Transcript UNRESTRAINED BEAM DESIGN - Structural Engineering Forum …

IS 800:2007
Section 8
Design of members
subjected to bending
Dr S R Satish Kumar, IIT Madras
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SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING
8.1
General
8.2
Design Strength in Bending (Flexure)
8.2.1 Laterally Supported Beam
8.2.2 Laterally Unsupported Beams
8.3
Effective Length of Compression Flanges
8.4
Shear
------------------------------------------------------------------------------------------------8.5
Stiffened Web Panels
8.5.1 End Panels design
8.5.2 End Panels designed using Tension field action
8.5.3 Anchor forces
8.6
Design of Beams and Plate Girders with Solid Webs
8.6.1 Minimum Web Thickness
8.6.2 Sectional Properties
8.6.3 Flanges
Cont...
Dr S R Satish Kumar, IIT Madras
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SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING
8.7
Stiffener Design
8.7.1 General
8.7.2 Design of Intermediate Transverse Web Stiffeners
8.7.3 Load carrying stiffeners
8.7.4 Bearing Stiffeners
8.7.5 Design of Load Carrying Stiffeners
8.7.6 Design of Bearing Stiffeners
8.7.7 Design of Diagonal Stiffeners
8.7.8 Design of Tension Stiffeners
8.7.9 Torsional Stiffeners
8.7.10 Connection to Web of Load Carrying and Bearing Stiffeners
8.7.11 Connection to Flanges
8.7.12 Hollow Sections
8.8
Box Girders
8.9
Purlins and sheeting rails (girts)
8.10
Bending in a Non-Principal Plane
Dr S R Satish Kumar, IIT Madras
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RESPONSE OF BEAMS TO VERTICAL LOADING
• Plastic hinge formation
• Lateral deflection and twist
• Local buckling of
i) Flange in compression
ii) Web due to shear
iii) Web in compression due to
concentrated loads
• Local failure by
i) Yield of web by shear
ii) Crushing of web
iii) Buckling of thin flanges
Dr S R Satish Kumar, IIT Madras
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LOCAL BUCKLING AND SECTION CLASSIFICATION
OPEN AND CLOSED SECTIONS
Strength of compression members depends on slenderness ratio
Dr S R Satish Kumar, IIT Madras
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LOCAL BUCKLING
(a)
(b)
Local buckling of Compression Members
Beams – compression flange buckles locally
Fabricated and cold-formed sections prone to local buckling
Local buckling gives distortion of c/s but need not lead to collapse
Dr S R Satish Kumar, IIT Madras
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BASIC CONCEPTS OF PLASTIC THEORY
w
Collapse mechanism
L
Plastic hinges
Plastic hinges
Mp
Mp
Bending Moment Diagram
Bending Moment Diagram
Formation of a Collapse Mechanism in a Fixed Beam
First yield moment My
Plastic moment
Mp
Shape factor S = Mp/My
Rotation Capacity
(a) at My (b) My < M<Mp (c) at Mp
Plastification of Cross-section under Bending
Dr S R Satish Kumar, IIT Madras
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SECTION CLASSIFICATION
Plastic
Mp
Compact
My
Semi-compact
Slender
y
u
Rotation 
Section Classification based on Moment-Rotation Characteristics
Dr S R Satish Kumar, IIT Madras
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SECTION CLASSIFICATION BASED ON
WIDTH -THICKNESS RATIO
Mp
My
SemiPlastic Compact Compact Slender
1
2
3
=b/t
Moment Capacities of Sections
For Compression members use compact or plastic sections
Dr S R Satish Kumar, IIT Madras
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Table 2 Limits on Width to Thickness Ratio of Plate Elements
Type of Element
Type of
Section
Class of Section
Plastic (1)
Outstand element of
compression flange
Internal element of
compression flange
Web
Angles
  250 f
y
Rolled
b/t  9.4
Compact
(2)
b/t  10.5
Welded
b/t  8.4
b/t  9.4
b/t  13.6
bending
b/t  29.3
b/t  33.5
b/t  42
not
applicable
b/t  42
d/t  84.0
d/t  105
d/t  126
b/t  9.4
b/t  10.5
b/t  15.7
not
applicable
b/t  15.7
(b+d)/t  25
D/t  442
D/t  632
D/t  882
Axial
comp.
NA at mid
depth
bending
Axial
comp.
Circular tube with
outer diameter D
Dr S R Satish Kumar, IIT Madras
Semi-compact (3)
b/t  15.7
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Condition for Beam Lateral Stability
• 1 Laterally Supported Beam
The design bending strength of beams, adequately
supported against lateral torsional buckling (laterally
supported beam) is governed by the yield stress
• 2 Laterally Unsupported Beams
When a beam is not adequately supported against lateral
buckling (laterally un-supported beams) the design
bending strength may be governed by lateral torsional
buckling strength
Dr S R Satish Kumar, IIT Madras
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Design Strength in Bending (Flexure)
The factored design moment, M at any section, in a beam due to
external actions shall satisfy
M  Md
8.2.1 Laterally Supported Beam
Type 1 Sections with stocky webs
d / tw  67
The design bending strength as governed by plastic strength, Md,
shall be found without Shear Interaction for low shear case
represented by
V <0.6 Vd
Dr S R Satish Kumar, IIT Madras
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8.2.1.3 Design Bending Strength under High Shear
• V exceeds 0.6Vd
Md = Mdv
Mdv= design bending strength under high
shear as defined in section 9.2
Dr S R Satish Kumar, IIT Madras
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Definition of Yield and Plastic Moment Capacities
Dr S R Satish Kumar, IIT Madras
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8.2 Design Strength in Bending (Flexure)
The factored design moment, M at any section, in a beam
due to
M M
external actions shall satisfy
d
8.2.1 Laterally Supported Beam
The design bending strength as governed by plastic
strength, Md, shall be taken as
Md = b Z p fy / m0  1.2 Ze fy / m0
8.2.1.4 Holes in the tension zone
(Anf / Agf)  (fy/fu) (m1 / m0 ) / 0.9
Dr S R Satish Kumar, IIT Madras
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Laterally Stability of Beams
Dr S R Satish Kumar, IIT Madras
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BEHAVIOUR OF MEMBERS SUBJECTED TO
BENDING
Mcr
Plastic Inelastic Elastic
Range Range
Range
Mp
My
Mo
Mo
L
Unbraced Length, L
Beam Buckling Behaviour
Dr S R Satish Kumar, IIT Madras
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LATERAL BUCKLING OF BEAMS
 FACTORS TO BE CONSIDERED
 Distance between lateral supports to the compression
flange.
 Restraints at the ends and at intermediate support
locations (boundary conditions).
 Type and position of the loads.
 Moment gradient along the unsupported length.
 Type of cross-section.
 Non-prismatic nature of the member.
 Material properties.
 Magnitude and distribution of residual stresses.
 Initial imperfections of geometry and eccentricity of
loading.
Dr S R Satish Kumar, IIT Madras
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SIMILARITY BETWEEN COLUMN BUCKLING
AND LATERAL BUCKLING OF BEAMS
Both have tendency to fail by buckling in their weaker plane
Column
Short span
Axial
compression
& attainment
of squash load
Long span Initial
shortening
and lateral
buckling
Pure flexural mode
Function of slenderness
Dr S R Satish Kumar, IIT Madras
Beam
Bending in the plane of
loads and attaining
plastic capacity
Initial vertical deflection
and lateral torsional
buckling
Coupled lateral
deflection and twist
function of slenderness
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SIMILARITY OF COLUMN BUCKLING AND BEAM BUCKLING -1
Y
M
P
X
Z
B
B
B
M
B

P
u
Section B-B
u
Section B-B
Column buckling
EA
l

EI y
l
3
Dr S R Satish Kumar, IIT Madras
Beam buckling
EIx >EIy
EIx >GJ
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LATERAL TORSIONAL BUCKLING OF
SYMMETRIC SECTIONS
Assumptions for the ideal (basic) case
• Beam undistorted
• Elastic behaviour
• Loading by equal and opposite moments in the
plane of the web
• No residual stresses
• Ends are simply supported vertically and laterally
The bending moment at which a beam fails by
lateral buckling when subjected to uniform end
moment is called its elastic critical moment (Mcr)
Dr S R Satish Kumar, IIT Madras
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(a) ORIGINAL BEAM (b) LATERALLY BUCKLED BEAM
A
M
y
Lateral
Deflection
M
A
Elevation
l
z
x
Section
θ
Section A-A
Plan
Twisting
(a)
Dr S R Satish Kumar, IIT Madras
(b)
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Mcr = [ (Torsional resistance )2 + (Warping resistance )2 ]1/2
Π
M cr 
L

 Π 2  E  I y  Γ 

E  I y  G  J   
2

L



1
2
or
1
Π
M cr  E  I y  G  J 2
L
  Π2 E Γ
1   2
  L G J



1
2
EIy = flexural rigidity
GJ = torsional rigidity
E = warping rigidity
Dr S R Satish Kumar, IIT Madras
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FACTORS AFFECTING LATERAL STABILITY
• Support Conditions
• effective (unsupported) length
• Level of load application
• stabilizing or destabilizing ?
• Type of loading
• Uniform or moment gradient ?
• Shape of cross-section
• open or closed section ?
Dr S R Satish Kumar, IIT Madras
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EQUIVALENT UNIFORM MOMENT FACTOR (m)
Elastic instability at M’ = m Mmax (m  1)
m = 0.57+ 0.33ß + 0.1ß2 > 0.43
ß = Mmin / Mmax (-1.0  ß  1.0)
Mmax
Mmin
Mmax
Mmin
Mmax
Mmin
Mmax
Mmin
Positive
Negative
also check Mmax  Mp
Dr S R Satish Kumar, IIT Madras
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8.2.2 Laterally Unsupported Beams
The design bending strength of laterally unsupported beam
is given by:
Md = b Zp fbd
fbd = design stress in bending, obtained as ,fbd = LT fy /γm0
LT = reduction factor to account for lateral torsional
buckling given by:
 LT 

1
[  LT   LT  LT

2

2 0.5
 1.0
]
LT  0.5 1   LT LT  0.2  LT
LT 
2

 b Z p f y / M cr
LT = 0.21 for rolled section,
LT = 0.49 for welded section
Dr S R Satish Kumar, IIT Madras
Cont… 26
8.2.2.1 Elastic Lateral Torsional Buckling Moment
2
  2 EI y  
 

EI
w
 GIt 
M cr  
2
2 


KL  
 KL   
 LT  EI y h 
2
M cr 
2( KL) 2
 KL / ry 
1
1  

 20  h / t f 

2 0.5




APPENDIX F ELASTIC LATERAL TORSIONAL BUCKLING
F.1
Elastic Critical Moment
F.1.1 Basic
F.1.2 Elastic Critical Moment of a Section Symmetrical about
Minor Axis
Dr S R Satish Kumar, IIT Madras
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EFFECTIVE LATERAL RESTRAINT
Provision of proper lateral bracing improves lateral stability
Discrete and continuous bracing
Cross sectional distortion in the hogging moment region
Discrete bracing
• Level of attachment to the beam
• Level of application of the transverse load
• Type of connection
Properties of the beams
• Bracing should be of sufficient stiffness to produce
buckling between braces
• Sufficient strength to withstand force transformed by
beam before connecting
Dr S R Satish Kumar, IIT Madras
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BRACING REQUIREMENTS
Effective bracing if they can resist not less than
1) 1% of the maximum force in the compression flange
2) Couple with lever arm distance between the flange
centroid and force not less than 1% of compression
flange force.
Temporary bracing
Dr S R Satish Kumar, IIT Madras
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Other Failure Modes
Shear yielding near support
Web buckling
Dr S R Satish Kumar, IIT Madras
Web crippling
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Web Buckling
Pwb  ( b1  n1 ) t fc
d/2
b1
L
0.7 d
 E 
n1
ry
450
d/2
ry 
ry
t3
t


A
12t 2 3
Iy
LE
2 3
d
 0.7 d
 2.5
ry
t
t
Effective width for web buckling
Dr S R Satish Kumar, IIT Madras
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Web Crippling
Pcrip  ( b1  n2 ) t f yw
b1
n2
1:2.5 slope
Root
radius
Stiff bearing length
Dr S R Satish Kumar, IIT Madras
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SUMMARY
• Unrestrained beams , loaded in their stiffer planes may undergo
lateral torsional buckling
• The prime factors that influence the buckling strength of beams
are unbraced span, Cross sectional shape, Type of end restraint
and Distribution of moment
• A simplified design approach has been presented
• Behaviour of real beams, cantilever and continuous beams
was described.
• Cases of mono symmetric beams , non uniform beams and
beams with unsymmetric sections were also discussed.
Dr S R Satish Kumar, IIT Madras
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