1c8 advanced design of steel structures prepared by josef machacek
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1C8 Advanced design
of steel structures
prepared by
Josef Machacek
2
List of lessons
1) Lateral-torsional instability of beams.2) Buckling of plates.3) Thin-walled steel members.4) Torsion of members. 5) Fatigue of steel structures.6) Composite steel and concrete structures. 7) Tall buildings. 8) Industrial halls.9) Large-span structures.10) Masts, towers, chimneys.11) Tanks and pipelines.12) Technological structures. 13) Reserve.
3
1. Lateral-torsional buckling
• Introduction (stability and strength).
• Critical moment.
• Resistance of the actual beam.
• Interaction of moment and axial force.
• Eurocode approach.
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
4
IntroductionObjectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Stability of ideal (straight) beam under bending
impulse
segment laterally supported in bending and torsion
Mcr
L
Strength of real beam (with imperfections 0, 0)
1M
yyLTRdb,
f
WM bifurcation under bending
strength
initial
yyLT fW
M
Mcr,1
0,0,
cr
yyLT
M
fW
reduction factor LT depends on:
5
Critical moment
Stability of ideal beam under bending (determination of Mcr)
"Basic beam" - with y-y axis of symmetry(simply supported in bending and torsion, loaded only by M)
Two equations of equilibrium (for lateral and torsional buckling)may be unified into one equation:
0d
d
d
d
z
2
2
2
t4
4
w EI
M
xGI
xEI
The first non-trivial solution gives M = Mcr:
L
GIEI
GIL
EI
L
GIEIM tz
crt
2w
2tz
cr 1
2wt
t2
w2
cr 11 GIL
EI
t
wwt GI
EI
L
where
Fz
S G
z
y
Fz
SG
z
y
Fz
S G
z
y
F z
hf
z
y
F z
S=G
z
y
Fz
S G
z
y
Fz
S G
z
y
F z
S G
z
y
F z
S G
z
yS=G
y y
z
z
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
6
Critical moment
Generally (EN 1993-1-1) for beams with cross-sections according to picture:
zs
(C )
S
G
(T )
z
y
zs
S
G
(T )
z
y
hf
zg
F z
zg
F z
(C )
zaza
hf
sh
=
Fz
S G
z
y
Fz
SG
z
y
Fz
S G
z
y
F z
hf
z
y
F z
S=G
z
y
Fz
S G
z
y
Fz
S G
z
y
F z
S G
z
y
F z
S G
z
yS=G
Fzza zg
z
yG
S
zs
j3g2
2j3g2
2wt
z
1cr 1 CCCC
k
C
t
w
LTwwt GI
EI
Lk
t
z
LTz
gg GI
EI
Lk
z
t
z
LTz
jj GI
EI
Lk
z
C1 represents mainly shape of bending moment,
C2 comes in useful only if loading is not applied in shear centre,
C3 comes in useful only for cross-sections non-symmetrical about y-y.
symmetry about z-z symmetry about k y-y, loading through shear centre
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Procedure to determine Mcr:
1. Divide the beam into segments of lengths LLT according to lateral support:
7
Critical moment
2. Define shape of moment in the segment:
from table factor C1 1:
1 ~1,77 ~2,56
lateral support (bracing) in bending and torsion (lateral support "near" compression flange is sufficient)
segment 1segment 2
e.g. segment 1: (LLT,1)
e.g. segment 2: a) usually linear distribution
b) almost never
(because loading here creates continuous support)
~1,13 ~1, 35
segment 3
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Critical moment
Other cases of k :
angle of torsion is zero
stiffener rigidin torsion (½ tube)
stiffener non-rigidin torsion
kz = 1kw = 1
kz = 1kw = 0,5
possible lateral buckling possible lateral buckling
3. Determine support of segment ends: (actually ratio of "effective length")
kz = 1 (pins for lateral bending)kw = 1 (free warping of cross section)
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
9
Critical moment
Cantilever: - only if free end is not laterally and torsionally supported (otherwise concerning Mcr this case is not a cantilever but normal beam segment),
- for cantilever with free end: kz = kw = 2.
konzervativní hodnotykz = 0,7kw = 1
kz = 1kw = 0,7 (conservatively 1)
conservatively(theor. kz = kw = 0,5)
structure torsionallyrigid torsionally
rigidtorsionallynon-rigid
possible lateral buckling
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
10
Critical moment
4. Formula for Mcr depends also on position of loading with respect to shear centre (zg):
- lateral loading acting to shear centre S (zg > 0) is destabilizing: it increases the torsional moment
- lateral loading acting from shear centre S (zg < 0) is stabilizing: it decreases the torsional moment
Factor C2 formoment shape M:(valid for I cross-section)
Mel 0,46 0,55 1,56 1,63 0,88 1,15Mpl (plast. hinges) 0,98 1,63 0,70 1,08
Come in useful for lateral loading (loading byend moments is considered in shear centre).
S
S
F
F
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
11
Critical moment
5. Cross-sections non-symmetrical about y-y
For I cross section with unequal flanges:
warping constant
parameter of asymmetry
2sz
2fw 21 )/h(I)(I
I+ I
I I ftfc
ftfcf
second mom. of area of compress. and tens. flange about z-z
ffA
h,dAz)zy(I
,zz 450
50 22
ysj
Factor C3 greatly depends on f and moment shape (below for kz = kw = 1):
f = -1 1,00 1,47 2,00 0,93
f = 0 1,00 1,00 0,00 0,53 f = 1 1,00 1,00 -2,00 0,38
Mcr =+1Mcr = 0 Mcr = -1
z s
(C )
S
G
(T )
z
y
z s
S
G
(T )
z
y
hf
zg
F z
zg
F z
(C )
zaza
hf
sh
=
Fzza zg
z
yG
S
zs hf
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
12
Critical moment
Cross sections with imposed axis of rotation
V (imposed axis)
suck
oftenSMcr is affected by positionof imposed axis of rotation(Mcr is always greater, holdingis favourable)
For a simple beam with doubly symmetric cross section and generalimposed axis:
coefficients for shape of M: 1 2
2,00 0,00
0,93 0,81
0,60 0,811
t
2
LTw
2
zw
cr
2
2
h
IGLk
hIEIE
M
vg2v1
t
2
LTw
2vzw
cr zzz
IGLk
zIEIE
M
z
zg zv
G ≡ S axis y
For suck loading applied at tension flange:
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
13
Critical moment
Approximate approach for lateral-torsional buckling
In buildings, the reduction factor forlateral buckling corresponding to"equivalent compression flange" (defined as flange with 1/3 ofcompression web)may be taken instead:
1
LT
zf,f
i
L
9931 ,f
E
y
Note: According to Eurocode the reduction factor is taken from curve c, but for cross sections with web slenderness h/tw ≤ 44 from curve d.The factor due to conservatism may be increased by 10%.
hw/6
impulse roughly
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
14
Critical moment
Practical case of a continuous beam (or a rafter of a frame):The top beam flange is usually laterally supported bycladding (or decking). Instead of calculating stability toimposed axis a conservative approach may be used, considering loading at girder shear centre and neglecting destabilizing load location (C2 = 0):
It is desirable to secure laterally the dangerous zones ofbottom free compression flanges against instability:
• by bracing in the level of bottom flanges,• or by diagonals (sufficiently strong) between bottom flange and crossing beams (e.g. purlins).
Length LLT then corresponds to the distance of supports).
Note:For suck stabilizing effect should be applied (C2, zg < 0).
Mel C1 = 2,23Mpl C1 = 1,21
Mel C1 = 2,58Mpl C1 = 1,23
according to M distribution(for different M may be usedgraphs by A. Bureau)
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
15
Critical moment
Beams that do not lose lateral stability:
1. Hollow cross sections Reason: high It high Mcr
2. Girders bent about their minor axis Reason: high It high Mcr
3. Short segment ( ) - all cross sections, e.g. Reason: LT 1
4. Full lateral restraint: "near" to the compression flange is sufficient ( approx. within h/4 )
40LT ,
compression flange loaded tension flange loaded
zg
zg
z v
z v
or higher
or anywhere higher
zv ≥ 0,47 zg zv ≤ 0,47 zg
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
16
Resistance of the actual beam (Mb,Rd)
Similarly as for compression struts: actual strength Mb,Rd < Mcr
(due to imperfections)
e.g. DIN: n = 2,0 (rolled)
= 2,5 (welded)
1/n2nLTRdpl,Rdb, 1
MM
Eurocode EN 1993:
The procedure is the same as for columns: acc. to is determined
LT with respect to shape of the cross section (see next slide).
Note: For a direct 2. order analysis the imperfections e0d are available.
LT
10,1
but 1
2LT
LT
LT
2LT
2LTLT
LT
215,0 LTLT,0LTLTLT
1M
yyLTRdb,
f
WM ... Wy is section modulus acc. to cross section class
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
17
Resistance of the actual beam (Mb,Rd)
Similarly as for compression struts: actual strength Mb,Rd < Mcr
(due to imperfections)
e.g. DIN: n = 2,0 (rolled)
= 2,5 (welded)
1/n2nLTRdpl,Rdb, 1
MM
Eurocode EN 1993:
The procedure is the same as for columns: acc. to is determined
LT with respect to shape of the cross section (see next slide).
Note: For a direct 2. order analysis the imperfections e0d are available.
LT
101
but 1
2LT
LT
LT
2LT
2LTLT
LT
,
2
150
LT
LT,0LTLTLT
,
1M
yyLTRdb,
f
WM ... Wy is section modulus acc. to cross section class
For common rolled and welded cross sections: = 0,75For non-constant M the factor may be reduced to LT,mod (see Eurocode).
400LT ,,
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Choice of buckling curve:
rolled I sections shallow h/b ≤ 2 (up to IPE300, HE600B) b
high h/b > 2 c
welded I sections h/b ≤ 2 c h/b > 2
d
18
Resistance of the actual beam (Mb,Rd)
Lm
greater residual stressesdue to welding
rigid cross section
For common rolled and welded cross sections: = 0,75For non-constant M the factor may be reduced to LT,mod (see Eurocode).
400LT ,,
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
19
Resistance of the actual beam (Mb,Rd)
Lm
In plastic analysis (considering redistribution of moments and"rotated" plastic hinges) lateral torsional buckling in hinges must beprevented and designed for 2,5 % Nf,Ed:
0,025 Nf,Edgirder in location of Mpl
strut providingsupport force in compression
flange
Complicated structures (e.g. haunched girders) may be verified using "stable length" Lm
(in which LT = 1) - formulas see Eurocode.
hh
hsL h
L yLm
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
20
Interaction M + N ("beam columns")
Lm
Always must be verified simple compression and bending in the moststressed cross section – see common non-linear relations.
In stability interaction two simultaneous formulas should be considered:
Usual case M + Ny:
1Rdy,LT
Edy,yy
Rdy
Ed
M
Mk
N
N
1Rdy,LT
Edy,zy
Rdz
Ed
M
Mk
N
N
1
11
M
Rkz,
Edz,Edz,yz
M
Rky,LT
Edy,Edy,yy
M1
Rky
Ed
M
MMk
M
MMk
NN
1
11
M
Rkz,
Edz,Edz,zz
M1
Rky,LT
Edy,Edy,zy
M
Rkz
Ed
M
MMk
M
MMk
NN
for class 4 only
factors kyy ≤ 1,8; kzy ≤ 1,4
(for relations see EN 1993-1-1, Annex B)
Note: historical unsuitable linear relationship (without 2nd order factors kyy, kzy)
1,b
Rdb,
Edy,Ed
M
M
N
N
Rd
MyMz
N
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
21
Interaction M + N ("beam columns")
Lm
Complementary note:
Generally FEM may be used (complicated structures, non-uniform members etc.) to analyse lateral and lateral torsional buckling.First analyse the structure linearly, second critical loading. Then determine:
ult,k - minimum load amplifier of design loading to reach characteristic
resistance (without lateral and lateral-torsional buckling);
cr,op - minimum load amplifier of design loading to reach elastic critical
loading (for lateral or lateral torsional buckling).
opcr,
kult,op
op = min(, LT)
opM1Rky,
Edy,
1MRk
Ed
M
M
N
N
Resulting relationship:
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Assessment
• Ideal and actual beam – differences.
• Procedure for determining of critical moment.
• Destabilizing and stabilizing loading.
• Approximate approach for lateral torsional buckling.
• Resistance of actual beam.
• Interaction M+N according to Eurocode.
22
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
Notes to users of the lecture
• This session requires about 90 minutes of lecturing.
• Within the lecturing, design of beams subjected to lateral torsional buckling is described. Calculation of critical moment under general loading and entry data is shown. Finally resistance of actual beam and design under interaction of moment and axial force in accordance with Eurocode 3 is presented.
• Further readings on the relevant documents from website of www.access-steel.com and relevant standards of national standard institutions are strongly recommended.
• Keywords for the lecture:
lateral torsional instability, critical moment, ideal beam, real beam, destabilizing loading, imposed axis, beam resistance, stability interaction.
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
24
Notes for lecturers
• Subject: Lateral torsional buckling of beams.
• Lecture duration: 90 minutes.
• Keywords: lateral torsional instability, critical moment, ideal beam, real beam, destabilizing loading, imposed axis, beam resistance, stability interaction.
• Aspects to be discussed: Ideal beam and real beam with imperfections. Stability and strength. Critical moment and factors which influence its determination. Eurocode approach.
• After the lecturing, calculation of critical moments under various conditions or relevant software should be practised.
• Further reading: relevant documents www.access-steel.com and relevant standards of national standard institutions are strongly recommended.
• Preparation for tutorial exercise: see examples prepared for the course.
Objectives
Introduction
Critical moment
Resistance of actual beam
Interaction M+N
Assessment
Notes
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