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12/14/11
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Topic 7 Laterally unrestrained beams
Prof Dr Shahrin Mohammad
Structural Steel and Timber Design SAB3233
Structural Steel Design
Topic 1 -‐ Overview
Topic 4 – Design of steel structures (BS EN 1993)
Topic 3 –AcLons on
Structures (BS EN 1991)
Topic 5 – Cross-‐secLon
classificaLon
Topic 2 -‐ Basis of Structural Design (BS EN 1990)
Topic 6 – Laterally restrained beams
Topic 7 – Laterally unrestrained beams
Topic 8 – Columns
Topic 9 – Trusses
Topic 10 -‐ ConnecLons
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Beam is a member predominantly subject to bending. A beam is a structural member which is subject to transverse loads, and accordingly must be designed to withstand shear and moment. Generally, it will be bent about its major axis
Lateral-‐torsional buckling
Dead weight load applied verLcally
Buckled posiLon
Unloaded posiLon
Clamp at root
§ In the case of a beam bent about its major axis, failure may occur by a form of buckling which involves both lateral deflecLon and twisLng.
§ Slender structural elements loaded in a sLff plane tend to fail by buckling in a more flexible plane.
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• Perfectly elasLc, iniLally straight, loaded by equal and opposite end moments about its major axis.
M M
L
ElevaLon SecLon
Plan
y
z x
u
f
§ Unrestrained along its length.
§ End Supports § TwisLng and lateral deflecLon prevented.
§ Free to rotate both in the plane of the web and on plan.
• The compression flange is not restrained from deflect laterally and rotate about the plan of the secLon, which is called lateral torsional buckling
• Three components of displacement i.e. verLcal, horizontal and torsional displacement
Unrestrained beam
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• Lateral restraint may be of along the span or at some points along the span.
Front view of the primary beam
P1 P2
Primary beam
Secondary beams
A B C D
Plan view
Secondary beams
Deform shape
Original shape
A B C D
Points A, B, C and D are restrained from deform laterally by the secondary beams and the connecAon at column
Examples : Unrestrained Beam
Steel slide
UB
Timber floor
UB
Crane railway
Steel column
This crane girder is not restrained laterally between two brackets.
Bracket This beam only laterally restrained on both ends.
Support
Water tank
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Design factors which will influence the lateral stability can be summarized as: • The slenderness of the member between adequate
lateral restraints; • the shape of cross-‐secLon; • the variaLon of moment along the beam; • the form of end restraint provided, • the manner in which the load is applied, i.e. to tension
or compression flange.
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ElasLc buckling of beams
Includes:
§ Lateral flexural sLffness EIz § Torsional and Warping sLffnesses GIt and Eiw
Their relaLve importance depends on the type of cross-‐secLon used.
⎥⎦
⎤⎢⎣
⎡+=
z2
t2
z
w2z
2
cr EIGIL
II
LEIM
ππ
CriLcal Buckling Moment for uniform bending moment diagram is
Effect of Slenderness
Non-‐dimensional moment resistance plot
0
Stocky Slender
pl
cr
1,0
1,0 M M
=
Intermediate
pl
cr M M
pl
M M
LT λ
Non-‐dimensional plot permits results from different test series to be compared
Intermediate slenderness -‐ adversely affected by inelasLcity and geometric imperfecLons
EC3 uses a reducLon factor χLT on plasLc resistance moment to cover the whole slenderness range
Stocky beams ( < 0,4) unaffected by lateral torsional buckling
Slender beams ( >1,2) resistance close to theoreLcal elasLc criLcal moment Mcr
LT λ
LT λ
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Design buckling resistance
The design buckling resistance moment Mb.Rd of a laterally unrestrained beam is calculated as
which is effecLvely the plasLc resistance of the secLon mulLplied by
the reducLon factor cLT
1Myy.plwLTRd.b /fWM γβχ=
0
Welded beams
Slenderness
c
Redu
cLon
factor
1.0 2.0
1.0
Lateral-‐torsional buckling reducLon factor
LT λ
ReducLon factor for LTB
χLTLT =1
ϕLT + ϕLT2 −λLT
2"# $%.0.5
ϕLT = 0.5 1+∝LT (λLT − 0.2)+λLT2#
$%
&
'(
where
α LT = 0.21 for rolled secLons
α LT = 0.49 for welded secLons
and
0
Welded beams
Slenderness
Redu
cAon
factor
1.0 2.0
1.0
Lateral-‐torsional buckling reducLon factor
LT λ
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Determining
LTλ
The non-‐dimensional slenderness crRdplLT MM /.=λ
calculated by calculaLng the plasLc resistance moment Mpl.Rd and elasLc criLcal moment Mcr from first principles
where
5.0
1 fyE⎥⎦
⎤⎢⎣
⎡=πλ
For any plain I or H secLon with equal flanges, under uniform moment with simple end restraints
25.02
f
z
zLT
t/hi/L
2011
i/L
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
=λ
5.0w
1
LTLT β
λλ
λ ⎥⎦
⎤⎢⎣
⎡=or using
18
Effect of load paeern on LTB
… which is increased from the basic (uniform moment) case by a factor C1=4.24/π=1.365
t
wtzcr GIL
EIGIEIL
M 2
21
ππ+=
The elasLc criLcal moment for a beam under uniform bending moment is
M M
t
wtzcr GIL
EIGIEIL
M 2
2124,4 π+=
The elasLc criLcal moment (mid-‐span moment) for a beam with a central point load is
M
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C1 factor
C1 appears:
• as a simple mulLplier in expressions for Mcr
• as 1/ C10.5 in expressions for λLT.
EC3 expresses the elasLc criLcal moment Mcr for a parLcular loading case as
p 2 1+ EI w 2 M = C p
L EI GJ cr 1 L GJ Mmax C1
M
M
M
FL/4
FL/8
FL/4
1.00
1.879
2.752
1.365
1.132
1.046
Loads Bending moment
M M
M
M -‐M
F
F
F F = = = =
t
wtzcr GIL
EIGIEIL
CM 2
2
1 1ππ
+=
20
End support conditions • Basic case assumes end conditions which prevent
lateral movement and twist but permit rotation on plan.
L
Elevation Section
Plan
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End support conditions
Choice of k is at the designer’s discretion
§ End conditions which prevent rotation on plan enhance the elastic buckling resistance
§ Can include the effect of different support conditions by redefining the unrestrained length as an effective length
§ Two effective length factors, k and kw. § Reflect the two possible types of end fixity, lateral bending
restraint and warping restraint. § Note: it is recommended that kw be taken as 1.0 unless special
provision for warping fixing is made. § EC3 recommends k values of 0,5 for fully fixed ends, 0,7 for one
free and one fixed end and of course 1,0 for two free ends.
22
Level of applicaLon of load
• Loads applied to top flange are destabilising
• Problem increases with depth of secLon and/or as span reduces
• EC3 introduces C2 factor into expressions for λLT
F
F
F
10 100 10001
a=d/2
a=0
a=d/2
F
L GIEI2
t
w
Equiv
ale
nt
uniform
mo
me
nt
mo
mentm
om
en
t fa
cto
rm
1,4
1,2
1,0
0,8
0,6
0,4
Effect of load position on buckling resistance
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Beams with intermediate lateral support
• If beams have lateral restraints at intervals along the span the segments of the beam between restraints must be treated in isolaLon
• beam design is based on the most criLcal segment • Lengths of beams between restraints should use an effecLve length factor k of 1.0
Design Procedure :
!
Start
Estimate support condition of the steel beam
Estimate all the load subjected onto the beam
Determine the maximum design of shear force, VEd and moment, MEd
Chose steel grade and cross section size that suitable for the design (EN 1993-1-1:2005, Table 3.1)
Classify the cross section of the steel beam (EN 1993-1-1:2005, Sheet 1, 2 and 3)
Determine the maximum shear force of the section, Vc, Rd (EN 1993-1-1:2005, Clause 6.2.6)
VEd ≤ Vc, Rd
Determine the maximum moment resistance, Mc, Rd (EN 1993-1-1:2005, Clause 6.2.5)
MEd ≤ Mc, Rd
No
Yes
B
Is shear buckling resistance checking for web need? (EN 1993-1-1:2005, Clause 6.2.6)
Check for shear buckling resistance of web (EN 1993-1-5, Section 5)
Yes
No
A
Yes
No
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Design Procedure :
!
Check for combined bending and shear force resistance (EN 1993-1-1:2005, Clause 6.2.8)
Checking for lateral torsional buckling resistance (EN 1993-1-1:2005, Clause 6.3.2)
B A
Adopt section
End
No
Is the combined bending and shear checking need?
VEd > 0.5Vc, Rd
Yes
No
Determine the allowable deflection (EN 1990:2002, Clause A1.4.3)
Actual deflection ≤ Allowable deflection
Yes
MEd ≤ MV, Rd No
Yes
Determine value of permanent load that gives effect to the beam deflection
MEd ≤ Mb, Rd No
Yes
Design Procedure :
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Example 1 : Design of an unrestrained beam
Thank You
