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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
NCCI: Determination of non-dimensional slenderness of I and H sections
This NCCI presents a method for determining the non-dimensional slenderness without explicit determination of Mcr. The basic, conservative method can be refined to take account of section geometry and bending moment distribution.
Contents
1. Simplified method 2
2. Economy from more complexity 3
3. Allowance for the effect of destabilizing loads 6
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
1. Simplified method For straight segments of hot-rolled doubly symmetric I and H sections with lateral restraints to the compression flange at both ends of the segment considered and with no destabilizing loads, the value of LTλ required by EN1993-1-1 §6.3.2.2 or §6.3.2.3 may be conservatively taken from Table 1.1.
S 235 S 275 S 355 S420 S 460
104z
LTiL
=λ 96
zLT
iL=λ
85z
LTiL
=λ 78z
LTiL
=λ 75
zLT
iL=λ
Table 1.1 LTλ for different grades of steel
where
L is the distance between points of restraint of the compression flange
iz is the radius of gyration of the section about the minor axis.
NOTES
Table 1.1 is derived from equation (1) taking C1 = 1,0, U = 0,9, V = 1,0 and wβ = 1,0.
Improved economy can be gained by increasing the complexity of the slenderness calculation. For beams designed as “simply supported”, there may be little gain, but for columns with large moments, the gain may be significant.
It is advisable to detail structures to avoid “destabilising” loading. This may be achieved by detailing so that the load and the beam flange are not free to move laterally. For example, where a floor acts as a horizontal diaphragm restraining the beam, the loading is not “destabilising”.
For further information, see also:
Economy from more complexity
Non-uniform bending moment distribution reduces LTλ by up to 40% where there is significant reversal of moment.
Section geometry reduces LTλ by up to 15%.
Lower yield strengths for thicker elements reduce LTλ by up to 5%.
Allowance for the effects of destabilising loads
Destabilising loads are rare but when they do exist the bending resistance is reduced. Destabilising loads need to be taken into account in design.
Background Theory Derivation of above simplified equations
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
2. Economy from more complexity A less conservative value of LTλ may be obtained by taking account of bending moment diagram, section geometry and lower yield strengths.
There is little economy to be gained for simply supported beams by use of 1
1C
, but in
columns with negative values of ψ (see Table 2.1) and large bending moments, the economy may be significant.
NOTE: For beams in “simple” construction (designed as Simple Supported beams), see EN1993-1-8 §5.1.1(2).
When the loading is not “destabilising”, LTλ is given by
wz1
w1
z
1LT
11 βλβλλλ UV
CUV
C==
(1)
where
C1 is a parameter dependent on the shape of the bending moment diagram. Values of
1
1C
for some bending moment diagrams are given in Table 2.1 and Table 2.2.
Values for other bending moment diagrams can be obtained from [SN003].
Conservatively, C1 = 1,0 (this value has been used in the simplified method above).
U is a parameter dependent on the section geometry and is given by:
w
zypl,
II
AgW
U =
In which g allows for the curvature of the beam if it has zero vertical deflection
before it is loaded and is given by ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
y
z1IIg or, conservatively, g = 1,0
Conservatively, U = 0,9 (this value has been used in the simplified method above).
V is a parameter related to the slenderness. Where the loading is not “destabilising”, it may be taken as:
either, conservatively, = 1,0 for all sections symmetric about the major axis,
or as
42
f
z2011
1
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
th
Vλ
for doubly symmetric hot rolled I and H sections
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
The exact definition of V, where the loading is not “destabilising”, is:
( )4
z
w
t
2
2z
2
w
1
II
IA
GEπk
kV
λ+⎟⎟
⎠
⎞⎜⎜⎝
⎛= If k = kw, then
( )4
z
w
t
2
2z1
1
II
IA
GEπ
Vλ
+
=
zz i
kL=λ ,in which
L is the distance between points of restraint to the compression flange
k is the effective length parameter and should be taken as 1,0 unless it can be demonstrated otherwise
ypl,
yw W
Wβ =
Wy is the modulus used to calculate Mb,Rd
For Class 1 and 2 sections Wy = Wpl,y
For Class 3 sections Wy = Wel,y
y1 f
Eπλ = in which fy is the yield strength appropriate to the thickness of the steel.
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
Table 2.1 Values of 1
1C
for end moment loading, to be used with k=1,0
1
1C
ψ
+1,00 1,00
+0,75 0,94
M ψ M
-1 ≤ ψ ≤ +1
+0,50 0,87
+0,25 0,81
0,00 0,75
-0,25 0,70
-0,50 0,66
-0,75 0,62
-1,00 0,63
Table 2.2 Values of 1
1C
for cases with transverse loading, to be used with k=1,0
Bending moment diagram 1
1C
Loading and support conditions
0,94
0,62
0.86
0,77
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
3. Allowance for the effect of destabilizing loads The effect of a ‘destabilising’ load may be taken into account by increasing the value of the non-dimensional slenderness.
3.1 Beams with destabilizing loads A beam with the load acting at a distance above the shear centre of the section is shown in Figure 3.1b. If both the load and the beam are free to move laterally, such a load is described as a “destabilising” load. The destabilising effect arises because when the beam buckles, deflecting laterally and twisting, the line of action of the load remains vertical but moves relative to the shear centre of the section. The load therefore applies an additional torque,
increasing the effect of lateral torsional buckling.
w
w
e
a) Load acting through b) Load acting at top flange shear centre (destabilising load)
Figure 3.1 An example of a destabilising load
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
3.2 Slenderness with destabilising loads Where the loading is “destabilising”, LTλ is given by
wz1
w1
z
1LT
11 βλUVDC
βλλUVD
Cλ ==
(2)
where
( ) ( )
25,0
w
z22
z
w
t
2
22
w
1
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++⎟⎟⎠
⎞⎜⎜⎝
⎛
=
IIzC
II
IA
GEk
k
V
gz
πλ
For doubly symmetric hot rolled I and H sections, V may be taken conservatively as:
( )4w
z22
2
f2011
1
IIzC
th
V
gz +⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=λ
C2 is a parameter dependent on the shape of the bending moment diagram. Values of C2 are given in SN003.
zg is the height of the “destabilising” load above the shear centre
5,0
w
zg2
21
1
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
II
zCV
D
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
Annex A Background Theory
The theoretical consistency between the simplified method and the explicit method using Mcr for calculating values of LTλ is demonstrated below.
The elastic critical buckling moment may be written:
( )( ) ( )
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−++⎟⎟
⎠
⎞⎜⎜⎝
⎛= g2
2g2
z2
t2
z
w2
w2
z2
1cr zCzCEIπ
GIkLII
kk
gkLEIπCM
where g is the correction factor for the increase in critical buckling moment caused by
increased curvature, which may be taken as ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
y
z1IIg , or conservatively as g =1,0.
EN 1993-1-1 defines the “non-dimensional” slenderness ascr
yyLT
MfW
=λ
( )( ) ( )
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−++⎟⎟
⎠
⎞⎜⎜⎝
⎛=
g22
g2z
2t
2
z
w2
w2
2
1 zCzCEIπ
GIkLII
kk
gkLEIπC
fW
z
yy
( )
( ) ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+⎟⎠⎞
⎜⎝⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
=
g2w
z2g2
w
z2
z
22
wz
wy
2
2
1
111
zCIIzC
II
EAπGI
AI
kLkk
IIf
EπAI
kLA
gWC
t
z
y
( )( ) ( )
( )( )
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡++⎟⎟
⎠
⎞⎜⎜⎝
⎛=
w
zg2
2g2
w
zt22
z
22
wz
w
y
22
2
1
1111
IIzC
IIzC
AIII
EπG
ikL
kk
II
fEπi
kLA
gWC
w
zz
y
defining y
1 fEπ=λ and
zz i
kL=λ
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
( )( )
( ) ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−++⎟⎟⎠
⎞⎜⎜⎝
⎛
=
w
zg2
w
z2g2
z
w
t
2
22
w
21
2z
y
1
111
IIzC
IIzC
II
IA
GEπλ
kk
λλ
II
AgW
C
z
w
z
defining ( ) ( )
w
z2g2
z
w
t
2
22
w
1
IIzC
II
IA
GEπλ
kk
V
z ++⎟⎟⎠
⎞⎜⎜⎝
⎛=
( )( )
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
w
zg22
21
2z
w
zypl,
pl.y
y
1LT
111
IIzC
Vλλ
II
AgW
WW
Cλ
defining ypl,
yw W
Wβ = and
w
zypl,
II
AgW
U =
( )( )
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
w
zg2
2
2
21
2z2
W1
LT
1
1
IIzCV
VλλUβ
Cλ
defining
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
w
zg2
21
1
IIzCV
D
( )( )
222
1
2z2
W1
LT1 DV
λλUβ
C=λ
W1
z
1LT
1 βλλUVD
Cλ =∴
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
V may be simplified as follows. Where k = kw and the load is applied through the shear centre of the section, V reduces to
( ) ( )4
z
w
t
2
2z
z
w
t
2
2z 1
1
1
1
II
IA
GEπλ
II
IA
GEπλ
V
+
=
+
=
For hot-rolled I-sections, 2
fz
w
t
220 ⎟⎟
⎠
⎞⎜⎜⎝
⎛≈
th
II
IA
GEπ
Therefore, for hot-rolled I-sections, and where the loads are not “destabilising”, V may be taken as:
42
f
z2011
1
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
thλ
V
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NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU
Quality Record RESOURCE TITLE NCCI: Determination of non-dimensional slenderness of I and H
sections
Reference(s)
ORIGINAL DOCUMENT
Name Company Date
Created by James Lim The Steel Construction Institute
Technical content checked by Charles King The Steel Construction Institute
Editorial content checked by D C Iles SCI 2/3/05
Technical content endorsed by the following STEEL Partners:
1. UK G W Owens SCI 1/3/05
2. France A Bureau CTICM 1/3/05
3. Sweden A Olsson SBI 1/3/05
4. Germany C Mueller RWTH 1/3/05
5. Spain J Chica Labein 1/3/05
Resource approved by Technical Coordinator
G W Owens SCI 21/04/06
TRANSLATED DOCUMENT
This Translation made and checked by:
Translated resource approved by:
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