Composite Beam I
-
Upload
stefan-raluca-asavoae -
Category
Documents
-
view
164 -
download
5
Transcript of Composite Beam I
Chiew SingChiew Sing--PingPing
School of Civil and Environmental EngineeringNanyang Technological University, Singapore
Design of Composite Beams Simply-supported beams
Sagging moments:Basic behaviour, concepts and codified design
2
Composite construction
4
Composite construction
Greater stiffness and higher load carrying capacities.
Fast erection of structural members.
Reduce height of a structure and offer further savings in associated features through integration with building services.
Good inherent fire resistance in slabs and columns.
Steel deckings as permanent formwork provide additional safety features during construction.
5
Prescriptive Codified Design Approach
Typical use and practical cross-section configurations
Modern design standards
Composite action in beams
Full and partial shear connection
Basic resistances
Design of a composite beam
Practical design procedures
Effective width / sagging moment resistance / shear resistance /moment resistance under high shear / transverse reinforcement / deflection / serviceability stress
Comparison between different design approaches
Scope
6
Composite beams with profiled steel deckings
7Composite beam with composite slab using profiled steel deckings
Composite beam with solid concrete slab
Beam span parallel to slab spanB
D
Transverse reinforcement
Ds
Profiled deckling
Beam span perpendicular to slab span
DpDsDp
D
B
Be
Profiled deckling
Transverse reinforcement
Transverse reinforcement
Be
B
D
8
The concrete slab works best in compression while the steel section works best in tension, hence, a large moment resistance is generated as a force couple.
Resistance mobilization in both the concrete slab and the steel section is limited by the shear resistance along the concrete interface.
Composite beams
Rc
Rq
Rs
9
Modern design codes
British Standards Institution.BS5950: Structural use of steelwork in building. Part 3 Section 3.1: Code of practice for design of composite beams.
British Standards Institution.BS EN1994-1-1 Eurocode 4: Design of Composite Steel and Concrete Structures.Part 1.1: General Rules and Rules for Buildings.
Standards Australia.Composite Structures. Part 1: Simply Supported Beams. Australian Standard AS2327.1 – 1996.
10
Buildings Department, Government of Hong Kong SARCode of Practice for the Structural Use of Steel 2005Chapter 10: Composite construction.
Composite slabsComposite beamsComposite columnsShear connection
Harmonized designBS 5400: Part 5, BS 5950: Parts 1 to 8, and EC3 & 4: Parts 1.1 & 1.2.
Modern design codes
11
The design of a composite beam is a two stage process:
At the construction stage, the steel section alone will resist the dead weight of the slab and the construction load, i.e. Steel Beam Design.
Moment capacities / lateral buckling / shear capacity / deflection
At the composite stage, the steel section and the concrete slab together will resist the loads resulting from the usage of the structure, i.e. Composite Beam Design.
Sagging & hogging moment capacities / degree of shear connection / shear resistance / transverse reinforcement / deflection / serviceability stress
Practical design of a composite beam
12
Composite action in beams ε σ
No composite action at the interface.
Composite action developed at specified locations at the interface.
a
a b
b
a-a
b-b
Fully developed composite action at the interface.
13
Free slippage at the concrete-steel interface.
Strain
Concrete slab and steel section each bends about its own neutral axis.
Controlled slippage at the concrete-steel interface.
Strain
Concrete slab and steel section bends about the neutral axis of the combined section.
Composite action in beams
14
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.
Full shear connection, partial shear connection, minimum degree of shear connection.
Current design methodology
15
Forces:
Rc = Compressive resistance in the concrete slab
Rs = Tensile resistance in the steel section
Rq = Shear resistance in the shear connectors
Basic resistances against sagging moment
Rc
Rq
Rs
16
Prescriptive design approach- Plastic section analysis
Various degree of shear connection
Assume a rigid plastic load-slippage curve of shear connectors.
(b) yp in steel flange
P.N.A
(c) yp in steel web
P.N.A
(a) yp in slab
Rs
P.N.A
0.45 fcu
py
Rc
17
Development of moment resistance along beam span
Rigid shear connectors
Compressive force
Tensile force
Sufficient shear connectors provided for full strength mobilization
Various degree of shear connection(c) yp in steel web
P.N.A
(a) yp in slab
P.N.A
(b) yp in steel flange
P.N.A
18
Full shear connection- Large concrete slab with small steel section
Full resistance mobilized in the steel section
Rc
Rs
Rq ≥ RS
Force equilibrium
Rc’ ≦ Rc
= Rs
Rs
Z
P.N.A
19
Full shear connection - Small concrete slab with large steel section
Full shear connection is achieved when
Rq ≧ Smaller of Rs and Rc
Full resistance mobilized in the concrete slab
Rc
Rs
Rq ≥ Rc
Force equilibrium (Rc + Rft + Rwt = Rwb + Rfb)
Rc
Rwb
Rft
Rfb
RwtP.N.A
20
Partial shear connection
When insufficient shear connectors are provided, the full resistance of neither the steel section nor the concrete slab is mobilized, i.e.
Partial shear connection is achieved when
Rq < Smaller of Rs and Rc
21
Example: Design of a composite beam
Grade 30 concrete with UB457x152 x 52 kg/m Grade S355
DimensionsL = 12 mB = 2.8 mDcs= 125 mm solid slab
Design data:
Shear connectors19 mm diameter headed shear studs95 mm as welded heightQk = 76.3kN (Characteristic value from design code)
B
Dcs Solid slab
22
Example: Design of a composite beamComposite stage designConsider the composite section at mid-span
As Rs ≦ Rc, the full tensile resistance of the steel section will be mobilized while only part of the concrete slab will be mobilized in compression.
Resistance of the steel sectionRs = A x py = 66.6 x 102 x 355 x 10-3 = 2364 kN
Resistance of the concrete slabRc = 0.45 x Be x fcu x Dcs = 0.45 x 2800 x 30 x 125 x 10-3 = 4725 kN
Resistance of a shear connector (headed shear stud)Qp = 0.8 x 76.3 = 61.0 kN (The coefficient 0.8 is adopted for shear connectors in sagging moment region.)
Effective width of the concrete slabBe = 12000 / 4
= 3000 mm > B=2800 mm∴Be=2800 mm
23
The depth of the compression zone, dc, in the concrete slab is
dc = Rs / (0.45 x fcu x Be)= 2364 x 103 / (0.45 x 30 x 2800)= 62.5 mm
Sagging moment resistance
Force equilibrium
or dc = Ds x c
s
R
R
As the depth of the compression zone in the concrete slab is smaller than the slab thickness of the concrete slab, the plastic neutral axis of the composite beam is located within the concrete slab.
24
The lever arm, z between the tensile resistance, Rs and the compressive resistance, Rc’ is given by:
Sagging moment resistance
mmcd
sD
Dz 7318
2
562125
2
8449
22.=)
.(+
.=)(+= --
=> M = Rs x z = 2364 x 318.7 x 10-3 = 753.4 kNm
Assume full shear connection.
Moment equilibrium
449.8
125
Rc’
Rs
dc = 62.5
Z
P.N.A
25
In order to fully mobilize the tensile resistance of the steel section,
Rq ≧ Rs = 2364 kN for full shear connection
n x Qp ≧ 2364 => n = 38.8 or 39 (min.)
Use 40 no. shear connectors at a spacing of 150mm over half span, i.e. 6000 mm, i.e. a total of 80 headed shear studs along the beam length.
Provision of shear connectors
Rc’ = Rs= 2364kN
Rs = 2364kN
Rq RS≧Z = 318.7
26
Steel beam: Ms = 389.0 kNm
Composite beam: Mcs = 754.3 kNm
=> Mcs : Mc = 754.3 : 389.0 = 1.94 : 1
An increase of 94% in the moment resistance in the beam is achieved with proper provision of shear connectors.
Moment resistances
Similar increase in the flexural rigidity, (EI)cp of the beam is found. However, in some cases, design of composite beam for partial shear connection is more economical, depending on the applied moment.
27
Prescriptive design approach- Simplified load slippage curve
Sh
ear
forc
e, F
s
Slippage, S
0.5QK
0.5 mm 5 mm 7 mm
Typical
Fs
s
QK
Not more than 20% decrease
R-72
Assume a rigid plastic load-slippage curve of shear connectors.
28
Section classification of composite cross-sectionMoment resistance with full shear connectionShear resistanceShear connectionMoment resistance with partial shear connectionTransverse reinforcement
Design procedures
For structural adequacy, the following checks should be satisfied:
Ultimate Limit State
Serviceability Limit StateDeflection
Serviceability stresses
29
For those composite beam with Rq ≥ the lesser of Rc and Rs
Sagging moment resistanceFull shear connection
Partial shear connection
For those composite beams with Rq < both Rc and Rs
Applicable only to symmetric I or H section with equal flanges
– [Case 1a] Plastic neutral axis in steel web
– [Case 2a] Plastic neutral axis in steel flange
– [Case 2b] Plastic neutral axis in concrete flange
– [Case 3a] Plastic neutral axis in steel web
– [Case 4] Plastic neutral axis in steel flange
30
The plastic moment capacity is expressed in terms of the resistance of the various elements of the beams as follows:
Resistance of Concrete Flange:
Resistance of Steel Beam:
Resistance of Shear Connection:
Resistance of Steel Flange:
Resistance of Overall Web Depth:
Plastic moment resistance of steel beam:
Plastic moment resistance of composite beam:
Rc = 0.45 fcu Be (Ds – Dp)
Rs = A py
Rq = Na Q
Rf = B T py
Rw = Rs – 2 Rf
Ms = py Sx or 1.2py Zx
Mc
Sagging moment resistance
Dp
Ds
Be
B
td
T
T
D
31
Full Shear Connection : Rc < Rw
[Case 1a] Plastic neutral axis in steel web
Sagging moment resistance
Full Shear Connection : Rc ≥ Rw
( )42
)(
2
2 T
R
RRDDR
DRM
f
cspscsc
−−
++=
( )⎭⎬⎫
⎩⎨⎧ −
−+=22
ps
c
sssc
DD
R
RD
DRM
[Case 2b] Plastic neutral axis in concrete slab (Rs ≤ Rc)
[Case 2a] Plastic neutral axis in steel flange (Rs > Rc)
Typicaldesign
( )42
2 d
R
RDDDRMM
v
cpscsc −
+++=
P.N.A
P.N.A
3232
Section classification in composite cross-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 are determined 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.
33
Contribution of the concrete slab
Allowance is made for the in-plane shear flexibility (shear lag) of a concrete slab by using the concept of effective width
Actual width
Effective width
Mean bending stress in concrete slab
Idealized stress
Actual stress
34
Beam span is perpendicular to slab spanbe = Lz /8 but not greater than b
Beam span is parallel to slab spanbe = Lz /8 but 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 = Σ bei
Lz = distance between points of zero moments
b = actual width
be1 be2
b
35
Effective breadth of the concrete slab in 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
36
It is assumed that the vertical shear due to factored loading isresisted by the steel section only.
The calculation of the shear resistance (Pv) should be with reference to BS5950: Part 1.
Shear resistance
Pv = 0.6py x Av where Av = shear area of the steel section
= D x t for rolled sections
= d x t for fabricated sections
37
Where the shear force Fv exceeds 0.5Pv, the moment capacity should be reduced to allow for the influence of shear. The reduced moment capacity Mcv should be determined from the following equation:
( )2
12
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−=
v
vfcccv P
FMMMM
Mc = Plastic moment capacity of composite beam
Mf = Plastic moment capacity of the remaining composite section after deducting the shear area (Av) of the steel section defined in BS5950: Part 1
Pv = Lesser of shear capacity and the shear buckling resistance, both determined from BS5950: Part 1
The above equation is only applicable for a web that is plastic and compact.
Moment resistance under high shear
38
Moment resistance under high shear
Sh
ear,
Fv
Moment, M
Mf0
0.5 Pv
Pv
Mc
Moment - shear interaction curve for a composite beam
The interaction between moment and shear is considered not to be significant.
Linear interaction
Non-linear interaction
39
Partial shear connection : Rq < Rw
[Case 3a] Plastic neutral axis in steel web
( )422
2d
R
RDD
R
RD
DRMM
v
qps
c
qsqsc −
⎭⎬⎫
⎩⎨⎧ −
−++=
Sagging moment resistance
Partial shear connection : Rq ≥ Rw
[Case 4] Plastic neutral axis in steel flange
( ) ( )422
2T
R
RRDD
R
RDR
DRM
f
qsps
c
qsqsc
−−
⎭⎬⎫
⎩⎨⎧ −
−+=
Rq = NaQp
P.N.A
P.N.A
40
Global transfer of force in slab
Transverse distribution of forces along the A-A
Support
Transverse tension
Transverse compression
Uniform compression in slab
Beam
AA
12345
3 4
Line of principle compression
Very large shear forces acting at the shear connectors
41
Transverse reinforcement refers to the reinforcement in the concrete slab running transversely to the span of the beam.
Sufficient transverse reinforcement should be used to enable theconcrete slab to resist the longitudinal shear transmitted by the shear connectors, both immediately adjacent to the shear connectors and elsewhere within its effective breadth, Be.
When profiled steel sheets are used, they may also act as transverse reinforcement.
Transverse reinforcement
42
The total longitudinal shear force per unit length (v) to be resisted at any point in the span of the beam should be determined from the spacing of the shear connectors by the following equation:
v = N Q / s
v ≤ vr
Longitudinal shear force
For structural adequacy, the longitudinal shear force, v, shouldnot be larger than the local shear resistance in the concrete slab, vr :
N = Number of shear connectors in a group
s = Longitudinal spacing of shear connectors
Q = Smaller of Qp and Qn
Qp = Resistance of shear connectors in sagging moment (positive)
Qn = Resistance of shear connectors in hogging moment (negative)
43
Longitudinal shear force
Section A-A
S
Longitudinal shear force
Q
Q
A A
Transverse reinforcement
v = N Q / s
44
The local shear resistance of the concrete slab is given by
fcu = characteristic cube strength of concrete in N/mm2 but ≤ 40 N/mm2
η = 1.0 for normal weight concrete and 0.8 for lightweight concrete
Acv = mean cross-sectional area, per unit length of the beam, of theconcrete shear surface under consideration = (Ds + Dp )/2
Asv = mean cross-sectional area, per unit length of the beam, of
both the top and bottom reinforcement crossing the shear surface
vp = contribution of the profiled steel sheeting, if any.
vr = 0.7 Asv fy + 0.03 η Acv fcu + vp
but vr ≤ 0.8 η Acv √ fcu + vp
Local shear resistance
45
d) Composite slab. Profiled decking spanning perpendicular to the beam
e) Composite slab. Profiled decking spanning parallel to the beam
Transverse area crossing the shear surfaces, Asv
Shear surface Asv1-1 (Ab+At)2-2 2Ab
3-3 At
a) Solid slabsAt
Ab2
1
1 2
3 Lap joint in profiled decking
3
At3
3
At3
3
Profiled decking
46
Profiled decking may be assumed to contribute to the transverse reinforcement provided that it is either continuous across the top flange of the steel beam or that it is welded to the steel beam by stud shear connectors.
d = nominal shank diameter of the stud
n = 4
pyp = design yield strength of profiled decking
tp = thickness of profiled decking
vp = tp pyp
a) Continuous profiled decking with ribs perpendicular to the beam span
vp = (N/s)(n d tp pyp) but vp ≤ tp pyp
b) Discontinuous profiled decking with studs welded to the steel beam
Contribution of profiled decking
47
Deflection needs to be checked under the serviceability limit state in both the construction and the composite stages. The deflections for both stages (δcon & δ ) are then added up to determine if any precamber is necessary.
Construction stage (δcon)
Dead loads (Steel beam and slab weight)
Deflection calculations based on steel beam
Composite stage (δ )Imposed loads only
Deflection calculations based on composite beam
Depends on the degree of shear connection
Deflection
48
An effective modular ratio, αe, is used to express the elastic section properties of the composite section in terms of an equivalent steel section. It is determined from the proportions of the both short and long term loadings.
αe = αs + ρl (αl – αs)
ρl = (Dead load + 1/3 Imposed load) / Total load
Short termαs
Long termαl
Normal weight concrete 6 18
Lightweight concrete 10 25
Deflection for composite stage
49
Deflection calculation is based on the gross uncracked composite section. Reference for the calculation of the second moment of area, Ig, can be made to Appendix B.3.1 for steel beam with equal flanges and any concrete within the depth of the ribs of the profiled decking is neglected.
( ) ( )( )( ){ }psee
pspse
e
psexg DDBA
DDDDDABDDBII
−+
++−+
−+=
αα 412
23
Deflection for composite stage
50
The increased deflection under serviceability loads arising frompartial shear connection should be determined from the following equations:
δs = Deflection of steel beam under dead loads = δcon
δc = Deflection under imposed loads of composite beam with full shear connection
δ = δc + 0.5(1 – Na / Np)(δs – δc)
Propped construction
Unpropped construction
δ = δc + 0.3(1 – Na / Np)(δs – δc)
Deflection for composite stage
51
Serviceability stresses need to be checked for the construction and composite stage. It is to be checked under the serviceability limit state with the allowable stresses.
Construction stage
Dead loads (Md)
Elastic properties based on the steel section
Composite stage
Imposed loads (Mi)
Elastic properties based on the composite section
Either cracked or uncracked sections should be considered, depending on the position of the neutral axis.
Serviceability stress
52
Construction stage
Dead loads – Self weights of steel section, concrete slab and profiled deckingElastic properties based on steel sectionCalculate bending stresses in the steel section only
x
d
Z
MstressBending =
Zx = Elastic section modulus of steel section
Serviceability stress
53
Composite stage
( )( ) ep
eps
DD
BDDA
α2
2
+
−≥
Need to check whether the concrete slab is cracked (Case 1) or uncracked (Case 2) – Appendix B.4.1
Case 1: Elastic neutral axis in the concrete slab
( )( ) ep
eps
DD
BDDA
α2
2
+
−<
Case 2: Elastic neutral axis in the steel section
Serviceability stress
A = cross-sectional area of steel section
54
Composite stage
Case 1: Elastic neutral axis in concrete slab
23
23⎟⎠⎞
⎜⎝⎛ −+++= es
e
eexp yD
DA
yBII
α
The concrete on the tension side of the elastic neutral axis is taken as cracked and the properties of the cracked section is used.
( )
( )5.0
211
2
⎭⎬⎫
⎩⎨⎧
+++
+=
se
e
se
DDAB
DDy
α
Concrete slab: Zp = Ipαe / ye
Bottom flange of steel section: Zs = Ip / (D + Ds – ye)
Serviceability stress
55
Elastic analysis of a composite beamBe / αe
Transformed section
Ds
D
Dp
Propped construction
Stress ≤ 0.5fcu
Composite stage
E.N.A
Unpropped construction
Ds
D
Stress ≤ 0.5fcu
Composite stage
E.N.ADp
Construction stage
56
Composite stage
Case 2: Elastic neutral axis in steel section
The concrete is uncracked and the gross section properties apply.
( ) ( )( ){ }psee
pseseg DDBA
DDBDDAy
−+
−++=
αα
2
2 2
Concrete slab: Zg = Igαe / yg
Bottom flange of steel section: Zs = Ig / (D + Ds – yg)
Serviceability stress
57
Bending stress at extreme fibre of member
Applied stressAllowable stress
Cl 2.4.3
Concrete slab (Compression)
Mi / Zp (Case 1)
Mi / Zg (Case 2)0.5 fcu
Steel section (Tension)
Md / Zx + Mi / Zs py
Assume composite beam is simply supported.
Serviceability stress
58
Comparison between different approaches
Prescriptive design approach in design codesSimplified design assuming rigid and ductile shear
connectors in strength calculation.Empirical formulae to allow for partial shear connection in
deflection calculation.
Performance-based design approachAllow non-linear deformation characteristic of shear connectors.Allow non-uniform distribution of forces and deformations
of shear connectors along beam length.Allow various failure criteria.Accurate prediction in deflection, allowing for actual deformationsof shear connectors.
Important in continuous composite beams with both hogging and sagging moment regions.
59
Conclusions (1)
1. Composite beam design is well established to achieve effective use of materials, i.e. the concrete slab in compression and the steel section largely in tension.
2. Composite action is achieved with proper provision of shear connectors.
3. Design rules for composite beams under sagging moment are provided.
4. Dimensional detailing on the installation of shear connectors should be carefully considered.
5. Transverse reinforcement should be provided to avoid longitudinal splitting in the concrete slab.
60
Conclusions (2)
6. Deflection calculation is very similar to that of reinforced concrete beam.
7. Serviceability stresses in both steel sections and concrete flanges may be readily evaluated.