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Buckling modes of double-channel cold-formed beams
Thin-WalledStructures19 (1994) 353-366 © 1994 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/94/$7.00 ELSEVIER
Bucklfng Modes of Double-Channel Cold-Formed Beams A. Ghersi, R. Landolfo & F. M. M a z z o l a n i University of Naples, Institute of Civil Engineering, Piazzale Tecchio, 1-80125 Naples, Italy
ABSTRACT The present paper starts from the analysis of previous experimental results on double-channel cold-formed beams subjected to local and lateraltorsional buckling. The scope of this reanalysis is to identify the range of definition of the buckling modes, which influence the ultimate behaviour of such beams. The verification approach of EC3 and AISI have been compared to the experimental results, giving a satisfactory agreement. The definition of coupled instability range has been set up by means of EC3 formulations. The proposed equations relate together the main geometrical parameters of the beam, giving useful indications to the designer.
NOTATION b E
fy G h
/t /w L m
Mer Me
Meet
Flange or web thickness Modulus of elasticity Yielding stress of steel Shear modulus Height of the section Torsion constant Warping constant Second moment of area about minor axis Length of beam between points which have lateral restraint Coefficient used in eqn (8) Elastic critical moment for lateral-torsional buckling Global collapse moment for inelastic lateral-torsional buckling Bending moment of the effective section for a given stress diagram Plastic moment 353
354 Mc,red My n
P t
0~LT
/3w fll & A,LT ALT
2M O'er
~LT ZLT
A. Ghersi, R. Landol/b, F. M. Mazzolani
Reduced global collapse moment Yielding moment Coefficient used in eqn (2) Limit difference between local and coupled buckling moment Element thickness Coefficient used in eqn (9) Shape factor Imperfection factor Coefficient used for eqn (3) Lower limit of coupled instability field Upper limit of coupled instability field Reduction factor by C E C M - E C C S defined by eqn (2) Geometrical slenderness ratio for lateral-torsional buckling Nondimensional slenderness for lateral-torsional buckling Nondimensional bending slenderness according to C E C M - E C C S Coefficient used for eqn (3) Stress corresponding to the critical moment Coefficient used in eqn (3) Reduction factor by Eurocode 3 defined by eqn (3)
1 INTRODUCTION This paper is part of a general research project devoted to the structural aspects of cold-formed thin-gauge members.l'2 As is well known, the local buckling of the compressed parts is the phenomenon which mainly affects the behaviour of such profiles. For this reason, the first goal of this project was to set up a numerical procedure based on a consistent physical model which is able to simulate the complete load-deflection (moment-curvature) history of this kind of section. 3 The analysis of the influence of the different theoretical approaches and the calibration of the proposed model (i.e. the definition of the most suitable physical interpretation of local buckling and the evaluation of the numerical parameters involved) was first carried out in the case of thin-gauge welded box-sections, 4'5 for which some experimental results were available. 6 In order to analyse in the same way the behaviour of double-channel thin-gauge sections, bending tests on a series of simply supported beams were performed at the 'Laboratorio Prove Materiali e Strutture' of the University of Ancona, Italy. 7`s In the case of stiffened sections the experimental results have been successfully used to calibrate the proposed model. 9'1° On the other hand, the behaviour of unstiffened specimens was affected, more or less significantly, by their lateral-torsional
Buckling modes of double-channel cold-formed beams
355
buckling. The results related to these last sections are here shortly recalled and used to analyse the coupled effects of local and lateral instability, in order to define behavioural zones according to the formulation of Eurocode 3, which involves the geometrical parameters of the members.
2 EXPERIMENTAL RESULTS The experimental investigation has been carried out on specimens made of Fe360 steel, composed by two back to back coupled cold-formed channels. Five sections without stiffening lips, named P6 to PI0, 7 are renamed A to E in the present analysis (Fig. 1). Cross-sections have been designed to cover a wide range of b/t ratio values, going from slender (A and B) and semicompact (C and D) to plastic (E), according to the classification of Eurocode 3.11 Monotonic tests have been performed on simply supported beams, 3.00 m of span, imposing displacements and measuring the corresponding forces, applied at two points 1.00 m distant across the midspan (Figs 2 and 3). The vertical load is transmitted to the beam by means of two rigid bars, which prevent torsional rotation. For this reason only the central part of the beam, 1.00 m long, is free to buckle laterally under increasing loads, as shown by the experimental behaviour of C, D and E specimens (Fig. 4). Geometrical data and experimental values of ultimate bending moment are given in Table 1.
3 B U C K L I N G BEHAVIOUR OF BEAMS
3.1 Lateral-torsional buckling The theory of elastic lateral-torsional instability of beams was developed more than 30 years ago. 12'~3 Under standard conditions of restraint at each end, the elastic critical moment mcr of a beam loaded through its shear centre and subject to uniform moment, as given by annex F of Eurocode 3, is A
B
C
200x lO0x25
200x 100x5
200x40x3
:z I
D 200x50x4
Z
JE Fig. 1. Specimens cross-section.
E 200x50x5
356
A. Ghersi, R. LandolJb, F. M. Mazzolani
T4
.
,,
Fig. 2. Testingsystemand instrumentation. ~2Elz Mcr -
L2
/Iw + L2GIt ,:ei_
(l)
An equivalent formulation is provided by the AISI Specifications H for bending about the symmetry axis. A more general expression, proposed by Pekoz and Winter,15 is given in the case of bending about the centroidal axis perpendicular to the symmetry axis, for singly symmetric sections. Furthermore, both codes provide modified formulations to take into account the effect of different conditions of restraint, of loads not applied in the shear centre of the beam, and of nonuniform moment distribution (none of these problems will be analysed in the present paper). The approach to the inelastic lateral buckling of beams, including the influence of imperfections, is usually based on the evaluation of the reduction in lateral and torsional stiffness due to yielding. 16Starting from theoretical and experimental results, the most important international codes impose evaluation of the global collapse moment Mc by reducing the yielding or the plastic moment by means of a coefficient which takes into account this phenomenon. The CECM-ECCS Recommendations 17 define a reduction factor 7/ given by rl=
(2)
1+
where
~M
~Y
M~pl
Mpl
=Va-~-=V~cTc~ ; with a suggested value of
a-My n =
2.5.
Buckling modes of double-channel cold-formed beams
357
Fig. 3. A step of the loading process.
In a similar way, the Eurocode 3 defines the reduction factor ZLT for lateral-torsional buckling given by )~LT =
with XLT --< 1
(3)
~LT + V/~T -- ~2T'
where ~LT = 0"511
+ ~LT(~LT--
0"2) + ~2T] ;
'~.LT ~LT -- 71 V ~ w
and ~LT = 0.21;
/---
2LT =
--
;
),l = =~/~y
with flw depending on the class of the section.
358
A. Ghersi, R. Landolfo, F. M. Mazzolani
A
B
C D
E
Fig. 4. Mode of failure of specimens.
Finally, in the case of 1 sections bent about the centroidal axis perpendicular to the web the AISI Specification evaluates the reduction factor as 1 for Mot > 2.78 My10 (1
-9
Mcr My
lOMy~
Mcr
36McrJ for2.78 > ~
> 0-56
(4)
for Mcr myy -< 0.56
The reduction coefficients M / M y are plotted in Fig. 5 versus the geometrical slenderness ratio 2LT, in order to compare the values provided by these codes in the case of a typical double-channel coldformed section.
Buckling modes of double-channel cold-formed beams
359
TABLE 1 Specimen Dimensions, Material Properties and Experimental Results
Section A B C D E
h (mm)
b (mm)
t (ram)
b/t Web
Flange
EC3 (class)
200 200 200 200 200
100 100 40 50 50
2.5 5.0 3.0 4.0 5.0
80 40 66-7 50 40
40 20 13.3 12.5 10
4 4 3 3 1
fy
mex p
(MPa) (kN m) 284.0 246.6 236.1 233.1 236.1
22.4 52.5 18.7 29.7 35.5
3.2 Local buckling The approach to the local buckling of thin-gauge sections is based on the theory of stability of plates developed by Von Karman. The actual nonlinear distribution of stress is usually substituted with a linear distribution acting on a reduced part (effective section concept). The effective width befr is evaluated by means of a formulation given by Winter, 18 which is rewritten in Eurocode 3 in the form: beff = pb
(5)
with p = 1 when 2 < 0.673 P--
1 - 0.22/2 •y - - ,~ 2 4- 0"18 ~.y -- 0"6'
p ~ 1 when 2 > 0.673
and ,~,-- ~ -
;
/],y - - " - - ~
The same relations are provided by the AISI Specification. European and American codifications differ only in the way in which they take into account the corners in the evaluation of the geometrical width b. An example of such difference, analysed in detail by Ghersi and Landolfo, 19 is shown in Fig. 6 through the Meff/My v e r s u s b/t curves.
4 C O U P L E D E F F E C T S OF LOCAL A N D L A T E R A L B U C K L I N G A theoretical approach to the evaluation of the coupled effects of local and lateral buckling of beams might be analogous to the one used for the
360
A. Ghersi, R. LandolJ~, F. M. Mazzolani
M My
- -
EC3
1 0.8 0.6
~ ' ~ .
elastic " ~ t e r a l buckling
°
0.4
h=2b
0.2 0 ' 20
' 40
' 60
' 8(3
' 100
' 120
' 140
' 160
' 180
' kLT
Fig. 5. Lateral-torsional buckling curves for a double-channel section.
M My
- 1
"
•
.
.
.
.
.
.
EC3 .
.
.
.
.
.
.
.
AISI
0.8 0.6 0.4 b
bit=40
b/t
0.2 1 0
Fig. 6. Local buckling curves for a double-channel section.
interaction of local and flexural buckling of columns. In that case the overall buckling load for a thin-gauge member might be determined as the critical load of the effective section, calculated at the overall buckling stress. This leads to an iterative procedure, because the buckling stress itself depends on the dimension of the effective section, z° Nevertheless, a preliminary numerical analysis on the same double-channel beams pointed out some incongruities connected to the use of a procedure based on the effective section concept. In particular, the progressive elimination of the end of the locally buckled flange gives a strong reduction of the second moment of area about the minor axis and, therefore, a severe lowering of the overall buckling load much greater than the experimental evidence. For the practical applications, both the Eurocode 3 and the AISI Specification provide a simplified approach which first requires the
Buckling modes of double-channel cold-formed beams
361
evaluation of the elastic lateral buckling load for the unreduced section. According to EC3, the coefficient ZLT is evaluated, assuming flw = We~/Wpl. The reduced global collapse moment Mc,rea is finally given by (6)
Mc,re d = ZLTMeff(fy)
where Me~(fy) is the yielding moment corresponding to the effective section determined for the yielding stress fy. The approach provided by the AISI Specification is slightly different. The stress acr induced on the unreduced section by the global collapse moment is first evaluated. The reduced global collapse moment is then given by Mc.re d = Meff(O'cr )
(7)
where Meff(O'cr) is the yielding moment corresponding to the effective section determined for the critical stress O'cr. The reduced global collapse moment given by Eurocode 3 and the AISI Specification are plotted in Fig. 7 as a function of the geometrical slenderness ratio 2LT, in case of a typical double-channel coldformed section.
5 IDENTIFICATION OF BUCKLING MODES A parametric analysis of the coupled effect of local and lateral buckling, evaluated according to Eurocode 3, has been performed both to compare M My
1
-
EC3
..............
AISI
0.8 AISI 0.6
~ ' ~ " ~ . . ~ EC3 ~ . . ~ .
~
elastic "~ateral buckling
0.4
0.2
0
h=2b
~'"
2; '.o '6o ' 8; ',oo',2o l~o'~;o'~;o'~L.
Fig. 7. Reduced global collapse moment for a double-channel section.
362
A. Ghersi, R. LandolJo, F. M. Mazzolani
code prescription and experimental results and to identify the limits of local, lateral and coupled buckling modes. Figures 8 and 9 show the relationship of ultimate moment versus bit ratio of flange for local buckling, inelastic lateral buckling and coupled buckling. The comparison is made for four different h/t ratios (80, 66-7, 50 and 40) and two values of the L/h ratio (5, corresponding to the tested specimens, and 30). We may note that the experimental data are in a good compliance with code values for coupled buckling, being always slightly greater (3-11%). It can be explained by the fact that, in the evaluation of the ultimate moment, the plastic reserve has been neglected. We may, furthermore, see that the bit range is divided into three fields:
b/t < ill, where the beam is subjected to lateral torsional buckling only: fll < bit < f12, where the beam is subjected to coupled buckling; bit > f12, where the beam is predominantly subjected to local buckling. The limit fl~ is univocally determined as the branching-off point of lateral and coupled buckling curves. On the other hand, local and coupled buckling curves are asymptotic and the limit f12 has to be defined referring to a percentage scatter in between. Referring to Eurocode 3, which allows one to neglect lateral buckling when 2CT < 0.4 (i.e. according to eqn (3) when Zcac> 0.953), a corresponding 4.7% limit difference has been assumed; a M
M'-~ r LocalBuckling InelasticLateralBuckling
K 1
0.8
0.8 0.6
o.4t..J 0.2 O
0
_M_
I
',
0.4
'
'
0.2
~ 10 I~L
1~2
30
40
b/t
0
IV, iv
D
1 0.8
0.8
0.6
0.6
0.4
10
20
30
20
30
40
b/t
E |
0.4
02
°o
h/t=66.7
h/t=50 . ". . l .b ". . . 2o
313
"
4o
0.2
b/t
0
hit=40 10
40
bit
Uh =5
Fig. 8. Ultimate moment for local, lateral and coupled buckling for a double-channel section with L/h = 5.
363
Buckling modes of double-channel cold-formed beams
i .NMy
M
M~, 0.8"
0.8
0.8-
0.6
0.4
•
0.4
0.2
•
0.2 rt
~7
.
.
.
20
40
6o
bff
60
0.8" 0.6" 0.4" 0.2" C b/t
°c
M
MI 0.8" 0.6"
I
0.4. 02 0
20
40
Lh/=30 20
40
60
b/t
Fig. 9. Ultimate moment for local, lateral and coupled buckling for a double-channel section with L/h = 30. smaller value of such percentage would lead to a wider extension of the coupled effect field. It is evident from Figs 8 and 9 that fll has a constant value (//1 = 13.6), while//2 strongly depends on h/t and L/h ratios. The coupled effect field is, therefore, wider, as h/t and L/h are larger, and it might even disappear (f12
3
(8)
The coefficient m is related to the
L/h ratio by means of the equation:
i/L ,~o.85 m = v~)
(9)
with 1Oge V = --2"311 + 0"01 14(1Ogep) 7/3]
(lO)
where p
MR -- Meff _
M~
× 1000 is the limit difference between local and coupled buckling moment.
364
A. Ghersi, R. Landol[o, F. M. Mazzolani
Equation (10) is available in the range l < p < 1 0 0 ; for the above mentioned limit scatter o f p =47 we obtain v = 0.05443. These equations may be used to define explicitly a condition which must be satisfied by the geometrical dimensions of the section in order to avoid lateral or coupled buckling: b-3t
--7>
0.05443/L, °85
(11)
Figure 10 shows the fields of local, lateral and coupled buckling for the limit L/h ratio equal to 5, which corresponds to the L/h ratio of the central unrestrained part of the tested beams. The location of the experimental points indicates that only the case C exhibits a coupled behaviour, as emphasized by testing. Figure 11 gives the variation of the coupled instability field as a function of the L/h ratio. 6 CONCLUSION The identification of buckling modes is a useful means in order to foresee the actual behaviour of thin-walled beams. The physical intuition suggests that the coupled instability field becomes wider as the beam slenderness (L/h) increases. This behaviour is confirmed by the numerical analysis performed and quantified by the proposed expressions. The use of eqn (11) may be recommended to the designer in order to prevent the lateral buckling effect
h t 8O
Lateral
Buckling Local Buckling
D 40
El b
,' J 0
0
;
10
20
3'o
4'0 bit
Fig. 10. Fields of local, lateral and coupled buckling for a double-channel section with
L/h = 5.
Buckling modes of double-channel cold-formed beams
365
,f 4O
20
40
60
b~
Fig. 11. Influence of L/h ratio of the instability fields. by assuming appropriate values of cross-section dimensions. The analysis of the available testing results gives confirmation of the reliability of the ultimate moments provided by Eurocode 3.
REFERENCES 1. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., Il comportamento flessionale di profili sottili sagomati a freddo: impostazione della ricerca. Acciaio, Sept. 1989, pp. 415-22. 2. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., I1 comportamento flessionale di profili sottili sagomati a freddo: ricerca teorico-sperimentale. XII Congresso C.T.A., Capri, Oct. 1989, pp. 535-47. 3. De Martino, A., Ghersi, A. & Mazzolani, F. M., Analisi dei parametri di influenza del comportamento flessionale delle sezioni a cassone in par~te sottile. XII Congresso C.T.A., Capri, Oct. 1989, pp. 521-34. 4. De Martino, A., Ghersi, A. & Mazzolani, F. M., Calibration of a bending model for thin walled steel box-section. Int. Colloquium on Stability of Steel Structures, Budapest, April 1990. 5. De Martino, A., Ghersi, A. & Mazzolani, F. M., On the bending behaviour of thin walled box-sections. Costruzioni Metalliche, 6 (1991) 347-60. 6. Ballio, G. & Calado, L., Sezioni inflesse in acciaio sottoposte a carichi ciclici. Sperimentazione e simulazione numerica. Construzioni Metalliche, 1 (1986). 7. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., II comportamento flessionale di profili sottili sagomati a freddo: indagine sperimentale. Acciaio, Dec. 1990, pp. 577-588. 8. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., Bending behaviour of double-C thin-gauge beams: experimental evidence versus codification, 4th Int. Colloquium on Structural Stability, Mediterranean Section, Istanbul, Sept. 1991. 9. De Martino, A., Ghersi, A. & Mazzolani, F. M., Bending behaviour of
366
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A. Ghersi, R. Landolfo, F. M. Mazzolani double-C thin walled beams. X Int. Specialty Conj. on Cold-Formed Steel Structures, St. Louis, MO, Oct. 1990, pp. 637~48. De Martino, A., Ghersi, A., Landolfo, R. & Mazzolani, F. M., Calibration of a bending model for cold-formed sections. XI Int. Specialty Conj'. on ColdFormed Steel Structures, St. Louis, MO, Oct. 1992. Eurocode 3, Design of Steel Structures, 1990. Timoshenko, S. & Gere, J. M., Theoo' of Elastic Stability, 2nd edn., McGraw-Hill, New York, 1961. Vlasov, V. Z., Thin-Walled Elastic Beams, 2nd edn., National Science Foundation, Washington, DC, 1961. American Iron and Steel Institute, Specification for the Design of ColdFormed Steel Structural Members. Cold-Formed Steel Design Manual, Aug. 1986, with Dec. 1989 Addendum. Pekoz, T. B. & Winter, G., Torsional-flexural buckling of thin-walled sections under eccentric load. J. Struct. Div., ASCE, (May) (1969) 941-63. Galambos, T. V., Inelastic lateral buckling of beams. J. Struct. Div., ASCE, (Oct.) (1963) 217~42. CECM-ECCS, European Recommendations Jor Steel Construction, 1978. Winter, G., Strength of Thin Steel Compression Flanges, ASCE, Vol. 112, 1947. Ghersi, A. & Landolfo, R., Aspetti innovativi dell'EC3 nella verifica delle sezioni in parete sottile. Costruzioni Metalliche, 5 (1992) 288-302. De Wolf, J. T., Pekoz, T. & Winter, G., Local and overall buckling of coldformed members. J. Struct. Div., ASCE, (Oct.) (1974) 2017 36.
Bucklfng Modes of Double-Channel Cold-Formed Beams A. Ghersi, R. Landolfo & F. M. M a z z o l a n i University of Naples, Institute of Civil Engineering, Piazzale Tecchio, 1-80125 Naples, Italy
ABSTRACT The present paper starts from the analysis of previous experimental results on double-channel cold-formed beams subjected to local and lateraltorsional buckling. The scope of this reanalysis is to identify the range of definition of the buckling modes, which influence the ultimate behaviour of such beams. The verification approach of EC3 and AISI have been compared to the experimental results, giving a satisfactory agreement. The definition of coupled instability range has been set up by means of EC3 formulations. The proposed equations relate together the main geometrical parameters of the beam, giving useful indications to the designer.
NOTATION b E
fy G h
/t /w L m
Mer Me
Meet
Flange or web thickness Modulus of elasticity Yielding stress of steel Shear modulus Height of the section Torsion constant Warping constant Second moment of area about minor axis Length of beam between points which have lateral restraint Coefficient used in eqn (8) Elastic critical moment for lateral-torsional buckling Global collapse moment for inelastic lateral-torsional buckling Bending moment of the effective section for a given stress diagram Plastic moment 353
354 Mc,red My n
P t
0~LT
/3w fll & A,LT ALT
2M O'er
~LT ZLT
A. Ghersi, R. Landol/b, F. M. Mazzolani
Reduced global collapse moment Yielding moment Coefficient used in eqn (2) Limit difference between local and coupled buckling moment Element thickness Coefficient used in eqn (9) Shape factor Imperfection factor Coefficient used for eqn (3) Lower limit of coupled instability field Upper limit of coupled instability field Reduction factor by C E C M - E C C S defined by eqn (2) Geometrical slenderness ratio for lateral-torsional buckling Nondimensional slenderness for lateral-torsional buckling Nondimensional bending slenderness according to C E C M - E C C S Coefficient used for eqn (3) Stress corresponding to the critical moment Coefficient used in eqn (3) Reduction factor by Eurocode 3 defined by eqn (3)
1 INTRODUCTION This paper is part of a general research project devoted to the structural aspects of cold-formed thin-gauge members.l'2 As is well known, the local buckling of the compressed parts is the phenomenon which mainly affects the behaviour of such profiles. For this reason, the first goal of this project was to set up a numerical procedure based on a consistent physical model which is able to simulate the complete load-deflection (moment-curvature) history of this kind of section. 3 The analysis of the influence of the different theoretical approaches and the calibration of the proposed model (i.e. the definition of the most suitable physical interpretation of local buckling and the evaluation of the numerical parameters involved) was first carried out in the case of thin-gauge welded box-sections, 4'5 for which some experimental results were available. 6 In order to analyse in the same way the behaviour of double-channel thin-gauge sections, bending tests on a series of simply supported beams were performed at the 'Laboratorio Prove Materiali e Strutture' of the University of Ancona, Italy. 7`s In the case of stiffened sections the experimental results have been successfully used to calibrate the proposed model. 9'1° On the other hand, the behaviour of unstiffened specimens was affected, more or less significantly, by their lateral-torsional
Buckling modes of double-channel cold-formed beams
355
buckling. The results related to these last sections are here shortly recalled and used to analyse the coupled effects of local and lateral instability, in order to define behavioural zones according to the formulation of Eurocode 3, which involves the geometrical parameters of the members.
2 EXPERIMENTAL RESULTS The experimental investigation has been carried out on specimens made of Fe360 steel, composed by two back to back coupled cold-formed channels. Five sections without stiffening lips, named P6 to PI0, 7 are renamed A to E in the present analysis (Fig. 1). Cross-sections have been designed to cover a wide range of b/t ratio values, going from slender (A and B) and semicompact (C and D) to plastic (E), according to the classification of Eurocode 3.11 Monotonic tests have been performed on simply supported beams, 3.00 m of span, imposing displacements and measuring the corresponding forces, applied at two points 1.00 m distant across the midspan (Figs 2 and 3). The vertical load is transmitted to the beam by means of two rigid bars, which prevent torsional rotation. For this reason only the central part of the beam, 1.00 m long, is free to buckle laterally under increasing loads, as shown by the experimental behaviour of C, D and E specimens (Fig. 4). Geometrical data and experimental values of ultimate bending moment are given in Table 1.
3 B U C K L I N G BEHAVIOUR OF BEAMS
3.1 Lateral-torsional buckling The theory of elastic lateral-torsional instability of beams was developed more than 30 years ago. 12'~3 Under standard conditions of restraint at each end, the elastic critical moment mcr of a beam loaded through its shear centre and subject to uniform moment, as given by annex F of Eurocode 3, is A
B
C
200x lO0x25
200x 100x5
200x40x3
:z I
D 200x50x4
Z
JE Fig. 1. Specimens cross-section.
E 200x50x5
356
A. Ghersi, R. LandolJb, F. M. Mazzolani
T4
.
,,
Fig. 2. Testingsystemand instrumentation. ~2Elz Mcr -
L2
/Iw + L2GIt ,:ei_
(l)
An equivalent formulation is provided by the AISI Specifications H for bending about the symmetry axis. A more general expression, proposed by Pekoz and Winter,15 is given in the case of bending about the centroidal axis perpendicular to the symmetry axis, for singly symmetric sections. Furthermore, both codes provide modified formulations to take into account the effect of different conditions of restraint, of loads not applied in the shear centre of the beam, and of nonuniform moment distribution (none of these problems will be analysed in the present paper). The approach to the inelastic lateral buckling of beams, including the influence of imperfections, is usually based on the evaluation of the reduction in lateral and torsional stiffness due to yielding. 16Starting from theoretical and experimental results, the most important international codes impose evaluation of the global collapse moment Mc by reducing the yielding or the plastic moment by means of a coefficient which takes into account this phenomenon. The CECM-ECCS Recommendations 17 define a reduction factor 7/ given by rl=
(2)
1+
where
~M
~Y
M~pl
Mpl
=Va-~-=V~cTc~ ; with a suggested value of
a-My n =
2.5.
Buckling modes of double-channel cold-formed beams
357
Fig. 3. A step of the loading process.
In a similar way, the Eurocode 3 defines the reduction factor ZLT for lateral-torsional buckling given by )~LT =
with XLT --< 1
(3)
~LT + V/~T -- ~2T'
where ~LT = 0"511
+ ~LT(~LT--
0"2) + ~2T] ;
'~.LT ~LT -- 71 V ~ w
and ~LT = 0.21;
/---
2LT =
--
;
),l = =~/~y
with flw depending on the class of the section.
358
A. Ghersi, R. Landolfo, F. M. Mazzolani
A
B
C D
E
Fig. 4. Mode of failure of specimens.
Finally, in the case of 1 sections bent about the centroidal axis perpendicular to the web the AISI Specification evaluates the reduction factor as 1 for Mot > 2.78 My10 (1
-9
Mcr My
lOMy~
Mcr
36McrJ for2.78 > ~
> 0-56
(4)
for Mcr myy -< 0.56
The reduction coefficients M / M y are plotted in Fig. 5 versus the geometrical slenderness ratio 2LT, in order to compare the values provided by these codes in the case of a typical double-channel coldformed section.
Buckling modes of double-channel cold-formed beams
359
TABLE 1 Specimen Dimensions, Material Properties and Experimental Results
Section A B C D E
h (mm)
b (mm)
t (ram)
b/t Web
Flange
EC3 (class)
200 200 200 200 200
100 100 40 50 50
2.5 5.0 3.0 4.0 5.0
80 40 66-7 50 40
40 20 13.3 12.5 10
4 4 3 3 1
fy
mex p
(MPa) (kN m) 284.0 246.6 236.1 233.1 236.1
22.4 52.5 18.7 29.7 35.5
3.2 Local buckling The approach to the local buckling of thin-gauge sections is based on the theory of stability of plates developed by Von Karman. The actual nonlinear distribution of stress is usually substituted with a linear distribution acting on a reduced part (effective section concept). The effective width befr is evaluated by means of a formulation given by Winter, 18 which is rewritten in Eurocode 3 in the form: beff = pb
(5)
with p = 1 when 2 < 0.673 P--
1 - 0.22/2 •y - - ,~ 2 4- 0"18 ~.y -- 0"6'
p ~ 1 when 2 > 0.673
and ,~,-- ~ -
;
/],y - - " - - ~
The same relations are provided by the AISI Specification. European and American codifications differ only in the way in which they take into account the corners in the evaluation of the geometrical width b. An example of such difference, analysed in detail by Ghersi and Landolfo, 19 is shown in Fig. 6 through the Meff/My v e r s u s b/t curves.
4 C O U P L E D E F F E C T S OF LOCAL A N D L A T E R A L B U C K L I N G A theoretical approach to the evaluation of the coupled effects of local and lateral buckling of beams might be analogous to the one used for the
360
A. Ghersi, R. LandolJ~, F. M. Mazzolani
M My
- -
EC3
1 0.8 0.6
~ ' ~ .
elastic " ~ t e r a l buckling
°
0.4
h=2b
0.2 0 ' 20
' 40
' 60
' 8(3
' 100
' 120
' 140
' 160
' 180
' kLT
Fig. 5. Lateral-torsional buckling curves for a double-channel section.
M My
- 1
"
•
.
.
.
.
.
.
EC3 .
.
.
.
.
.
.
.
AISI
0.8 0.6 0.4 b
bit=40
b/t
0.2 1 0
Fig. 6. Local buckling curves for a double-channel section.
interaction of local and flexural buckling of columns. In that case the overall buckling load for a thin-gauge member might be determined as the critical load of the effective section, calculated at the overall buckling stress. This leads to an iterative procedure, because the buckling stress itself depends on the dimension of the effective section, z° Nevertheless, a preliminary numerical analysis on the same double-channel beams pointed out some incongruities connected to the use of a procedure based on the effective section concept. In particular, the progressive elimination of the end of the locally buckled flange gives a strong reduction of the second moment of area about the minor axis and, therefore, a severe lowering of the overall buckling load much greater than the experimental evidence. For the practical applications, both the Eurocode 3 and the AISI Specification provide a simplified approach which first requires the
Buckling modes of double-channel cold-formed beams
361
evaluation of the elastic lateral buckling load for the unreduced section. According to EC3, the coefficient ZLT is evaluated, assuming flw = We~/Wpl. The reduced global collapse moment Mc,rea is finally given by (6)
Mc,re d = ZLTMeff(fy)
where Me~(fy) is the yielding moment corresponding to the effective section determined for the yielding stress fy. The approach provided by the AISI Specification is slightly different. The stress acr induced on the unreduced section by the global collapse moment is first evaluated. The reduced global collapse moment is then given by Mc.re d = Meff(O'cr )
(7)
where Meff(O'cr) is the yielding moment corresponding to the effective section determined for the critical stress O'cr. The reduced global collapse moment given by Eurocode 3 and the AISI Specification are plotted in Fig. 7 as a function of the geometrical slenderness ratio 2LT, in case of a typical double-channel coldformed section.
5 IDENTIFICATION OF BUCKLING MODES A parametric analysis of the coupled effect of local and lateral buckling, evaluated according to Eurocode 3, has been performed both to compare M My
1
-
EC3
..............
AISI
0.8 AISI 0.6
~ ' ~ " ~ . . ~ EC3 ~ . . ~ .
~
elastic "~ateral buckling
0.4
0.2
0
h=2b
~'"
2; '.o '6o ' 8; ',oo',2o l~o'~;o'~;o'~L.
Fig. 7. Reduced global collapse moment for a double-channel section.
362
A. Ghersi, R. LandolJo, F. M. Mazzolani
code prescription and experimental results and to identify the limits of local, lateral and coupled buckling modes. Figures 8 and 9 show the relationship of ultimate moment versus bit ratio of flange for local buckling, inelastic lateral buckling and coupled buckling. The comparison is made for four different h/t ratios (80, 66-7, 50 and 40) and two values of the L/h ratio (5, corresponding to the tested specimens, and 30). We may note that the experimental data are in a good compliance with code values for coupled buckling, being always slightly greater (3-11%). It can be explained by the fact that, in the evaluation of the ultimate moment, the plastic reserve has been neglected. We may, furthermore, see that the bit range is divided into three fields:
b/t < ill, where the beam is subjected to lateral torsional buckling only: fll < bit < f12, where the beam is subjected to coupled buckling; bit > f12, where the beam is predominantly subjected to local buckling. The limit fl~ is univocally determined as the branching-off point of lateral and coupled buckling curves. On the other hand, local and coupled buckling curves are asymptotic and the limit f12 has to be defined referring to a percentage scatter in between. Referring to Eurocode 3, which allows one to neglect lateral buckling when 2CT < 0.4 (i.e. according to eqn (3) when Zcac> 0.953), a corresponding 4.7% limit difference has been assumed; a M
M'-~ r LocalBuckling InelasticLateralBuckling
K 1
0.8
0.8 0.6
o.4t..J 0.2 O
0
_M_
I
',
0.4
'
'
0.2
~ 10 I~L
1~2
30
40
b/t
0
IV, iv
D
1 0.8
0.8
0.6
0.6
0.4
10
20
30
20
30
40
b/t
E |
0.4
02
°o
h/t=66.7
h/t=50 . ". . l .b ". . . 2o
313
"
4o
0.2
b/t
0
hit=40 10
40
bit
Uh =5
Fig. 8. Ultimate moment for local, lateral and coupled buckling for a double-channel section with L/h = 5.
363
Buckling modes of double-channel cold-formed beams
i .NMy
M
M~, 0.8"
0.8
0.8-
0.6
0.4
•
0.4
0.2
•
0.2 rt
~7
.
.
.
20
40
6o
bff
60
0.8" 0.6" 0.4" 0.2" C b/t
°c
M
MI 0.8" 0.6"
I
0.4. 02 0
20
40
Lh/=30 20
40
60
b/t
Fig. 9. Ultimate moment for local, lateral and coupled buckling for a double-channel section with L/h = 30. smaller value of such percentage would lead to a wider extension of the coupled effect field. It is evident from Figs 8 and 9 that fll has a constant value (//1 = 13.6), while//2 strongly depends on h/t and L/h ratios. The coupled effect field is, therefore, wider, as h/t and L/h are larger, and it might even disappear (f12
3
(8)
The coefficient m is related to the
L/h ratio by means of the equation:
i/L ,~o.85 m = v~)
(9)
with 1Oge V = --2"311 + 0"01 14(1Ogep) 7/3]
(lO)
where p
MR -- Meff _
M~
× 1000 is the limit difference between local and coupled buckling moment.
364
A. Ghersi, R. Landol[o, F. M. Mazzolani
Equation (10) is available in the range l < p < 1 0 0 ; for the above mentioned limit scatter o f p =47 we obtain v = 0.05443. These equations may be used to define explicitly a condition which must be satisfied by the geometrical dimensions of the section in order to avoid lateral or coupled buckling: b-3t
--7>
0.05443/L, °85
(11)
Figure 10 shows the fields of local, lateral and coupled buckling for the limit L/h ratio equal to 5, which corresponds to the L/h ratio of the central unrestrained part of the tested beams. The location of the experimental points indicates that only the case C exhibits a coupled behaviour, as emphasized by testing. Figure 11 gives the variation of the coupled instability field as a function of the L/h ratio. 6 CONCLUSION The identification of buckling modes is a useful means in order to foresee the actual behaviour of thin-walled beams. The physical intuition suggests that the coupled instability field becomes wider as the beam slenderness (L/h) increases. This behaviour is confirmed by the numerical analysis performed and quantified by the proposed expressions. The use of eqn (11) may be recommended to the designer in order to prevent the lateral buckling effect
h t 8O
Lateral
Buckling Local Buckling
D 40
El b
,' J 0
0
;
10
20
3'o
4'0 bit
Fig. 10. Fields of local, lateral and coupled buckling for a double-channel section with
L/h = 5.
Buckling modes of double-channel cold-formed beams
365
,f 4O
20
40
60
b~
Fig. 11. Influence of L/h ratio of the instability fields. by assuming appropriate values of cross-section dimensions. The analysis of the available testing results gives confirmation of the reliability of the ultimate moments provided by Eurocode 3.
REFERENCES 1. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., Il comportamento flessionale di profili sottili sagomati a freddo: impostazione della ricerca. Acciaio, Sept. 1989, pp. 415-22. 2. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., I1 comportamento flessionale di profili sottili sagomati a freddo: ricerca teorico-sperimentale. XII Congresso C.T.A., Capri, Oct. 1989, pp. 535-47. 3. De Martino, A., Ghersi, A. & Mazzolani, F. M., Analisi dei parametri di influenza del comportamento flessionale delle sezioni a cassone in par~te sottile. XII Congresso C.T.A., Capri, Oct. 1989, pp. 521-34. 4. De Martino, A., Ghersi, A. & Mazzolani, F. M., Calibration of a bending model for thin walled steel box-section. Int. Colloquium on Stability of Steel Structures, Budapest, April 1990. 5. De Martino, A., Ghersi, A. & Mazzolani, F. M., On the bending behaviour of thin walled box-sections. Costruzioni Metalliche, 6 (1991) 347-60. 6. Ballio, G. & Calado, L., Sezioni inflesse in acciaio sottoposte a carichi ciclici. Sperimentazione e simulazione numerica. Construzioni Metalliche, 1 (1986). 7. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., II comportamento flessionale di profili sottili sagomati a freddo: indagine sperimentale. Acciaio, Dec. 1990, pp. 577-588. 8. De Martino, A., De Martino, F. P., Ghersi, A. & Mazzolani, F. M., Bending behaviour of double-C thin-gauge beams: experimental evidence versus codification, 4th Int. Colloquium on Structural Stability, Mediterranean Section, Istanbul, Sept. 1991. 9. De Martino, A., Ghersi, A. & Mazzolani, F. M., Bending behaviour of
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10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A. Ghersi, R. Landolfo, F. M. Mazzolani double-C thin walled beams. X Int. Specialty Conj. on Cold-Formed Steel Structures, St. Louis, MO, Oct. 1990, pp. 637~48. De Martino, A., Ghersi, A., Landolfo, R. & Mazzolani, F. M., Calibration of a bending model for cold-formed sections. XI Int. Specialty Conj'. on ColdFormed Steel Structures, St. Louis, MO, Oct. 1992. Eurocode 3, Design of Steel Structures, 1990. Timoshenko, S. & Gere, J. M., Theoo' of Elastic Stability, 2nd edn., McGraw-Hill, New York, 1961. Vlasov, V. Z., Thin-Walled Elastic Beams, 2nd edn., National Science Foundation, Washington, DC, 1961. American Iron and Steel Institute, Specification for the Design of ColdFormed Steel Structural Members. Cold-Formed Steel Design Manual, Aug. 1986, with Dec. 1989 Addendum. Pekoz, T. B. & Winter, G., Torsional-flexural buckling of thin-walled sections under eccentric load. J. Struct. Div., ASCE, (May) (1969) 941-63. Galambos, T. V., Inelastic lateral buckling of beams. J. Struct. Div., ASCE, (Oct.) (1963) 217~42. CECM-ECCS, European Recommendations Jor Steel Construction, 1978. Winter, G., Strength of Thin Steel Compression Flanges, ASCE, Vol. 112, 1947. Ghersi, A. & Landolfo, R., Aspetti innovativi dell'EC3 nella verifica delle sezioni in parete sottile. Costruzioni Metalliche, 5 (1992) 288-302. De Wolf, J. T., Pekoz, T. & Winter, G., Local and overall buckling of coldformed members. J. Struct. Div., ASCE, (Oct.) (1974) 2017 36.