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Analysis and Design of Precast Concrete Structures to Eurocode 2
Precast and Prestressed Floors
and Composite Floors
www.civil.utm.my
Precast & Prestressed Floors and Composite Floors
www.civil.utm.my
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• Hollow core floor units & slab fields
• Double tee units
• Half-slab (precast + in-situ topping)
• Composite floors
• Load vs. span data
Fixing Rates at 2000 m2 per week?
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40 11 feet Wide HCU onto Precast Walls at MGM Hotel, Las Vegas, 1992
Hollowcore Units
Daily production: 200 m2 – 1500 m2
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Double Tee Units
Twice the price but up to 4 capacity than hollow core
Prestressed Half Slab
Popular for housing and awkward shapes
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Prestressed Half Slab
Jett
y at
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Prestressed Half Slab
Propping required for 5 m
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Prestressed Hollow Core Floor Units
• 400 – 3600 mm wide; typically 1200 mm
• 90 – 730 mm deep; typically 150, 200, 250, 300 mm
• Self weight 1.5 to 5.0 kN/m2
• Void ratio 40 – 60 % of solid section
• Spans 6 – 20 m (economical range)
Longitudinal pretensioning strand or wire
1195 mm
No shear or torsion links
30 mm flanges and webs
Shear key profile
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Splitting cracks due to sawing restraints as prestress in transferred
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Too much plasticiser!! in 450 deep units
Actually air-entrainment agent is used by several producers as a plasticiser
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Strand Pull-in:
An important indicator of success
Should be about 1 mm, irrespective of length
Theoretical pull-in limit for zero prestress:
𝑷𝑳
𝑨𝑬
* Use a linear scale between the extremes
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Bearing onto neoprene or mortar for spans more than about 15 m onto insitu or masonry
20 m long HCU
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Section for Analysis for Prestressed
• Pretension and losses (about 18 – 25%)
• Service moment (bottom tension critical)
• Ultimate moment (usually Msr 1.5)
• Ultimate shear uncracked & flexurally cracked
• End bearing and transmission length
• Deflection and camber (long-term, creep)
• Live load deflection after installation
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• 𝑀𝑠𝑟 = 𝑓𝑏𝑡 + 𝑓𝑐𝑡 𝑍𝑏
• 𝑀𝑢𝑟 = 0.87𝑓𝑝𝑏𝐴𝑝𝑠 𝑑 − 𝑑𝑛
• 𝑉𝑐𝑜 =𝐼∙𝑏𝑤
𝐴∙𝑦 𝑓𝑐𝑡𝑑
2 + 𝛼𝑙𝜎𝑐𝑝𝑓𝑐𝑡𝑑
Section for Analysis for Prestressed P
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Load vs Span Curve
Allowable span
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Load vs Span Curve
Allowable span
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Bearing limit
Handling limit span/depth = 50
Load vs Span Curve
Allowable span
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Bearing limit
Handling limit span/depth = 50
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Load vs Span Curve
Allowable span
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Shear
Service moment
Deflection Handling
Service moment control
Deflection control
Possibly shear ?
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Example 1 Calculate Msr for the 203 mm deep prestressed HCU shown below. The initial prestressing force may be taken as 70% of characteristics strength of the standard 7-wire helical strand. Take the final long-term losses as 24%. Geometric and data given by the manufacturer are as follows: Ac = 135 103 mm2, I = 678 106 mm4, yt = 99 mm, fck = 50 N/mm2, Ec = 30 kN/mm2, fck (t) = 35 N/mm2, Eci = 27 kN/mm2, fp0,1k = 1750 N/mm2, Eps = 195 kN/mm2, Aps = 94.2 mm2 per strand ( = 12.5 mm) and cover = 40 mm.
Section Properties
Zb = I/yb = 678 106/104 = 6.519 106 mm3
Zt = I/yt = 678 106/99 = 6.848 106 mm3
Eccentricity, e = 104 – 40 – 12.5/2 = 57.7 mm
Initial prestress in tendons, fpi = 0.70 1750 = 1225 N/mm2
Initial prestressing force, Pi = fpi Aps = 1225 7 94.2 10-3 = 807.8 kN
Example 1 - Solution
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Example 1 - Solution
Check Stress at Transfer
fbc = (807.8 103/135 103) + (807.8 103 57.75/6.519 106)
= +13.14 N/mm2 (compression) 0.6fck (t) (21 N/mm2)
ftt = (807.8 103/135 103) (807.8 103 57.75/6.848 106)
= 0.83 N/mm2 (tension) fctm (t) (3.20 N/mm2)
Final Prestress in Bottom and Top
fbc = +13.14 (1 – 0.24) = +9.97 N/mm2 (compression)
ftt = 0.83 (1 – 0.24) = 0.63 N/mm2 (tension)
OK
Service Moment
At the bottom fibre, Msr is limited by the tensile stress limit of 0 N/mm2
Msr = (fbc)Zb = (9.97) 6.519 106 10-6
= 65.0 kNm
At the top fibre, Msr is limited by the compressive stress limit of 0.6fck = 30 N/mm2
Msr = (ftt + 0.6fck)Zt = (0.63 + 30) 6.848 106 10-6
= 209.7 kNm 65.0 kNm
Example 1 - Solution
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Calculate Mur for the section given in Example 1. Manufacturer’s data gives the breadth of the top of the HCU as b = 1168 mm.
Effective depth, d = 203 – 40 – 12.5/2 = 156.75 mm
Stress in steel after losses = (1 – 0.24) 0.70 1750 = 931 N/mm2
Therefore, strain in steel after losses = 𝑓𝑠
𝐸𝑠=
931
205×103= 0.0045
which is less than y, the yield strain.
Example 2 - Solution
Example 2 - Solution
𝒇𝒑𝟎,𝟏𝒌
𝜸𝒎=𝟏𝟕𝟓𝟎
𝟏. 𝟏𝟓= 𝟏𝟓𝟐𝟐
𝜺𝒚 =𝟏𝟓𝟐𝟐
𝟐𝟎𝟓 × 𝟏𝟎𝟑= 𝟎. 𝟎𝟎𝟕𝟒𝟐
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As a first attempt, try x = 75 mm, approximately equal to 0.5d
(a) Steel Strains
Final stress strain, s = Prestress Strain + Bending Strain, ’s
Bottom layer at the tendon:
sb = 0.0045 + ’sb
= 0.0045 +𝑑−𝑥
𝑥𝜀𝑐𝑐
= 0.0045 +156.75−75
75× 0.0035 = 0.008315 y
Take sb = 0.00742
cc = 0.0035
’sb
x
d
Example 2 - Solution
(b) Steel Stresses Bottom layer at the tendon: fsb = sb Es
= 0.00742 205 103
= 1522 N/mm2 (c) Forces in Steel & Concrete Steel tensile forces, Fst =fs.Aps = (fsb) 7 94.2 = 1003 103 N Concrete compressive forces, Fcc= 0.567fckb 0.8x = 0.567 50 1168 0.8 75 = 1987 103 N Since Fcc Fst, a smaller depth of neutral axis, x must be tried.
Example 2 - Solution
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Example 2 - Solution Try x = 38 mm 28 mm (upper flange) and then carry out the previous analysis. Final Forces in Steel & Concrete Steel tensile forces, Fst =fs.Aps = (fsb) 7 94.2 = 1003 103 N Concrete compressive forces, Fcc= 0.567fckb 0.8x = 0.567 50 1168 0.8 38 = 1006 103 N Fst OK Ultimate Moment of Resistance: Mur = Fstz = Fst (d – 0.4x) = 1003 103 (156.75 – 0.4 38) 10-6 = 142 kNm
Shear searches out the weakest web
Shear failure in 400 deep units
Shear Analysis
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Calculate the uncracked shear capacity, Vco for the section in Example 1.
𝜎𝑐𝑝 =𝛾𝑐𝑝𝜂𝑃𝑖
𝐴=
0.9×(1−0.24)×0.70×1750×7×94.2
135×103 = 4.09 N/mm2 0.133fck (6.65 N/mm2)
𝑓𝑐𝑡𝑑 =𝑓𝑐𝑡𝑘,0.05𝛾𝑚
=2.90
1.5= 1.40 𝑁/𝑚𝑚2
𝛼1 =𝑙𝑥𝑙𝑝𝑡2
=800 + 203/2
961= 0.94 ≤ 1.0
Example 3 - Solution
𝒍𝒑𝒕𝟐 = 𝟏. 𝟐𝒍𝒑𝒕 = 𝟏. 𝟐𝜶𝟏𝜶𝟐∅𝝈𝒑𝒎𝟎
𝒇𝒃𝒑𝒕= 𝟏. 𝟐
𝟏. 𝟎 × 𝟎. 𝟐𝟓 × 𝟏𝟐. 𝟓 × 𝟏𝟐𝟐𝟓
𝟒. 𝟕𝟖
= 𝟗𝟔𝟏 𝒎𝒎 1 = 1.0 2 = 0.25 pm0 = 0.70 1750 = 1225 N/mm2
= 12.5 mm lpt (transmission length) = 800 mm
𝑓𝑏𝑝𝑡 = 𝜂𝑝1𝜂1𝑓𝑐𝑡𝑑 𝑡 = 𝜂𝑝1𝜂1𝑓𝑐𝑡𝑘,0.05 (𝑡)
𝛾𝑚= 3.2 × 1.0 ×
2.20
1.5= 4.78 𝑁/𝑚𝑚2
Example 3 - Solution
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Total breadth at centroid, bw = 1168 – (6 150) = 268 mm
Area above the centroidal axis, A = 67500 mm2
y’ = 64.5 mm (calculate from geometry)
𝑉𝑐𝑜 =𝐼 ∙ 𝑏𝑤𝐴 ∙ 𝑦
𝑓𝑐𝑡𝑑2 + 𝛼1𝜎𝑐𝑝𝑓𝑐𝑡𝑑
=678 × 106 × 268 × 10−3
67500 × 64.51.42 + 0.94 × 4.09 × 1.40
= 113.1 kN
Example 3 - Solution
Watch out for flexural-shear (Vcr) failure !
Shear Analysis
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Vco
Vcr
Decompression point
Vcr = [0.12k (100lfck)1/3 + 0.15cp] bwd
Vcr, min = [0.035k3/2 (fck)1/2 + 0.15cp] bwd
Flexural-Shear Formula, Vcr P
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UDL
SF diagram
Vcr failure here
Flexural-Shear Formula, Vcr
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Calculate the minimum value of Vcr for the slab in Example 1.
𝜎𝑐𝑝 =𝛾𝑐𝑝𝜂𝑃𝑖
𝐴=
0.9×(1−0.24)×0.70×1750×7×94.2
135×103 = 4.09 N/mm2 0.133fck (6.65
N/mm2)
𝑘 = 1 +200
𝑑= 1 +
200
156.75= 2.12 ≤ 2.0 (From Example 2, d = 156.75
mm) From Example 3, cp = 4.09 N/mm2
Vcr, min = [0.035k3/2 (fck)
1/2 + 0.15cp] bwd = [0.0035(2.0)3/2 (50)1/2 + 0.15 4.09] 268 156.75 10-3
= 28.7 kN
Example 4 - Solution
• Refer to Cl. 10.9.5
• Refers to contact bearing pressure at the bearing ledge
• Non-isolated – components are connected to other components with a secondary means of support
• Isolated – components rely entirely on their own bearing for total support
Ultimate bearing capacity, FEd = fRd b1 a1
Bearing Capacity
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𝒂 = 𝒂𝟏 + 𝒂𝟐 + 𝒂𝟑 + ∆𝒂𝟐𝟐 + ∆𝒂𝟑
𝟐
Bearing Definitions P
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a1 must NOT be less than the value in Table 10.2
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Net Bearing Length, a1 (Perpendicular to the Floor)
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Limiting Value of a2 P
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Limiting Value of a3
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Limiting Value of a2 and a3 P
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Net Bearing Width, b1
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Ultimate Bearing Strength, fRd P
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Calculate the bearing capacity of the HCU in Example 1. The unit has an actual 75 mm dry bearing length onto a reinforced concrete. Assumed that the unit is 6 m long and has secondary support on steel beams.
Ultimate bearing strength, fRd = 0.4fcd = 0.4 ×50
1.5 = 13.33 N/mm2
Net bearing width, b1 = 1200 mm b1, max = 600 mm Use b1 = 600 mm Net bearing length for non-isolated component, a1:
From a2 = 0 mm a2 = 10 mm a = 75 mm a3 = 5 mm a3 = 2.4 mm
Example 5 - Solution
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Refer to Table 10.2 for the minimum values of a1
The ultimate bearing capacity:
FEd = 13.33 60.3 600 10-3 = 482.3 kN > Vco (109.3 kN) OK
Note:
𝜎𝐸𝑑 =𝑉𝑐𝑜 (𝐹𝑟𝑜𝑚 𝐸𝑥𝑎𝑚𝑝𝑙𝑒 4)
1200 × 75= 1.25 𝑁/𝑚𝑚2
𝜎𝐸𝑑𝑓𝑐𝑑
=1.25
501.5
= 0.0375
Example 5 - Solution
𝒂𝟏 = 𝒂 − 𝒂𝟐 − 𝒂𝟑 − ∆𝒂𝟐𝟐 + ∆𝒂𝟑
𝟐
𝒂𝟏 = 𝟕𝟓 − 𝟎 − 𝟓 − 𝟏𝟎𝟐 + 𝟐. 𝟒𝟐 = 𝟔𝟎. 𝟑 𝒎𝒎 > 𝟐𝟓 𝒎𝒎
Composite Floors Required for Double Tee, but Optional for Hollowcore
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Surface Laitance due to Cutting Slurry
50 mm minimum at the highest point, increasing (with slab and beam cambers) to about 80 mm
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Composite Construction
• Minimum thickness of topping 40 mm
• Average depth of topping allowances for camber should be made – allowing span/300 will suffice
• Concrete grade – C25 to C30
• Minimum mesh reinforcement area = 0.13% Concrete area
Composite Construction
• Advantages to composite construction:
(a) Increase bending resistance and flexural stiffness
(b) Improve vibration, thermal & acoustic performance
(c) Provide horizontal diaphragm action
(d) Provide horizontal stability ties across floors
(e) Provide a continuous and monolithic floor finish
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with 75 mm topping
Composite Design
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• Selfweight of slab + in-situ concrete topping + construction traffic allowance (1.5 kN/m2)
• Section properties = Precast unit
• Selfweight of slab + in-situ concrete topping + construction traffic allowance (1.5 kN/m2)
• Section properties = Precast unit
Stage 1
• Superimposed load
• Section properties = Composite section
• Superimposed load
• Section properties = Composite section
Stage 2
Composite Design -2 Stage Approach Se
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– 2
Sta
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+
_
+
_
+
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+
+
+ =
_
Topping
Precast
Prestress Imposed Stage 1 stresses
Imposed Stage 2 stresses
Total stresses
0.6
f ck
limit
f ct l
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Stress Diagram at Serviceability
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Bottom fibre stress of the HCU:
𝒇𝒃𝟏 = 𝒇𝒃𝒄 −𝑴𝟏
𝒁𝒃𝟏< +𝟎.𝟔𝒇𝒄𝒌 (𝒕) 𝒕𝒓𝒂𝒏𝒔𝒇𝒆𝒓 & 𝟎 (𝒔𝒆𝒓𝒗𝒊𝒄𝒆)
Top fibre stress of the HCU:
𝒇𝒕𝟏 = 𝒇𝒕𝒄 −𝑴𝟏
𝒁𝒕𝟏< 𝒇𝒄𝒕𝒎 𝒕 𝒕𝒓𝒂𝒏𝒔𝒇𝒆𝒓 & + 𝟎. 𝟔𝒇𝒄𝒌 (𝒔𝒆𝒓𝒗𝒊𝒄𝒆)
Serv
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Stage 1 – Service Moment, M1
Bottom fibre stress of the HCU:
𝒇𝒃𝟐 = −𝑴𝟐
𝒁𝒃𝟐
Top fibre stress of the HCU:
𝒇𝒕𝟐 = +𝑴𝟐
𝒁𝒕𝟐
Top fibre stress of the in-situ topping:
𝒇′𝒕𝟐 = +𝑴𝟐
𝒁′𝒕𝟐
beff = b Ec’/Ec where b is the full breadth
Stage 2 – Service Moment, M2
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Adding Stage 1 + Stage 2:
fb =fb1 + fb2 = fbc – M1/Zb1 – M2/Zb2 0
ft = ft1 + ft2 = ftc + M1/Zt1 + M2/Zt2 +0.6fck
f’t = f’t2 = + M2/Z’t2 +0.6fck, in-situ
Service Moment, Msr
Bottom of the slab:
Ms2 = M2 (fbc)Zb2 – [M1(Zb2/Zb1)]
Top of the slab:
Ms2 = M2 (0.6fck – ftc)Zt2 – [M1(Zt2/Zt1)] Serv
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Total Service Moment, Ms
Calculate the Stage 2 service bending moment that is available if the HCU in Example 1 has a 50 mm minimum thickness structural topping. The floor is simply supported over an effective span of: (a) 4.0 m; (b) 8.0 m. The precamber of the HCU may be assumed as span/300 without loss of accuracy. Use fck, in-situ = 30 N/mm2 and Ec, in-
situ = 26 kN/mm2. Self weight of concrete = 25 kN/m3. Selfweight of HCU = 3.24 kN/m run. What is the minimum imposed loading for each span?
Example 6
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Example 6 - Solution
Effective breadth of topping, beff = 1200 26/30 = 1040 mm
Total depth of composite section = 203 + 50 = 253 mm
Depth to neutral axis from top of composite section, yt2
=1040×50×25 +(135000× 50+99 )
135000+(1040×50)= 114.5 mm
Second moment of area of composite section, I2 = 1266 106 mm4
Example 6 - Solution
𝑍𝑏2 =1266×106
(253−114.5)= 9.14 × 106 mm3
𝑍𝑡2 =1266×106
114.5= 19.63 × 106 mm3
Zb2/Zb1 = 9.14 106 /6.519 106 = 1.40
Zt2/Zt1 = 19.63 106 /6.848 106 = 2.86
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(a) 4.0 m
Precamber = 4000
300= 13 mm
Maximum depth of topping at supports = 50 + 13 = 63 mm
Average depth of topping = (50+63)
2= 57 𝑚𝑚
Self weight of topping = 0.057 25 1.2 = 1.71 kN/m run for 1.2 m wide unit
Self weight of HCU = 3.24 kN/m run
Stage 1 moment, M1 = (3.24+1.71)×42
8= 𝟗. 𝟗𝟎 𝒌𝑵𝒎
Example 6 - Solution
Bottom of the slab:
𝑀𝑠2 > 𝑓𝑏𝑐𝑍𝑏2 − 𝑀1
𝑍𝑏2𝑍𝑏1
𝑀𝑠2 > 9.97 × 9.14 × 10−6 − 9.90 × 106 1.40 × 10−6
= 77.46 kNm
Example 6 - Solution
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Top of the slab:
𝑀𝑠2 = 𝑀2 > 0.6𝑓𝑐𝑘 − 𝑓𝑡𝑡 𝑍𝑡2 − 𝑀1
𝑍𝑡2𝑍𝑡1
𝑀𝑠2 > 30 − (−0.63 ) × 19.63 × 106 − 9.90 × 106 2.86 × 10−6
= 573.35 kNm
The allowable maximum imposed load = 8 77.46/4.02 = 38.73 kN/m
* Additional notes:
Total Ms = Ms1 + Ms2 = 9.90 + 77.46 = 87.36 kNm
Example 6 - Solution
Ultimate Limit State - 2 Stage Approach
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Total area of reinforcement, Aps = Aps1 + Aps2
Effective breadth, beff = b fcu in-situ/fcu
Stage 1
fpdAps1 = 0.567fck b(0.8X) but dn1 = 0.4X or X = dn1/0.4
then dn1 = fpdAps1/1.134fckb
then Mu1 = fpdAps1(d – dn1) ---------------- (1)
Ultimate Limit State - 2 Stage Approach
Stage 2
Aps2 = Aps – Aps1
Mu2 = fpdAps2 (d + hs – dn2) ---------------- (2)
or by replacing (1) into (2):
Mu2 = fpd [Aps – Mu1/fpd(d – dn1)] (d + hs – dn2)
One Step Approach
Mu = fpdAps (d + hs – dn)
Ultimate Limit State - 2 Stage Approach
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Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
In the neutral axis is below the interface, X hs
VEd = 0.567fckbeffhs
In the neutral axis is above the interface, X hs, VEd = 0.567fckbeff0.9X The shear stress at the interface should satisfy: vEdi vRdi
Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
Design shear stress at the interface; 𝑣𝐸𝑑𝑖 =𝛽𝑉𝐸𝑑
𝑧𝑏𝑖
Design shear resistance at the interface; vRdi = cfctd +n + fyd ( sin + cos ) 0.5vfcd
If vEdi vRdi ; provide shear reinforcement (per 1 m run) projecting from the precast unit into the structural topping. The amount of steel required:
𝑨𝒇 =𝟏𝟎𝟎𝟎𝒃𝝉𝒉𝟎. 𝟖𝟕𝒇𝒚𝒌
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Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
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Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
Interface Shear Stress in Composite Slabs (Cl. 6.2.5)
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• 2400 – 3000 mm wide
• 300 – 2000 mm deep; typically 400 – 800 mm
• self weight 2.6 to 10 kN/m2
• void ratio 70 – 80 % of solid section
• spans 8 – 30 m (economical range)
Double Tee Floor Units
Mostly prestressed, but RC if manufacturer prefers
Bearing pads required 150 x 150 x 10 mm
Double Tee Floor Units