SN030
Short Description
SN030...
Description
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending bending and axial compression SN030a-EN-EU
NCCI CCI: Mono-symmetrical uni form members under bendin g and axial compression This NCCI gives a method for the elastic verification of mono-symmetrical uniform members under under bending and axial compression compression
Contents
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
1.
Scope
2
2.
Notations and geometrical properties of the cross-section
2
3.
Member resistance according to EN 1993-1-1
4
4.
Evaluation of the elastic critical moment
9
5.
Evaluation of the non dimensional slenderness
12
6.
Information about LT Beam freeware to calculate the elastic critical moment
13
7.
References
14
Page 1
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
1.
Scope
This NCCI provides information for dealing with mono-symmetrical uniform members subjected to bending and axial compression satisfying the following conditions: The verification is restricted to the elastic behaviour of the member The cross-section is symmetrical about the weak axis The flanges and the web are made of the same steel grade The loads create bending moments about the strong axis only The axial load is expected to be applied at the centroid of the cross-section The web is made of a solid plate of constant thickness The effects of the fillet welds are not taken into account Note 1: Such a mono-symmetrical cross-section is susceptible to torsional-flexural buckling |3|. Note 2: This kind of cross-section can be found, for instance, in composite structures where the upper flange of the beam is connected to a composite slab by means of shear connectors. Then, the following calculations are required in the non-composite stage when the fresh concrete acts only as an external load. In this case, the smaller flange is generally mainly in compression. This kind of cross-section can be found also in welded cross-sections when a higher resistance to torsional-flexural buckling is needed for the member. In this case, the smaller flange is generally mainly in tension. Note 3: Cellular beams or beams made of two different hot-rolled profiles are not covered by this NCCI.
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
2. Notations and geometrical properties of the cross-section The dimensional characteristics of the cross-section are shown in Figure 2.1.
Page 2
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
b1
z t1
y
1 tw
y
G S
zG
hw
hs
h
t2
zSC
2 z b2
Key:
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
1
Top fibre
2
Bottom fibre
Figure 2.1
Notations
The geometrical properties are the followings |2|, |8|, |10|: Area A = b1 t 1 + hw
zG
I y
+ b2 t 2 (1)
t w
Position of the centroid from the bottom fibre of the cross-section
=
b1 t 1 (h −
t 1
2
) + hw t w (t 2 +
hw
2
) + b2
2
t 2
2
(2)
A
Second moment of area about the strong axis y-y =
b1 t 13 + b2 t 23 + t w hw3
12
2
t ⎞ + b1 t 1 ⎛ ⎜ h − 1 − zG ⎟ + ... ⎝ 2 ⎠ 2
h ⎞ ⎛ t ⎞ ... + hw t w ⎛ ⎜ t 2 + w − zG ⎟ + b2 t 2 ⎜ 2 − zG ⎟ 2 ⎝ 2 ⎠ ⎝ ⎠
I z
2
(3)
Second moment of area about the weak axis z-z =
3 b13t 1 + b23t 2 + hw t w
(4) 12 Elastic section modulus:
W el, y, top
=
W el, y, bottom
I y h − z G
=
I y z G
(5) (6) Page 3
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
Position of the shear centre S from the bottom fibre of the cross-section:
=
St Venant torsional constant
I w
2
+ hs
b13 t 1
zSC
I T
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s e 0 r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
t 2
b23 t 2
(7)
+ b13 t 1
b1 t 13
+ b2 t 23 + hw t w3 = 3 Warping constant: =
hs2 I z
(8)
(b13t 1 b23t 2 ) (b13t 1 + b23t 2 )2
(9)
3.
Member resistance according to EN 1993-1-1
3.1
General
As bending is expected to occur about the strong axis, the verification of member stability is based on Clause (6.3.3) of EN 1993-1-1 |4| with M z,Ed = 0. Nevertheless, the method given in Clause (6.3.3) is restricted to uniform members with double symmetric cross-sections. But, it may be extended to uniform members with monosymmetric cross-sections (symmetrical about the weak axis) when the following conditions are satisfied:
the elastic resistance is the only one to be considered (elastic resistance of the whole section for Class 1, 2 or 3 cross-sections and elastic resistance of the effective crosssection for Class 4 cross-sections),
χ y and χ z must be replaced by χ min
= min( χ y , χ z , χ TF ) in (6.61) and (6.62) if χ y and χ z are the reduction factors due to flexural buckling and χ TF is the reduction factor for torsional-flexural buckling (see Section 5.2), In Table A.2 of Annex A, the equivalent uniform moment factors must be limited to:
C my ,0
≥ 1−
N Ed
(10)
N cr, y
Therefore, such members should satisfy |1|, |9|: N Ed χ min N Rk
+ k yy
χ LT
γ M1 N Ed χ min N Rk
γ M1
M y,Ed
+ k zy
M y,Ed
χ LT
+ Δ M y,Ed M y,Rk
≤1
(11)
≤1
(12)
γ M1
+ Δ M y,Ed M y,Rk
γ M1
Page 4
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
where:
and M y,Ed are respectively the design values of the compression force and the maximum moments about the y-y axis along the member, Δ M y,Ed is the moment due to the shift of the centroidal axis in the case of Class 4 cross-sections (see Section 3.3),
N Ed
and M y,Rk are respectively the characteristic resistance to normal force and the characteristic moment resistance about the y-y axis,
N Rk
γ M1 is the partial factor for resistance of members to instability , χ min is the relevant reduction factor: χ min
= min( χ y , χ z , χ TF ) ,
χ LT is the reduction factor due to lateral torsional buckling (see Section 5.1), k yy
and
k zy
are interaction factors.
and k zy have been derived from two alternative approaches provided in Annex A (alternative Method 1) and Annex B (alternative Method 2) of EN 1993-1-1 |4|. Method 1 (see Section 3.3) has been established to provide an accurate fully-theoretical derived set of formulae. Method 2 is simpler than Method 1. It has been established to be a user-friendly set of formulae. The National Annex may give a choice from alternative Methods 1 or 2. k yy
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s e 0 r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
Note:
It is important to notice that, in both cases, the resistance of the cross-sections must be checked at each end of the member.
3.2
Susceptibil ity to torsion al deformatio ns
Some moment factors depending on the susceptibility of the member to experience torsional deformations, it is necessary to clarify the boundary of this phenomenon. The susceptibility to torsional deformations depends on the value of
λ 0 ,
the non dimensional
slenderness for lateral torsional buckling due to uniform bending moment. The limiting value λ 0,lim is the following:
λ 0,lim
= 0,2
where:
C 1 4 N cr, z N cr,TF
⎛ N Ed ⎞⎛ N Ed ⎞ ⎜1 − ⎟⎜ ⎟ ⎜ N cr,z ⎟⎜1 − N cr,TF ⎟ ⎝ ⎠⎝ ⎠
(13)
is the elastic flexural buckling force about the z-z axis, is the elastic torsional-flexural buckling force (see Section 5.2),
is a factor depending on the bending moment distribution and end restraint conditions (see Section 4). C 1
If λ 0 ≤ λ 0,lim , the member is not susceptible to torsional deformations. In that case, lateral torsional buckling is also prevented and χ LT = 1 .
If
λ 0
> λ 0,lim , the member is susceptible to torsional deformations.
Page 5
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
3.3 Elastic resistance obtained from alternative Method 1 (Annex A)
Class 1, 2 and 3 cross-sections
For Class 1, 2 and 3 cross-sections, the member should satisfy: N Ed A f y
χ min
χ min
χ LT
k yy
k zy
W el,y f y
μ y
≤1
(15)
W el,y f y
γ M1
= C my C mLT
= C my C mLT
=
=
y
1−
N Ed
(16)
(17)
N cr,y z
1−
N Ed N cr,y
N Ed N cr, y
1 − χ y 1−
μ z
(14)
γ M1
1− in which:
≤ 1
M y,Ed
+ k zy
γ M1
where:
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s e 0 r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
χ LT
γ M1
N Ed A f y
M y,Ed
+ k yy
N Ed
(18)
(19)
N cr,y
N Ed N cr,z
1 − χ z
N Ed N cr,z
N cr, y
is the elastic flexural buckling force about the y-y axis,
N cr, z
is the elastic flexural buckling force about the z-z axis,
and C mLT are uniform moment factors depending on the susceptibility of the member to torsional deformations (see hereinafter). C my
Class 4 cross-sections
For Class 4 cross-sections, equations (14) and (15) become: N Ed Aeff f y
χ min
γ M1
+ k yy
M y,Ed
χ LT
+ e N, y N Ed W eff,y f y
≤1
(20)
γ M1
Page 6
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
N Ed Aeff f y
χ min
where:
+ k zy
M y,Ed
χ LT
γ M1
+ e N,y N Ed W eff, y f y
≤1
(21)
γ M1
is the shift of the centroid of the effective area calculated under pure compression, e N, y
is the effective area determined under pure compression,
Aeff
W eff,y
is the effective section modulus about the y-y axis determined under bending
only, and
k yy
k zy
are interaction factors defined by equations (16) and (17).
Influence of the susceptibility to torsional deformations
•
If the member is not susceptible to torsional deformations: = C my,0
C my
(see Table A.2 in EN 1993-1-1 |4|)
C mLT = 1,0 t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
and:
• then:
χ LT = 1
If the member is susceptible to torsional deformations:
= C my,0 + (1 − C my,0 )
C my
C mLT
2 = C my
and: where:
ε y
=
ε y
=
or: and:
aLT
ε y aLT
1 + ε y
aLT
(see Table A.2 in EN 1993-1-1 |4|)
aLT
⎛ N Ed ⎞⎛ N Ed ⎞ ⎜1 − ⎟⎜ ⎟ ⎜ N cr,z ⎟⎜1 − N cr,TF ⎟ ⎝ ⎠⎝ ⎠
M y,Ed A N Ed W el, y
for Class 1, 2 and 3 cross-sections
M y,Ed Aeff N Ed W eff, y
= 1−
I T I y
for Class 4 cross-sections
≥0
Page 7
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
3.4 Elastic resistance obtained from alternative Method 2 (Annex B)
Class 1, 2 and 3 cross-sections
For Class 1, 2 and 3 cross-sections, the member should satisfy: N Ed A f y
χ min
χ min
W el, y f y
χ LT
≤1
(22)
≤1
(23)
γ M1
M y,Ed
+ k zy
γ M1
W el,y f y
γ M1
⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ N Ed N Ed ⎜ ⎟ ⎜ ⎟ (24) k yy = C my 1 + 0,6λ y C my 1 + 0,6 ≤ ⎜ ⎜ A f y ⎟ A f y ⎟ χ min χ min ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ γ γ ⎝ M1 ⎠ M1 ⎠ ⎝ k zy depends on the susceptibility of the member to torsional deformations (see below).
where: t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
χ LT
γ M1
N Ed A f y
M y,Ed
+ k yy
Class 4 cross-sections
For Class 4 cross-sections, equations (14) and (15) become: N Ed Aeff f y
χ min
where:
M y,Ed
χ LT
γ M1
N Ed Aeff f y
χ min
+ k yy
+ k zy
γ M1
M y,Ed
χ LT
+ e N,y N Ed W eff,y f y
≤1
(25)
≤1
(26)
γ M1
+ e N,y N Ed W eff, y f y
γ M1
⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ N Ed N Ed ⎜ ⎟ ⎜ ⎟ k yy = C my 1 + 0,6λ y C my 1 + 0,6 ≤ ⎜ ⎜ Aeff f y ⎟ Aeff f y ⎟ χ min χ min ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ γ γ ⎝ M1 ⎠ M1 ⎠ ⎝ k zy
(27)
depends on the susceptibility of the member to torsional deformation (see below).
The equivalent moment factor C my is given in Table B.3 in EN 1993-1-1 |4|.
Influence of the susceptibility to torsional deformations
Members not susceptible to torsional deformations: k zy
= 0,8 k yy
(28)
Therefore, because the first terms of equations (22) and (23) or (25) and (26) are the same, in that case, the most critical relationship is the first one of each set respectively. Page 8
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
Members susceptible to torsional deformations: k zy
= 1−
0,05 λ z (C mLT − 0,25)
N Ed A f y
χ min
γ M1
0,05 ≥ 1− (C mLT − 0,25)
k zy
= 1−
0,05 λ z (C mLT − 0,25)
N Ed A f y
χ min
(29)
for Class 1, 2 or 3 cross-sections
γ M1
N Ed Aeff f y
χ min
γ M1
0,05 ≥ 1− (C mLT − 0,25)
N Ed Aeff f y
χ min
(30)
for Class 4 cross-sections
γ M1
The equivalent moment factor C mLT is given in Table B.3 in EN 1993-1-1 |4|. t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
4.
Evaluation of the elastic critical moment
In the case of a member of uniform cross-section symmetrical about the weak axis, the critical moment for lateral torsional buckling is: 2 ⎧ ⎫ 2 k z ⎞ I w (k z L ) GI T π2 EI z ⎪ ⎛ 2 ⎜ ⎟⎟ + + (C 2 zg − C 3 z j ) − (C 2 zg − C 3 z j )⎪⎬ (31) M cr = C 1 2 ⎨ ⎜ 2 π EI z (k z L ) ⎪ ⎝ k w ⎠ I z ⎪⎭ ⎩ where: L is the length of the member between points where lateral restraint is provided,
C 1 , C 2
and C 3 are factors depending on the loading and the end restraint conditions (see Table 4.1 and Table 4.2), k z
is the effective length factor that refers to the end rotation about the z axis,
k w
is the effective length factor that refers to the end warping,
zg
= za − zs
z j = zs −
0,5
2 2 z (( y + z )dA (approximations are given hereinafter |6|) ∫ I y A
za
and:
is the coordinate of the point of load application,
is the coordinate of the shear centre: ( zs = zG − zSC according to notations given in Figure 2.1).
zs
Page 9
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
The effective length factors, k z and k w , take the following values: • 0,5 full restraint at both ends,
• 0,7 one end fixed and one end free, • 1,0 no restraint, the normal conditions of restraint at each end being: k z = 1,0 free to rotate about the z axis and restraint against lateral movements, • •
k w
= 1,0 free to warp but restraint against rotation about the longitudinal axis.
Sign conventions for z , za , z g and z j
The sign conventions for z , za , zg and z j are defined in Figure 4.1. They are the followings:
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
• •
z is positive from the centroid of
•
z g is
•
is positive when the flange with the larger value of I z is in compression at the point of the larger bending moment.
za
the cross-section to the compression flange,
is positive when the loads have a destabilising effect,
positive when the loads act towards the shear centre from their point of application.
z j
1
z j>0 z 2
za>0
3
zg>0
G
zs
y
y
G
zs
S
S
3
2
za 40 mm, then α TF = 0,76 (buckling curve d) t f min
= min(t sup , t inf ) .
6. Information about LTBeam freeware to calculate the elastic critical moment It is to be noted that, in order to help in solving the calculation of M cr for different loading and support conditions, a freeware is available. Named LT Beam, it may be downloaded from CTICM site (www.cticm.com). This software allows determining the elastic critical moment for mono-symmetrical uniform members with various loading cases including the effect of the position of the applied load. A short presentation in English in available in Chapter 7.3 of |5| and in French in |7|.
Page 13
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
7.
References
1 N. Boissonnade, R. Greiner & J.P. Jaspart “Rules for Member stability in EN 1993-1-1. Background documentation and design guidelines” - ECCS Technical Committee 8 “Stability” (to be published).
2
A. Bureau “Résistance plastique en flexion composée d’une section en I mono-symétrique” – Construction Métallique, n°1-1997, pp. 41-52.
3
A. Bureau “Flambement par torsion et par flexion-torsion d’une barre comprimée” – Construction Métallique, n°2-2004, pp. 39-54.
4
EN 1993-1-1:2004 “Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings”
5 t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
ECSC Steel RTD Programme “Lateral Torsional Buckling in Steel and Composite Beams” N° 7210-PR-183 (1999-2002) Final Technical Report – Book 2: “Design Guide” - Chapters 3 and 7.3.
6
Eurocode 3 – “Calcul des structures en acier – Partie 1-1: Règles générales et règles pour les bâtiments” Eyrolles Paris, 1996.
7
Y. Galéa “Moment critique de déversement élastique de poutres fléchies. Présentation du logiciel LTBEAM” – Construction Métallique, n°2-2003, pp. 47-76.
8
M. Pignataro, N. Rizzi and A. Luongo “Stability, bifurcation and post-critical behaviour of elastic structures” – Development in Civil Engineering, vol. 39, Elsevier, 1991.
9
M. Villette “Analyse critique du traitement de la barre comprimée et fléchie et propositions de nouvelles formulations” – PhD Thesis, University of Liège, Belgium, 14 January 2005.
10 B.Z. Vlassov “Pièces longues et voiles minces” – 2ème édition, Éditions Eyrolles, Paris, 1962.
Page 14
NCCI: Mono-symmetrical uniform members under bending and axial compression
NCCI: Mono-symmetrical uniform members under bending and axial compression SN030a-EN-EU
Quality Record RESOURCE TITLE
NCCI: Mono-symmetrical uniform members under bending and axial compression
Reference(s) ORIGINAL DOCUMENT Name
Compan y
Date
Created b y
Jean-Pierre Muzeau
CUST
21/12/2005
Technical content checked by
Alain BUREAU
CTICM
21/12/2005
Editorial content checked by
D C Iles
SCI
March 2006
1. UK
G W Owens
SCI
10/3/06
2. France
Alain Bureau
CTICM
10/3/06
3. Sweden
A Olsson
SBI
10/3/06
4. Germany
C Müller
RWTH
10/3/06
5. Spain
J Chica
Labein
10/3/06
G W Owens
SCI
24/6/06
Technical content endorsed by the following STEEL Partners:
t n e m e e r g A e c n e c i L l e e t S s s e c c A e h t f o s n o i t i d n o c d n a s m r e t e h t o t t c e j b u s s i t n e m u c o d s i h t f o e s U . d e v r 8 e 0 s 0 e r 2 s , t 9 h g 1 i r r l e l b a m e t t h p i g e r S y , p y o a c d s i r i l F a i n r o t e d a e t m a s i e r h C T
Resource approved by Technical Coordinator TRANSLATED DOCUMENT This Translation made and checked by: Translated resource approved by:
Page 15
View more...
Comments
