Design of members subjected to bending

DESIGN OF MEMBERS SUBJECTED TO BENDING

General

Design of Laterally Supported Beam

Design of Laterally Unsupported Beams

LOCAL BUCKLING AND SECTION CLASSIFICATION

OPEN AND CLOSED SECTIONS

Strength of compression members depends on slenderness ratio

(b)(a)

Local buckling of Compression Members

LOCAL BUCKLING

Beams – compression flange buckles locally Fabricated and cold-formed sections prone to local bucklingLocal buckling gives distortion of c/s but need not lead to collapse

L

Bending Moment Diagram

Plastic hinges

Mp

Collapse mechanism

Plastic hinges

Mp

Formation of a Collapse Mechanism in a Fixed Beam

w

Bending Moment Diagram

BASIC CONCEPTS OF PLASTIC THEORY

First yield moment MyPlastic moment MpShape factor S = Mp/MyRotation Capacity (a) at My (b) My < M<Mp

(c) at Mp

Plastification of Cross-section under Bending

SECTION CLASSIFICATION

Mp

Rotation

My

y u

Slender

Semi-compact

Compact

Plastic

Section Classification based on Moment-Rotation Characteristics

Moment Capacities of Sections

My

Mp

1 2 3 =b/t

Semi-Compact SlenderPlastic Compact

SECTION CLASSIFICATION BASED ON WIDTH -THICKNESS RATIO

For Compression members use compact or plastic sections

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 middepth

d/t 84.0 d/t 105 d/t 126

Angles bending    

Axial comp.

   

Circular tube with outer diameter D

  D/t 442 D/t 632 D/t 882

Table 2 Limits on Width to Thickness Ratio of Plate Elements

yf250

b/t 9.4 b/t 10.5 b/t 15.7

not applicable b/t 15.7(b+d)/t 25

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

(A) Design Strength in Bending (Flexure) Clause 8.2.1 pg 52

The factored design moment, M at any section, in a beam due to

external actions shall satisfy

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

dMM

(1) Design Strength in Bending (Flexure) Clause 8.2.1 pg 52

If V <0.6 Vd (Clause 8.2.1.2 pg 53)

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----- for simply supported beam

1.5 Ze fy / m0---------- for cantilever beam

b= 1 for plastic & compact sections

= Ze/Zp for semi compact sections

(2) Design Bending Strength under High Shear Clause 9.2.2 , pg 70

B) Design of shear Strength in Bending (Flexure) Clause 8.4 pg 59

C) Deflection Limit (Table: 6)

Maximum deflection should not exceed the limit given in table 6 of

the code

Laterally Stability of Beams

Beam bucklingEIx >EIy EIx >GJ

SIMILARITY OF COLUMN BUCKLING AND BEAM BUCKLING -1

M

u

M

Section B-B

uP

P

Section B-B

B

B B

B

Y

XZ

Column buckling

3l

yEI

l

EA

(a) ORIGINAL BEAM (b) LATERALLY BUCKLED BEAM

M

Plan

Elevationl

M

Section

(a)

θ

Lateral Deflection

y

z

(b)

Twisting

x

A

A

Section A-A

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 ?

Other Failure Modes

Shear yielding near support

Web buckling Web crippling

Web Buckling: at support web acts as column

450

d / 2

d / 2 b1 n1

Effective width for web buckling

t

d5.2

t

32d7.0

yrEL

32

t

t12

3t

AyI

yr

yr

d7.0

yrEL

Where : fcdw= web buckling strength b1= width of the stiff bearing on flange n1= h/2, h= depth of the section tw= thickness of the web Effective legth of web = 0.7 depth of web column Slenderness ratio= λ ≈ 2.5d / tw

Web Crippling

b1 n2 1:2.5 slope

Root radius

Stiff bearing length

Where : fw=web crippling strength b1= stiff bearing length nc= length obtained by dispersion trougth flange to web junction at slope 1 : 2.5 to plane of flange Fyw= yeild stress of web

Design Steps of Laterally supported Beam

Problem

A simply supported beam 5m span carries UDL load of 40 KN/m. In addition beam carries a central point load of 50 KN. The beam is laterally supported. Design the

section & check section for the shear and deflection.

50 KN

2.5m 2.5m

40KN/m

Step 1: Load calculation

Factored UDL= 1.5 x 40 = 60KN/m

Factored pt load= 1.5 x 50 = 75KN/m

Step 2: calculation of plastic modulus (Zp rquired)

Step 3: selection of the section

Step 4: Check for shear strength

Step 5: Check for design moment capacity

Step 6: Check for Web Buckling

450

d / 2

d / 2 b1 n1

Effective width for web buckling

Assume b1= width of the stiff bearing on flange=75mm, d= h1= 340.5mm n1= h/2=400/2=200 , tw= 8.6mm ,

Slenderness ratio= λ ≈ 2.5d/tw= 2.5 X 340.5/8.6= 98Find fcd

Step 7: Check for Web Crippling

b1 n2 1:2.5 slope

Root radius

Stiff bearing length

Step 8: check for deflection

Problem

A laterally supported beam of span 4 m is subjected to factored column load of

400KN. Load is transferred through base plate of 200mm length . Check limit states.

Section available is ISMB 400

CURTAILMENT OF FLANGE PLATES

When beam subjected to UDL then, maximum bending moment occurs at the centre. Since the values of BM are decreases towards the supports from center .Therefore the flange plates may be curtailed at a distance from the centre of span greater than the distance .It gives economy as regards to the material and cost. At least one flange plate should be run for the entire length of the girder.

CURTAILMENT OF FLANGE PLATES

Mcr = [ (Torsional resistance )2 + (Warping resistance )2 ]1/2

2

1

2

y2

ycr L

ΓIEΠJGIE

L

ΠM

2

1

2

2

2

1

ycr JGL

ΓEΠ1JGIE

L

ΠM

or

EIy = flexural rigidityGJ = torsional rigidityE = warping rigidity

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)

MminMmax

Mmin

Positive

Mmax Mmin

Mmin

Negative

Mmax

Mmax

also check Mmax Mp

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…

0.1][

15.022

L TL TL T

L T

22.015.0 L TL TL TL T

crypbL T MfZ /

8.2.2.1 Elastic Lateral Torsional Buckling Moment

2

2

2

2

K L

E IG I

K L

E IM w

ty

cr

5.02

2

2

/

/

2 0

11

)(2

f

yyL Tcr th

rK L

K L

hE IM

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

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

8.2.1.4 Holes in the tension zone

(Anf / Agf) (fy/fu) (m1 / m0 ) / 0.9

dMM

EFFECTIVE LATERAL RESTRAINT

Provision of proper lateral bracing improves lateral stabilityDiscrete and continuous bracingCross 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

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

BRACING REQUIREMENTS

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.