 37.16 26.74

I5 0 1200 1200I

5083 - O 22.05 45.01I6 0 1200 1200

6082 - T6 37.16 75.84

R6 0 1200 1200 47.75 65.23

IL51 1200 1200 1200

I

5083 - O 13.80 61.70

IL53 1200 1600 800 12.40 72.90

IL61 1200 1200 1200

6082 - T6

23.20 103.90

IL63 1200 1600 800 21.00 122.90

RL61 1200 1200 1200

29.80 89.30

RL63 1200 1600 800 26.90 105.70

Fig. 3. The finite element model.

Euler–Bernoulli beam elements have been used to model the

structural system. Lateral displacements and twisting out of the

vertical plane have been prevented assuming in-plane geometry

without longitudinal and cross-sectional imperfections. Toobtain the output data in an adequate number of cross-sections

(continuous curvature diagram), the mesh density, which is

constant along each span, increases for the inner span, where

the highest stress gradient is localised. A preliminary study

on mesh refinement has been performed and the mesh density

depicted in Fig. 3 has been finally adopted. The external load

has been applied by imposing the vertical displacement to

the loaded section. For solving the non-linear problem Riks’s

method has been employed.

As far as the mechanical behaviour of the material is

concerned, the Ramberg–Osgood model defined according to

Eq. (1) has been used:

ε =σ

E + ε0

σ

f ε0

n

(1)

where E is the Young’s modulus at the origin, f ε0

is the

conventional limit of elasticity, ε0 is the residual deformation

(usually assumed equal to 0.2%) corresponding to f ε0 and n is

a shape coefficient representing the material hardening degree.

In particular, the value of the n exponent has been determined

according to the following relationship, which is valid in the

range of large deformations [18]:

n = ln εU

0.002

ln

f U

f 0.2

(2)

where εU is the residual deformation corresponding to ultimate

maximum stress f U , which is assumed as the peak stress of the

material behaviour.




Fig. 4. Comparison between experimental and numerical results (simple supported beams).

Fig. 5. Comparison between experimental and numerical results (continuous beams).

4.3. Comparison between numerical and experimental results

For all tested beams, the comparison between FE analyses

and experimental test results shows a very good agreement in

terms of dimensionless load–displacement behaviour (Figs. 4

and 5). However, sometimes the predicted load-bearing

capacity deviates to some extent from the experimental one.

Nevertheless, the scatter, which is more significant for non-

symmetrical loading conditions, is always less than 8%. Also,

it can be observed that the differences are more remarkable for

high deformation levels, which usually are outside the range

of practical applications. Therefore, it can be concluded that




the proposed numerical model is able to correctly simulate

the behaviour of tested beams accounting for the inelastic

behaviour of material and the spreading of plasticity throughout

the beam length.

5. The parametric study

5.1. Main definitions

In order to characterise the inelastic response of a complex

structural system, the latter can be considered as an assemblage

of standard simple supported beams on which both the required

and the available rotational capacity should be evaluated [19].

There are two different types of standard beams (Fig. 6(a)):

the first one (SB1), which is subjected to concentrated load

in the mid-span, represents a beam under moment gradient;

the second one (SB2), which is loaded by either a uniform

load or by two concentrated forces, reproduces the constant

moment condition. The inflection points, therefore, allow thedefinition of the standard beams and, hence, the length on

which to evaluate the rotational capacity. They allow solving

an indeterminate system by evaluating the behaviour of simple

systems (i.e. simple supported beams), which are commonly

investigated by means of numerical and experimental tests.

In this paper, symmetrical and non-symmetrical continuous

beams subjected to a concentrated load are investigated

(Fig. 6(b)). For the given structural configuration subjected

to a fixed load distribution it is possible to locate the

inflection points based on the bending moment diagram.

Actually, the length of standard beams should be evaluated

according to the relevant collapse mechanism. Hence, in

order to face the application of the procedure, the ultimate

moment distribution is needed. To this purpose, reference to

an equivalent elastic–perfectly plastic material can be made

and the plastic hinge location can be easily defined. Therefore,

the inflection points may be fixed conventionally assuming a

bending moment diagram whose values in all critical sections

are posed equal to the conventional plastic moment ( M PL =

f 0.2 ·W PL). This simplification has only marginal effects on the

actual determination of the inelastic rotations.

Once the inflection points (O and O) have been defined, the

area subtended to the curvature diagram within such points can

be evaluated. It represents the sum of the absolute rotations

of the supports of the corresponding standard beams. Then,the rotation capacity for such a standard beam can be defined

according to Eq. (3) (see Fig. 6(b)):

β =(θ OO + θ OO)U

(θ OO + θ OO)0.2− 1. (3)

Depending on the assumed ultimate condition, which in

Eq. (3) is defined by the subscript U , the parameter β may

assume different meanings. In fact, it represents the required

rotational capacity (βrequired) to be demanded of a cross-section

if the rotation (θ OO + θ OO)U is measured when a pre-defined

ultimate load level is reached, while it corresponds to the

available rotational capacity (βavailable) if it is evaluated when

(a) Definition of standard beams.

(b) Analysed cases.

Fig. 6. Investigated structural scheme.

a load level corresponding to the attainment of a conventional

ultimate curvature level of applied cross-section is attained.

Since it is always possible to calculate an ultimate load

level by means of the plastic hinge method applied by using

a conventional moment capacity M PL, in the following, the re-

quired rotational capacity necessary to attain such a load level

measured on the actual load–displacement curve obtained by a




Fig. 7. Definition of the rotational capacity (required and available).

step-by-step analysis will be explicitly indicated by β (EPP).

It represents a reference value. On the contrary, the ductility

demand evaluated for different load levels will be indicated by

βrequired. It has a more general meaning, representing the mini-

mum required rotational capacity of a cross-section to allow the

structural scheme to actually attain that specific load level.

On the other hand, the cross-section available ductility

(βavailable) is defined as a function of cross-section curvature

χ and is indicated by the symbol β(χ x ). It is defined when

on the actual load–displacement curve obtained by a step-by-step analysis the cross-sectional curvature limit (χ x ) is attained,

where the suffix x represents a multiplier of the conventional

elastic limit (χ x = x · χ0.2). In this paper, attention will be

focused on the values of χ x corresponding to the conventional

ultimate curvature limits provided by EC9, namely χ5 and χ10

for brittle and ductile alloys, respectively.

As it appears from the previous considerations, the approach

employed to measure the ductility of the structural scheme

is different from the one adopted in previous studies [12,

13], which defines the material deformation on the basis of

the rotation of a plastic hinge having a preliminarily fixed

dimension. This hypothesis appears to be too strong for strain

hardening materials. On the contrary, the method used in

this study takes into account the actual curvature distribution

throughout the beam length, which depends on the hardening

degree of the material, leaving the concentrated plastic hinge

hypothesis, therefore providing indications on the actual cross-

section local deformation.

5.2. The analysed cases

The continuous beam is a structural scheme for which

Eurocode 9 allows the implementation of the plastic hinge

method. When instability phenomena are excluded, the

inelastic behaviour of this scheme is influenced by the shape

Table 3

Range of values for the parametric analysis

Length

ratio

Shape

factor

Conventional ultimate

curvature

Hardening degree

Lr α χU /χ0.2 n

0.5 1.10 5 5

1.0 1.15 10 10

2.0 1.20 15 15

1.30 20 20

30 30

3540

factor of the cross-section, the structural configuration and

the material properties. All these parameters are assumed

as variable in the performed numerical analysis defined

according to Table 3. In order to consider the effect of the

reduced deformation capacity of the material, several levels of

ultimate cross-section curvature (χ5, χ10, χ15, χ20, χ30) have

been considered.

Four different cross-sections covering the whole range of

possible shape factors (α) are investigated. For each value of

the α factor and for a fixed value of the conventional yieldstress f 0.2, five different hardening degrees (n) are considered.

Moreover, in order to provide different length ratios ( Lr ) and

hence different levels of ductility requirements, three values of

the outside span length are assumed. In particular, the length

ratio L r is defined as the ratio between the length of the external

span and the length of the internal span, which is of interest in

the collapse mechanism, and is therefore representative of the

ratio between maximum and minimum bending moment in the

member where the plastic mechanism will form.

For each structural configuration (base material, cross-

section and length ratio), two types of inelastic analyses

have been performed (Fig. 7): (a) incremental procedure by




FEM analysis and (b) plastic hinge analysis. The obtained

load–displacement curves are normalised to the corresponding

elastic values v0.2 and F 0.2, which are reached as soon as the

f 0.2 stress value is attained in the highly stressed fibre of the

cross-section. The comparison between the actual behaviour

(incremental curve) and the conventional one, defined by

the plastic hinge method (piecewise curve), allows the direct

definition of the simplified stress factor η to be adopted when

extending the conventional plastic hinge method to hardening

materials.

5.3. The obtained results

5.3.1. Rotational capacity evaluation

In Fig. 8, the obtained numerical results are shown. In

particular, for different geometrical configurations ( Lr and α )

and material hardening (n), the available rotational capacity

defined for different levels of cross-section ultimate curvature

is compared with the one required for the attainment of the

ultimate load level F EPP determined by the plastic hinge methodβ (EPP). Both symmetric (3 span beams) and non-symmetric

(2 span beams) load conditions are considered [20]. For a fixed

value of Lr and α (Fig. 8(a)), the obtained values of rotation

capacity can be well-fitted by the following equations:

βavailable = A1 · n− B1 (4a)

βrequired = A2 · n B2 (4b)

in which Ai and Bi are positive constants depending on

the length ratio of the structural scheme and the shape

factor of the cross-section. Fig. 8(a) shows that the required

rotational capacity β (EPP) is significantly greater in the case

of symmetric beams, while the available rotational capacityappears to be only slightly influenced by the applied loading

condition. Besides, it is apparent that the required rotational

capacity of three span beams ranges between 0.7 and 4.5, while

for two span beams between 0.6 and 3. This outcome confirms

that for cross-section classification purposes, the assumption of

required rotation capacity equal to 3, as implicitly assumed for

defining the class 1 cross-section limit [15], can be considered

appropriate also in the case of aluminium alloys, independently

from the material hardening.

Also, it can be observed that for each value of limit curvature

χ x , the variation of available rotation capacity βavailable due

to parameter Lr is limited (Fig. 8(b)). Therefore, for the

sake of simplicity, the mean line could be considered as therepresentative curve for all limit curvature values. Obviously,

this gives rise to some deviations from the actual values (see

Fig. 8(c)), but such a scatter, which is depending on the material

hardening, is generally less than 20% and it is significant only

for large values of the hardening parameter n.

It is important to observe that the available rotation

capacity strongly increases for higher hardening levels. This

can be easily explained considering the curvature distribution

throughout the beam length. In Fig. 9 it is shown that higher

material hardening causes the spreading of the plasticity over

a larger portion of the beam. As a consequence, for a given

value of the maximum curvature of cross-section, the rotations

at the supports of the relevant standard beams increase with the

material hardening, since the area subtended to the curvature

diagram amplifies (Fig. 9(b)). Because the elastic rotations are

not independent of the material hardening, the ratio between

ultimate and elastic rotation increases for those alloys which

are characterised by higher strain hardening levels, giving rise

to higher values of the available rotational capacity.

On the contrary, the required rotational capacity reduces

significantly for increasing material hardening, because a more

spread plastic curvature distribution requires less beam support

rotation to reach a given load level.

In Fig. 10, the comparison between the required and

available rotational capacity is schematically depicted. Since

the corresponding curves present an intersection point for a

given value of the hardening parameter (let us say n∗), it can

be concluded that for n < n∗ the ductility demand is smaller

than the cross-section capacity. In other words, in the range

n < n∗, the application of the standard plastic hinge method up

to the complete development of the plastic mechanism provides

a strength level of the structure compatible with the availableductility of the material and therefore is conservative. On the

contrary, for n > n∗ the application of the plastic hinge

method would lead to unsafe results. In such a case, to extend

its applicability also to hardening materials characterized by

limited ductility, the material conventional elastic strength

f 0.2 should be properly reduced by a factor η lower than

unity.

5.3.2. Strength evaluation

The comparison between required and available rotational

capacity (Fig. 10) allows also the definition of the factor η,

which is used either to amplify or to reduce the nominalconventional limit stress f 0.2 to be adopted when applying an

equivalent plastic hinge method, giving rise to an ultimate load

level for which cross-section ductility demand corresponds to

the available cross-section rotational capacity. In particular, for

a fixed structural scheme ( Lr ) and cross-section shape factor

(α), the factor η can be assessed by equating the ultimate load

corresponding to the attainment of the cross-section ultimate

curvature F (χU ) to the ultimate load level F ( M PL) obtained

by means of a modified plastic hinge method implemented

according to a modified value of yield stress η f 0.2. I t i s

important to remark that this is a different definition of the

factor η with respect to previous studies, where it was obtained

by dividing the load F U for the one corresponding to the

same displacement (vU /v0.2) but evaluated on the piecewise

curve [12].

In Fig. 11, for fixed values of L r and α, the obtained values

for the η factor (circle points) are provided as a function of

χU and n. The values of η are also fitted by the following

relationships:

η = A · e Bn + C · e Dn with A, B, C , D = f (χU , α, L r )

(5)

in which A, B, C , D are four constants depending, for a fixed

structural scheme ( Lr and α), on the available material ductility




Fig. 8. The obtained results for continuous (symmetric and non-symmetric) beams.

(χU ). In Table 4, the obtained values for such constants are

specified for some combinations of the parameters Lr and α,

namely ( Lr , α)min, ( Lr , α)mid, ( Lr , α)max, which, for specific

values of the ultimate curvature (χ5 and χ10), define the

maximum, middle and minimum values of the coefficient η,

respectively (see Fig. 10).




Fig. 9. Curvature distribution for different material hardening levels.

Fig. 10. Comparison between required and available rotational capacity.

Table 4

Numerical values of A , B, C , D coefficients

A B C D

( Lr , α)min

χ5 1.0260 −0.2201 1.0860 −0.003427

χ10 1.7090 −

0.2370 1.1730 −

0.003014( Lr , α)mid

χ5 0.9602 −0.2356 0.9859 −0.004564

χ10 1.5720 −0.2615 1.1500 −0.006382

( Lr , α)max

χ5 0.8830 −0.2369 0.8486 −0.004629

χ10 1.2750 −0.2043 0.9317 −0.004741

In Fig. 12, with reference to the curvature limit χ5, the

influence of the shape factor α is proposed. It is evident that

the influence of factor α is practically negligible for weak

hardening alloys and is less than 10% for reduced values of

n, resulting in its secondary importance with respect to length

ratio L r (see also Fig. 13).

5.3.3. Comparison with Eurocode 9

For evaluating the η factor, EC9 provides the following

relationship (see Table 5):

η = 1/(a + b · nc) where a, b, c = f (χU ,α). (6)

Two ranges of variation for the parameter α are defined:

1.1 ÷ 1.2 and 1.4 ÷ 1.5, which in the following are labelled

as EC9[α=1.1÷1.2] and EC9[α=1.4÷1.5], respectively. Therefore,

the influence of the structural configuration, herein defined by

means of the parameter L r , is not taken into account.

In Fig. 13, a comparison between the curves of η factor

provided by EC9 and the values obtained by the above

parametric study for a fixed value of α is shown. Although

EC9 provides values that on average are comparable with the

numerical ones, a non-negligible scatter is evident for different

values of the Lr parameter. Therefore it is apparent that,




Fig. 11. Evaluation of η factor.

contrarily to EC9 assumptions, the factor η depends on the

adopted structural scheme also, rather than on the mechanical

properties of the alloy and the geometric characteristics of

the cross-section only. In order to avoid the unsafe behaviour

provided by the EC9 formulation and also to simplify the

application of the proposed method, this approach could be

applied by assuming the values of the η factor based on

the parameter combination ( Lr , α)mid, which provides average

results.

In Fig. 14, for the sake of example, a comparison among

different methods of analysis is provided for a given structural

scheme (three span beam) and two aluminium alloys having




(a) Three span beams. (b) Two span beams.

Fig. 12. Influence of shape factor on η(χ5) for three and two span beams — L r = 1.0.

(a) Three span beams. (b) Two span beams.

Fig. 13. Comparison between the obtained results (α = 1.15) and the EC 9 provisions.

Fig. 14. Application of the procedure: comparison among different methods.




(a) Available (χ5) rotational capacity. (b) Available (χ10) rotational capacity.

(c) Required rotational capacity.

Fig. 16. Obtained results for portal frames.

beam schemes (Fig. 16(a) and (b)), while required rotational

capacity (Fig. 16(c)) is remarkably smaller (see Fig. 8(a)).

In particular, Fig. 16 highlights as the irregularities in the

structural configuration, also in relation to higher values of

the moment ratio ( M max/ M min), produce an increase of the

required plastic deformations. On the other hand, the Fig. 17

highlights as the values obtained for analysed portal framesare included in the range of variability defined in the case of

continuous beams and that also in this case Eurocode 9 may

provide unsafe results.

6. Conclusions

In this paper the inelastic flexural strength of aluminium

alloy structures has been investigated by using a numerical

approach. On the basis of a comparison between required and

available ductility, which have been assessed by considering

cross-section rotational capacity evaluated on simple supported

beams duly taken out from the considered structural scheme,

the possibility to apply the plastic hinge method to structures

whose constitutive material is of round-house type has

been evaluated. In particular, continuous beams subjected

to symmetric and non-symmetric load conditions and one-

storey portal frames have been considered. Some important

conclusions have been drawn.

In particular, as far as the rotational capacity is concerned,the obtained results confirm the correctness of the assumptions

made to define the cross-section slenderness limits of the

ductile cross-sections according to EC9 when structural plastic

analysis is considered.

Then, on the basis of a comparison between available

ductility and cross-section plastic deformation demand, the

numerical factor η, which is used as a modifying factor for

the conventional yield stress limit in order to apply the plastic

hinge method according to Eurocode 9 provisions, has been

revised. The proposed formulation provides a range of variation

of the η factor larger than the one presently prescribed by

EC9, which in some cases appears to be not conservative. On




Fig. 17. η factor for continuous beams and portal frames.

the other hand, it is worth noticing that the approach already

proposed in the ENV version of EC9 has been implementedalso in the EN version of EC9 due to its simplicity, which

corresponds to disregarding the effect of some parameters,

whose influence has been emphasised by the results of this

paper. Therefore, the method here proposed should be intended

as an alternative approach, more accurate and conservative but

also more complicated.

Acknowledgements

This research has been developed within the activity of

CEN/TC250-SC9 Committee devoted to the preparation of the

EN version of Eurocode 9 “Design of aluminium structures”.The authors gratefully acknowledge the helpful contribution

provided by Prof. Federico M. Mazzolani, who chaired this

Committee, supporting and encouraging the present research

activity.

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