Composite Beam II

8/2/2019 Composite Beam II

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Chiew SingChiew Sing --PingPing

School of Civil and Environmental Engineering

Nanyang Technological University, Singapore

Design of Composite BeamsContinuous beams

Hogging moments and moment redistribution:

Basic behaviour, concepts and codified design

22

Composite construction


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Long span composite floor system

Full integration with building services.

44

Composite construction

Greater stiffness and higher load carrying capacities.

Fast erection of structural members.

Reduce height of a structure and offer further savings inassociated features through integration with building services.

Good inherent fire resistance in slabs and columns.

Steel deckings as permanent formwork provide additional safetyfeatures during construction.


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Composite beam with composite slab using profiled steel deckings

Composite beam with solid concrete slab

Beam span parallel to slab spanB

D

Transversereinforcement

Ds

Profiled

deckling

Beam span perpendicular to slab span

DpDsDp

D

B

Be

Profileddeckling



Be

B

D

66

Composite beams under hogging moments.

Continuous composite beams with moment re-distribution.

Understanding on structural behaviour of composite beams.

Design of composite beams to codified methods.

Scope


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Composite beams with profiled steeldeckings

Deck

perpendicular to

secondary beam

Deck

parallel to

primary beam

1010

Prescriptive design approach

Moment capacities according plastic stress blocks.

Sagging moment capacities with full or partial shear connection.

Hogging moment capacities with full shear connection.

Minimum degree of shear connection.

Rigid shear connectors with a elastro-plastic load slippage curve.

Prescribed percentage of moment re-distribution.

Current design methodology


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Prescriptive design approach- Simplified load slippage curve

Shearforce,

Fs

Slippage, S

Fs

s

PK

R-72

Assume a rigid plastic load-slippage curve of shear connectors.

Assume ductile behaviour

1212

Forces:

Rr= Tensile resistance of reinforcement

Rs = Tensile resistance in the steel section

Rq = Shear resistance in the shear connectors

Basic resistances against hoggingmoment

Rr

Rq

Rwb

Rfb

RwtRft


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Prescriptive design approach

- Plastic section analysis

Various degree of shear connection

Assume a rigid plastic load-slippage curve of shear connectors.

RrRrP.N.A

Rr

(a) yp outside steelsection (unlikely inpractice)

Rs

0.87 fy

py

P.N.A

(c) yp in steel web

P.N.A

0.87 fy

(b) yp in steel flange

0.87 fy

1414

Development of moment resistancealong beam span

Rigid shear

connectors

Sufficient shear connectors provided for full strength mobilization

Compressive

force

Tensile

force

Full shear connection

0.87 fy

(a) yp in steel flange

py

P.N.A

(c) yp in steel web (free end)

P.N.A

(b) yp in steel web

P.N.A


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3 cases to be considered for hogging moments

Hogging moment applied in a composite section where the steel sectionhas two equal flanges and a compact web.

[Case 5a ] Plastic neutral axis in web

[Case 6ai] Plastic neutral axis in steel flange

[Case 6aii] Plastic neutral axis outside steel section

Composite beams subjected to hogging moments should have full shear connection.

Hogging moment resistance

1616

The plastic moment capacity is expressed in terms of the resistance ofthe various elements of the beams as follows:

Resistance of Steel Beam:

Resistance of Steel Flange:

Resistance of Clear Web Depth:Resistance of Reinforcement:

Plastic moment resistance of steel beam:

Plastic moment resistance of composite beam:

Rs = A py

Rf= B T py

Rv= d t py

Rr= 0.87 fyAr

Ms = pySxor 1.2pyZx

Mc


Dp

Ds

Be

B

td

T

T

Dr

Ar

D


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[Case 5a] Plastic neutral axis in steel web : Rr< Rw

42

2d

R

RD

DRMM

v

rrrsc

++=


[Case 6ai] Plastic neutral axis in steel flange : RrRw

[Case 6aii] Plastic neutral axis outside steel section : Rr

Rs

( )42

2T

R

RRDR

DRM

f

rsrrsc

+=

+= rsc D

DRM

2

Dr= Distance from top of steel section to centroid of reinforcement

P.N.A

P.N.A

Typical

designP.N.A

1818

Section classification in compositecross-sections

In general, the moment capacities of composite cross-sections are

limited by local buckling in the steel web or in the steelcompression flange.

For composite cross-sections of either class 1 plastic or class 2

compact, the moment capacities of composite beams aredetermined with rigid plastic theory, i.e. rectangular stress blocks.

The section classification of a composite cross-section is oftensimilar to that of the steel beam.


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For beam subjected to uniform loads, the total number of shear

connectors (Nn) required to develop the negative plastic moment

capacity of the section under full shear connection can be determined

from the equation:

Nn = Number of shear connectors between points of zero

and maximum hogging moment

Qn = Shear resistance shear connectors at hogging

moment region

Fn

= Longitudinal compressive force at the point of

maximum hogging moment

Nn = Fn/ Qn

Fn = Smaller of Rcand Rs


2020

Beam span is perpendicular to slab spanbe = Lz/8but not greater than b

Beam span is parallel to slab spanbe = Lz/8but not greater than 0.8b

Beam at edgebe = Lz/8+ projection of slab beyond

centreline of beam

Effective width of the concrete slab

Effective width, Be ,is calculated as follows:

Be = bei

Lz

= distance between

points of zero

moments

b = actual width

be1 be2

b


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Distance between points of zero momentsin continuous beams

0.8L1 0.7L2 0.8L3 - 0.3L4

but 0.7L3

L2L1 L3 L4

0.25(L1 + L2) 0.25(L1+ L2) 1.5L4

but L4 + 0.5L3

2222

Section classification of composite cross-section

Moment resistance with full shear connection

Shear resistance

Shear connection

Moment resistance with partial shear connection

Transverse reinforcement

Practical design procedures

For structural adequacy, the following checks should be

satisfied:

Ultimate Limit State

Serviceability Limit State

Deflection

Serviceability stresses


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Analysis onA Single Span Composite Beam

under Hogging Moment

Reference: Loh, H.Y., Uy, B. and Bradford, M.A. The effects of partial shear connection in the hoggingmoment regions of composite beams. Part 1: Experimental study , Journal of ConstructionalSteel Research, 2004, 60(6), 897-919.

2424

Single span composite beam under hogging moment

Beam CB2. py = 400.0 N/mm2, fy = 500 N/mm

2, Ar = 1206 mm2,

pc = 27.0 N/mm2 and fcu = 33.8 N/mm

2.

P

250UB25.7

2500

600

500248

120

B

B417 typ

Section B-B

515

256

120

124 8.0

124 8.0

5.0

100 8.0

50 8.0

50

Y1652


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Resistance of the steel sectionRs = A x py = 4304 x 400 x 10

-3 = 1722 kN

Resistance of a shear connector ( hogging moment region )Rq = 105 x 6 = 630 kN > Rr It is full shear connection.

Resistances of various elements of the beam

Resistance of the reinforcementRr= fy x At = 500 x 1206 x 10

-3 = 603 kN

Section properties of steel beamA = 4304 mm2, Zx = 291.9 x 10

3 mm3 , Sx = 362.8 x 103 mm3

Ms

= py

x Sx

or 1.2 x py

x Zx

= 145.1 kNm or = 140.1 kNm

Ms = 140.1 kNm

2626

Location of P.N.ACompression = 320 + 396.8 + 144 + ( 240 y1) x 5 x 400 / 1000

Tension = 396.8 + 603 + (y1 8 ) x 5 x 400 / 1000

y1 = 89.25 mm, R4t = 301.5 kN, R4b = 162.5 kN


Self weight of composite beam= 1.48 (concrete slab) + 0.33 (steel beam) = 1.81 kN/m

Hogging moment resistance= ( 320 x 162.75 + 396.8 x 154.75 ) x 10-3 +

(144 x 96.75 + 301.5 x 75.38 ) x 10-3 +

(162.5 x 40.625 + 396.8 x 85.25 + 603 x 184.25 ) x 10-3

= 301.7 kNm

Moment due to self-weight= 1.81 x 2.52 / 8 = 1.41 kNm

95

256

100

P.N.A.

R3 = 144 kN

R5 = 396.8kN

Rq = 603.0kN

R4b

R4t

y1

R1 = 320 kNR2 = 396.8 kN

124

5

Ultimate load to composite beam

= (301.7 1.41 ) x 4 / 2.5 = 480.4 kN


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-3 = 1528 kN

Ms = py x Sx or 1.2 x py x Zx= 128.9 kNm or = 124.5 kNm

Ms = 124.5 kNm

Resistance of a shear connector ( hogging moment region )Rq = 0.6 x ( 2 x 72 )x 3 = 259.2 kN < Rr It is partial shear connection. ( degree of psc = 0.54 )


Resistance of the reinforcementRr= 0.87 x fy x At = 0.87 x 460 x 1206 x 10

-3 = 482.6 kN

Section properties of steel beamA = 4304 mm2,py= 355 N/mm

2, Zx = 291.9 x 103 mm3 , Sx = 362.8 x 10

3 mm3

Section classification of steel sectionFor flange, b / T = 7.75 < 9 => Flange is compact sectionFor web, d / t = 46.4 < 80 => Web is plastic section It is a compact section

2828

Location of P.N.ACompression = 284 + 352.2 + 127.8 + ( 240 y1) x 5 x 355 / 1000

Tension = 352.16 + 259.2 + (y1 8 ) x 5 x 355 / 1000

y1 = 167.0 mm, R4t = 129.6 kN, R4b = 282.2 kN


Self weight of composite beam= 1.48 (concrete slab) + 0.33 (steel beam) = 1.81 kN/m

Hogging moment resistance= ( 284 x 85.01 + 352.16 x 77.01) x 10-3 +

( 127.8 x 19.01 + 129.6 x 36.51) x 10-3 +

( 282.2 x 79.49 + 352.16 x 162.99 + 259.2 x 261.99 ) x 10-3

= 206.2 kNm

Moment due to self-weight= 1.81 x 2.52 / 8 = 1.41 kNm

95

256

100

P.N.A.

R3 = 127.8kN

R5 = 352.2kN

Rq = 259.2kN

R4b

R4t

y1

R1 = 284.0 kNR2 = 352.2 kN

124

5

Ultimate load to composite beam

= (206.2 1.41 ) x 4 / 2.5 = 327.7 kN


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Elastic linear analysis gives

large hogging moment

small sagging moment

Design methods for continuouscomposite beams

However, in composite beams, there are

small hogging moment resistances (top reinforcements over supports), but

large sagging moment resistances (large concrete flange near mid-span).

3030

Moment redistribution allowed forimproved structural performance

Question:

How to evaluate both the hogging and the sagging moments after re-distribution

with minimum effort but still recognizing the real behaviour of a composite beam?

i.e. a) Cracked section over hogging moment region

b) Rotational capacity over supports depending on section classification of

composite beams

Moment re-distribution


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Section classification in compositecross-sections

In general, the moment capacities of composite cross-sections are

limited by local buckling in the steel web or in the steel

compression flange.

For composite cross-sections of either class 1 plastic or class 2

compact, the moment capacities of composite beams aredetermined with rigid plastic theory, i.e. rectangular stress blocks.

The section classification of a composite cross-section is oftensimilar to that of the steel beam.

3232

Section classification in compositecross-sections of continuous beams

The section classification of composite cross-sections

governs the maximum moment re-distribution in continuouscomposite beams.

By considering the attachment effect to the steel compression

flange of the composite cross-section, it is possible to up-grade the section classification if needed.


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Analysis methods for continuouscomposite beamsAccording to the relevant conditions, the moments in continuous

composite beams may be determined using any of the following

methods:a. Simplified method

Based on certain assumptions, moment coefficients are givenaccording to simplified analysis rules.

b. Global elastic analysisStructural analyses on composite beams are required according todifferent assumptions on members:

- Uncracked sections over hogging moment regions

- Cracked sections over hogging moment regionsc. Global plastic analysis

Plastic hinge analyses may be adopted if the composite sections

are classified as class 1 plastic or shown to possess sufficient

ductility against rotations.

3434

Simplified method

Simplified method can be employed if the followingconditions are satisfied:

The steel beam should be of uniform section with equal flanges and

without any haunches.

The steel beam should be of the same section in each span.

The loading should be uniformly distributed.

The unfactored imposed load should not exceed 2.5 times the unfactored

dead load.

No span should be less than 75% of the longest.

End spans should not exceed 115% of the length of the adjacent span.

There should not be any cantilevers.


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Simplified method

Double span beam

Triple span beam

Multi span beam

For composite beams with class 1 plastic compression steel flanges in negativemoment region:

0.56

1.0

0.64

0.8

0.20

0.62

0.86

0.29

0.57

0.75

0.61

0.80

0.57

0.56

0.80

0.57

0.65

0.5

Moment redistribution coefficients to be multiplied by WL/8

3636

Table of moment coefficients (to be multiplied by WL/8)

LocationNumber of

spans

Classification of compression flange

in negative moment region

Class 1: plastic Class 2:compactGenerally Non-reinforced

Middle of end span2 0.75 0.79 0.71

3 or more 0.80 0.82 0.80

First internal support

2 0.61 0.50 0.71

3 or more 0.57 0.48 0.67

Middle of internalspans

3 0.56 0.63 0.52

4 or more 0.65 0.67 0.65

Internal supportsexcept the first

4 or more 0.50 0.42 0.58

Simplified method


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Uncracked section- The properties of the uncracked section are used throughout, and the

analysis is not dependent on the amount of reinforcement over supports.

- For equal spans, standard moment coefficients may be used.

Global elastic analysis

EIu EIu

Cracked section- For a length of15% of the span on each side of internal supports, the

section properties are those of the cracked section under negative moments.

- Outside the15% length, the section properties are those of the uncrackedsection, and will be calculated using the mid-span effective breadth for the

concrete flange but ignoring any longitudinal reinforcement.

EIu EIcEIu

0.85L0.15L

3838

P CL

P

Before redistribution

After redistribution Mhog

: Percentage ofmoment redistribution

Mhog

Class of cross-section in

hogging moment region

Class 1

Plastic

Class 2

Compact

Class 3

Semi-compact

Class 4

Slender

Cracked section analysis 30% 20% 10% 0%

Uncracked section analysis 40% 30% 20% 10%

Re-distribution in global elastic analysis


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P P

Msag

Mhog

Mo

Msag + Mhog / 2 = Mo

From equilibrium

Assume plastic hinges are formed over internal supports and near themid-span.

Global plastic analysisEstablish the ultimate load resistance from equilibrium consideration

Msag

It is important to ensure that

the ductility requirements at

various cross-sections are met

satisfactorily, i.e. section

classification of composite

cross-sections.

for continuous beams under

point loads.

4040

Analysis ofA Double Span Composite Beam

with Moment Re-distribution

Reference: Ansourian, P. Experiments on continuous composite beams. Proceeding of Institute ofCivil Engineering, Part 2, 1981, 71, 25-51.


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4141

Double span composite beam with a solid concrete slab

Details of test specimen

35

7

2250

35

7

P P

2250

IPB200

22502250

3 28 @ 320 c/c

A

A

Beam CTB4. pyf= 236.0 N/mm2, pyw = 238.0 N/mm2, fy = 430 N/mm

2, pc =

27.2 N/mm2 , fcu = 34 N/mm2

Section A-A

200 10190

100

200 10

800

6.5

Art = 804 mm2

Arb = 767 mm2

4242

Design to codified method

Design to codified method



Global elastic analysis

Uncracked section analysis

Cracked section analysis

Global plastic analysis


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Compression steel flange

py = 235 N/mm2

=> The compression steel flange is Class 2

compact, and hence, the composite cross-section

is classified as Class 2 compact.

1.08235

275 ==

b/T = 100 / 10 = 10 10= 10.8

=>

In addition, the composite cross-section is upgraded to Class 1 plastic as

the compression steel flange is restrained with effective attachment to a

solid concrete flange by shear connectors.

10

190

200

6.5

100


4444

Section properties of steel beamA = 5105 mm2, py= 235 N/mm

2, Zx = 369.4 x 103 mm3 , Sx = 407.0 x 10

3 mm3



-3 = 1199.7 kN

Ms = py x Sx or 1.2 x py x Zx= 95.6 kNm or = 104.2 kNm

Ms = 95.6 kNm

Section classification of steel sectionFor flange, b / T = 10 < 10 => Flange is compact sectionFor web, d / t = 26.2 < 80 => Web is plastic section It is a compact section

Effective width of the concrete slab

Span coefficient for sagging moment region = 0.8Bc = 0.8 x 4500 / 4 = 900 mm > 800 mm

Be = 800 mm

Resistance of the concrete slabRc = 0.45 x fcu x Bc x(Ds Dp)= 0.45 x 30 x 800 x (100 0) x 10

-3 = 1080 kN


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Resistance of the steel webRw = Rv = Rs 2 R f= 1199.7 2 x 470 = 259.7 kN

Resistance of shear connectors (sagging moment region)R

q= 7x(0.8x3x72) = 1209.6 kN Min. value of R

s(=1199.7kN) and R

c(=1080kN)

It is full shear connection.

Resistance of the reinforcementRrt = 0.87 x fy x Art = 321.8 kN

Rrb = 0.87 x fy x Arb = 307.0 kN

Resistance of shear connectors (hogging moment region)Rq = (14 - 10 ) x ( 0.6 x 3 x 72 ) = 518.4 kN Sum of (Rrt and Rrb) = 628.8kN It is partial shear connection. (degree of psc = 0.82)


Resistance of the steel flangeRf= B x T x pyf= 200 x 10 x 235 10

-3 = 470.0 kN

46

4

)(

2

)(

2

2 T

R

RRDD

RD

RMf

csps

csc

+

+= = 167.9 kNm

46

For sagging moment region,Rc > Rw & Rs > Rc => P.N.A in steel flange.

Sagging moment resistance ( full shear connection )


For hogging moment region,Rr> Rw & Rr< Rs => P.N.A in steel flange.

Hogging moment resistance ( full shear connection )

( )42

2

TR

RRDRDRMf

rsrrsc

+= = 143.7 kNm


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Uncracked and cracked section analyses

P P

0.156PL0.156PL

0.188PL

Bending moment from linear elasticanalysis with prismatic beam

P P

0.194PL0.194PL

0.113PL

Bending moment aftermoment redistribution at 40%

Uncracked sectionClass 1 plastic composite cross-section

P P

0.167PL0.167PL

0.164PL

Bending moment from linear elasticanalysis with non-prismatic beam

P P

0.192PL0.192PL

0.115PL


Cracked sectionClass 1 plastic composite cross-section

4848


PS2 = 137.8 / (0.113 x 4.5)

= 271.0 kN => 2PS2 = 542.0 kN

40% moment redistribution cannot be attained.

However, additional check shows that

Ms = 0.194 x PS2 x L=0.194 x 271.0 x 4.5

= 236.6 kN > Msag = 164.6 kNm

Hence, not good.

P P

0.194PL0.194PL

0.113PL

137.8 kNm

PS2

236.6 kNm


Class 1 plastic composite cross-section


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PS2 = 137.8 / (0.115 x 4.5)

= 266.3 kN => 2PS2 = 532.6 kN

30% moment redistribution cannot be attained.

However, additional check shows thatMs = 0.192 x PS2 x L

= 0.192 x 266.3 x 4.5

= 230.1 kN > Msag = 164.6 kNm

Hence, not good.

P P

0.192PL0.192PL

0.115PL

137.8 kNm

PS2

230.1 kNm


Class 1 plastic composite cross-section

50

164.6 + 137.8 / 2 = P x 4.5 / 4

=> P = 207.6 kN

50

P P

164.6 kNm

137.8 kNm

Free moment

Msag + Mhog / 2 = P x L / 4

From equilibrium

Establish the applied load, P, through equilibrium consideration at failure

Global plastic analysis


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Percentage of moment re-distribution


175.6 kNm

0.188

Mhog,el = 175.6 kNm

Msag, el = 145.7 kNm

CL

P = 207.6 kN

Elastic analysis

Bending moment from linear elastic analysis

145.7 kNm

0.156

175.6 kNm

Percentage of moment redistribution

= (175.6 137.8) / 175.6

= 21.5

CL

137.8 kNm

P = 207.6 kN

Elastic analysis

Nonlinear analysis

Bending moment after moment re-distribution

145.7 + 0.5*(175.6 - 137.8)

= 164.6 kNm

52

Percentage of moment re-distribution


153.2 kNm153.2 kNm

Mhog,el = 153.2 kNm

Msag,el = 156.0 kNm

CL

P = 207.6 kN

Elastic analysis

Bending moment from linear elastic analysis

156.0 kNm

0.164

0.167

Percentage of moment redistribution

= (153.2 137.8) / 153.2

= 10.0

CL

137.8 kNm

P = 207.6 kN

Elastic analysisNonlinear analysis

Bending moment after moment re-distribution

156.0 + 0.5*(153.2 - 137.8)

= 163.7 kNm


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In prescriptive codified design approach, the load carrying capacityof a composite beam depends largely on the hogging and thesagging moment capacities as well as the amount of moment re-

distribution permitted, whenever applicable.

The prescriptive design approach is considered to be a goodmanual design procedure which is simple and conservative.

Larger percentage of moment re-distribution in continuouscomposite beams is permitted according to the proposed model.

It should be noted that larger deformation capacity is required inheaded shear connectors installed in long span composite beamswith deep steel sections.

Conclusions

Documents

Composite Beam II