beam-6.pdf

1

Basic principles of steel structures

Dr. Xianzhong ZHAO

[email protected]

www.sals.org.cn

Bending members (beam)Outlines

Introduction

Resistance of cross-section of beams

Lateral torsional buckling of beams

Local buckling of plate element in beams

Deflection of beams

Bending members (beam)introduction

state of loading- bending moment- shear force- axial force- combination of above

- bending about one axis- bending about both axes (bi-axial bending)

roof purlin crane beam

Bending members (beam)introduction

boundary conditions and spans

simply-support both-end fixed cantilever

multi-span continuous beam

structural system and load transfer

roof panel →secondary beams →main beams →columns → foundations

Bending members (beam)introductionsection types

solid-web section(normal / light gauged)

castellated beam

composite beam

Bending members (beam)introductionsection types

solid-web section

open-web section

non-uniform section(height, width, strength)

laced built-up section(truss)

2

Bending members (beam)introduction: failure modes

strength failure (cross-section resistance)yield, fracture, fatigue

Lateral-torsional-buckling (LTB) of beam

Local buckling of plates in beam

flexural-torsional buckling

flange: compressionweb: compression, bending and shear

excessive deflection

Less rigidity of bending

Cross-sectional resistance of beamsbending moment resistance (1)

assumption- perfect elasto-plastic model- cross section remains plane during bending

σ

ε

yf

yεdistribution of normal strain and stress

y1 f<σ y1 f=σ y1 f=σ

y2 f=σ

y1 εε <

y2 εε <

y1 εε = y1 εε >

y2 εε =

y1 εε >

y2 εε > ∞⇒2ε

∞⇒1ε


Criteria 1: yielding at extreme fibre

exx MM ≤ dxn

x fWM

≤=σ

Criteria 2: yielding on full section

x

px

ex

pxpx W

WMM

==γ

pxx MM ≤d

pxn

x fWM

≤

Criteria 3: yielding on partial section

exxx γ≤ MM dxnx

x fW

M≤

γpxx1 γγ <<

elastic net (effective) section modulus

plastic net section modulus

elastic net section modulus

shape factor

plastic adaptation factor

Memory: tension members with bendingconcept of full plastic moment

Assumption of stress distribution- subjected to bending only, no tension- stress at each point reach yield point- yield point under tension and

compression is the same

yyypx )( fyAyAyfAyfAM −−++−−++ +=+=

0=N 0yy =− −+ fAfA −+ = AA

−−++ += yAyAWpx ypxpx fWM =

y1 f=σ

y2 f=σ

x

y+A

−A −y+y

Plastic neutral axis

The plastic neutral axis divides the cross section into two equal areas, and it may not coincide with the centroidal axis

(Full) plastic moment

plastic modulus of a section:

Review of section modulusshape factor & plastic adaptation factor of cross-section

shape factor

plastic adaptation factor


Bi-axial bending:

Criteria 2: yielding on full section

Criteria 3: yielding on partial section

Criteria 1: yielding at extreme fibre

yx ,MM

dyn

y

xn

x fWM

WM

≤+=σ

dpyn

y

pxn

x fWM

WM

≤+

dyny

y

xnx

x fW

MW

M≤+

γγ

3

Cross-sectional resistance of beamsshear resistance

tISV

tISV

y

yx

x

xy +=τ

shear stress:

shear stress under bi-axial shear forces

shear resistance

mechanics of material:tI

SV

x

xy=τ

w

y

AV

=τapproximate value: (for I-shape or channel shape only)

vdf≤τ

algebraic or vector add?

Note: gross section or net section?

Cross-sectional resistance of beamstransverse force resistance

local stress due to the transverse force

dwz

ft

Fc ≤

⋅=σ

z

a

z y R5 {2 }a h h= + +

Fyh :distance from load surface to

the upper edge of the effective web width

— place, thickness and width of stiffeners— strength and stability of stiffeners and web around

w15tw15t

(height of rail)

Stress check

Design of stiffeners

Cross-sectional resistance of beamstransverse force resistance

local stress due to the transverse force

illustration of yh

Cross-sectional resistance of beamsstate of stresses and equivalent stress

state of combined stresses (SCS)

2c

2c

2zs 3τσσσσσ +⋅−+=

— stress state of existence of two or more stress components at same point / same load condition

Criteria of elasticityCriteria of partially plastic adaptation

— sign of stress component: tension positive and compression negative

d1zs f⋅≤ βσ 11 =β11 >β

maxσmaxτ

2max

2maxzs 3τσσ += ？

Which section has SCS?

Which point has SCS?

Equivalent stress of beam

practical design equation

Cross-sectional resistance of beamsdesign procedure of beam strength

Analytical model of the structure

Internal force diagram under different load casesLoading pattern and value, boundary conditions

Ascertain the computing point

Ascertain the section to be checked

Strength check

Bending moment and shear forces

Calculate the sectional properties

Calculate the nominal stress and equivalent stress

Lateral-torsional buckling of beamspreparation: shear centre

concept of shear centre

hbtIbhV

M z ××=22f

x

y

eVM z y= x

f22

4Ithbe =

4

Lateral-torsional buckling of beamspreparation: shear centre

centroid and shear centre

Lateral-torsional buckling of beamspreparation: free torsion

free torsion1. shear stress on section due to free torsion2. uniform torsional angle along the member

'θtk GIM =

tItM z=τ tA

M z

02=τ

∑=

=n

iiitbI

1

3t 31

∫=

ts

AI d4 2

0t

r0

δ

Tr0

δ

T

open-section closed-section

Lateral-torsional buckling of beamspreparation: restrained torsion

''2yyω 5.0 θhEIhMB −==

restrained torsion and warping

θhu 5.0=''''

y 5.0 θφ hu ==''

y''

yy 5.0 θhEIuEIM −=−=

'''yy 5.0/ θhEIdzdMV −==

'''2yyω 5.0 θhEIhVM −==

24/5.0 2f

32yω htbhII ≈=

'''ωω θEIM −=

θ

ft

b

h

u

Tω MMM k =+

L

x

y'θtk GIM =

Example:cantilevered I-shape beam under end torsional moment

''ωω θEIB −=

Lateral-torsional buckling of beamsdifferential equations for elastic LTB of beam

equilibrium and deflection of LTB of beam

00''

x =−+ θNxNvvEI00

''y =−+ θNyNuuEI

0)( '20 =−+ θRNr

'0

'0

''''ω uNyvNxGIEI t +−− θθ

zy

xMxM

0x''

x =+ MvEI

0x''

y =+ θMuEI

0'x

''''ω =+− uMGIEI tθθ

zx

xM

xM

v

u'u

x

y

uv

θ

xM

xM

θxM

'xuM

Simply-supported beamEqual bending momentSecond-order, small deflection

differential equations: flexural-torsional buckling

Lateral-torsional buckling of beamssolution of LTB for simply-supported beam with uniform bending

0x''

y =+ θMuEI0'

x'

t'''

ω =+− uMGIEI θθ (6-46c)Substituting Equ.(-b) into Equ.(-c), we get differential equation about

0)/( y2''

tIV

ω =−− θθθ EIMGIEI x

)/sin( znC πθ =boundary conditions: 0''''

00 ==== ll θθθθ

(6-47)

(6-46b)

θ

(a)

Substituting Equ.(a) into Equ.(6-47), we get

0sin)(y

2x

2

22t

4

44ω =⋅⋅−+

znCEIMnGInEI πππ (b)

Then we have

)1(ω

2

2t

y

ω2y

2

crxx EIGI

IIEI

MMπ

π+=⇒ (6-48)

Lateral-torsional buckling of beamscritical bending moment

critical bending moment

)1(ω

2

2t

y

ω2y

2

crx EIGI

IIEI

Mπ

π+=

wf ,,, tthb

Assume web thickness is small but have enough stiffness to keep its shape, then we have the same eqution of critical bending moment

crxM

,25.024 y2

f23

2ω I

hthb

hI

=⋅

=

)1(ω

2

2t

2y

ω2y

2crx

1 EIGI

hIIEI

hMN

ππ

+=≈

Flange width, height, thickness of flange and web for I-shape

1N

,039.02 ≈E

Gπ 22

22f

f23

23f

ω

2t 1624

32

hbt

thbbt

II

=≈

2y

2

1 64.0EI

Nπ

≈if 122

22f ≈hb

t2y

2

5.0EIπ

>？

5

Lateral-torsional buckling of beamscritical bending moment with different boundary and loading

Effect of boundary conditions

])(

[)( ω

2

2yt

2ω

2y

y

ω2

y

y2

crx EIGI

IIEI

Mπ

μμμ

μπ

+=Table 6-2 in pp.163

ωy ,μμ

ocrx1crx MM ⋅= β

: critical bending moment for beam under uniform bending

ocrxM

0.11 =β 13.11 =β 35.11 =β 65.21 =β

Effect of loading pattern

Lateral-torsional buckling of beamscritical bending moment with effects of section type and loading

Effect of section types and loading point

a

])1()([ω

2

2t

y

ω2y32y322

y2

1crx EIGI

IIBaBa

EIM

πββββ

πβ +++++=

yB

2β

—distance from load point to shear centre

—parameter indicating asymmetric degree for section

∫ −+= 022

xy )(21 ydAyxyI

B

— bending:0; UDL:0.46; CL at mid-span 0.55

3β — bending:1; UDL:0.53; CL at mid-span 0.40

load

deflection

S

Same direction with deflection for load to S，a<0

loaddeflection

S

Reverse direction with deflection for load to S，a>0

Lateral-torsional buckling of beamsparameters affect the LTB of beams

Critical bending moment

])1()([ω

2

2t

y

ω2y32y322

y2

1crx EIGI

IIBaBa

EIM

πββββ

πβ +++++=

])(

[)( ω

2

2yt

2ω

2y

y

ω2

y

y2

crx EIGI

IIEI

Mπ

μμμ

μπ

+=

—— rigidity of section—— distance of lateral supports—— section types (width of compressive flange)—— loading pattern (bending moment)—— loading point—— boundary conditionsHow about the initial imperfection?

Lateral-torsional buckling of beamselasto-plastic lateral-torsional buckling of beams

)1(ω

2

2t

y

ω2y

2

crx EIGI

IIEI

Mπ

π+=

Elastic LTB

})(])[(1{

)()()(

ω2

2t

y

ω2y

2

crxt

t

t

tt

EIKGI

EIEIEI

Mπ

π ++=

Elasto-plastic LTB using tangent modulus theory

• short beam• large residual stress

Lateral-torsional buckling of beamsultimate capacity of beam with initial imperfections

xMxM

,u θ

xM

crxM

Lateral-torsional buckling of beamsdesign equations

dy

y

xb

x fWM

WM

≤+ϕ

Design equation for beams

yxbyxy

crxex

ex

crxcrxx fWfW

fM

MMMM ϕσ

===≤

dxb

x fW

M≤⇒

ϕ : stability coefficient for beamsbϕ

xW

←bi-axial bending

: gross section modulus

Conditions no need to check LTB of beams1. A rigid decking is securely connected to compressive flange of beams

How about the bottom flange under compression?2.The max ratio of unsupported length to flange width is less than values

listed in Table 6-3 in pp167

21

b 2x y

4320 23514.4

yb b

y

tAhW h f

λϕ β η

λ

⎡ ⎤⎛ ⎞⎢ ⎥= ⋅ ⋅ + + ⋅⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

6

Local buckling of plates in beamsintroduction

Local buckling of plates in beamflange: compressionweb: compression, bending and shear

22

2

tcr )()1(12 b

tEkυ

πχψσ−

=

Critical local buckling stress

1. ← flange having larger stiffness in beams

2. ← outstand flange of I-section, b is flange outstand

← flange supported along both edges of box-sectionb is the supported part in both edges

Local buckling of plates in beamscritical local buckling stress of flange

2

2

2

2

cr )1(12 btEk ⋅

−⋅=

μπχσ

1=χ

425.0=k

stress distribution of flange in beam

- shear stress is small- normal stress is nearly uniform in flange

critical local buckling stress

0.4=k

Local buckling of plates in beamscritical local buckling stress of web

local buckling of plate under uneven compressive stress

2

2

2

2

cr )1(12 btEk ⋅

−⋅=

μπχσ

χ,

)5.01/(4 α−=k

tb,

crσ : max. compressive stress( ) while plate buckles

: height and thickness ofplate (web)

maxσ

minσ

maxσ)474.01/(1.4 α−=k

26α=k

max

minmaxσ

σσα −=

3/20 ≤≤α4.13/2 ≤< α

44.1 ≤< α

k : stability factor of plate 24,2 == kα =1.61ww , th

max crσ σ→

critical local buckling stress of plate simply supported at 4 edges


local buckling of plate in shear

τ

why plate buckles in shear?


critical local buckling stress of plate in shear

τ

2minmax )/(

434.5 +=k

χ =1.24

min

max

2

2

2

2

cr )1(12 btEk ⋅

−=

μπσ

2min

2

2

2

cr )1(12tEk ⋅

−=

μπτ

2

w

w2

2

cr )()1(12 h

tEkμ

πχτ−

⋅=

2w )/(434.5 ahk +=

4)/(34.5 2w += ahk

while 1/w ≤ah

while 1/w >ah

wh

a

critical local buckling stress of web of I-section in shear


local buckling and critical local buckling stress of plate under local compressive stress

cσcrc,c σσ ⇒

2 2c

cr c,cr cr

( ) ( ) 1σ σ τσ σ τ

+ + ≤

2 2c

cr c,cr cr

( ) ( ) 1σ σ τσ σ τ

+ + ≤

Equ.(6-67)

2 2c

cr c,cr cr

( ) ( ) 1σ σ τσ σ τ

+ + ≤ Equ.(6-67a)

2c

cr c,cr cr

( ) 1σ σ τσ σ τ

+ + ≤ Equ.(6-68)

Equ.(6-68a)

21crc, )100(

htC=σ

critical local buckling stress of plate under combined stress

7

Local buckling of plates in beamsdesign criteria of preventing local buckling of plates

Criteria 1: critical local buckling stress is larger than yield point

ycr f>σ

ybcr f⋅> ϕσ

σσ >cr

Criteria 2: critical local buckling stress is larger than critical overall buckling stress of member

Criteria 3: critical local buckling stress is larger than actual stress in plate

Discussion: which criteria is the severest for local buckling prevention?

Local buckling of plates in beamsmethod to prevent plates from local buckling

how to promote the critical local buckling stress?

1. Modify boundary condition2. Modify width-to-thickness ratio

2

2

2

2

cr )1(12 btEk ⋅

−=

μπσ

- increase thickness- setup stiffeners- which is better?

transverse stiffener

longitudinal stiffener

short stiffener

decrease the actual stress in members (criteria 3)Increase the height of section, thus makes the decrease of actual stress is faster than that of critical local buckling stress

Local buckling of plates in beamsstiffeners

transverse stiffener

longitudinal stiffener

short stiffener

Local buckling of plates in beamsdesign against local buckling of I-section

outstand flange under compression

consider the residual stress and initial imperfection,

y

2

2

2

cr 95.0)1(12

fbtEk

≥⎟⎠⎞

⎜⎝⎛

−=

υπσ

yy2

52

y2

2 2358.1895.0)3.01(121006.2425.0

95.0)1(12 fffEk

tb

=×−

×××≤

×−≤

πυπ

yy2

52

y2

2 23513)3.01(12

1006.225.0425.0)1(12 fffEk

tb

=×−

××××≤

×−≤

πυηπ

yy2

52

y2

2 2354.15)3.01(12

1006.25.0425.0)1(12 fffEk

tb

=×−

××××≤

×−≤

πυηπ

consider the residual stress and initial imperfection, plus partial plasticity


web under bending, shear, local compression, respectively

1. subjected to bending moment

y

2

2

52

cr )3.01(121006.22461.1 f

ht

w

w ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−

××××=

πσy

235174ft

h

w

w ≤

Q/A: how about for the unsymmetric I-section with larger compressive flange?

2. subjected to pure shear

y

235104ft

h

w

w ≤yvy

2

2

52

cr 58.0)3.01(12

1006.2)134.5(24.1 ffht

w

w =≥⎟⎟⎠

⎞⎜⎜⎝

⎛−

×××+×=

πτ

3. subjected to local compression

y

2

crc, 100166 fht

w

w ≥⎟⎟⎠

⎞⎜⎜⎝

⎛××=σ

y

23584ft

h

w

w ≤

1.61, 170, restrained

1.23, 150, unrestrained

a / h=2.0

a / h=2.0


design procedure for real steelwork (no local buckling allowed)

1. For hot-rolled sections

2. For welded built-up sections

3. subjected to local compression

No need to check

y

23580ft

h

w

w ≤ O.K.yf

f 23515ft

b≤

yf

f 23515ft

b≥ modify flange section

y

23580ft

h

w

w ≥ modify web section, or setup stiffeners

yy

235)150(17023580ft

hf w

w ≤< setup transverse stiffeners

y

235)150(170ft

h

w

w ≥ setup transverse, longitudinal stiffeners, plus short stiffeners if necessary

8

1)()( 2

crc,

c2

crcr≤++

σσ

ττ

σσ


design procedure for real steelwork (no local buckling allowed)

4. ascertain the space of stiffenerstransverse stiffeners longitudinal stiffenersstability check for each grid

Method：Equ.(6-76)-(6.86)

ww 25.0 hah ≤≤

w1w 25.02.0 hhh ≤≤ a

1h

1)()(crc,

c2

cr

2

cr≤++

σσ

ττ

σσ

1b

1t

5. design of stiffenersdimension, requirement of strength, rigidity and stability

Local buckling of plates in beamsdesign of bending members using post-buckling strength

Mechanism of post-buckling strength in bending— subjected to bending moment

— subjected to shear force

Local buckling of plates in beamsdesign of bending members using post-buckling strength

design principles— elastic design allowing local buckling— subjected to static load

— subjected to bending momentconcept of effective sectionbending resistance on effective section, Equ.(6-87~90) pp177

design method

— subjected to shear forceconcept of tension field and truss-beamdiscount the shear strength, Equ.(6-91~94) pp178-179

— subjected to combined bending and shearEqu.(6-95~100) pp179-180

Local buckling of plates in beamsapplication of post-buckling strength

Portal frame

Local buckling of plates in beamsapplication of post-buckling strength

National Stadium bird nest

1000×1000×20×20

600×600×10×10

Local buckling of plates in beamslimit of width-to-thickness ratio and design method of beam

width-to-thickness height-to-thicknessof flange of web

plastic design 9 70

partially plastic design 13 (150)

elastic design 15 (170)

design using (20) 250~300post-buckling strength

9

deflection of beamscalculation of elastic deflection and rigidity

uniformly-distributed-load (UDL)

EIqL

3845 4

=δ

EIFL48

3=δ

EIML10

2≈δ

[ ], [ ] /1200 ~ /150L Lδ δ δ≤ =

q

F

nF

concentrated load at mid-span (CL)

multiple concentrated load (MCL)

requirement of rigidity

deflection of beamsconcept of deflection capacity

xMxMθ

p/θθ

pxx /MM pxx /MM

p/θθ

1

1

1

1

Design of bending memberssummary

Selection of sectionCalculation of section resistance (strength)

Calculation of width-to-thickness ratio, setup stiffeners, design of stiffenersCalculation of beams using post-buckling strength

Calculation of overall stability

Calculation of deflection

normal stress, shear stress, local compressive stress, combined stressUsing net section, except shear stress check

Calculation of local buckling of plates

Need to check the overall stability?Ascertain the critical LTB bending momentUsing gross section

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