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ThinWalk,d Structures Vol. 27, No. I. pp. 103 116. 1997 Copyright ~' 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 02638231/97 $15.00
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I 46
Prediction of the Rotation Capacity of Aluminium Alloy Beams
F, M. Mazzolani" & V. Piluso ~' "Department of Structural Analysis and Design, University of Naples, Italy bDepartment of Civil Engineering, University of Salerno, Italy
ABSTRACT In this paper the theoretical steps of a semiempirical method.[or eva/uating the rotation capacity of aluminium alloy members subjected to nonun([brm bending are outlined. The approach is represented by the c.vtension to aluminium alloy members of the semiempirical methods proposed .[br evaluating the rotation capacity o[" steel members. The moment curvature relationship of aluminium alloy members can be conveniently represented by means o]'a RambergOsgood type relation. This allows. with reference to the classical threepoint bending test, the deriwttion in closed form of the curvature diagram. Furthermore, a closedff'orm integration of the curvature diagram can be per[ormed, providing a relation jor evaluating the rotation corresponding to the occurrence ~[' local buckling. The rotation capacity is then computed. The.final ring q[' this link is represented by the experimental evaluation o[ the nondimensional stress corresponding to the attainment o[" the local bucklh~g. The test&g needs .[or this evaluation are outlined, and both the preliminarl' test results and the planned activity are presented. Copyright ~ 1996 Elsevier Science Ltd.
1 INTRODUCTION Within the activities of CENTC250 SC9 Project Team PT1, the lack of knowledge regarding the influence of local buckling phenomena on the loadbearing capacity of aluminium alloy members has been recognized, 103
104
F. M. Mazzolani, V. Piluso
especially in assessing the behavioural classes. This gap should be covered in the future drafts of Eurocode 9, but to date the required experimental data are insufficient or totally missing. For this reason a theoretical and experimental research has been planned in order to find out the influence of the widthtothickness (b/t) ratios on the behaviour of aluminium alloy members subjected to local buckling. In this paper, the theoretical aspects concerning the evaluation of the rotation capacity of aluminium alloy members are outlined and a semiempirical method for its evaluation is delineated. Semiempirical methods have been widely exploited for computing the rotation capacity of steel members. ~ s They are based on the theoretical prediction of the momentcurvature relationship, which can be expressed in a closedform solution in the case of the simplified model represented by the ideal twoflange section. The integration of the curvature diagram allows one to compute the ultimate rotation provided that the conditions corresponding to the attainment of local buckling are known. These conditions are outlined by means of the analysis of the experimental data relating the nondimensional ultimate stress to the slenderness parameter of the flanges and of the web. These data are the results of classical stub column tests performed on various sections characterized by different widthtothickness ratios of the flanges and of the web. This type of data is available for steel sections in sufficient amount, but it is insufficient or completely missing in the case of aluminium alloy sections. Therefore, the experimental part of the planned research program is based on the execution of a great number of stub column tests aimed at the evaluation of the link between the stress corresponding to the occurrence of local buckling and the slenderness parameters of the plates which compose the section. As this experimental activity is currently in progress only the preliminary test results will be presented. Notwithstanding, the link between the theoretical part and the experimental part will be underlined in order to allow the full comprehension of the features of the proposed approach.
2 CROSSSECTION C L A S S I F I C A T I O N In limit design of structures, it is required that plastic hinges must have a sufficient rotation capacity without losing the bending capacity of the sections. In the case of steel members, the bending capacity of the sections is represented by the fully plastic m o m e n t Mp = J l Z . On the contrary, in aluminium alloy sections, being the material of roundhouse type, it is not logical to define a fully plastic m o m e n t because the
Rotation capacity q[aluminium alloy beams
105
fibres never reach a plateau characterized by yielding. Therefore, reference is usually made to the conventional elastic moment M0.2, corresponding to the conventional elastic limit stress fo.2, This means that, in the case of steel members, the rotation capacity represents the ability of a member to withstand large plastic rotations before its bending resistance falls below the fully plastic moment of the section: on the contrary, in the case of aluminium alloy members, the rotation capacity has to be interpreted as the ability to sustain large plastic rotations before the bending resistance falls below the conventional elastic moment. The rotation capacity of metal members is undermined by the occurrence of local buckling of the plate elements which constitute the member crosssection and, if torsional restraints are not provided, by the occurrence of lateral torsional buckling. In order to design crosssections able to provide sufficient rotation capacity, the local buckling phenomenon has to be controlled. In particular the occurrence of local buckling in the elastic range has to be absolutely avoided; therefore, geometrical properties of the crosssections have to guarantee the development of the plastic behaviour before buckling. For this reason, a very important concept in the design of metal structures, which has been introduced for the first time in Eurocode 3J> is represented by the subdivision of the structural sections into different behavioural classes according to their capacity to exploit the plastic reserves. An analogous classification can be introduced in case of aluminium alloy members with reference to four limit states: elastic buckling limit state, elastic limit state, plastic limit state and collapse limit state. The elastic buckling limit state is related to the attainment of the elastic local buckling of the compressed parts which compose the section. The elastic limit state corresponds to the occurrence of the conventional elastic limit .[i~._~ of the material in the most stressed fibres of the section. The plastic limit state is represented by the attainment of a bending resistance corresponding to perfectly plastic behaviour of the material with a limit value equal to the conventional elastic limit. Finally, due to the hardening behaviour of the stressstrain law, aluminium alloy sections experience increasing stresses for increasing values of detbrmation until failure is reached in most stressed fibres; therefore, the ultimate limit state is referred to the attainment of a bending resistance M,< = z~5Mo.2 corresponding to a conventional limit curvature equal to 5Z02 (Zo.2 being the curwtture corresponding to the attainment of the limit stress/i~.2 in the most stressed fibres). The limit value of the curvature has been chosen taking into
106
F. M. Mazzolani, V. Piluso
account the most unfavourable failure conditions for the less ductile materials.7 9 According to the above limit states, the bending behaviour of aluminium alloy members subjected to local buckling, can be classified in the following four groups (Fig. l): class l: ductile behaviour; class 2: compact behaviour; class 3: semicompact behaviour; class 4: slender behaviour. Members belonging to the first class are characterized by the capability of developing the conventional ultimate resistance M, = o~5Mo.2 before the occurrence of local plastic buckling and, therefore, exhibiting a high rotation capacity. Secondclass members are able to provide their conventional plastic flexural strength Zjo.2 ( Z being the plastic section modulus which has, in the case of aluminium alloy members, only a geometrical meaning), before local plastic buckling occurs, with an appreciable deformation capacity. Structural members fall in the third class when the bending moment leading to the attainment of the conventional elastic limit ./'o.2 in the most stressed fibres can be developed but, due to elasticplastic local buckling, plastic redistribution is not possible. These members provide very poor rotation capacity. Finally, members belonging to the fourth class are not able to develop their total elastic flexural strength due to the premature attainment of elastic local buckling of the compressed parts of the section, so that failure occurs in elastic range.
M
A
BEHAVIOURALCLASSES
Mo.2
(~)
1  ductile
c( 5
2  compact (x o
1
~"
3  semicompact 4  slender
w
0/00, 2 Fig. I. Behavioural classes of aluminium alloy members.
Rotation capacity of aluminium alloy beams
107
By introducing the parameter: ~v

mtl
(1)
M0.2 the following classification is obtained: ductile behaviour: ~,vt >~ :~5: compact behaviour: ~0 ~ ~M < ~s; semicompact behaviour: 1 ~< ~,vt < :q): slender behaviour: ~M < 1; where ~0 = Z / W is the geometrical shape factor (W being the elastic section modulus). It is clear that the parameter governing the above behaviour and, therefore, which defines the class of the structural sections, is the widthtothickness ratio (b/t) of the compressed plates which constitute the section.
3 EVALUATION OF ROTATION CAPACITY Beams and columns of rigid frames subjected to horizontal forces have to withstand double curvature bending, which can be simulated by an assembly of configurations of cantilever beams. This model can also interpret to a good approximation m a n y structural schemes in different loading conditions, by considering the part of the structure between the point of zero m o m e n t and the critical section where the plastic hinge is located. Moreover, the rotation capacity of cantilever beams can be compared with that of centrally loaded beams which are usually adopted as test specimens (Fig. 2). Rotation capacity can be defined as the ratio between the plastic rotation at the collapse state 0p = 0,,  0o2 to the limit elastic one 002 (Fig. 3): Op _ 0,  00.2 _ 0,, R  00.2 00.2 00.2
I
(2)
where 0,, is the ultimate rotation and 00.2 the rotation corresponding to the attainment of the limit stress./~.2. It is useful to point out that rotation capacity can be divided into two parts. The first one is related to the stable increasing branch of the m o m e n t rotation diagram, while the second is related to the unstable softening branch describing the postbuckling behaviour. The stable part of the rotation capacity is given by: R0
0,,, 
00.2
1
(3)
where 0,,, is the rotation corresponding to the occurrence of local buckling.
108
F. M. Mazzolani, V. Piluso
q~
,•F
, J
L
@
, L
f
~ ~ ~ ~ ~ ~ ~ ~ ~ ½~ ~ ~ ~ ~ ~ {
P
'
~ ~ v ~ /
i
I L [ L I
II
x+t.
Fig. 2. Threepoint bending test and equivalent cantilever.
In order to evaluate the rotation capacity of metal members by means of closedform relations, the existing methods can be subdivided into three groups: theoretical methods, semiempirical methods and empirical methods. Theoretical methods are based upon the approximated theoretical evaluation of the moment~zurvature relationship of the member crosssection and upon the theoretical analysis of buckling phenomena. Semiempirical methods differ from theoretical ones due to the fact that local buckling phenomena are taken into account by means of relations provided by experimental evidence. To this scope stub column tests can be performed. Both theoretical and semiempirical methods are able to clarify the role of the parameters affecting the rotation capacity, but they are limited to the analysis of the stable part of the actual plastic deformation capacity. M
" I
1
=i
Om
OU
~o.2
eo.2
Fig. 3. Rotation capacity definitions.
"
Rotation capacity of aluminium alloy beams
109
Empirical methods are based upon the direct analysis of experimental data of fullscale member tests or upon the analysis of the numerical simulation data. In both cases, they can take into account, by means of experimental or numerical evidence, both the precritical and the postbuckling behaviour of metal members. A detailed analysis of the above methods, with reference to steel members, is presented elsewhere. I° In the following, the semiempirical approach is applied for evaluating the stable part of the rotation capacity of aluminium alloy beams subjected to local buckling under nonuniform bending.
4 MOMENTCURVATURE RELATIONSHIP The stressstrain law of aluminium alloy members is characterized by a continuity between the quasielastic and the inelastic hardening behaviour. When the ultimate capacity of the crosssection is concerned, it is possible to use the RambergOsgood law to represent the stressstrain relationship, provided that its parameters are calibrated on the base of the point of" rupture of the material, i.e. the rupture deformation e,t and the ultimate strength.ft. Taking into account that the stressstrain law of aluminium alloys is usually expressed as: rr (o'~" ~: = ~ + 0002 \ . ~ /
(4)
this means that: ~:t
~ + 0002
.~
(5)
providing, for the parameter n the following relation:
n=
log log
.1;/E 0002
(.[i/[i~.2)
(6)
The momentcurvature M  Z relationship of a general crosssection subjected to bending and compression can be evaluated under the hypotheses that the crosssection remains plane, the material constitutive law is expressed through eqn (4) with the parameter n given by eqn (6), and by assuming that local unloadings are elastic. The problem is usually numerically solved by discretizing the crosssection into small elementary areas which are considered as concentrated
F. M. Mazzolani, V. Piluso
110
at their centres o f gravity. The compatibility equations are written in the following form: e,,i = e,o  gYi
(7)
where e~ is the strain of the ith elementary area, e0 the strain o f a fibre at the gravity centre level and Yi the distance between the ith elementary area and the gravity centre (Fig. 4). The equilibrium equations are given by: II,,
Z aiAi = N
(8)
i
II c
Z
criAi)'i ~ M
(9)
i
where ai is the stress of the ith elementary area Ai, N the axial force and M the bending moment. The evaluation of the m o m e n t  c u r v a t u r e relationship has been carried out, including also the influence of mechanical imperfections, for a great n u m b e r o f different crosssection shapes and with reference to different aluminium alloys. F r o m these analyses, 9" ~1 is has been recognized that the m o m e n t  c u r v a t u r e relationship can be accurately represented by means of a relation arranged in the R a m b e r g  O s g o o d form: 7. _ M " M ""' Zo.2 M0~ + k ( ~ )
(lO)
t111111
G
x
Eo
m
Jrll
II
/
y, Fig. 4. Section discretization.
Rotation capaci O, o[aluminium alloy beams
1 11
being: m=
k 
log [(10  ~10)/(5  ~5)] log (cq0/~s) 5  ~5 ~5 m
(11)
10  cq0   
(12)
0~10m
where ¢z5 and el0 are the socalled generalized shape factors which correspond, with reference to a limit curvature equal to 5Z0.2 and 10Z0.2 respectively, to the ratio between the ultimate limit m o m e n t and the conventional elastic m o m e n t M0.2. For a given cross section, characterized by the geometrical shape factor ~o = Z / W , and a given aluminium alloy, whose constitutive law is defined through the exponent n of the R a m b e r g  O s g o o d law, the generalized shape factors can be computed by means of the following relations: ~5 = 5 
3.889 + 0.001897n
(13)
5¢0(0.2696 + 0.0014n)
~10 ~ ~00211°g(lOOOn) 107.955x I0 2 8085×10 21og(n/lO)
(14)
5 ROTATION CAPACITY In order to evaluate the stable part of the rotation capacity, it is necessary to compute the rotation corresponding to the attainment of local buckling, i.e. the rotation corresponding to the m a x i m u m flexural resistance M~, = ~M Mo.2. With reference to the threepoint bending test (Fig. 2) and to the ultimate conditions, the bending m o m e n t diagram can be represented through the following relation: M, M(x) = ~ (L  x)
(15)
which is valid for half span (i.e. 0 ~< x ~< L being x  0 for the midspan section). Therefore, the corresponding curvature diagram is given by the following relation:
x x) Zo.2

M. L  x) Mo.2L
k /I +
\
M , ~ ( L  . ~ ) ~x ' " "
Mo.2L
J
(16)
112
F. M. Mazzohmi, V. Piluso
which provides: _
x
x~],,
;(0.2 The rotation corresponding to the attainment of the maximum flexural resistance corresponding to the occurrence of local buckling is given by:
/
.L
0,, =
Z(x) dx
(18)
. 0
which, through eqn (7), provides: 0,,, = Zo.2
[c )
{cq~ (1   c / L ) + k[c~M(1  x / L ) ] ' " }dx
(19)
The integration of the above equation provides the following relation expressing the rotation corresponding to the occurrence of local buckling: 0,,,   Z ° 2 L
2
M
7.M 1 +
(20)
m+ 1
Therefore, taking into account that the rotation corresponding to the attainment of the conventional elastic limit in the most stressed fibres can be expressed as: Oil.2
Zo.2 L
(21 )
2
the stable part of the rotation capacity can be expressed as: 0,,, Ro  0o.2
( I = ~xM
2k~t'" 1+
nl
') 1
1
(22)
6 LOCAL B U C K L I N G The relation for evaluating the rotation capacity of aluminium alloy members presented in the previous section can be practically used provided that the conditions corresponding to the occurrence of the local buckling of the compressed parts, which compose the section, are expressed as a function of the slenderness parameters of the flange and of the web. In semiempirical methods, which have already been applied successfully in case of steel members j 4, this step is accomplished by means of the analysis of the experimental data obtained from simple stub column tests. Therefore, the aim of these tests is to derive a relation
I13
Rotation capacity oi aluminium alloy beams
between the nondimensional buckling stress and the slenderness parameters of the compressed parts of the section: ~Lj
:~i .Ii~.2 .f(2t, 2,,.)
(23)
where: h
at,,,
(24)
are the slenderness parameters of the flange and of the web respectively (h being the flange width, d,. the web depth, E the modulus of elasticity and t~ and t,,. the thickness of the flange and of the web, respectively. An experimental program dealing with this topic, i.e. aiming to develop eqn (23) in explicit form, is currently in progress. The programmed stub column tests are devoted to square tubes, rectangular tubes, channels and Tsections in 6060T6 aluminium alloy (Fig. 5). The specimens cover a wide range of widthtothickness ratios: 5 ~ bit <<. 16 and 10 ~< hit <<.45 for rectangular tubes; 75 ~< a/t <~ 25 for square tubes; 5 ~< bit <~ 22 and 5 ~ hit <~ 29 for channels; 8 ~< bit <~ 15 and 8 ~< hit <~ 15 for Tsections. As the whole experimental program includes a great number of specimens. it has not yet been completed. For this reason, in this section only some preliminary test results are presented.
a
h
h
Fig. 5. Sections of specimens.
h
F. M. Mazzolani, V. Piluso
114
In Fig. 6 the results of six stub column tests, which have been carried out under displacement control, on square hollow sections are presented. The height of each specimen is equal to three times the width of the section. It is clear that these preliminary tests are not sufficient to render eqn (23) in explicit form, because they refer only to the particular case 2r = 2,,.. Even though the test results of Fig. 6 are not presented in nondimensional form, it can be seen that, obviously, the ratio aLS//0.2 decreases as far as the widthtothickness ratio bit increases (the specimen with b/t : 175 has a lower value of the measured conventional elastic limit j0.2). The fabrication of aluminium alloy profiles involves a technological phase to reduce the initial geometrical imperfections (longitudinal curvature) to within the tolerance limit (usually the highest camber has to be less than L/1000). In the case of aluminium alloy extruded profiles, the straightening process is usually done by 'tractioning'. As during tractioning high plastic deformations are required to reduce the initial curvature to the tolerance value, the strain hardening of the material occurs. This produces the increase of the tensile strength and relieves the residual stresses caused by heat treatment, but, on the other hand, due to the Bauschinger effect a reduction of the loadbearing capacity under compression occurs ~l. For this reason, additional experimental tests are aimed at the evaluation of the conventional elastic limit under compresSQUARE HOLLOW SECTIONS 350
.....................................................":..................................................................... [
300 25O
~ 200 Z
,"
 :.,,.\
'!
"\ .................
150
'...,..
"... "\
~
.........
....... i b / t = 1 6 . 6 7 "',. ~ . : .
....
i
......................................................................
~ .......................
"'...._ "'"'+ b / t = 1 7 . 5 b / t = 20 ...........................................................................................................................
[.. 100
50 ..... ] 1 5 x 1 5 x 2 0
,
0
I
2.5
40x40x4 ,
50x50x3 I
5
,
50x50x4 I
i
70x70x4 I
7.5 10 AXIAL STRAIN (%)
Fig. 6. P r e l i m i n a r y test r e s u l t s .
i
8oxsox4 I
.......
I
12.5
15
Rotation capacity of aluminium alloy beams
115
sionf'0.2. By means o f these tests, it is possible to recognize the part of the strength reduction under compression due to the occurrence of local buckling and the part due to the Bauschinger effect caused by tractioning. In fact, the nondimensional buckling stress of eqn (23) can be written in the following form: ale
Ji,2
_ a l e f'0.2
(25)
'!
.1 o.2 ,/i~2
The first term of the righthand side of eqn (25) represents the actual effect of local buckling, while the second term provides the influence of tractioning.
7 CONCLUSIONS The issues concerning the evaluation of the rotation capacity of aluminium alloy beams subjected to local buckling under nonuniform bending have been faced. In addition, the theoretical basis for its evaluation according to a semiempirical m e t h o d have been laid. Finally, the research needs regarding the problem of evaluating the local buckling resistance of aluminium alloy members accounting for the interaction between flange and web slenderness have been evidenced.
REFERENCES 1. Kato, B., Rotation capacity of steel members subject to local buckling. 9th World Cot~[~ Earthquake Engineering, Vol. IV, paper 623, Tokyo Kyoto. 1988. 2. Kato, B., Rotation capacity of Hsection members as determined by local buckling. J Constr. Steel Res., 13 (1989) 95109. 3. Kato, B., Deformation capacity of steel structures. ,I. Constr. Steel Res.. 17 (1990) 3394. 4. Mazzolani F. M. & Piluso V., Evaluation of the rotation capacity of steel beams and beamcolumns. 1st Cost C1 Workshop, Strasbourg, 1992. 5. Mazzolani F. M. & Piluso V., Member behavioural classes for steel beams and beamcolumns, X X V I C.T.A., Giornate ltaliane della Costruzione in A cciaio, Viareggio, 1993. 6. Commission of the European Communities. Eurocode 3: Design of Steel Structures, 1992. 7. Cappelli M., De Martino A. & Mazzolani F. M., Ultimate bending moment evaluation for aluminium alloy members. A comparison among different definitions. Proe. of the Int. Con/~ Steel and Aluminium Structures. ed. R. Nardyanan, Elsevier Applied Science, London, 1987.
116
F. M. Mazzolani, V. Piluso
8. Mazzolani F. M., Cappelli M. & Spasiano G., Plastic analysis of aluminium alloy members in bendings. Aluminium (1985) p 61. 9. Mazzolani F. M., Plastic' Design of Aluminium Alloy Structures. Verba Volant, Scripta Manent, Liegi, 1984, pp. 295313. 10. Mazzolani F. M. & Piluso V., Theory and Design ~?['Seismie Resistant Steel Frames. E & FN Spon, Chapman & Hall, London, 1996. 11. Mazzolani F. M., Alummium Alloy Structures. Chapman & Hall, London, 1995.