1. IS 800:2007 Section 8 Design of members subjected to bending

2. 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
End Panels design
End Panels designed using Tension field action
8.5.3 Anchor forces
8.6 Design of Beams and Plate Girders with Solid Webs
Minimum Web Thickness
Sectional Properties
Flanges Cont...

3. SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING
8.7 Stiffener Design
General
Design of Intermediate Transverse Web Stiffeners
Load carrying stiffeners
Bearing Stiffeners
8.7.5 Design of Load Carrying Stiffeners
8.7.6 Design of Bearing Stiffeners
Design of Diagonal Stiffeners
Design of Tension Stiffeners
Torsional Stiffeners
Connection to Web of Load Carrying and Bearing Stiffeners
Connection to Flanges
Hollow Sections
8.8 Box Girders
Purlins and sheeting rails (girts)
8.10 Bending in a Non-Principal Plane

4. 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

5. LOCAL BUCKLING AND SECTION CLASSIFICATION
OPEN AND CLOSED SECTIONS
Strength of compression members depends on slenderness ratio

6. 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

7. BASIC CONCEPTS OF PLASTIC THEORYw
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

8. SECTION CLASSIFICATIONMp
Rotation  My y u
Slender
Semi-compact
Compact
Plastic
Section Classification based on Moment-Rotation Characteristics

9. SECTION CLASSIFICATION BASED ON WIDTH -THICKNESS RATIOMy Mp 1 2 3
=b/t
Semi-
Compact
Slender
Plastic
Moment Capacities of Sections
For Compression members use compact or plastic sections

10. Table 2 Limits on Width to Thickness Ratio of Plate Elements
Type of Element
Type of Section
Class of Section
Plastic (1)
Compact
(2)
Semi-compact (3)
Outstand element of compression flange
Rolled
b/t  9.4
b/t  10.5
b/t  15.7
Welded
b/t  8.4
b/t  9.4
b/t  13.6
Internal element of compression flange
bending
b/t  29.3
b/t  33.5
b/t  42
Axial comp.
not
applicable
b/t  42
Web
NA at mid
depth
d/t  84.0
d/t  105
d/t  126
Angles
bending
b/t  9.4
b/t  10.5
b/t  15.7
Axial comp.
not
applicable
b/t  15.7
(b+d)/t  25
Circular tube with outer diameter D
D/t  442
D/t  632
D/t  882

11. 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

12. Design Strength in Bending (Flexure)
The factored design moment, M at any section, in a beam due to
external actions shall satisfy
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

13. 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

14. Definition of Yield and Plastic Moment Capacities

15. 8.2 Design Strength in Bending (Flexure)
The factored design moment, M at any section, in a beam due to
external actions shall satisfy
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
Holes in the tension zone
(Anf / Agf)  (fy/fu) (m1 / m0 ) / 0.9

16. Laterally Stability of Beams

17. BEHAVIOUR OF MEMBERS SUBJECTED TO BENDING
Plastic
Range
Inelastic
Elastic Mp My
Mcr
Unbraced Length, L Mo L
Beam Buckling Behaviour

18. 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.

19. SIMILARITY BETWEEN COLUMN BUCKLING AND LATERAL BUCKLING OF BEAMS
Both have tendency to fail by buckling in their weaker plane
Column
Beam
Short span Axial
compression
& attainment
of squash load
Bending in the plane of
loads and attaining
plastic capacity
Long span Initial
shortening
and lateral
buckling
Initial vertical deflection
and lateral torsional
Pure flexural mode
Function of slenderness
Coupled lateral
deflection and twist
function of slenderness

20. SIMILARITY OF COLUMN BUCKLING AND BEAM BUCKLING -1P X Z B B B B M  P u u
Section B-B
Section B-B
Column buckling
Beam buckling
EIx >EIy
EIx >GJ

21. 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)

22. (a) ORIGINAL BEAM (b) LATERALLY BUCKLED BEAM
Plan
Elevation l
Section
(a) θ
Lateral
Deflection y z
(b)
Twisting x A
Section A-A

23. Mcr = [ (Torsional resistance )2 + (Warping resistance )2 ]1/2
EIy = flexural rigidity
GJ = torsional rigidity
E = warping rigidity

24. 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 ?

25. EQUIVALENT UNIFORM MOMENT FACTOR (m)
Elastic instability at M’ = m Mmax (m  1)
m = ß + 0.1ß2 > 0.43
ß = Mmin / Mmax (-1.0  ß  1.0)
Mmin
Mmax
Positive
Negative
also check Mmax  Mp

26. 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 = 0.21 for rolled section,
LT = 0.49 for welded section
Cont…

27. APPENDIX F ELASTIC LATERAL TORSIONAL BUCKLING
Elastic Lateral Torsional Buckling Moment
APPENDIX F ELASTIC LATERAL TORSIONAL BUCKLING
F.1 Elastic Critical Moment
F.1.1 Basic
F Elastic Critical Moment of a Section Symmetrical about
Minor Axis

28. 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

29. 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

30. Shear yielding near support
Other Failure Modes
Shear yielding near support
Web buckling
Web crippling

31. Effective width for web buckling
450
d / 2 b1 n1
Effective width for web buckling

32. Web Crippling b1 n2
1:2.5 slope
Root radius
Stiff bearing length

33. 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.