Transcript Slide 1

1C8
Advanced design
of steel structures
prepared by
Josef Machacek
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.
2
Objectives
Introduction
1. Lateral-torsional buckling
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
• Introduction (stability and strength).
• Critical moment.
Notes
• Resistance of the actual beam.
• Interaction of moment and axial force.
• Eurocode approach.
3
Objectives
Introduction
Introduction
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
Stability of ideal (straight) beam under bending
impulse

Notes
L

segment laterally supported in
bending and torsion
Mcr
Strength of real beam (with imperfections 0, 0)
M
bifurcation under bending
Mcr,1
 LTW yf y
strength
0,0
initial
 ,
Mb,Rd   LTW y
reduction factor LT
depends on:
 LT 
fy
 M1
W y fy
Mcr
4
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Stability of ideal beam under bending (determination of Mcr)
Interaction M+N
F"Basic
F
Fz
Fz
z
beam" - zwith y-y axis of symmetry
z supported in bending
z and torsion, loaded
(simply
z only by M) z
Fz
zz
Assessment
Notes
hf
y
y S
S=G
Two equations of equilibrium (for lateral and torsional buckling)
G
may be unified into one equation:
G
S
y
G
d
d M

GI

 0
t
dx 4
dx 2 EI z
4
EI w
G
S
y
2
2
y
S
z
The first non-trivial solution gives M = Mcr:
M cr 
 EI zGI t
L
1
where cr  1 
 2EI w
L2GI t
 2EI w
2
L GI t
 cr
 EI zGI t
2
 1   wt
L
 wt 
 EI w
L GI t
5
Objectives
Introduction
Critical moment
Critical moment
Generally (EN 1993-1-1) for beams with cross-sections according to picture:
Interaction M+N
Assessment
cr 
Notes

C1 
2
1   wt
 C2 g  C3 j


kz
 wt 
F
zazF zzzg Fz F z
z
(C)
z
z
z
z
a
g
S
S
y S y G
zs G y y
S=G
G
zs
S
2
2 g
EI w
GIt

 C3 j 

g 
 zg
k z LLT
EI z
GIt
j 
Fz
Fz
z
(C)
G
 zj
k z LLT
Fz
Fz
z
Fz
z
zg
z
S
hf = hs
hf
zs
za

k w LLT
  C 
z
EI z
GIt
Fz
z
y
G
S
hf
Resistance of
actual beam
y
G
y
S
G
y
S
G y
S G y
S
(T)
(T)
symmetry about z-z
symmetry about k y-y, loading through shear centre
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.
6
Objectives
Introduction
Critical moment
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:
segment 3
segment 2
e.g. segment 1: (LLT,1)
segment 1
lateral support (bracing) in bending and torsion (lateral
support "near" compression flange is sufficient)
from table factor C1  1:
2. Define shape of moment in the segment:
 

1
e.g. segment 2:
~2,56
~1,77
a) usually linear distribution
b) almost never
~1,13
~1, 35
(because loading here creates continuous support)
7
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
Notes
3. Determine support of segment ends:
(actually ratio of "effective length")
kz = 1 (pins for lateral bending)
kw = 1 (free warping of cross section)
Other cases of k :
kz = 1
kw = 1
angle of torsion
is zero
stiffener non-rigid
in torsion
possible lateral buckling
kz = 1
kw = 0,5
stiffener rigid
in torsion (½ tube)
possible lateral buckling
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
Notes
kz = 0,7 conservatively
konzervativní
hodnoty
(theor. kz =
kw = 0,5)
kw = 1
kz = 1
kw = 0,7 (conservatively 1)
structure torsionally
rigid
torsionally
torsionally rigid
non-rigid
possible lateral buckling
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.
9
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Interaction M+N
4. Formula for Mcr depends also on position of loading with respect
to shear centre (zg):
Come in useful for lateral loading (loading by
end moments is considered in shear centre).
Assessment
F
Notes
- lateral loading acting to shear centre S (zg > 0)
is destabilizing: it increases the torsional moment
S
S
- lateral loading acting from shear centre S (zg < 0)
is stabilizing: it decreases the torsional moment
F
Factor C2 for
Mel
0,46 0,55
moment shape M:
(valid for I cross-section) Mpl (plast. hinges)
1,56 1,63 0,88 1,15
0,98 1,63 0,70 1,08
10
Objectives
Introduction
Critical moment
Critical moment
Interaction M+N
5. Cross-sections non-symmetrical about y-y
F
For
I cross
section with unequal flanges:
z
zaz zzg FF z
zz
z
z
Assessment
a
a
g
zs
G
G
(C)
S
S
yy
hf = hs
zs
S
z
warping constant
Notes
zs
I w  ( 1   f2 )I z ( hs / 2 ) 2
zg
(C)
y
G
hf
hf
Resistance of
actual beam
parameter of asymmetry
f 
I fc  I ft
I fc + I ft
(T)
second mom. of area of compress. and tens. flange about z-z
0,5
2
2
 ( y  z ) z dA  0,45 f hf
Iy A
Factor C3 greatly depends on f and moment shape (below for kz = kw = 1):
(T)
z j  zs 
Mcr
=+1
Mcr
= 0
Mcr
= -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
11
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Cross sections with imposed axis of rotation
Interaction M+N
suck
Assessment
Notes
S
often
V
(imposed
axis)
Mcr is affected by position
of imposed axis of rotation
(Mcr is always greater, holding
is favourable)
For a simple beam with doubly symmetric cross section and general
imposed axis:

z
zg
zv
G≡S
axis y
Mcr
Mcr


For suck loading applied at tension flange:
2

 h  
E I w  E I z    
 2    k w LLT



h
1
2
2
 
2 

  G I t
E I w  E I z zv 
 k w LLT 

  1z v   2 zg  z v
2

  G I t

coefficients 
for shape of M:

1
2
2,00
0,00
0,93
0,81
0,60
0,81
12
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Approximate approach for lateral-torsional buckling
Interaction M+N
Assessment
Notes
impulse
roughly
hw/6
In buildings, the reduction factor for
lateral buckling corresponding to
"equivalent compression flange"
(defined as flange with 1/3 of
compression web)
may be taken instead:
f 
LLT
i f, z 1
1  
E
 93 ,9
fy
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%.
13
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
Notes
Practical case of a continuous beam (or a rafter of a frame):
The top beam flange is usually laterally supported by
cladding (or decking). Instead of calculating stability to
imposed axis a conservative approach may be used,
considering loading at girder shear centre and neglecting
destabilizing load location (C2 = 0):
Mel
Mpl
C1 = 2,23
C1 = 1,21
Mel
Mpl
C1 = 2,58
C1 = 1,23
according to M distribution
(for different M may be used
graphs by A. Bureau)
Note:
For suck stabilizing effect should be applied (C2, zg < 0).
It is desirable to secure laterally the dangerous zones of
bottom 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).
14
Objectives
Introduction
Critical moment
Critical moment
Resistance of
actual beam
Beams that do not lose lateral stability:
Interaction M+N
Assessment
1. Hollow cross sections
Reason: high It  high Mcr
Notes
2. Girders bent about their minor axis
Reason: high It  high Mcr
3. Short segment (  LT  0,4 ) - all cross sections, e.g.
Reason: LT  1
4. Full lateral restraint: "near" to the compression flange
is sufficient ( approx. within h/4 )
compression flange loaded
tension flange loaded
zv ≥ 0,47 zg
zg
zv ≤ 0,47 zg
or higher
zv
zv
zg
or anywhere higher
15
Objectives
Introduction
Resistance of the actual beam (Mb,Rd)
Critical moment
Resistance of
actual beam
Interaction M+N
Similarly as for compression struts: actual strength Mb,Rd < Mcr
(due to imperfections)
e.g. DIN: Mb,Rd
Assessment
 
 Mpl,Rd 1   LT

2n 1/n
Notes


n = 2,0 (rolled)
= 2,5 (welded)
Eurocode EN 1993:
The procedure is the same as for columns: acc. to  LT 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.
Mb,Rd   LTW y
 LT 
fy
... Wy is section modulus acc. to cross section class
 M1
  LT  1,0

but    1
2
 LT  LT

1
2
2
 LT   LT
   LT


 LT  0,5 1   LT  LT   LT,0    LT 

2

16
Objectives
Introduction
Resistance of the actual beam (Mb,Rd)
Critical moment
Resistance of
actual beam
Interaction M+N
Similarly as for compression struts: actual strength Mb,Rd < Mcr
(due to imperfections)
e.g. DIN: Mb,Rd
Assessment
Notes
 
 Mpl,Rd 1   LT

2n 1/n


n = 2,0 (rolled)
= 2,5 (welded)
Eurocode EN 1993:
The procedure is the same as for columns: acc. to  LT 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.
Mb,Rd   LTW y
 LT 
fy
 M1
... Wy is section modulus acc. to cross section class

1
2
2
 LT   LT
   LT

1   LT  LT   LT,0  
  LT  1,0


but    1  LT  0,5 
2
    LT

2
 LT  LT
For common rolled and welded cross sections:  LT,0  0,4
 = 0,75
For non-constant M the factor may be reduced to LT,mod (see Eurocode).
17
Objectives
Introduction
Resistance of the actual beam (Mb,Rd)
Critical moment
Resistance of
actual beam
Interaction M+N
For common rolled and welded cross sections:  LT,0  0,4
 = 0,75
For non-constant M the factor may be reduced to LT,mod (see Eurocode).
Assessment
Notes
Choice of buckling curve:
rolled I sections
shallow
high
welded I sections
greater residual stresses
due to welding
rigid cross section
h/b ≤ 2 (up to IPE300, HE600B)
h/b > 2
h/b ≤ 2
h/b > 2
b
c
c
d
18
Objectives
Introduction
Resistance of the actual beam (Mb,Rd)
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
In plastic analysis (considering redistribution of moments and
"rotated" plastic hinges) lateral torsional buckling in hinges must be
prevented and designed for 2,5 % Nf,Ed:
Notes
strut providing
support
girder in location of Mpl
force in compression
flange
0,025 Nf,Ed
hs
Complicated structures (e.g. haunched girders)
hh
may be verified using "stable length" Lm
(in which LT = 1) - formulas see Eurocode.
Lh
Ly
19
Objectives
Introduction
Interaction M + N
("beam columns")
Critical moment
Resistance of
actual beam
Interaction M+N
Assessment
Notes
Always must be verified simple compression and bending in the most
stressed cross section – see common non-linear relations.
In stability interaction two simultaneous formulas should be considered:
for class 4 only
M y,Ed  M y,Ed
M
 M z,Ed
NEd
 k yy
 k yz z,Ed
1
 y NRk
 LT M y,Rk
M z,Rk
 M1
 M1
N
 M1
M y,Ed  M y,Ed
M
 M z,Ed
NEd
 k zy
 k zz z,Ed
1
 z NRk
 LT M y,Rk
M z,Rk
 M1
 M1
Usual case M + Ny:
M y,Ed
NEd
 k yy
1
 y NRd
 LT M y,Rd
M y,Ed
NEd
 k zy
1
 z NRd
 LT M y,Rd
 M1
Mz
My
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)
M y,Ed
NEd

1
Nb,Rd Mb,Rd
20
Objectives
Introduction
Interaction M + N
("beam columns")
Critical moment
Resistance of
actual beam
Complementary note:
Interaction M+N
Assessment
Notes
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).
 op 
ult,k
 cr,op
op = min(, LT)
Resulting relationship:
NEd
NRk  M1

M y,Ed
M y,Rk  M1
  op
21
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
Notes to users of the lecture
Critical moment
Resistance of
actual beam
Interaction M+N
•
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:
Assessment
Notes
lateral torsional instability, critical moment, ideal beam, real
beam, destabilizing loading, imposed axis, beam resistance,
stability interaction.
Objectives
Introduction
Notes for lecturers
Critical moment
Resistance of
actual beam
•
Subject: Lateral torsional buckling of beams.
Interaction M+N
•
Lecture duration: 90 minutes.
Assessment
•
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.
Notes
24