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Some specific aspects of elastic-plastic behaviour of profiled steel sheeting and decking
ELSEVIER
PII:
Thin-Walled Structures Vol. 29, Nos. 14, pp. 10l-112, 1997 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263-8231/97 $17.00 + .00 S0263-823 1 (97)00017-7
Some Specific Aspects of Elastic-Plastic Behaviour of Profiled Steel Sheeting and Decking
Leopold Sokol PAB Services, D6veloppement des Produits, 93 Rue des Trois Fontanot, 92000 Nanterre, France
ABSTRACT The non-linear behaviour of continuous sheeting and decking is greatly influenced by the complexity of the moment-rotation relationship in the region over the internal support. The rotation of the hinge created in this region modifies the deflexion and also causes the redistribution of moments and forces. Eurocode 3, Part 1.3, allows the use of plastic deformation in a global analysis for ultimate states, provided that the relationship between the support moment and the corresponding hinge rotation is obtained by testing. The ultimate limit state is defined as that which is created when plastic hinges occur at the support and also in the span. However, in order to ease the difficulties in calculation, some simplifying assumptions are adopted. The aim of this paper is to examine if these assumptions ensure sufficient accuracy in different practical conditions, and, in conclusion, to propose improvements to the assumptions. © 1998 Elsevier Science Ltd. All rights reserved
NOTATION Symbols E Y o u n g modulus
f
Deflexion
I L M q R 0
Second area o f section Span length Support m o m e n t Transversal load Support reaction Rotation 101
L. Sokol
102
Subscripts a p s
Allowed moment-reaction interaction condition Plastification condition Serviceability (limit deflexion) condition
INTRODUCTION When the combination of moment and reaction at an internal support in continuous sheeting or decking reaches its ultimate value, a plastic hinge appears in the section over the support. The rotation of this hinge modifies the deflexion and also causes the redistribution of moments and forces. Eurocode 3, Part 1.31 allows the use of plastic deformation in the global analysis for ultimate states, provided that the relationship between the support m o m e n t and the corresponding hinge rotation is obtained by testing. The ultimate limit state is defined as that created when a mechanism occurs with plastic hinges at the support and in the span. As the calculations are quite tedious, in order to make these easier, Ref. 1 proposes some simplifying assumptions. The aim of the present paper is to examine if these assumptions ensure sufficient accuracy and homogeneity of the degree of safety in different practical conditions, and, in conclusion, to propose some improvements to the assumptions.
D E F I N I T I O N OF T H E P R O B L E M Let us consider a continuous profiled sheeting under uniformly distributed, progressively increasing loading. While this loading is small, all sections are fully effective and the global behaviour of the system is elastic and linear, that means, the deformations are proportional to the load. We will define then this first stage as the
elastic-linear phase. Next, the most highly loaded sections become partially ineffective, although no plastic deformations occur. The deformations become nonlinear but still remain elastic, then we will define this second stage as the
elastic-non-linear phase. Afterwards, the first plastic stresses and deformations appear in the most highly loaded sections, situated at and near the internal support, because of the effect of interaction between m o m e n t and reaction as well as between m o m e n t and shear forces. A plastic hinge is created over the support and the system enters the third, plastic phase which remains, of course, non-linear.
Some specific aspects of elastic-plastic behaviour
103
The property of the hinge which occurs at the support is quite different from a classic hinge in thick, hot rolled sections. Here the moment-carrying capacity reduces as the rotation increases. When the loading increases, the support moment decreases and the span moments and deflexions increase in amplified proportion. Finally, when a second hinge is created in the span, collapse occurs by creation of a mechanism.
ANALYSIS TO E U R O C O D E 3, PART 1.31 The behaviour of the section situated over the support is defined in Ref. 1 by two items: • the resistance under combined support moment and support reaction, called 'moment-reaction (M-R) interaction', may be obtained either by calculation or by testing; • the moment-rotation (M-O) relationship may be obtained only by testing. However, in the literature some theoretical approaches are proposed.Z, 3 The M - R interaction is treated in a simplified way and shall satisfy the following relation that we present in a slightly different, but more general form than given in the Eurocode: M
R
CM_ R ~- --~00"~--~00 ~ F a M < Mmax R < Rma x
(1)
where M0 and R0 are the points of intersection of the inclined line (Fig. 1) with, respectively, the M (moment) and the R (reaction) axis, and F a is a safety (reduction) coefficient of the resistance at the support. The relationship between the support moment and the corresponding plastic hinge rotation obtained in practice is shown in Fig. 2. Figures 1 and 2 are taken from actual test results. The points marked by circles in Fig. 1 are the maximum design values (after a statistical treatment), corresponding to the maximal values reached by curves from the Fig. 2. Reference 1 states that, in order to avoid excessive plastic deformation under service conditions (serviceability limit state), the moment-reaction interaction should not exceed 0.9 times the combined design resistance. The two following phases of calculation are distinguished: • first, elastic (treated as linear) phase, governed by eqn (1) (Fig. 1), up to 0.9 times the combined design resistance of the section above internal support;
104
L. Sokol
~].. 300
.-. 2 ~ E
E "0
150
E 0
100
50
0
200
400
600
1000
1200
1400
1600
I~ton(eaqr@ Fig. 1. Internal support test. Moment-reaction interaction.
moment (daN*rdm 350
250 ¢ 200
0
L=0.680 m
[]
L=0.9055 m
.~
L=1.130m
X
L=1.3575 m
X
L=I .,580 m
150 100
,50 0! 0.05
0.1
0.15
0.2
rotation (rod)
Fig. 2. Internal support test. Moment-rotation relationship for different length of specimens.
Some specific aspects of elastic-plastic behaviour
105
• second, plastic (non-linear) phase, govemed by the M-0 relationship (Fig. 2), starting when the plastic hinge is created in the support and continuing until the creation of another plastic hinge in the span. In other words, the possible non-linearity under the limit conditions in the first phase is neglected and the domain between the end of first and the beginning of the second phase, is not considered.
THEORETICAL ANALYSIS OF SYSTEM IN EACH SUCCESSIVE PHASE General
Let us take one graph from Fig. 2 and present it in a more detailed manner in Fig. 3. All phases specified in Section 2 may be distinguished in this figure: • phase 1 lasts from point 0 to point 1, • phase 2 lasts from point 1 to point 2, • phase 3 lasts from point 2 to point 3.
moment (daN*m) 35O
%l 2 "e,
.,-~ ~....
__
a
I 0
~
0.05
0.1
0.15
0.2
0.25
rotation (rad)
Fig. 3. Internal support test. Moment-rotation graph.
0.3
106
L. Sokol
Point 1 corresponds to the linear limit moment Ml and point 2 corresponds to the limit moment inducing plastification, Mp. Note, that phase 2 may be simulated by an elastic, non-linear hinge over the internal support that we will call the 'equivalent conventional hinge'. Plastification load
Let us consider the system shown in Fig. 4. From the static equilibrium equation of this system: M p --
q~L2
3EIOp
-
2L
(2)
we get directly:
8MpL + 12EIOp qp ----
(3)
L3
where Mp is the plastification moment (see Fig. 3), qp is the plastification load, Op is the rotation corresponding to the plastification moment Mp (see Fig. 3) and L is the span length. The load qp may be calculated from (3) and (1) by iteration, starting with any arbitrary Mp value (with its corresponding rotation Op) and going in each step towards the ultimate limit value of interaction (1). The deflexionfp due to the load qp is: 5qpL 4
gpL 2
fP- 48EI
16EI
(4)
Limit deflexion load
When, under the limit deflexionfs, the support moment Ms is less than the limit linear moment Ml (Fig. 3), the load qs may be found by classic, elastic-linear calculation. "~ ®p •
/ deflexionline
Fig. 4. Two span system under the plastification load.
Some specific aspects of elastic-plastic behaviour
When, under the limit deflexion fs, M I < following conditions shall be satisfied: • equilibrium condition, similar to eqn (3): 8MsL + 12EIOs.
qs
=
L3
Ms
107
< Mp (Fig. 3), the two
(5)
,
• deformation condition: 5qsL 4
A - 48EI
Ms L2
(6)
16EI "
By putting (5) into (6), we get: fs-
M s L 2 5OsL 24-----E-I q- 3--T-
(7)
In eqn (7): • the limit deflexion f~ is given; • Ms and Os are not known, but they are linked by the known M-0 relationship (Fig. 3), so, we can solve by iteration. However, we can propose here a more elegant, direct, although approached procedure: after examination of a number of test results we have observed that between points 1 and 2 (Fig. 3) the moments and rotations vary in near elliptic function (plotted in dotted line), which the half axes are equal, respectively: Op for the rotation (horizontal); Mp-M~ for the moment (vertical);so, from the corresponding equation:
Op j
~
~
(8)
--1
we get:
-
Op
~-
1
+ ~t~.
(9/
By putting (9) into (7), we get the following equation, giving directly an approximate value of Os, replacing the iterative procedure: ab + C20p - c ~/2abOp + c202p - a 2 O~ = b2 + c2
)
/24EI where: a = ~ z - ~ f s - M 1 0 p a _ > 0
(10)
L. Sokol
108
b=
15EI _
~-~Op
c=M v -M 1 0 _< Os < Op
MI < Ms < Mp. Allowed interaction 1 load As we noted in Section 3, Ref. 1 states that, in order to avoid excessive plastic deformation under service conditions (serviceability limit state), the moment-reaction interaction should not exceed 0.9 times the combined design resistance, that we call 'allowed M-R interaction'. We note that the Pa value in eqn (1) becomes here equal to 0.9. The allowed load qa corresponding to this allowed M-R interaction is defined by the following equation, similar to (3):
8MaL + 12EIOa qa =
(11)
L3
where, as in Section 4.3, the rotation Oa corresponding to Ma may be taken directly from the test results (Fig. 3), or, alternatively, the approximate value can be found from eqn (8), by replacing the subscript 's' by 'a':
o p[1J1 /
/ Ma > MI-
(12)
Putting (12) into (11) and taking count of the allowed moment-reaction interaction (1), the load qa is calculated from (11) by iteration, in similar way as in Section 4.1. The support moment M~ and deflexion f~ are defined, respectively, by the equations equivalent to (9) and (7), by replacing the subscript 's' by 'a':
Ma
M p~)p _M,;I_
(Oa ~pp-- 1)2 + MI
(13)
Ma L2 5OaL (14) fa-- 24EI ~ 32 For Ma<_M~ the rotation Oa=0, then the load qa and the deflexionfa may be found by classic elastic-linear calculation.
Some specific aspects of elastic-plastic behaviour
109
EXAMPLES
Example 1 Data Two-span system, span length L = 1.80 m Mmax = 314 d a N * m M m i n :-- 235 d a N * m Rma x -- 1568 d a N Rmin -- 794 d a N I = 43.84 cm 4 M1 = 0 . 8 0 M p
Limit deflexion f~ = L/200 = 0.9 cm ~p=0.018.
Results (by lm width) P l a s t i f i c a t i o n c o n d i t i o n - - c a l c u l a t i o n w i t h safety ( r e d u c t i o n ) f a c t o r Fa = 1.0 The calculation with the rotation Op = 0.018 corresponding to the plastification m o m e n t Mp gives (see Section 4.2): plastification load (by iteration, from eqn (3)): qp = 942 d a N / m support m o m e n t (eqn (2)) Mp = 243 d a N * m support reaction (eqn (1)) Rp = 1966 d a N deflexion at mid-span (eqn (4)): fp = 0.86 cm = L/208
Calculation with safety (reduction) factor Fa = 0.9 The corresponding rotation, f o u n d from eqn (12), is Oa = 0.0024. With this rotation we get (see Section 4.4):
110 from from from from
L. Sokol
eqn eqn eqn eqn
(11) (13) (1) (14)
qa = 640 d a N / m M a = 240 d a N * m Ra = 1416 d a N fa = 0.42 cm = L/428 < f s (satisfying)
The linear (neglecting the rotation) calculation according to Ref. 1 (see Section 3), gives the following, inaccurate results: qa = 602 d a N / m Ma = 243 d a N * m R a = 1355 d a N fa = 0.36 cm = L/503 < f s
Example 2
Data T w o - s p a n system, span length L = 1.50 m Mmax = 310 d a N * m Mmi n = 205 d a N * m Rma x = 2387 d a N Rmin = 795 d a N I = 19.43 cm 4 M 1 = 0.65 Mp Limit deflexion fs = L/200 = 0.75 cm Op = 0.035.
Plastification condition--calculation with safety (reduction) factor Fa = 1.0 The c o r r e s p o n d i n g r o t a t i o n , f o u n d f r o m eqn (12), is: r o t a t i o n we get (see Section 4.2):
O a =
0.035. W i t h this
qp = 1280 d a N / m Mp = 217 d a N * m Rp = 2208 d a N Up = 1.32 cm = L/114 > f s (not satisfying) A linear (neglecting the rotation) calculation, gives the following, m u c h inaccurate results: qp = 898 d a N / m Mp = 252 d a N * m Rp = 1684 d a N fp = 0.58 cm = L/258
Some specific aspects of elastic-plastic behaviour
111
Calculation with safety (reduction) factor ])a 0.9 =
The corresponding rotation, found from eqn (12), is: this rotation we get (see Section 4.4): qa =
Oa=0.0105.
With
923 d a N / m
M a = 217 d a N * m Ra = 1673 d a N fa = 0.74 cm = L/202
The linear (neglecting the rotation) calculation according to Ref. i (see Section 3), gives the following inaccurate results: qa = 808 d a N / m Ma = 227 daN*m Ra-- 1515daN f~ = 0.52 cm = L/287
It should be noted that the above examples are realistic, calculated with actual characteristics of sections, determined by testing. However, the relatively small span values have been adopted with intent to demonstrate the differences between the results obtained with the simplified calculation according to Ref. ~ and the calculation taking into account the actual behaviour of the section over the support. The differences become more emphasised when the governing conditions are those of resistance, which occur rather for small spans with large loads. On the other hand, when the governing conditions are those of deformations, which occur rather for large spans with small loads, a reserve of resistance necessarily exists, so the behaviour of the system remains more or completely linear and the simplified assumptions of Ref. become more accurate. CONCLUSIONS The calculation according to Ref. 1, neglecting the non-linearity of the system under the allowed interaction condition, leads necessarily to the inaccurate results, when the rotation of the conventional equivalent hinge, defined in Section 4.1, is a non-zero value. This is more typical for small spans with large loads, which is the case of the sheeting for composite steel and concrete slabs. The larger is the rotation, the larger are the inaccuracies. In these circumstances, as a general rule:
112
• • • •
L. Sokol
the the the the
deflexion is under-estimated; support m o m e n t is over-estimated; support reaction is under-estimated; corresponding allowed load is under-estimated.
In conclusion, we consider that the taking account o f non-linear behaviour under the allowed interaction condition is r e c o m m e n d e d for improving the reliability o f the results o f calculation.
REFERENCES 1. Eurocode 3, ENV 1993, Design of steel structures. Part 1.3: cold formed thin gauge members and sheeting (final draft). 2. Tsai, Y.-M. Comportement sur appuis de trles minces formres/t froid. Thrse no. 689, Ecole Polytechnique Frdrrale de Lausanne, 1987. 3. Luure, P. and Crisinel, M., Redistribution of moments at the interior support for cold formed sheeting. ICOM 288, Ecole Polytechnique Frdrrale de Lausanne, October 1993.
PII:
Thin-Walled Structures Vol. 29, Nos. 14, pp. 10l-112, 1997 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263-8231/97 $17.00 + .00 S0263-823 1 (97)00017-7
Some Specific Aspects of Elastic-Plastic Behaviour of Profiled Steel Sheeting and Decking
Leopold Sokol PAB Services, D6veloppement des Produits, 93 Rue des Trois Fontanot, 92000 Nanterre, France
ABSTRACT The non-linear behaviour of continuous sheeting and decking is greatly influenced by the complexity of the moment-rotation relationship in the region over the internal support. The rotation of the hinge created in this region modifies the deflexion and also causes the redistribution of moments and forces. Eurocode 3, Part 1.3, allows the use of plastic deformation in a global analysis for ultimate states, provided that the relationship between the support moment and the corresponding hinge rotation is obtained by testing. The ultimate limit state is defined as that which is created when plastic hinges occur at the support and also in the span. However, in order to ease the difficulties in calculation, some simplifying assumptions are adopted. The aim of this paper is to examine if these assumptions ensure sufficient accuracy in different practical conditions, and, in conclusion, to propose improvements to the assumptions. © 1998 Elsevier Science Ltd. All rights reserved
NOTATION Symbols E Y o u n g modulus
f
Deflexion
I L M q R 0
Second area o f section Span length Support m o m e n t Transversal load Support reaction Rotation 101
L. Sokol
102
Subscripts a p s
Allowed moment-reaction interaction condition Plastification condition Serviceability (limit deflexion) condition
INTRODUCTION When the combination of moment and reaction at an internal support in continuous sheeting or decking reaches its ultimate value, a plastic hinge appears in the section over the support. The rotation of this hinge modifies the deflexion and also causes the redistribution of moments and forces. Eurocode 3, Part 1.31 allows the use of plastic deformation in the global analysis for ultimate states, provided that the relationship between the support m o m e n t and the corresponding hinge rotation is obtained by testing. The ultimate limit state is defined as that created when a mechanism occurs with plastic hinges at the support and in the span. As the calculations are quite tedious, in order to make these easier, Ref. 1 proposes some simplifying assumptions. The aim of the present paper is to examine if these assumptions ensure sufficient accuracy and homogeneity of the degree of safety in different practical conditions, and, in conclusion, to propose some improvements to the assumptions.
D E F I N I T I O N OF T H E P R O B L E M Let us consider a continuous profiled sheeting under uniformly distributed, progressively increasing loading. While this loading is small, all sections are fully effective and the global behaviour of the system is elastic and linear, that means, the deformations are proportional to the load. We will define then this first stage as the
elastic-linear phase. Next, the most highly loaded sections become partially ineffective, although no plastic deformations occur. The deformations become nonlinear but still remain elastic, then we will define this second stage as the
elastic-non-linear phase. Afterwards, the first plastic stresses and deformations appear in the most highly loaded sections, situated at and near the internal support, because of the effect of interaction between m o m e n t and reaction as well as between m o m e n t and shear forces. A plastic hinge is created over the support and the system enters the third, plastic phase which remains, of course, non-linear.
Some specific aspects of elastic-plastic behaviour
103
The property of the hinge which occurs at the support is quite different from a classic hinge in thick, hot rolled sections. Here the moment-carrying capacity reduces as the rotation increases. When the loading increases, the support moment decreases and the span moments and deflexions increase in amplified proportion. Finally, when a second hinge is created in the span, collapse occurs by creation of a mechanism.
ANALYSIS TO E U R O C O D E 3, PART 1.31 The behaviour of the section situated over the support is defined in Ref. 1 by two items: • the resistance under combined support moment and support reaction, called 'moment-reaction (M-R) interaction', may be obtained either by calculation or by testing; • the moment-rotation (M-O) relationship may be obtained only by testing. However, in the literature some theoretical approaches are proposed.Z, 3 The M - R interaction is treated in a simplified way and shall satisfy the following relation that we present in a slightly different, but more general form than given in the Eurocode: M
R
CM_ R ~- --~00"~--~00 ~ F a M < Mmax R < Rma x
(1)
where M0 and R0 are the points of intersection of the inclined line (Fig. 1) with, respectively, the M (moment) and the R (reaction) axis, and F a is a safety (reduction) coefficient of the resistance at the support. The relationship between the support moment and the corresponding plastic hinge rotation obtained in practice is shown in Fig. 2. Figures 1 and 2 are taken from actual test results. The points marked by circles in Fig. 1 are the maximum design values (after a statistical treatment), corresponding to the maximal values reached by curves from the Fig. 2. Reference 1 states that, in order to avoid excessive plastic deformation under service conditions (serviceability limit state), the moment-reaction interaction should not exceed 0.9 times the combined design resistance. The two following phases of calculation are distinguished: • first, elastic (treated as linear) phase, governed by eqn (1) (Fig. 1), up to 0.9 times the combined design resistance of the section above internal support;
104
L. Sokol
~].. 300
.-. 2 ~ E
E "0
150
E 0
100
50
0
200
400
600
1000
1200
1400
1600
I~ton(eaqr@ Fig. 1. Internal support test. Moment-reaction interaction.
moment (daN*rdm 350
250 ¢ 200
0
L=0.680 m
[]
L=0.9055 m
.~
L=1.130m
X
L=1.3575 m
X
L=I .,580 m
150 100
,50 0! 0.05
0.1
0.15
0.2
rotation (rod)
Fig. 2. Internal support test. Moment-rotation relationship for different length of specimens.
Some specific aspects of elastic-plastic behaviour
105
• second, plastic (non-linear) phase, govemed by the M-0 relationship (Fig. 2), starting when the plastic hinge is created in the support and continuing until the creation of another plastic hinge in the span. In other words, the possible non-linearity under the limit conditions in the first phase is neglected and the domain between the end of first and the beginning of the second phase, is not considered.
THEORETICAL ANALYSIS OF SYSTEM IN EACH SUCCESSIVE PHASE General
Let us take one graph from Fig. 2 and present it in a more detailed manner in Fig. 3. All phases specified in Section 2 may be distinguished in this figure: • phase 1 lasts from point 0 to point 1, • phase 2 lasts from point 1 to point 2, • phase 3 lasts from point 2 to point 3.
moment (daN*m) 35O
%l 2 "e,
.,-~ ~....
__
a
I 0
~
0.05
0.1
0.15
0.2
0.25
rotation (rad)
Fig. 3. Internal support test. Moment-rotation graph.
0.3
106
L. Sokol
Point 1 corresponds to the linear limit moment Ml and point 2 corresponds to the limit moment inducing plastification, Mp. Note, that phase 2 may be simulated by an elastic, non-linear hinge over the internal support that we will call the 'equivalent conventional hinge'. Plastification load
Let us consider the system shown in Fig. 4. From the static equilibrium equation of this system: M p --
q~L2
3EIOp
-
2L
(2)
we get directly:
8MpL + 12EIOp qp ----
(3)
L3
where Mp is the plastification moment (see Fig. 3), qp is the plastification load, Op is the rotation corresponding to the plastification moment Mp (see Fig. 3) and L is the span length. The load qp may be calculated from (3) and (1) by iteration, starting with any arbitrary Mp value (with its corresponding rotation Op) and going in each step towards the ultimate limit value of interaction (1). The deflexionfp due to the load qp is: 5qpL 4
gpL 2
fP- 48EI
16EI
(4)
Limit deflexion load
When, under the limit deflexionfs, the support moment Ms is less than the limit linear moment Ml (Fig. 3), the load qs may be found by classic, elastic-linear calculation. "~ ®p •
/ deflexionline
Fig. 4. Two span system under the plastification load.
Some specific aspects of elastic-plastic behaviour
When, under the limit deflexion fs, M I < following conditions shall be satisfied: • equilibrium condition, similar to eqn (3): 8MsL + 12EIOs.
qs
=
L3
Ms
107
< Mp (Fig. 3), the two
(5)
,
• deformation condition: 5qsL 4
A - 48EI
Ms L2
(6)
16EI "
By putting (5) into (6), we get: fs-
M s L 2 5OsL 24-----E-I q- 3--T-
(7)
In eqn (7): • the limit deflexion f~ is given; • Ms and Os are not known, but they are linked by the known M-0 relationship (Fig. 3), so, we can solve by iteration. However, we can propose here a more elegant, direct, although approached procedure: after examination of a number of test results we have observed that between points 1 and 2 (Fig. 3) the moments and rotations vary in near elliptic function (plotted in dotted line), which the half axes are equal, respectively: Op for the rotation (horizontal); Mp-M~ for the moment (vertical);so, from the corresponding equation:
Op j
~
~
(8)
--1
we get:
-
Op
~-
1
+ ~t~.
(9/
By putting (9) into (7), we get the following equation, giving directly an approximate value of Os, replacing the iterative procedure: ab + C20p - c ~/2abOp + c202p - a 2 O~ = b2 + c2
)
/24EI where: a = ~ z - ~ f s - M 1 0 p a _ > 0
(10)
L. Sokol
108
b=
15EI _
~-~Op
c=M v -M 1 0 _< Os < Op
MI < Ms < Mp. Allowed interaction 1 load As we noted in Section 3, Ref. 1 states that, in order to avoid excessive plastic deformation under service conditions (serviceability limit state), the moment-reaction interaction should not exceed 0.9 times the combined design resistance, that we call 'allowed M-R interaction'. We note that the Pa value in eqn (1) becomes here equal to 0.9. The allowed load qa corresponding to this allowed M-R interaction is defined by the following equation, similar to (3):
8MaL + 12EIOa qa =
(11)
L3
where, as in Section 4.3, the rotation Oa corresponding to Ma may be taken directly from the test results (Fig. 3), or, alternatively, the approximate value can be found from eqn (8), by replacing the subscript 's' by 'a':
o p[1J1 /
/ Ma > MI-
(12)
Putting (12) into (11) and taking count of the allowed moment-reaction interaction (1), the load qa is calculated from (11) by iteration, in similar way as in Section 4.1. The support moment M~ and deflexion f~ are defined, respectively, by the equations equivalent to (9) and (7), by replacing the subscript 's' by 'a':
Ma
M p~)p _M,;I_
(Oa ~pp-- 1)2 + MI
(13)
Ma L2 5OaL (14) fa-- 24EI ~ 32 For Ma<_M~ the rotation Oa=0, then the load qa and the deflexionfa may be found by classic elastic-linear calculation.
Some specific aspects of elastic-plastic behaviour
109
EXAMPLES
Example 1 Data Two-span system, span length L = 1.80 m Mmax = 314 d a N * m M m i n :-- 235 d a N * m Rma x -- 1568 d a N Rmin -- 794 d a N I = 43.84 cm 4 M1 = 0 . 8 0 M p
Limit deflexion f~ = L/200 = 0.9 cm ~p=0.018.
Results (by lm width) P l a s t i f i c a t i o n c o n d i t i o n - - c a l c u l a t i o n w i t h safety ( r e d u c t i o n ) f a c t o r Fa = 1.0 The calculation with the rotation Op = 0.018 corresponding to the plastification m o m e n t Mp gives (see Section 4.2): plastification load (by iteration, from eqn (3)): qp = 942 d a N / m support m o m e n t (eqn (2)) Mp = 243 d a N * m support reaction (eqn (1)) Rp = 1966 d a N deflexion at mid-span (eqn (4)): fp = 0.86 cm = L/208
Calculation with safety (reduction) factor Fa = 0.9 The corresponding rotation, f o u n d from eqn (12), is Oa = 0.0024. With this rotation we get (see Section 4.4):
110 from from from from
L. Sokol
eqn eqn eqn eqn
(11) (13) (1) (14)
qa = 640 d a N / m M a = 240 d a N * m Ra = 1416 d a N fa = 0.42 cm = L/428 < f s (satisfying)
The linear (neglecting the rotation) calculation according to Ref. 1 (see Section 3), gives the following, inaccurate results: qa = 602 d a N / m Ma = 243 d a N * m R a = 1355 d a N fa = 0.36 cm = L/503 < f s
Example 2
Data T w o - s p a n system, span length L = 1.50 m Mmax = 310 d a N * m Mmi n = 205 d a N * m Rma x = 2387 d a N Rmin = 795 d a N I = 19.43 cm 4 M 1 = 0.65 Mp Limit deflexion fs = L/200 = 0.75 cm Op = 0.035.
Plastification condition--calculation with safety (reduction) factor Fa = 1.0 The c o r r e s p o n d i n g r o t a t i o n , f o u n d f r o m eqn (12), is: r o t a t i o n we get (see Section 4.2):
O a =
0.035. W i t h this
qp = 1280 d a N / m Mp = 217 d a N * m Rp = 2208 d a N Up = 1.32 cm = L/114 > f s (not satisfying) A linear (neglecting the rotation) calculation, gives the following, m u c h inaccurate results: qp = 898 d a N / m Mp = 252 d a N * m Rp = 1684 d a N fp = 0.58 cm = L/258
Some specific aspects of elastic-plastic behaviour
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Calculation with safety (reduction) factor ])a 0.9 =
The corresponding rotation, found from eqn (12), is: this rotation we get (see Section 4.4): qa =
Oa=0.0105.
With
923 d a N / m
M a = 217 d a N * m Ra = 1673 d a N fa = 0.74 cm = L/202
The linear (neglecting the rotation) calculation according to Ref. i (see Section 3), gives the following inaccurate results: qa = 808 d a N / m Ma = 227 daN*m Ra-- 1515daN f~ = 0.52 cm = L/287
It should be noted that the above examples are realistic, calculated with actual characteristics of sections, determined by testing. However, the relatively small span values have been adopted with intent to demonstrate the differences between the results obtained with the simplified calculation according to Ref. ~ and the calculation taking into account the actual behaviour of the section over the support. The differences become more emphasised when the governing conditions are those of resistance, which occur rather for small spans with large loads. On the other hand, when the governing conditions are those of deformations, which occur rather for large spans with small loads, a reserve of resistance necessarily exists, so the behaviour of the system remains more or completely linear and the simplified assumptions of Ref. become more accurate. CONCLUSIONS The calculation according to Ref. 1, neglecting the non-linearity of the system under the allowed interaction condition, leads necessarily to the inaccurate results, when the rotation of the conventional equivalent hinge, defined in Section 4.1, is a non-zero value. This is more typical for small spans with large loads, which is the case of the sheeting for composite steel and concrete slabs. The larger is the rotation, the larger are the inaccuracies. In these circumstances, as a general rule:
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L. Sokol
the the the the
deflexion is under-estimated; support m o m e n t is over-estimated; support reaction is under-estimated; corresponding allowed load is under-estimated.
In conclusion, we consider that the taking account o f non-linear behaviour under the allowed interaction condition is r e c o m m e n d e d for improving the reliability o f the results o f calculation.
REFERENCES 1. Eurocode 3, ENV 1993, Design of steel structures. Part 1.3: cold formed thin gauge members and sheeting (final draft). 2. Tsai, Y.-M. Comportement sur appuis de trles minces formres/t froid. Thrse no. 689, Ecole Polytechnique Frdrrale de Lausanne, 1987. 3. Luure, P. and Crisinel, M., Redistribution of moments at the interior support for cold formed sheeting. ICOM 288, Ecole Polytechnique Frdrrale de Lausanne, October 1993.