Composite Beam II
Transcript of 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
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Composite construction
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Long span composite floor system
Full integration with building services.
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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
Transversereinforcement
Transversereinforcement
Be
B
D
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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
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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
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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
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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
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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
Hogging moment resistance
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
++=
Hogging moment resistance
[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
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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 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
Hogging moment resistance
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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
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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.
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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
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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
Resistances of various elements of the beam
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
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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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Resistance of the steel sectionRs = A x py = 4304 x 355 x 10
-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 )
Resistances of various elements of the beam
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
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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
Resistances of various elements of the beam
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
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100
P.N.A.
R3 = 127.8kN
R5 = 352.2kN
Rq = 259.2kN
R4b
R4t
y1
R1 = 284.0 kNR2 = 352.2 kN
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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).
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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.
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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.
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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
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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
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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.
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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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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
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Design to codified method
Design to codified method
Section classification of composite cross-section
Resistances of various elements of the beam
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
Section classification of composite cross-section
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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
Resistances of various elements of the beam
Resistance of the steel sectionRs = A x py = 5105 x 235 x 10
-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)
Resistances of various elements of the beam
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 )
Resistances of various elements of the beam
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
Bending moment aftermoment redistribution at 30%
Cracked sectionClass 1 plastic composite cross-section
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Uncracked section analysis
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
Bending moment aftermoment redistribution at 40%
Class 1 plastic composite cross-section
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Cracked section analysis
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
Bending moment aftermoment redistribution at 30%
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
Uncracked section analysis
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
Cracked section analysis
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