Composite Construction Design (ULS Only)
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Transcript of Composite Construction Design (ULS Only)
Date or reference
Composite DesignComposite Design
Beams and SlabsBeams and Slabs
2
British Standards to Eurocode
Composite Design hasn’t really changed significantly – just a few new formulae to consider.
(and a little bit more work as usual!)
3
Some Useful References
1. R.P.Johnson. Composite structures of steel and concrete
3rd Ed 2004, Blackwell.
2. W.H.Mosley, J.H.Bungey, R.Hulse, Reinforced concrete design to Eurocode 2, 2007, Palgrave Macmillan.
4
Composite Systems
Typical composite sections
Shear Studs
5
Composite Systems
6
Composite Systems
Composite Profiled Slimdec System
7
Composite Systems
Composite Slab with Steel Decking
8
Design Procedure
Apart from EN1990 (Actions), the following codes of practice
are required for the design of Composite Beams.
(a)EC2, (EN1992-1-1) for the design of concrete structures
(b)EC3, (EN 1993-111) for the design of steel structures
(c)EC4, (EN 1994-1-1) for the design of composite steel and
concrete structures
Also the National Annexes may be required.
9
Effective Width of Concrete Flange
EC4 CL 5.4.1.2
Effective breadth beff =>
b0 is the distance between centres of adjacent shear studs
bei is the effective width of the concrete flange on each side of the steel web.
bei = Le/8 but < half the distance to the centre of the adjacent beam
Le is the approximate distance between points of zero bending moment (L/2 for midspan of a continuous beams or L for a single simply supported beam.)
oeieff bbb + Σ=
10
Effective Width of Concrete Flange:
Basic Example
Continuous beam, with spans equal to 12m, with adjacent beams at 4m centres.
The effective breadth of the concrete flange is:
beff = 2 x Le/8 = 2 x 0.5 x 12/8 = 1.5m
This is less than half the distance between adjacent beams (2m), so this is o.k.
If the beam was simply supported, the effective
breadth would be 3m
11
Design Procedure
Principal Stages in the Design are:
1. Preliminary sizing – the depth of the beam (UB) initially approximated as:
Simply supported => Span/20Continuous => Span/24
The yield strength and classification should then be determined to EC3.
12
Design Procedure
2. During construction (Unpropped only)
Loads considered =>
Beam self weight
Shuttering/steel decking weight
Weight of the wet concrete
Imposed construction load => usually 0.5~0.75 kN/m2
Check Bending, Shear at ULS, check deflection at SLS
13
Design Procedure
3. Bending and shear of composite section at ULS
Compare the moment of resistance of the composite section with the ultimate design moment.
Check the shear strength of the steel beam (alone)
4. Shear connectors and Transverse steel at ULS
Design shear connectors at the concrete/steel beam
interface, for either full or partial interaction
Provide transverse reinforcement to resist the longitudinal shear in the concrete flange.
14
Design Procedure
5. Deflection
Check the deflection of the beam to protect against cracking of architectural finishes.
Note that for slabs, deflection due to pondingshould also be considered.
15
Design Procedure
Design of the steel beam during construction =>
At ULS bending,
The plastic section modulus Wpl,y for the steel beam may be calculated
from:
Where
MEd is the ultimate design momentfy is the design strength of the steel (EC3 table 3.1)
This assumes the compression flange for the steel beam is adequately restrained against buckling by the steel decking and the steel section used can be classified as a plastic or compact section as defined in EC3 sections 5.5 and 5.6
y
Ed
yplf
MW =,
16
Design Procedure
Design of the steel beam during construction =>
At ULS Shear
The shear (as usual) is considered to be carried by the steel beam alone at the construction stage and also for the final composite stage.
The ultimate shear strength of a rolled I-beam is based on the following shear area Aw
Aw = Aa – 2btf + (tw + 2r)tf but not less than ηhwtw
where Aa is the cross sectional area of the steel beam and hw is the overall height of the web. η can be taken as 1.0
The other dimensions are shown in the next figure.
17
Design Procedure
Design of the steel beam during construction =>
At ULS Shear
18
Design Procedure
Design of the steel beam during construction =>
At ULS Shear
However, you can still take the shear area conservatively as
the web area
Av = d x tw
where d is the depth of the straight portion of the web.
The design plastic shear resistance is:3
,
MO
yv
Rdpl
fAV
γ=
MOγ = 1.0
19
Design Procedure
Design of the steel beam during construction =>
At SLS Deflection
Deflections at midspan are calculated, i.e.
EI
wL
384
5 4
=δ
These deflections at the construction stage due to the permanentloads are locked into the beam as the concrete hardens.
20
Design Procedure
Example => Design of steel beam for construction loads
21
Design Procedure
Example => Design of steel beam for construction loads
Steel strength and Classification (EC3 tables 3.1 and 5.2)
Web thickness tw = 9mm
Flange thickness tf = 14.4mm
Both < 40mm
From EC3 section 3.2 table 3.1, yield strength fy = 355 N/mm2
From EC3, section 5.6, table 5.2
3.58723.450.9
6.40781.0
235=×<===== εε
wy t
d
t
c
f
Therefore the steel section is class 1
22
Design Procedure
Example => Design of steel beam for construction loads
Loading at Construction=>
Average depth of concrete slab and ribs = 90 + 50/2 = 115 mm
Weight of concrete =0.115 x 25 x 3 =8.62 kN/mSteel deck =0.15 x 3 =0.45 kN/mSteel beam =74 x 9.81 x 10-3 =0.73 kN/m
Total dead load =9.8 kN/m
Imposed construction load = 0.75 x 3 =2.25 kN/m
Ultimate load = 1.35Gk + 1.5Qk = (1.35 x 9.8 + 1.5 x 2.25) = 16.6 kN/m
23
Design Procedure
Example => Design of steel beam for construction loads
Bending.
Maximum bending moment = wL2/8 = (16.6 x 92)/8 = 168 kNm
Moment of resistance of steel section = Wpl,yfy = 1653 x 355 x 10-3
= 587kNm > 168 kNm OK
24
Design Procedure
Example => Design of steel beam for construction loads
Shear
Maximum shear force V = wL/2 = 16.6 x 9/2 = 74.7 kN
Shear resistance of section =
Using conservative approach, with web depth d = 407.6mm, web thickness t=9mm
Av = dtw = 407.6 x 9 = 3.67 x 103 mm2
Shear resistance of section =
(Capacity should be considerably larger here - OK)
3,
MO
yv
Rdpl
fAV
γ=
kNkNV Rdpl 7.747521030.1
3551067.3 33
, >=××
××= −
25
Design Procedure
Composite Section at ULS
Check moment capacity and shear strength. Define tensile and compressive strength of the elements as follows (no major difference to BS approach):
Resistance of the concrete flange Rcf = 0.567fckbeff(h-hp)Resistance of the steel section Rs = fyAa
Resistance of the steel flange Rsf = fybtfResistance of overall web depth Rw = Rs-2Rsf
Resistance of clear web depth Rv = fydtwResistance of the concrete above the neutral axis Rcx = 0.567fckbeffxResistance of the steel flange above the neutral axis Rsx = fybx1
Resistance of the web over distance x2 Rwx = fytwx2
26
Design Procedure
Composite Section at ULS
General Section dimensions =>
(depth between fillets)
27
Composite Section at ULS
Neutral axis in the concrete flange = >
28
Composite Section at ULS
Neutral axis in the steel flange = >
29
Composite Section at ULS
Neutral axis in the steel web = >
30
Composite Section at ULS
Equations for Moment Capacity =>
( ) ( )sf
fcfspcfasc
R
tRRhhRhRM
422
2−
−+
+=
( )v
cfpacf
scR
dRhhhRMM
42
2
−++
+=
Neutral axis in concrete flange =>
Neutral axis in steel flange =>
Neutral axis in steel web =>
( )
−
−+=22
p
cf
sa
sc
hh
R
Rh
hRM
31
Composite Section at ULS
Example – Moment of Resistance of Composite Section
32
Composite Section at ULS
Example – Moment of Resistance of Composite Section
(1)From first principles:
Resistance of concrete flange:
Rcf = 0.567fckbeff(h-hp) = 0.567 x 25 x 3000 x (140 – 50) x 10-3 = 3827kN
Resistance of steel beam: Rs = fy Aa = 355 x 9460 x 10-3 = 3358kN
As Rs < Rcf, the neutral axis is within the concrete flange:
Neutral axis depth: 0.567fckbeffx = Rs = 3358kN
Therefore: x = (3358 x 103)/(0.567 x 25 x 3000) = 79mm
33
Composite Section at ULS
Example – Moment of Resistance of Composite Section
Moment of resistance =>
Lever arm z to the centre of the steel section is:
z = (ha/2 + h - x/2) = 457/2 + 140 – 79/2 = 329 mm
Therefore
Mc = Rsz =3358 x 329 x 10-3 = 1105 kNm
34
Composite Section at ULS
Example – Moment of Resistance of Composite Section
(2) Alternatively, using the design equations =>
( ) ( )kNm
hh
R
Rh
hRM
p
cf
sa
sc 1105102
50140
3827
3358140
2
4573358
22
3 =
−
−+=
−
−+= −
Nothing has really changed here, the basic equilibriumequations are the same
35
Composite Section at ULS
Shear strength VRd of the composite section
The resistance to vertical shear is assumed to be taken by the steel beam alone (as for the construction stage).
Shear resistance =>
Shear area Av is given by:
(For most cases this is conservatively taken as the web area, d x tw)
3,
MO
yv
Rdpl
fAV
γ=
Av = Aa – 2btf + (tw + 2r)tf
36
Composite Section at ULS
Up to this stage everything is (almost) as
before in terms of the design procedures, with
a very few formula changes to EC
37
Composite Section at ULS
Design of Shear Connectors
Design shear resistance PRd of automatically welded shear studs is the lesserof the following two equations (cl. 6.6.1.3):
38
Composite Section at ULS
Design of Shear Connectors
39
Composite Section at ULS
Design of Shear Connectors – reduction factors (orientation of steel sheeting)
Further reduction factors kl and kt to be applied to PRd (CL 6.6.4)
Depends on whether the ribs of the profiled sheet are parallel or transverse
to the supporting beam.
For ribs parallel to the supporting beam:
For ribs transverse to the supporting beam:
40
Composite Section at ULS
Design of Shear Connectors – reduction factors (orientation of steel sheeting)
There is an upper limit kt,max given in table 6.2 of EC4
41
Composite Section at ULS
Design of Shear Connectors – reduction factors (orientation of steel sheeting)
There is an upper limit kt,max given in table 6.2 of EC4:
42
Composite Section at ULS
Degree of Shear Connection – Full Connection
The change in horizontal shear (between zero and maximum moment)
will be the lesser of Rs or Rc.
To develop the full moment of resistance of the composite section,
the number of shear connectors nf required over the half span is the lesser of:
where
43
Composite Section at ULS
Degree of Shear Connection – Partial Connection
Sometimes the full shear connection is not required to provide adequate
moment capacity. Hence the number of shear connectors can be reduced
providing partial shear connection (normally for simpler stud layout/detailing)
Degree of shear connection =>
n is the number of shear connectors for full shear connection over a length of beam
nf is the number of shear connectors provided in that length
44
Composite Section at ULS
Degree of Shear Connection – Partial Connection
Cl 6.6.1.2 provides limits to the degree of shear connection according to the distance
Le, the distance in sagging between the points of zero moment.
1. The nominal diameter d of the shank of the headed stud is within the range
16mm < d < 25mm, and the overall length of the stud after welding is > 4d
2. The nominal diameter d of the shank of the headed stud is d=19mm
and the overall length of the stud after welding is > 76mm
For Le <25
For Le <25
( ) 4.003.075.0355
1 ≥−
−≥ ηη e
y
Lf
( ) 4.004.000.1355
1 ≥−
−≥ ηη e
y
Lf
45
Composite Section at ULS
Ultimate Moment of Resistance – Partial Shear Connection
Other conditions are presented in Cl 6.6.1.2, but the moment capacity for partial
shear connection is derived from stress blocks.
In the analysis the depth of the concrete stress blocks sq is:
effck
q
qbf
Rs
567.0=
Where Rq is the shear resistance of the studs provided.
46
Composite Section at ULS
Ultimate Moment of Resistance – Partial Shear Connection
1. Neutral axis in the steel flange h < x < h+tf : Rs > Rq > Rw
47
Composite Section at ULS
Ultimate Moment of Resistance – Partial Shear Connection
2. Neutral axis in the steel web x > h+tf : Rc < Rw
48
Composite Section at ULS
Ultimate Moment of Resistance – Partial Shear Connection
The moment capacities are:
Neutral axis in steel flange: (a)
Neutral axis in steel web: (b)
( ) ( )sf
fqs
cf
pq
q
as
cR
tRR
R
hhRhR
hRM
422
2−
−
−−+=
( )v
q
cf
pqa
qscR
dR
R
hhRh
hRMM
422
2
−
−−++=
49
Composite Section at ULS
Ultimate Moment of Resistance – Partial Shear Connection
Good News – can still use the linear interaction method (method b)
50
Composite Section at ULS
Concentrated Loads – Shear Connection
Shear connectors need to be spaced closer together between the
concentrated load and adjacent supports.
Number of shear connectors between
Load and adjacent support
Where:
Nt is the total number of shear connectors required between the support and
the point of maximum moment
Mi is the bending moment at the concentrated load
Ms is the capacity of the steel member
Mc is the moment capacity of the composite section
( )( )sc
sit
iMM
MMNN
−
−=
51
Composite Section at ULS
Concentrated Loads – Shear Connection
52
Composite Section at ULS
Example – Partial Shear Connection
Using previous example for the composite section ULS =>
Span = 9 m
80 shear studs in pairs at 225 mm (19mm dia, 100mm height)
Wpl,y = 1653 N/mm2
fy = 355 N/mm2
fck = 25 N/mm2
Calculate the degree of shear resistance and moment of resistance of thecomposite section based upon the shear connectors provided.
53
Composite Section at ULS
Example – Partial Shear Connection
Design shear resistance PRd is the lesser of the equations in Cl.6.6.1.3, withfu = 450 N/mm2
kNdf
Pv
u
Rd 7.811025.1
4/194508.04/8.0 322
=××××
== −π
γ
π
Calculate the reduction factor kt (transverse sheeting) with an upper limit of 0.8 from table 6.2 EC4 for the profiled sheeting
79.0150
100
50
80
2
7.01
7.0 0 =
−×=
−=
p
sc
pr
th
h
h
b
nk
54
Composite Section at ULS
Example – Partial Shear Connection
Hence the design shear resistance PRd of a stud = 0.79 x 81.7 = 64.5 kN
For full shear connection, the number of studs required over the half span is:
525.64
3358===
Rd
s
fP
Rn
Hence for full shear connection, the total number of studs required overthe whole span = 104
55
Composite Section at ULS
Example – Partial Shear Connection
The degree of shear connection, η, is
77.0104
80==η
( ) ( ) OKLf
e
y
77.052.0903.075.0355
355103.075.0
3551 <=×−
−=−
−≥η
The lower limit for η is calculated from:
56
Composite Section at ULS
Example – Partial Shear Connection
Use the linear interaction method to calculate the moment of resistance Mp
For the steel beam
Hence
( ) sscp MMMM +−= η
kNmfWM yypls 587103551653 3
, =××== −
( ) kNmM p 986587587110577.0 =+−×=
57
Composite Section at ULS
Longitudinal Shear Resistance for Steel Sheeting
Use either:
1) m-k method
2) Partial interaction method
Both deal with parameters concerned with empirical behaviour
(i.e. lots of tests in the lab!)
58
Composite Section at ULS
Longitudinal Shear Resistance for Steel Sheeting
For the m-k method Cl. 9.7.3 applies (mechanical and frictional interlock)
The maximum design vertical shear in a slab of width b should not exceedthe design shear resistance given by:
59
Composite Section at ULS
Longitudinal Shear Resistance for Steel Sheeting
The partial interaction method should only be used when the longitudinalshear failure is ductile (EC4 contains caveats).
60
Composite Section at ULS
Vertical Shear Resistance
For partially anchored sheeting, vertical shear resistance is provided by
Equation 6.2b) in EC2 (for resistance of members not requiring designshear reinforcement =>
Normally σcp can be taken as zero for simply supported members
21
ck2
3
min 035.0 fkv =
61
Composite Section at ULS
Transverse Reinforcement
Transverse reinforcement is supplied to resist the longitudinal shear in a flanged beam, and follows the variable strut inclination method as outlined in EC2
Uses the rules in Cl.6.2.4 EC2
Longitudinal shear stress =
62
Composite Section at ULS
Transverse Reinforcement
63
Composite Section at ULS
Transverse Reinforcement
∆x is half the distance between point of maximum moment and zero moment
Transverse reinforcement per unit length:
For compression flanges, 26.50 < θ <450 ,
so for minimum reinforcement take θ = 26.50
64
Composite Section at ULS
Transverse Reinforcement
To prevent crushing of the concrete flange, the following must be satisfied:
(6.22)
65
Deflections
1) Calculate deflections at the construction stage
Deflection due to permanent load plus variable loading is checked.Permanent load deflection is locked into the beam at this stage
2) Calculate deflections for the composite stage based on a transformedsection (not shown, refer to Ref 1 for example)
Deflection due to composite stage plus construction stage gives totalDeflection.
Check < span/250 (for longer spans a pre camber can be incorporated)
66
Example:
Simply Supported Composite Beam
6m Long Composite Beam, UDL at 3m centres:
67
Example:
Simply Supported Composite Beam
Profiled Sheeting => Comflor 51 (Corus 2002)
68
Example:
Simply Supported Composite Beam
Design Data:
Beam Spacing => L = 6mBeam Spacing => s = 3mTotal Slab Depth => h = 130 mmDepth of Concrete above profile => hc = 79 mmDeck Profile Height => hp = 51 mmWidth of the bottom trough => bbot = 122.5 mmWidth of the top trough => btop = 112.5 mm
Average width of rib = 30mm (equates to 6.55 ribs per metre)
69
Example:
Simply Supported Composite Beam
Shear Connectors:
Diameter => d = 19 mm
Overall height before welding =>hsc = 100mm
Height after welding => 95 mm
70
Example:
Simply Supported Composite Beam
Material Properties of Steel (EN1993-1-1, Table 3.1)
Structural Steel:
Assume S275, maximum thickness < 40mm
Yield Strength, fy = 275 N/mm2
Ultimate Strength, fu= 430 N/mm2
Steel Reinforcement:
Yield Strength fy = 500N/mm2
71
Example:
Simply Supported Composite Beam
Material Properties of Concrete (EN1992-1-1, Table 3.1)
Normal Weight concrete, strength class C25/30
Wet Density => 26 kN/m2
Dry Density => 25 kN/m2
Cylinder strength => fck = 25 N/mm2
Secant modulus of elasticity => Ecm = 31 kN/mm2
72
Example:
Simply Supported Composite Beam
Actions – Permanent Loading (gk)
Self weight of concrete slab =>
Wet UDL => [(130x1000)-(7x30x51)]x26x10-6 = 3.10 kN/m2
Dry UDL => [(130x1000)-(7x30x51)]x25x10-6 = 2.98 kN/m2
Construction stage Composite stage
Concrete slab => 3.10 kN/m2 2.98 kN/m2
Steel Deck => 0.15 kN/m2 0.15kN/m2
Steel Beam=> 0.2 kN/m2 0.2 kN/m2
-----------------------------------------------------------------------------------------Totals => 3.45 kN/m2 3.48 kN/m2
73
Example:
Simply Supported Composite Beam
Actions – Variable Loading (qk)
Construction stage:
Construction loading => 0.5kN/m2
Composite stage:
Floor Loading => 3,8kN/m2 (from structural arrangement)
74
Example:
Simply Supported Composite Beam
Ultimate Limit State – Construction Stage
Combination of actions (EN1990, Eqn 6.10b)
Distributed load on beam (0.85 x 1.35 x 3.45) + (1.5 x 0.5) = 4.71 kN/m2
Total Load Fd = 4.71 x 6.0 x 3.0 = 84.78 kN
Maximum design moment at midspan
My,Ed = FdL/8 = 84.78 x 6 / 8 = 63.59 kNm
Maximum shear => beam assumed to take shear from composite loading, therefore not calculated here.
75
Example:
Simply Supported Composite Beam
Ultimate Limit State – Composite Stage
Combination of actions (EN1990, Eqn 6.10b)
Distributed load on beam (0.85 x 1.35 x 3.48) + (1.5 x 3.8) = 9.69 kN/m2
Total Load Fd = 9.69 x 6.0 x 3.0 = 174.42 kN
Maximum design moment at midspan
My,Ed = FdL/8 = 174.42 x 6 / 8 = 130.82 kNm
Maximum design shear
VEd = Fd/2 = 174.42 / 2 =87.21 kN
76
Example:
Simply Supported Composite Beam
Partial Factors for Resistance
Structural Steel (EN1993-1-16.1(1)) γM0=1.0
Concrete (EN 1992-1-1, Table 2.1N) γC=1.5
Reinforcement γS=1.15
Shear connectors (EN 1994-2.4.1.2) γv=1.25
Longitudinal shear γvs=1.25
77
Example:
Simply Supported Composite Beam
Steel beam – trial sectionRequired Plastic Modulus (construction stage loading) =>
Wpl,y = My,EdγM0/fy = 63.6x103x1.0/275 = 231cm3
Try => 254x102x22 UKB, S275 Wpl,y = 259cm3
From Section Tables =>
ha = 254.0 mm tf = 6.8mm
b = 101.6 mm r = 7.6mm
d = 225.2 mm Aa = 28cm3
tw = 5.7mm Wpl,y = 258cm3
Elastic Modulus E = 210kN/mm2
Section is Class 1 under bending (EN 1993-1-1,3,2,6(1))
78
Example:
Simply Supported Composite Beam
Composite stage member resistance checks
Compressive resistance of concrete slab:
Effective width at midspan of compression flange (EN1994 5.4.1.2) =>
beff = b0 + Σ bei ; bei = Le/8 = L/8 = 6/8 = 0.75m (for SS beam)
Assume single shear studs, hence b0 = 0m
beff = 0 + (2 x 0.75) = 1.50m<3m (beam spacing)
Compression resistance (EN1994 6.2.1.2) =>
Nc,slab = 0.85fckbeffhc/γc = 0.85x25x1500x79x10-3/1.5 = 1679 kN
79
Example:
Simply Supported Composite Beam
Composite stage member resistance checks
Tensile resistance of steel section:
Npl,a = fdAa =fyAa/γM0 = 275 x 28 x102/1.0 = 770 kN
Since Npl,a < Nc,slab, the plastic neutral axis lies in the concrete flange
Design bending resistance - full shear connection (EN1994 6.2.1)
kNmh
N
Nh
hNM c
slabc
apla
aplRdpl 184102
79
1679
770130
2
254770
22
3
,
,
,, =×
×−+=
×−+= −
80
Example:
Simply Supported Composite Beam
Composite stage member resistance checks
Bending moment at mid span My,Ed = 131 kNm
Hence, design bending resistance of the composite beam is adequate
0.171.0184
131
,
,<==
Rdpl
Edy
M
M
81
Example:
Simply Supported Composite Beam
Shear connector resistance (EN1994 6.6.3.1)
Design shear resistance of a single shear connector is the smaller of:
andv
cmck
Rd
EfdP
γ
α 229.0=
( )v
u
Rd
dfP
γ
π 4/8.0 2
=
26.519
100==
d
hsc0.4>
d
hsc 0.1=αAs Then
82
Example:
Simply Supported Composite Beam
Shear connector resistance
So
or
As 73.7 kN < 81.7kN, Hence PRd = 73.7 kN
kNPRd 7.731025.1
103125190.129.0 332
=×××××
= −
( )kNPRd 7.8110
25.1
4/194508.0 32
=××××
= −π
83
Example:
Simply Supported Composite Beam
Influence of deck shape (EN1994 6.6.4.2)
Ribs are transverse to beam,
Also assume 1 stud per trough, nr = 1.0
Reduction factor
Hence, Kt =1.0, no reduction in shear resistance i.e. PRd = 73.7 kN
0.117.0 0 ≤
−
=
p
sc
pr
th
h
h
b
nk
48.1151
100
51
5.112
1
7.0=
−
=tk > 1
84
Example:
Simply Supported Composite Beam
Number of shear studs in half span
Using 1 shear connector per trough =>
Stud spacing along the beam = 152.5 mm
Assuming either a primary beam width or column width of 254mm say, then:
( )18
5.152
2/2543000=
−=n Shear stud connectors
per half span
85
Example:
Simply Supported Composite Beam
Degree of shear connection
Hence, full shear connection is provided, and no reduction in bending resistance is required.
kNPR Rdq 13277.731818 =×==
0.17.1770
1327
,
>==apl
q
N
R
86
Example:
Simply Supported Composite Beam
Shear buckling resistance of the uncased web
For unstiffened webs if the shear buckling
resistance of the web should be checked.
Where:
εη
72>
t
hw
92.0275
235235===
yfε
With η = 1.0 ( ) mmthh 4.2408.622542 faw =×−=−=
2.6692.00.1
7272=×
=ε
η2.42
7.5
4.240
w
ww ===t
h
t
h
As 42.2 < 66.2 the shear buckling resistance of the web does not need to be checked.
87
Example:
Simply Supported Composite Beam
Resistance to vertical shear
Shear resistance of the composite beam is:( )
3
M0
yvRda,pl,Rdpl,
γ
fAVV ==
( )rttbtAA 22 wffv ++−= wwthη
[ ])6.72(7.58.6)8.66.1012(2800v ×+×+××−=A 1560v =A
For rolled I and H sections loaded parallel to the web:
but not less than
mm2
0.1=η 13707.54.2400.1ww =××=thη(Conservatively)
1560 mm2 > 1370 mm2 Therefore, Av = 1560 mm2 247kN 100.13
2751560 3-
Rdpl, =××
×=V
88
Example:
Simply Supported Composite Beam
Resistance to vertical shear
35.0247
87
Rdpl,
Ed ==V
V
< 1.0, Therefore the design resistance to vertical shear is adequate.
As there is no shear force at the point of maximum bending moment (mid span) no reduction (due to shear)in bending resistance is required
89
Example:
Simply Supported Composite Beam
Design of Transverse Reinforcement
f
ydsf
s
fA
θcotfEdhv
f
sf
s
Aθcotyd
fEd
f
hv
The area of reinforcement (Asf) can be determined using the following equation:
therefore,
Neglect the contribution of the decking and check the resistance of the concrete flange to splitting:
> >
4351.15
500
s
yyd ===
γ
ff
where: hf is the depth of concrete above the metal decking, therefore, hf = hc = 79 mm
For compression flanges 26.5° ≤ θ ≤ 45°
N/mm2
90
Example:
Simply Supported Composite Beam
Design of Transverse Reinforcement
The longitudinal shear stress is the stress transferred from the steel beam to the
concrete. This is determined from the minimum resistance of the steel, concrete and
shear connectors. In this case, the plastic neutral axis lies in the concrete flange,
and there is full shear connection, so the plastic resistance of the steel section to
axial force needs to be transferred over each half-span. As there are two shear
planes (one on either side of the beam, running parallel to it), the longitudinal shear
stress is:
83.13000702
1000770
2 f
apl,
EdL, =××
×==
xh
Nv
∆N/mm²
91
Example:
Simply Supported Composite Beam
Design of Transverse Reinforcement
For minimum area of transverse reinforcement assume θ = 26.5°
f
sf
s
A14710
6.52cot435
7983.1
cot
3
yd
fEdL,=×
×
×=
θf
hv≥
Therefore, provide A193 mesh reinforcement (193mm2/m) in the slab.
mm2/m
92
Example:
Simply Supported Composite Beam
Crushing of the concrete flange
Where
ffcdEdL, cossin θθνfv ≤
25016.0 ckf
- 54.0250
25 -16.0 =
59.35.26cos5.26sin5.1
2554.0cossincd =×××=fff θθν
83.1EdL, =v
Verify that: fcd = fck/γc
ν =
N/mm²
N/mm² < 3.59 N/mm²
Therefore the crushing resistance of the concreteis adequate.
where
=
93
Example:
Simply Supported Composite Beam
Serviceability limit state
Performance at the serviceability limit state should be verified. However, no verification is included here. The National Annex for the country where the building is to be constructed should be consulted for guidance.Considerations would be:
Short-term, long-term and dynamic modular ratiosServiceability combinations of actionsComposite bending stiffness of the beamTotal deflection and deflection due to imposed loadsStresses in steel and concrete (to validate deflection assumptions)Natural frequency.