inelastic flexural strength of aluminium alloys structures.pdf
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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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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
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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.
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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,
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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.
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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
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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
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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
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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
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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).
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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,
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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
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M. Manganiello et al. / Engineering Structures 28 (2006) 593–608 605
(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.
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(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
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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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