Page 1

Bridge Beam Design Manual =

3 4.1

a MP

1st Edition

y ikel re l a s ion Mpa ndit o c 29 . d 1 n bo = ood g e er h s w d m ran mm 0m (t) e st 5 r i 2 f d t w 7 n a η1 c 7 .7 th nd 935 = η p1 3 a greater : r y f bpt o b = f n ths re; 3. 2 give ϒc dep s whe i / r ) o h (t = 1f ngt η p1 .7 f ctm n le o 0 i s s = α ct smi η1 ran = t ) t e th f ctd( e of u w: l a / f bpt belo cv i e s 0 a h σ pm b of t 2ø The mm nds ble a 1α a r r α t u s = o 2 e 6 av l pt mm wir 48. unf e: 7 7 r e 2 e r d o 2.9 an wh = 1 he m 112 r3 t o = f s m a 1 a m α = .19 ed MP = 0 ulat 9 c 0 8 l . a .7 α2 c 95 15 h is 1, 2 ngt ø= e l = sion σ pm0 mis s n tra sign e d The 0.8l pt Precast concrete specialists = l pt1 . 2l pt 1 = l pt2

M. Slevin E. Stack


Acknowledgements Banagher Precast Concrete would like to thank the Concrete Bridge Development Group (CBDG) for asking us to present our Bridge Beam Manual at their annual conference in Oxford. We would also like to thank Abhishek Das in MIDAS for his help with the initial bridge model and his continued support throughout and a special thanks to our expert reviewers and researchers for their comments and help.

Irish Plant

UK Plant

Disclaimer This manual gives Banagher Precast Concretes view on prestressed bridge beam design and in particular our W-beam which we developed in 2005. Please note that the calculations presented in this design manual are for information only. Banagher Precast Concrete Ltd. do not accept any liability for the use of any presented material.

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Contents 1. Introduction 2 2. Design standards required 2 3. Scheme design 3 4. Materials 4 4.1 Concrete 4 4.2 Prestressing Steel 4 4.3 Reinforcing Steel 4 4.4 Cement 4 4.5 Modular Ratio 4 5. Cover 5 6. Structural model and analysis 6 6.1 Modelling Assumptions 6 6.2 Loads 7 6.3 Boundary Conditions 7 6.4 Construction Stages 8 7. Section Properties 9 8. Calculation of Loads 10 9. Combinations of actions 13 10. Differential Shrinkage 14 11. Temperature Effects 15 11.1 Uniform Temperature Component 15 11.2 Temperature Difference Component 16 11.3 Non Linear Component 17 12. Prestress Design 18 13. Prestress Losses 20 13.1 Immediate Losses 20 13.1.1 Relaxation Loss 20 13.1.2 Elastic Shortening Loss 20 13.2 Time Dependent Losses 21 13.2.1 Relaxation Loss 21 13.2.2 Creep 21 13.2.3 Shrinkage 23 13.3 Summary of Long Term Prestressing Losses 24 13.4 Effective Prestressing Force 24 13.5 Transmission Length 24 14. Serviceability Limit State (SLS) 25 14.1 Decompression Check at Transfer of Prestress 25 14.2 Revised Stresses With Debonding 26 14.2.1 At End of Transmission Length 26 14.2.2 At End of Debonded Length 26 14.3 Stress Check at Construction Stage 26 14.4 Decompression and Stress Check at Service (midspan max) 27 14.5 Stress Check at Service (midspan min) 28 14.6 Decompression and Stress Check at Service (end of diaphragm) 29 14.7 Decompression and Stress Check at Service (end of transmission length) 30 14.8 Decompression and Stress Check at Service (end of different debonded lengths) 31 14.9 SLS Stress Summary 33 15. Ultimate Limit State (ULS) 35 15.1 Ultimate Limit State Flexure Check 35 15.2 Minimum Reinforcement Requirement Check 35 15.3 Global Vertical Shear Design (beam ends) 36 15.4 Shear at The Interface Between the Beam and The Slab 37 15.5 Shear Check Between The Web and The Flange of The Composite Section 39 16. Beam camber estimates 40 17 Creep Induced Sagging Moment at Supports 40 Appendix A 42 Appendix B 45 LIST OF SYMBOLS 46

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1. Introduction Banagher Precast Concrete is pleased to introduce its bridge beam design manual. Prestressed concrete bridge beams are firm favourites in the short and medium span bridge market, of up to 50 metre spans and have been in successful use for the last 60 years. The Bridges for which the industry and of course Banagher Precast Concrete manufacture beams require considerable expertise in design. The design is usually carried out with a Contractor employed Engineer taking the project from planning to working drawings of the bridge for which the beams would be required. The design of the beams may be carried out by this engineer, by the engineer in cooperation with Banagher Precast Concrete the supplier or completely by Banagher Precast Concrete using the engineer’s analysis output as the starting point. In the last few years, the design regulations have been harmonised throughout Europe as part of the Common Market for Goods and Services with the introduction of a suite of Eurocodes and European Product Standards. Each Member State of the EU has the responsibility for Structural Safety and the Eurocodes therefore come with a National Annex in which National preferences with respect to safety factors etc are given. The designs in this manual have used the UK National Annexes which are very similar to the Irish National Annexes. Where the Irish regulation would be different, it is noted in the text. The coherence of the new regulations allow, with the use of Harmonised Product Standards, a common approach which is used as the basis of the CE Marking of Bridge Beams, in turn enabling them to be part of an open pan European Market.

This manual gives Banagher Precast Concretes view of the new design process for Prestressed Bridge Beams to the Eurocode and it is hoped will promote more use of Bridge Beams and a common and agreed understanding of the many new clauses in the codes. The manual considers the design of the innovative Banagher W beam in a typical two span road bridge. The internal beam is designed in detail and as the loading code is also new, the approach to the use of this code is also demonstrated. Other areas of the whole bridge design are commented upon, particularly where they can have an important influence on beam design. These are mentioned at appropriate places where they would naturally occur. Suggestions for further reading are also given. The format is for the manual to have the commentary and the design information followed by the example calculations. This design manual should be read in conjunction with our “Bridge Beam Manual” which includes our full range of precast prestressed beams and their associated span tables along with all other relevant information required by a bridge engineer in choosing a precast section.

2. Design standards required The following list contains the relevant standards that are required for the design of this bridge. 1. Eurocode - Basis of Structural Design - BS EN 1990 : 2002 2. Eurocode 1 Actions on structures - Part 1 and Part 2 - BS EN 1991-1 and BS EN 1991-2 3. Eurocode 2 Design of concrete structures - BS EN 1992-1-1 : 2004 4. Eurocode 2 Design of concrete structures. Concrete bridges. Design and detailing rules - BS EN 1992-2:2005 5. BS EN 206-1 : 2000 Concrete – Part 1: Specification, performance, production and conformity 6. BS 8500-1 : 2006 - Concrete – Complementary British Standard to BS EN 206-1 – Part 1: Method of specifying and guidance for the specifier 7. PD 6694-1 Recommendations for the design of structures subject to traffic loading to BS EN 1997-1 : 2004 8. BS EN 15050:2007+A1 : 2012 Precast concrete products - Bridge elements

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9. BS EN 13369 : 2013 Common rules for precast concrete products 10. EN 10138 - Prestressing Steel Note: The standards referenced in this design example are those relevant at the time of print. Please make yourself aware of ammendments before proceeding with this design example.


3. Scheme design The foundations for the abutment are modelled as strip foundations 1m deep and 4m wide. The sizing of the foundations is taken from experience and for this example will suffice as the W11 beams are all that is being looked at in detail not the supporting structure. The precast columns are fixed into place with dowel bars projecting up from the foundation below. The columns will need to be propped and the sleeves filled with non shrink grout which will then need to be left to strengthen before placing of the precast crosshead.

The design is for a two-span integral bridge, with each span having a length of 30.75m from centre of abutment to centre of pier giving an actual beam length of 30.50m as per fig 3.2. The bridge carries a 6.0m wide carriageway with 1.5m wide footways on either side as per fig 3.1. The superstructure consists of six Banagher Precast Concrete prestressed W11-beams with a 230mm structural in-situ reinforced concrete deck slab 200mm over the top of the beam cast on ribbed fibre reinforced concrete (FRC) semi participating permanent shutter. There are in-situ diaphragms at the abutments and pier. A 500mm gap between the precast beams is used at the pier to allow projecting links from the crosshead. There is a 1500mm wide precast crosshead spanning between two precast columns which makes up the pier. The bridge beams span from abutment to crosshead with 500mm bearing/embedment and without the need for temporary support.

500

1500

10000 9000

1500

500

140

110

80

1500

1450

2000

7 1466.624

1700

200

600

230

350 350

600

250

15

1500

It is recommended that a number of transverse holes are located at various depths in the precast beam at the abutment and pier diaphragms. Whilst the number and position of these diaphragms will be at the discretion of the designer in conjunction with the precast manufacturer, a sufficient number of holes should be provided in order to ensure anchorage of the precast beams and that the pier and / or abutment diaphragms act as a torsional beam.

730

3000

2000

3 NO. W11 BEAMS AT 3000mm CENTRES = 6000

Figure 3.1 - Section through bridge deck showing W-11 beams at 3.0m centres

ABUTMENT

PIER

100

100

500

29500

400

30500 OVERALL BEAM LENGTH

400

100

500

500

1500 CROSSHEAD

100

29500

400

500

500

30500 OVERALL BEAM LENGTH

400

ABUTMENT

62500

1000

900 300 PIER COLUMN 300 500

500

29800 30750

450

450

29800

500

500

30750

Figure 3.2 - Elevation of bridge

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4. Materials 4.1 Concrete Assumed properties of concrete are given in BS EN 1992-1-1, clause 3.1. BS EN 1992 uses cylinder strengths throughout but Table 3.1 gives the corresponding cube strengths, allowing the continued use of cubes as control specimens. The concrete strength class used in this design is C50/60 for the precast beams, C40/50 for the in-situ deck slab and diaphragms and C45/55 for the precast columns and crosshead. 4.2 Prestressing Steel The proposed European standard for prestressing steel is prEN 101383 and it is often referred to in BS EN 1992-1-1. prEN 10138-3, however, has since been voted down, though it is likely that it will be rewritten and published at a future date. In the meantime, BS 5896: 1980 has been amended to cover those products currently on the market for which no specification would otherwise exist.

4.3 Reinforcing Steel The European standard for reinforcing steel for concrete is BS EN 10080. However, BS EN 10080 does not define steel grades and rather inconveniently leaves it to the designer to specify its properties. In the UK this void is filled by BS 4449:2005 which specifies the required properties for standardised grades. 4.4 Cement Cement type class R is used by Banagher Precast Concrete in the design calculations as rapid hardening cement is generally used in production. 4.5 Modular Ratio A modular ratio of 1.0 is used for this example in accordance with common UK practice (BS 5400) which allowed a modular ratio of 1.0 if the difference between the precast strength & in-situ strength does not exceed 10 MPa.

Precast concrete

Characteristic cylinder strength fck 50.00 MPa Characteristic cube strength fck cube 60.00 MPa Mean compressive strength fcm 58.00 MPa Mean value of axial strength fctm 4.06 MPa Characteristic axial tensile strength of concrete fctk,0.05 2.84 MPa Design tensile strength of concrete fctd 1.90 MPa Modulus of elasticity Ecm 37.28 GPa Age of beam at transfer 3.00 days Characteristic cylinder strength at transfer fck @ transfer 28.00 MPa Characteristic cube strength at transfer fck cube @ transfer 35.00 MPa Mean compressive strength at transfer fcm @ transfer 36.00 MPa Mean value of axial strength at transfer fctm @ transfer 2.77 MPa Characteristic axial tensile strength of concrete at transfer fctk,0.05 @ transfer 1.94 MPa Design tensile strength of concrete at transfer fctd @ transfer 1.29 MPa Modulus of elasticity at transfer Ecm @ transfer 32.31 GPa

Insitu concrete

Characteristic cylinder strength fck Characteristic cube strength fck cube Mean compressive strength fcm Mean value of axial strength fctm Characteristic axial tensile strength of concrete fctk,0.05 Design tensile strength of concrete fctd Modulus of elasticity Ecm

40.00 MPa 50.00 MPa 48.00 MPa 3.51 MPa 2.46 MPa 1.64 MPa 35.22 GPa

Coefficients and strains

Modular ratio m 1.00 -

Prestressing steel type:

Diameter dia 15.70 mm Area per strand Aps 150.00 mm2 Ultimate strength fpu 1860.00 N/mm2 % Ultimate strength applied fpu% 75.00 % Characteristic value of max Force Fm 279.00 kN

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Initial force applied Pi Initial prestress applied fpi Elastic modulus Ep Relaxation of strand ( Class 2) - Mill Certs ρ1000

209.25 kN 1395.00 N/mm2 200.00 kN/mm2 1.50 %

Reinforcing steel:

Characteristic yield strength fyk 500.00 MPa

5. Cover

In the Republic of Ireland the National Roads Authorities (NRA) version of BD57/10 “Design for Durability” December 2010 version is a very useful booklet and has exposure class diagrams in Appendix A. At present in the UK there are no such diagrams available in any of the their documentation therefore use BS8500 “Concrete Complementary British Standard to BS EN 206-1”.

The diagram in figure 5.1 below shows the UK values with the NRA values in brackets where different. The exposure classes are specified in table 4.1 of BS EN 1992-1-1 and also in other documents. This bridge example is of a road over road bridge.

The nominal cover is worked out for the exposure class and the concrete grade. The most severe result has been used so that the worst case scenario is taken, e.g in Ireland XD1 with concrete grade C50/60 is 35mm minimum cover plus 5mm ΔC for precast = 40mm nominal. In the UK XD1 for the same concrete is 30mm minimum plus 5mm for ΔC = 35mm nominal.

Please note ΔC is taken as 5mm for precast and 10mm for the insitu in this example. It is also worth noting figure 5.2 which shows the differences in Ireland and the UK in relation to exposure class when dealing with de-icing salts.

XC4/XD3 50mm (55mm)

XC4/XD3 50mm (55mm) XC3 - 40mm (35mm) (IF DECK IS WATERPROOFED) XD3 - 50mm (55mm) (IF DECK IS NOT WATERPROOFED) INSITU DECK & PARAPET

XC3 35mm (30mm)

PRECAST W BEAM

XC3 35mm (30mm)

XC3 35mm (30mm)

PRECAST W BEAM

XC4/XD1 35mm (40mm)

PRECAST W BEAM

PRECAST CROSSHEAD BEAM

PRECAST COLUMN

XD3 - 50mm

PRECAST COLUMN

CONCRETE GRADES: ALL RC PRECAST - C45/55 PRESTRESSED BEAMS - C50/60 INSITU - C40/50 OVERBRIDGE STRUCTURE - EXPOSURE CLASS AND NOMINAL COVER DIAGRAM This is based on BD57/10 in ROI and BS8500 in the UK. The values are based on the UK with the ROI values in brackets where different

Figure 5.1 - Exposure Class & Nominal Cover Diagram. Based on UK values with ROI values in brackets where different.

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Outside This Envelope = XD1

Inside This Envelope = XD3

Carraigeway Level 8000 Rep of Ireland (NRA)

5000

7500

Outside This Envelope = XD1

Inside This Envelope = XD3

Carraigeway Level 10000 UK

Figure 5.2 - Exposure Class due to de-icing salts

6. Structural model and analysis 6.1 Modelling Assumptions Modelling of the 2nr span continuous integral bridge was carried out in 3D using MIDAS Civil, with a grillage model representing the deck, line beams for intermediate piers and 2D FE plate elements for the abutments on either side of the bridge. Concrete diaphragms, modelled as line elements are provided at the 2nr abutments and at the central intermediate pier location. Use of beam elements for the deck ensures direct extraction of design forces for the longitudinal beams. Longitudinal beams are modelled as composite sections with the effective width of the slab assigned to the composite section to take care of the shear lag effects. For simplicity each longitudinal section of the grillage consists of one precast beam and the effective width of

Figure 6.1 - Full Fleshed Model

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the slab on top of it. MIDAS composite section type beam elements allow extraction of discrete force and stress results for both the slab and the precast beams separately for a composite main beam section. For W beams it is common practice to provide one longitudinal grillage member per beam web (i.e. two per beam). Modelling a single line per beam does not accurately model the load transfer from deck slab to beam webs and does not accurately model load sharing between adjacent beams. However for this example this action was not taken for simplicity.


6.2 Loads Load type

Midas Civil function type

Load Value

Permanent Self Weight of beam

Auto self weight feature

In-situ concrete slab

Element Beam Load (UDL)

18.75 kN/m per precast beam

Parapet

Element Beam Load (UDL)

7.5 kN/m on the edge beams

Surfacing

Element Beam Load (UDL)

8.28 kN/m per precast beam

Earth Pressure on abutments

Hydrostatic Pressure

13.6-61 kN/m2 top to bottom

Pretensioning

Tendon Prestress

75% = 209.25 kN per tendon

Element Temperature

20 degrees

Load Model 1

Standard Eurocode Vehicle

Tandem system and co-existent UDL

Load Model 3

Standard Eurocode Vehicle

SV 196

Footway

Standard Eurocode Vehicle

5kN/m2

Variable Temperature Difference Variable Traffic Actions

Table 6.1 - Midas Civil Input Parameters

Figure 6.2 - Vehicle Position for gr5 for Max Sagging Moment

Figure 6.3 - Bridge Cross Section taken from MIDAS Civil showing tendon input locations 6.3 Boundary Conditions: Fixed supports under piers and vertical soil springs below the abutments

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Figure 6.4 - Skeletal Model Showing Boundary Conditions 6.4 Construction Stages: It is assumed that the slab will be poured in a single stage on the main precast beams. In a single analysis model, the following construction stages have been defined. Stage 1: All W11 precast beams are simply supported on the abutments and crossheads with prestress and beam self weight only considered. Stage 2: The in-situ wet concrete load is added to the precast beams with the beams again being simply supported. Stage 3: The composite structure is completed in both spans and the beams are now continuous. Grillage model is activated with all transverse elements, edge beams etc. Surfacing and parapet loads are applied when the slab attains the 28 days strength. Earth Pressure load is also applied at this stage. Stage 4: Final stage, considering the bridge at the end of its design life

Figure 6.5 - MIDAS Civil Screenshot Showing Construction Stages Input

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accounting for all long term time dependant effects, including creep & shrinkage. Time dependent analysis for concrete is done within the construction stages for both slab and beam. Creep and shrinkage behaviour is inputted as per Eurocode. Compressive strength gain of concrete is also applied as per Eurocode to account for short term and long term elasticities. To keep the analysis simple, cracking in the deck is not assumed at the intermediate support. Live Load Analysis: Highway Live Load Analysis was carried out based on MIDAS influence line analysis. 2nr Notional lanes and 2nr footway lanes were defined. The following live load combinations were analysed as per EN 1992-1-1 and its national annex. 1.       Gr1a: LM1 characteristic + 0.6 x Footway Characteristic 2.       Gr5: LM1 frequent (Psi=0.75) + LM3 (SV196) straddling between 2nr notional lanes


7. Section Properties The W11 - beam has 40mm wide by 50mm deep recesses at the top of the webs, this is in order to facilitate the placing and positioning of permanent shutter on site. This shutter is fibre reinforced concrete, FRC, and is manufactured by Banagher Precast Concrete. For this example FRC 50/20 will be used which has a 20mm bottom flange (ignored in calculating the deck slab thickness) and 30mm upward projecting ribs which participate structurally with the deck and is included in the composite section properties. The overall height of the composite section is 1700mm. The composite section properties are calculated by assuming the section is made up

from the W11 beam and a 3m rectangular slab. The code permits stiffnesses to be represented on the gross concrete section ignoring the reinforcement or strand. This is the most straightforward way since the reinforcement or prestressing strand has not yet been calculated. Composite section properties are used in the calculation of stresses resulting from loads applied after the structure is made integral (e.g. traffic loads, surfacing, parapets, string courses, finishes etc.)

Section Properties - Precast Prestressed Beam Section

Depth (mm)

Area (mm2)

Yc (mm)

ZT (mm3x106)

ZB (mm3x106)

Ixx (mm4x109)

Self Weight (kN/m)

Overall W (mm)

W11

1500

846920

631.30

248.01

341.33

215.46

21.17

1900.40

Table 7.1 - W11 Beam Section Properties

Maximum height above beam 200 mm Rebate depth less formwork 30 mm Total depth of slab 230 mm Beam centres 3000 mm The depth of carriagway surfacing has been taken as: 120 mm Modular ratio = Ecm(slab)/Ecm(beam) 1.00 -

Composite Section Properties Section

D (mm)

A (mm2)

Yc (mm)

Zinterface (mm3x106)

ZB beam (mm3x106)

ZT slab (mm3x106)

ZB slab (mm3x106)

Ixx (mm4x109)

Comp.

1700

1521500

1055.15

1262.25

532.16

870.76

1262.25

561.51

Table 7.2 - W11 Beam and Slab Composite Section Properties

6mm GFRP rod 98

50

148 20

22

0 R2

8

chamfer 10x10

50

8

56

20

15

138

148

148

148

138

15

750

Figure 7.1 - Section through Banagher Precast Concrete FRC 50/20 shutter

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SPAN PERMANENT SHUTTER

SPAN PERMANENT SHUTTER

SPAN PERMANENT SHUTTER

BRIDGE BEAM

Figure 7.2 - Section showing Banagher Precast Concrete FRC shutter in place prior to in-situ deck slab pour

8. Calculation of Loads Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges is the main document used to calculate loadings on bridges in Ireland and the UK. This document is to be used along with its national annex.

Transient Loads:

Definitions:

Notional Lanes:

Dead Loads:

The carriageway must be divided into notional lanes as specified in BS EN 1991-2, Table 4.1. For widths of 6.0 m and above, the carriageway is divided into an integer number of 3.0 m wide lanes. Any excess width is known as the ‘remaining area’. This example has a 6m wide carraigeway split into 2nr notional lanes of 3.0m width.

The weight of beam, deck slab and permanent shutter Superimposed Dead Loads: The weight of the road surfacings & parapets. The variablility of surfacing see EN 1991-1-1 Clause 5.2.3 (3) is ignored for simplicity. Live Loads:

Temperture, shrinkage and creep

All of this information is put into a computer programme, in this case MIDAS Civil, and analysed for the worst load effect by positioning the vehicles in the most onerous position on the bridge deck using influence lines/surfaces. This design example has no remaining area.

Loads due to vehicular and pedestrian traffic, LM1 & LM3 SV196

Dead Load: W11 Beam alone: Area = 846920 mm2 (value taken from “BPC Bridge Beam Manual”) Weight = 21.173 kN/m (value taken from “BPC Bridge Beam Manual”) Composite section: Area = 1521500 mm2 Weight = 38.04 kN/m Superimposed dead load: This loading is applied to the composite beam and slab structure

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Carraigeway: Asphalt surfacing - for simplicity assume maximum thickness of 120mm over whole carriageway. This includes allowance for waterproofing. SDL carr = 0.12m x 23kN/m3 = 2.76 kN/m2 To get this value in kN/m length of beam it is multiplied by beam centres, in this case 3m. = 2.76kN/m2 x 3.0m = 8.28 kN/m per beam SDL foot = 0.24m x 25kN/m3 = 6 kN/m2 (with 240mm being the average thickness of the footway)

Notional Lanes Carraigeway width, w =

6 m

Width of notional lanes, w1 =

3 m

Number of notional lanes, n1 =

2 nr

Width of remaining area, wr =

0 m

Table 8.1 - Notional lane results as per EN1991-2:2003 4.2.4 Load Model 1 (LM1) - Clause 4.3.2 + NA.2.12 A double-axle load called the Tandem System is applied in each traffic lane in conjunction with a uniformly distributed load called the UDL System.

Only one tandem system is applied to each lane, symmetrically around the centreline of the lane but no more than 0.5m from the TS in the opposite lane and in the position that causes the most severe effect on the element being considered.

LM1 consists of two parts: 1. A double-axle loading, referred to as the tandem system, or TS. Each axle has a weight of αQQk, where αQ is a nationally determined adjustment factor. 2. A uniformly distributed load (UDL) having a weight per square metre of αqqk, where αq is a nationally determined adjustment factor.

The tandem systems and UDL’s should only be applied in the unfavourable parts of the influence surface, both longitudinally and transversely. The nationally determined adjustment factors for the UDL have been set by the UK NA so that a UDL of 5.5 kN/m2 is applied to all lanes and the remaining area, irrespective of the number of nominal lanes, simplifying the input of loading into the analysis model.

Load Model 1 Tandem system Qik (kN)

αQi

UDL system αQiQik (kN)

qik (or qrk) (kN/m2)

αqi (or αqr)

αqi qik

(kN/m2)

Lane 1

300

1

300

9

0.61

5.5

Lane 2

200

1

200

2.5

2.2

5.5

Table 8.2 - Load Model 1 results as per EN1991-2:2003 4.3.2 & Table 4.2

Load Model 2 (LM2) - Clause 4.3.3 + NA.2.15 A single-axle load is applied anywhere on the carriageway. The UK use a 400kN axle load which includes a factor to allow for dynamic amplification effects. This load model is more predominant on short spans, < 7.0m and the transverse design of the deck slab.

The wheel contact shapes for LM1 and LM2 are 400mm square in the Republic of Ireland and the distribution through surfacing is 30° and through the concrete slab is 45°. The UK national annex has a different wheel size for LM1,2 & 3 and can be found in its NA.

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Load Model 2 Axle Weight, Qak =

400 kN

Applied axle weight, βQQak

400 kN

Adjustment factor, βQ = αQ1

1

Table 8.3 - Load Model 2 results as per EN1991-2:2003 4.3.3

Load Model 3 (LM3) - Clause 4.3.4 + NA.2.16

Load Model 4 (LM4) - Clause 4.3.5

In this example SV196 model vehicles will be used. The SV196 loading is combined with a reduced value of LM1, known as the ‘frequent’ value. Figure 8.1 below shows the SV196 vehicle from the UK national annex to EN1991-2:2003, the only difference between the UK and the ROI is that instead of the last 3nr loads being 180kN, 180kN & 100kN in the ROI they are all 165kN.

A uniformly distributed load of 5kN/m2 used to represent crowd loading and may be applied to both road bridges and footway/ cycleway bridges.

Load Model 3 Basic axle load (kN)

Dynamic amplification factor

Design axle weight (kN)

100

1.2

120

130

1.16

150.8

165

1.12

184.8

180

1.1

198

225

1.07

240.75

 SV196

Table 8.4 - Dynamic amplification factors for SV and SOV vehicles (Table NA.2 EN 1991-2:2003)

165 kN 1.2m

165 kN

165 kN

165 kN 1.2m

1.2m

165 kN

165 kN

1.2m

1.2m

0.35m 3.0m

3.0m

165 kN

165 kN 1.2m

1.2m

180 kN

180 kN 4.0m

1.6m

100 kN 4.4m

Direction of Travel

0.35m Overall Vehicle Width

165 kN

Critical of 1.2m or 5.0m or 9.0m

Fig 8.1 - SV 196 vehicle (Fig NA.1© SV 196 vehicle - EN1991-2:2003)

Groups of Traffic Loads (UK National Annex Table NA.3) Load Models 1 to 4 may be combined to form ‘Groups’ of traffic loads. The Groups are referenced gr1a, gr1b, gr2, gr3, gr4, gr5 and gr6 and the load models used in each group are listed in Table N.A.3 of the UK NA (this overwrites Eurocode EN1991-2 Table 4.4a). For this example gr1a & gr5 have been looked at.

Group gr5

The ‘Frequent’ value of Load Model 1 is combined with the ‘Characteristic’ value of Load Model 3. The Frequent value of LM1 is obtained by multiplying axle loads and UDL by 0.75 (Ψ1 = 0.75 from Table NA.A2.1 in NA to BS EN 1990:2002+A1:2005). Tandem Systems can be interchanged if a worse load effect is achieved. Loading from LM1 is omitted from the lane, or lanes, occupied by LM3 for a distance Group gr1a within 5 metres of the front and rear axles. This loading arrangement is shown in figure NA.5 of the UK NA to EN 1991-2. LM 1 is combined with footway loading. The footway loading is reduced to 3kN/m2 (0.6 x 5kN/m2). The vehicle load optimiser in MIDAS Civil places the LM1 and LM3 vehicles on the bridge and gives results for the worst possible position of same.

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Load type

Carriageway

Footways and cycle tracks

Vertical forces

Horizontal forces

Vertical forces only

Reference

4.3.2

4.3.3

Annex A

4.3.5

4.4.1

4.4.2

5.3.2.1 Equation (5.1)

Load system

LM1 (TS and UDL)

LM2 (Single axle)

LM3 (Special vehicles)

LM4 (Crowd loading)

Braking and accelaration forces

Centrifugal & transverse forces

Uniformly distributed load

Groups of loads

gr1a

Characteristic

gr1b gr2

0.6 times Characteristic Characteristic

Frequent(4)

Characteristic

Characteristic Characteristic

gr3

(1)

gr4 gr5

Characteristic Frequent(4)

Characteristic

Characteristic

gr6

Characteristic

Characteristic

Characteristic

Dominant component action (the group is sometimes designated by this component for convenience). (1) This group is irrelevant if gr4 is considered (2) Characteristic value obtained from 5.3.2.1. (3) This is a reduced value accompanying the characteristic value of LM1 and should not be factored by Ѱ1. However when gr1 is combined with leading non-traffic actions this value should be facted by Ѱ0 (4) The Ѱ1 factors should be taken from the UK National Annex to BS EN 1990

Table 8.5 - Assessment of groups of traffic loads (Table NA.3 EN 1991-2:2003)

9. Combinations of actions There are three combinations of actions that must be considered at the serviceability limit state(SLS): 1. the characteristic combination, which can be considered the most severe loading to which the structure should be subjected to 2. the frequent combination, which is the most severe load case to which the structure should be subjected to on a regular basis 3. the quasi-permanent loadcase, or the loading to which the structure is subjected to most of the time.

These are defined as follows: EQU - Loss of static equilibrium of the structure or any part of it when considered as a rigid body. STR - Internal failure or excessive deformation of the structure or structural member. GEO - Failure or excessive deformation of the ground where the strengths of soil are significant.

FAT - Fatigue failure of the structure or structural members. Four ultimate limit states (ULS) are defined in BS EN 1990, namely EQU, STR, GEO and FAT. Serviceability limit state (SLS) Action

Groups of loads

Load component

Ψ0

Ψ1

Ψ2

0.6

0.6

0.5

Tandem system

0.75

0.75

0

UDL

0.75

0.75

0

Pedestrian loads

0.4

0.4

0

SV vehicle

0

0

0

Thermal actions Traffic actions

gr1a

gr5

Table 9.1 - Values taken from the recommended values of Ψ for road bridges (Table NA.A2.1 EN 1990:2002)

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Ultimate limit state (ULS) Group of loads

γsup

γinf

γ0

Dead load

1.35

0.95

Superimposed DL

1.2

0.95

0

0

Temp. diff.

1.5

0

0

Prestress

1.1

0.9

0

Traffic groups

1.35

0

NA

Table 9.2 - Values taken from the design values for γ for road bridges (Table NA.A2.4(B) EN 1990:2002)

10. Differential Shrinkage When the insitu slab is cast onto the precast W11 beams some of the shrinkage of the beams has already occurred. Hence differential shrinkage occurs between the precast and the insitu concrete and this results in the development of internal stresses meaning the deck slab itself will shrink by a relatively greater amount. This relative shrinkage will compress the top of the beam causing axial force and sagging moment in it while generating tension in the deck slab. It is reasonable to assume that half of the beams shrinkage has occurred at the time of casting the top slab. The effects of differential

A sample calculation for working out the differential shrinkage stress in the top of the beam is shown across, using this logic the shrinkage stress in the bottom of the beam, the bottom of the slab and the top of the slab can all be worked out. All of the answers are shown in table 10.1 below and fig 10.1 shows them plotted in graph format.

Beam Btm (MPa)

Beam Top (MPa)

Slab Btm (MPa)

Slab Top (MPa)

Value

-0.35

1.1

-0.41

-0.22

H (mm)

0

1500

1500

1700

Table 10.1 - Long Term Differential Shrinkage Stresses

Figure 10.1 - Differential shrinkage diagram

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shrinkage will be reduced by creep. Allowance is made for this in the across sample calculation by using a reduction coefficent with a value of 0.43. Note differential shrinkage is only considered in the SLS.

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Example calculation for the stress in the top of the beam: Beam Top Differential Shrinkage Stress = Force/Area of the composite section + Moment/Ztp = 1.10 MPa where; Restraining Moment = Force x eccentricity = 5.44E+08 Nmm Restraining Force = 0.43 x differential shrinkage strain x Ecm x Total Area of insitu topping /Modular ratio = 1.51 N / mm2 Z top precast beam = 1.3E+09 mm3 Area of the composite section = 1.52E+06 mm2 Eccentricity = (area of top slab x lever arm / Total Area of insitu topping)-Height to the Centre of Gravity of the Composite Section = 532.14 mm

11. Temperature Effects 11.1. Uniform Temperature Component, ΔTu: The first stage in determining the uniform temperature component is to determine the minimum and maximum shade air temperatures for the location where the bridge is to be built. For the UK these can be determined from the maps of isotherms given in Figures NA.1 and NA.2 of the National Annex to BS EN 1991-1-5, with an adjustment 1. the uniform temperature component which causes expansion or being made for the effect of altitude. contraction of the deck The minimum and maximum shade temperatures are then converted into minimum and maximum uniform bridge temperatures, Te,min and 2. the temperature difference component which leads to curvature of the bridge and Te,max. For this example these values were taken as -9 ° C and 29 ° C 3. the non-linear temperature component, which causes local stresses within the structure. The maximum contraction that the bridge will experience will depend on the difference between the minimum uniform bridge temperature and the uniform bridge temperature at the time when the bridge is first made continuous with its abutments, T0. As the temperature at the time of construction cannot be known in advance the example takes T0 to be 15°C when considering contraction and 5°C for expansion as per the Irish National Annex, NA2.21. Daily and seasonal fluctuations in shade air temperature, solar radiation, etc. cause changes in the temperature of a bridge superstructure, thereby causing movement of that structure. Depending on the restraint conditions, this movement can lead to stresses in the structure. This effect can be divided into three components;

Figure 11.1 - Diagrammatic representation of constituent components of a temperature profile (Fig 4.1 - EN1991-1-5:2003)

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BS EN 1991-1-5 Cl.6.1.1(1) - Bridge is type 3 concrete deck Surfacing = 120 mm Altitude = 13.34 m Fig NA.1 - Minimum shade air temperature = -9 ° C Fig NA.2 - Maximum shade air temperature = 29 ° C Annex A1 note 2 - The minimum shade air temperature may be reduced by 0.5 ° C per 100m above sea level Annex A1 note 2 - The maximum shade air temperature may be reduced by 1.0 ° C per 100m above sea level Fig 6.1 - Uniform minimum bridge temperature, Te,min = Tmin + 8 = -1.07 ° C Fig 6.1 - Uniform maximum bridge temperature, Te,max = Tmax + 2 = 30.87 ° C NA2.21 - T0,con = 15 ° C NA 2.21 - T0,exp = 5 ° C Maximum contraction range ΔTN,con = T0 - Te,min = 16.07 ° C Maximum expansion range ΔTN,exp = Te,max - T0 = 25.87 ° C Table C1 - Coefficent of linear expansion for concrete, α = 0.00001/ ° C Length of beam = 30.5 m No. of continuous spans = 2 nr Maximum contraction = α.ΔTN,con.L = 9.80 mm Maximum expansion = α.ΔTN,exp.L = 15.78 mm

11.2. Temperature Difference Component, ΔTM: (- integral bridges only) Corrections to fig 11.2 are given in clause As well as uniform temperature changes, which cause uniform NA.2.9 of BS EN 1991-1-5 changes in length, variations in temperature through the thickness of the deck must be considered. Both heating and cooling temperature differences are considered. The values for temperature difference distributions given in Figure 6.2c of BS EN 1991-1-5 assume a depth of surfacing of 100 mm.

Figure 11.2 - Temperature differences for bridge decks - Type 3: Concrete Decks

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Heating

Cooling

Temperature (°C)

Depth (mm)

Temperature (°C)

Depth (mm)

13.5

0

-8.4

0

3

150

0

400

-0.5

250

0

1480

0

450

2.5

1700

0

1250

-1

1450

-6.5

1700

Table 11.1 The resulting restrained forces can be calculated using MIDAS: Moment (kNm)

Axial Force (kN)

Heating

173

428

Cooling

-21

-372

Table 11.2

Table 11.3 The uniform temperature components and linearly varying temperature difference components may be calculated as follows: ΔTu = (N / Ac) / (Eα)

ΔTM = (Mh / l) / (Eα)

ΔTu (°C)

ΔTM (°C)

Heating

1.75

1.41

Cooling

-1.52

-0.17

Table 11.4 11.3. Non linear temperature component, ΔTE: The non-linear temperature difference component is self-equilibrating, so it does not need to be included in the global model. However, it sets up local stresses that need to be assessed for SLS. The non-linear

temperatures are the difference between the original temperature distribution and the uniform and linearly varying temperatures.

Heating

Cooling

ΔTE (°C)

σE (MPa)

Depth (mm)

ΔTE (°C)

σE (MPa)

Depth (mm)

11.05

4.12

0

-6.8

-2.53

0

0.55

0.21

150

1.1

0.41

250

-2.45

-0.91

400

1.6

0.6

450

-1.04

-0.39

1480

1.43

0.53

1250

1.46

0.54

1700

0.43

0.16

1450

-5.07

-1.89

1700

Table 11.5

Table 11.6

Design value of thermal loading: For the characteristic combination of actions; Ψ0,1 = 0.60 For the frequent and quasi-permanent combination of actions; Ψ2 = 0.50

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Figure 11.3 - Heating and cooling graphs for temerature stresses.

12. Prestress Design The aim of prestress design is to limit tensile stresses and hence flexural cracking in the concrete under normal working conditions. The design is therefore based on the requirements of the serviceability limit state but the ultimate limit state for bending and shear must also be considered. Since concrete is strong in compression the material that is used in the beam will be most efficently used if it can be kept in compression, this compression force is provided by means of high strength steel wired strand anchored against the concrete at either end of the stressing line. The concrete is cast around the strands in steel moulds and when the required strength has been reached in the concrete (transfer strength) the strand is released and the force is transferred into the concrete by bond.

The across W11 beam showing strand input is the preferred choice however several iterations were performed: 1. A 60nr strand layout: Pro’s - Lowest amount of strand required for beam to work Con’s – In order to achieve decompression at the diaphragm face due to the hogging moment stress 18nr strands (30% of total) would need to be debonded which exceeds the ACI 25% recommendation that Banagher Precast Concrete uses as a guide. 2. A 62nr strand layout – (adding 2nr at 1230mm above soffit):

The prestressing tendon layout, and the resulting calculation of prestressing losses were initially based on an assumed layout guestimated from previous experience with similar beams. The tendon positions were selected from the standard positions available for W beams.

Pro’s – Debonding can be reduced to 12nr strands(20% of total), 6nr less than the first design. Con’s - 2nr extra strands to be used compared to the first design

Requirements for prestressing and the calculation of the effective prestressing force are given in BS EN 1992-1-1, clause 5.10. The applied prestress in the strands during tensioning should not exceed 80% of the characteristic tensile strength or 90% of the characteristic 0.1% proof stress, whichever is the lesser. This allows an applied prestress of up to 79% of the characteristic however we recommend a figure of 75% in accordance with long standing practice. This gives a limit of 1395 MPa for this example. Note that in pretensioning it is possible to eliminate “draw - in” losses by extending the strand 5-10mm beyond the theoretical strand extension, this means that in practice the prestress applied will be slightly more than the 75% depending on the length of the production line.

Pro’s - Debonding can be reduced to 8nr strands(12.5% of total), 10nr less than the first design. Con’s - 4nr extra strand to be used compared to the first design

The stress in the strands immediately after transfer is also limited – this time to the lesser of 75% of the characteristic tensile strength or 85%

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of the characteristic 0.1% proof stress.

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3. A 64nr strand layout – (adding 2nr at 980mm above soffit):

4. A 66nr strand layout– (adding 2nr at 1430mm above soffit): Pro’s – No debonding required Con’s - 6nr extra strand to be used compared to the first design and an increase in transfer strength from C28/35 to C35/45 plus an increase in 28day strength from C50/60 to C57/70 due to beam bottom compression from the pier hogging moment. From an examination of the above designs a 62nr strand arrangement was chosen as the preferred as it was both an extremely efficient design and suited factory production best as debonding strands is


relatively inexpensive and fairly straight forward in a factory controlled environment once the quantity is not excessive. This is why thought and several iterations should be carried out by the bridge designer plus input from the precaster to achieve the best fit. The first selection that a computer programme comes up with is not always the best one.

Prestressing Strand Parameters Diameter (mm)

Cross sectional area (mm2)

Ultimate strength (MPa)

%Ultimate strength applied (%)

Cracteristic value of max Force (kN)

Initial force applied (kN)

Initial prestress applied (MPa)

Elastic modulus (GPa)

Relaxation of strand (%)

15.7

150

1860

75

279

209.25

1395

200

1.5

Table 12.1 - Prestressing strand parameters.

194

71.5

150

200

50

71.5

4 No. 15.7Ø & 2B12-05

328

92

B12-07 4 No. 15.7Ø 2 No. 15.7Ø 07

07

328

07

07

100x100 transverse holes 2 No. 15.7Ø

1430

2 No. 15.7Ø

250

1500

B12-02

1115 1230

1172

B12-01

120x120 transverse holes 730

265

790

880

2 No. 15.7Ø B10-03-300

2 No. 15.7Ø

275

40

70

42.5 42.5 52.5 52.5 52.5 50 50 50 50 50 50 50 50

75

Length debonded bottom row

CL

50 50 50 50 50 50 50 50 52.5 52.5 52.5 42.5 42.5 70

480

60 110

12 No. 15.7Ø 28 No. 15.7Ø

340

4 No. 15.7Ø

B10-04

210

540

CL

Denotes fully bonded strand Denotes partially debonded strand

26mm O.D. Weephole

6000

6000

4000

4000

3000

3000

2000

2000

1500

1500

1000

1000

12 no. debonded both ends

Debonding both ends Debonding symmetrical about centreline

Figure 12.1 - Section through W11 beam showing strand layout.

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Strand Layout Row

No. strand used & H/O soffit

10

4 @ 1430 mm

9

4 @ 1230 mm

8

2 @ 1115 mm

7

2 @ 880 mm

6

2 @ 730 mm

5

2 @ 480 mm

4

2 @ 340 mm

3

4 @ 210 mm

2

12 @ 110 mm

1

28 @ 60 mm

Initial Prestressing Force = 12973500 N Eccentricity = 283.4 mm Moment = 3677000000 Nmm

62 @ 348 mm

Table 12.2 - Strand Layout.

13. Prestress Losses 13.1. Immediate Losses The prestressing force does not actually achieve its initial value. The prestress transferred to the beams after they are cast is less than the force initially jacked into the strands.

13.1.1. Relaxation Loss The loss due to the relaxation of the tendons during the period which elapses between the tensioning of the tendon and the prestressing of the concrete beam.

Prestress losses which occur at or before transfer are due to: 1. Relaxation of the strands 2. Elastic shortening of the beam under the prestressing force

Class 2: Δσpr / σpi = 0.66 x ρ1000 x e9.1μ(t/1000)0.75(1-μ) = 8.79 MPa - (Exp 3.29) = 0.63 % - (Cl 3.3.2(5)) where; Ratio of initial prestress to tensile strength, μ = 0.75 time, t at transfer = 140 hrs Relaxation of strand ( Class 2) - Mill Certs, ρ1000 = 1.5 %

13.1.2. Elastic Shortening Loss The loss due to the elastic deformation of the concrete beam as a result of the action of the pre-tensioned tendons when they are released from the anchorages. The elastic loss at transfer is calculated at the centroid of the tendons and is due to the compression arising from the prestressing force, after the initial relaxation loss prior to transfer and the self weight of the beam. The latter is included because the beam cambers during transfer and hence has to carry its selfweight as a simply

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supported beam. The methodology used in the “Designers Guide to EN 1992-2” by Hendy & Smith has been adopted with particular reference to equation D5.10-4. Using the denominator from this equation the losses due to elastic deformation at a given time can be modified.


Using the denominator from D5.10-4 we can work out the elastic deformation modification factor (MF) as follows: MF = 1 + Ep/Ecm(t) x Ap/Ac x (1+Ac/Ic x z2cp) = 1.0894 where; Modulus of elasticity of the prestressing steel, Ep = 200 GPa Short term modulus of elasticity for the concrete, Ecm(t) = 32.31 GPa Cross sectional area of the tendons, Ap = 150 mm2 Area of concrete section, Ac = 846920 mm2 Second moment of area of the concrete section, Ic = 215.46 x109 mm4 Eccentricity of the tendons, zcp = 283.4 mm Prestress at Centroid of section, σc = 15.59 MPa as per table 13.2 Loss of prestress = σc(Ep /Ecm(t))/MF = 88.71 MPa = 6.36 %

13.2. Time Dependent Losses

Further losses of prestress occur with the passage of time. Long term prestressing losses are due to: Further relaxation of the strands Creep of the concrete due to the permanent compressive stresses Shrinkage of the concrete as it cures These losses need to be evaluated to find the final (long term)

prestressing force. The final prestress is usually about 25% less than the initial prestress and it is the final value that must be used in the long term SLS calculations for the beams. 13.2.1. Relaxation Loss The loss due to the relaxation of the tendons checked at different times. The code sets out checks at transfer, construction, open to traffic and long term. All of these are checked across and tabulated.

Relaxation Loss Stage (days)

Loss (MPa)

Loss %

Transfer

3

7.76

0.56%

Construction

30

11.95

0.86%

Open to traffic

100

14.98

1.07%

Long term

20833

40.77

2.92%

Table 13.1 - Relaxation Loss Summary.

13.2.2 Creep Creep causes a number of effects that have to be considered at various stages in the design of prestressed concrete structures. Perhaps the most significant is the loss of prestress that results from creep and the effect that this will have on the SLS criteria.

BS EN 1992-1-1, clause 3.1.4 and Annex B describe how to calculate creep and is outlined in the calculations below.

Parameters required: Relative Humidity of the ambient environment in %, RH = 80.00 % α1 = 0.70 α2 = 0.90 α3 = 0.78 h0 = 384.44 mm Mean compressive strength, fcm = 58.00 MPa Curing period, Δti = 72.00 hrs T(Δti) = 20.00 °C Temperature adjusted age of concrete at loading in days, t0,T = 3 days For cement class R, α = 1

(Exp (Exp (Exp (Exp

B.8c) B.8c) B.8c) B.6)

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tT = e-(4000/[273+T(Δti)]-13.65).Δti = 3.00 days β(fcm) = 16.8/fcm1/2 = 2.21 Age of concrete at loading in days, t0 = t0,T(9/2+t0,T1/2 + 1)α ≥ 0.5 = 7.71 days Factor to allow for the effect of concrete age at loading, β(t0) = 1/(0.1+t00.20) = 0.62 φ0 = φRH x β(fcm) x β(t0) = 1.48 βc(t,t0) = [(t-t0)/(βH+t-t0)]0.3 = 0.99 -

(Exp (Exp (Exp (Exp (Exp (Exp

B.10) B.4) B.9) B.5) B.2) B.7)

Creep Coefficent: φ(t,t0) = φ0 x βc(t,t0) = 1.46 - (Exp B.1) Creep Strain: εcc(∞,t0) = φ(∞,t0).(σc/Ec) = 5.25 x 10-1 - (Exp 3.6) Creep Loss: Long Term Creep loss = Ep. εcc = 104.9 MPa = 7.52 %

Compressive Stress Stage (days)

P (N)

Zc (mm3)

σc (MPa)

Transfer

3

12051778

7.60E+08

15.58946219

Constr.

30

11880191

7.60E+08

15.09189229

Open to traffic

100

11749674

7.60E+08

14.88913387

long term

25550

11183839

7.60E+08

14.01010687

Table 13.2 - Compressive Stress Results Note: σc = stress at centroid of the strands due to prestress, self weight and all other quasi permanent actions at the time under consideration. σc = P/A ± Pe/Zc ± MQP/Zc where; P = Prestress force less losses at time being considered. A = Cross sectional area of beam. e = eccentricity Section modulus of centroid of strands about nuetral axis of section, Zc = Ixx / eccentricity

At transfer - 3 days

At construction - 30 days

At opening - 100 days

Long term - 25550 days

MQP = Msw

MQP = MDL+MSDL+MCONST.

MQP = MDL+MSDL

MQP = MDL+MSDL

NA MQP/Zc CENTROIDS OF STRANDS

P

Pe/Zc

Figure 13.1 - Calculating stress at the centroid of the strands (σc) due to prestress, self-weight and all other quasi permanent actions at the time under consideration

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Summary Table - Calculation of Creep Coefficent Stage (days)

fcm (MPa)

α1

α2

α3

ΦRH

Transfer

3

43

0.98

0.99

0.99

1.26

Constr.

30

58

0.70

0.90

0.78

1.08

Open to traffic

100

58

0.70

0.90

0.78

1.08

long term

25550

58

0.70

0.90

0.78

1.08

Summary Table - Calculation of Creep Coefficent Continued…. Stage (days)

β(fcm)

βH

βc(t,t0)

Φ0

Φ(t,t0)

Transfer

3

2.80

1099.73

0.00

2.20

0.00

Constr.

30

2.21

1047.43

0.33

1.48

0.49

Open to traffic

100

2.21

1047.43

0.48

1.48

0.71

long term

25550

2.21

1047.43

0.99

1.48

1.46

Table 13.3 - Creep Calculation parameters Creep Loss Summary Stage (days)

Ec (GPa)

σc (MPa)

Φ(t,t0)

εcc

Creep Loss (MPa)

Creep Loss %

Transfer

3

33.92

15.59

0.00

0.00E+00

0.00

0.00%

Construction

30

39.14

15.09

0.49

1.89E-01

37.88

2.72%

Open to traffic

100

39.14

14.89

0.71

2.69E-01

53.82

3.86%

long term

25500

39.14

14.01

1.47

5.25E-01

104.90

7.52%

Table 13.4 - Creep Loss Summary Table Long term σc of 14.01 MPa exceeds 0.45fck(t0), 12.6MPa, therefore non linear creep should have been taken into account as per (Exp 3.7) from Cl. 3.1.4 which will increase the creep coefficent and therefore the creep losses by a factor of exp(1.5(Kσ-0.45)) = 1.0785 13.2.3 Shrinkage In the calculation of the shrinkage of a section, a notable difference from previous practice is that the total shrinkage strain is decomposed into a drying shrinkage component and an autogenous shrinkage component.

Autogenous shrinkage is particularly important for high-strength concrete and high-performance concrete. Since autogenous shrinkage is independent of the size of the concrete member and the relative humidity of the ambient environment, the shrinkage in bulk concrete members exposed to humid environments is dominated by autogenous shrinkage.

Drying shrinkage strain: εcd,0 = 0.85[(220+110αds1).exp(-αds2.fcm/fcmo)].10-6.βRH = 299 x 10-6 - (Exp B.11) where; αds1 (based on Class R cement) = 6 αds2 (based on Class R cement) = 0.11 fcm = 58 MPa fcmo = 10 MPa βRH = 1.55[1-(RH/RH0)3] = 0.7564 - (Exp B.12) where; RH is the ambient relative humidity (%) = 80% for outdoor exposure based on Irish and UK meteorological data. RH0 is 100% = 100 % Autogeneous shrinkage strain: εca(∞) = 2.5(fck-10)10-6 = 100.0 x 10-6 - (Exp 3.12) where; fck = 50 MPa

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Drying shrinkage strain: εcd(t) = βds(t,ts).kh.εcd,0 = 215 x 10-6 - (Exp 3.9) where; Kh is a coefficent depending on the notional size, h0 according to table 3.3 = 0.7289 h0 = 384.44 mm - (Exp B.6) Note: The general formulae for working out u for W Beams is: 1515+2.02(Depth-70) in mm. Note: Internal void not included as it is not exposed to the atmosphere. ßcd(t,ts) = 0.99 Total shrinkage strain: εcs = εcd + εca = 315 x 10-6 Shrinkage loss: Long Term Shrinkage loss = Ep. εcs = 63.07 MPa = 4.52 %

13.3 Summary of Long Term Prestressing Losses: The design stresses in the precast W beam are the stresses due to the final prestress after all losses have been taken into account. The table below tabulates these losses. The design stresses when worked out are then compared to the allowable tensile and compressive stresses from the Eurocode to see if they meet the limits set out. Summary of Long Term Prestressing Losses Relaxation

Shrinkage

Elastic Shortening

Creep

Total

Loss MPa

Loss %

Loss MPa

Loss %

Loss MPa

Loss %

Loss MPa

Loss %

Loss MPa

Loss %

40.77

2.92

63.07

4.52

88.71

6.36

104.90

7.52

297.45

21.32

Table 13.5 - Long Term Prestressing Losses Note: (Exp 5.46) from Cl.5.10.6 may be applied to reduce the time dependent losses, (relaxation, shrinkage and creep), further. 13.4 Effective Prestressing Force: The effective prestressing force is shown below before any losses, at transfer, during construction, opening to traffic and long term. The different losses are subtracted each time to give a new force. The long term force corresponds to the long term loss as calculated earlier. These forces are used later in the calculations. Stage (days)

% of initial prestressing force after losses

Effective prestressing force (N)

Before Losses

0

100.00

12973500

Transfer

3

92.90

12051778

Construction

30

88.86

11527895

Opening

180

86.71

11249141

Long term

25550

78.68

10207201

Table 13.6 - Effective Prestressing Force Summary Table. 13.5 Transmission Length The transmission length of a tendon is the length over which the prestressing force is fully transmitted to the concrete. It is assumed that the transfer of stress from the tendon to the concrete is via a constant bond stress, fbpt such that there is a linear transfer of prestress from the tendon to the concrete beam.

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The calculations across are taken from clause 8.10.2.2 and in particular Exp(8.15),(8.16), (8.17) and (8.18) of BS EN 1992-1-1. For design purposes the transmission length is taken as being either 20% higher or lower than the calculated value, whichever is more unfavourable for the given situation.


fbpt = ηp1 η1 fctd(t) = 4.13 MPa - (Exp 8.15) where; ηp1 = 3.2 for 3 and 7 wire strands η1 = 1 for depths greater than 250mm where good bond conditions are likely fctd(t) = 1.29 MPa - (Exp 3.16) The basic value of the transmission length is given by: lpt = α1 α2 ø σpm0 / fbpt = 935.77 mm - (Exp 8.16) where: α1 = 1 α2 = 0.19 for 3 and 7 wire strands ø = 15.70 mm σpm0 = 1,295.89 MPa The design transmission length is calculated as the more unfavourable of the below: lpt1 = 0.8lpt = 748.62 mm - (Exp 8.17) lpt2 = 1.2lpt = 1122.92 mm - (Exp 8.18)

14. Serviceability Limit State Checks (sls) 14.1 Decompression check at Transfer of Prestress (at end of transmission length) The end of the transmission length is the critical location for the decompression check at transfer as the moment due to self-weight, which is favourable to maintain compression in the top fibre, is at a minimum towards the beam ends. As you can see that while the tension limit of 0.75fctm transfer is met the compression limit of 0.7fck transfer is not, therefore debonding is required. These limits are interpolated from Cl.5.10.2.2(5) Beam Top (MPa)

Beam Btm (MPa)

P/A

14.23

14.23

Pe /Z

-13.77

10.01

0.92

-0.67

Σ

1.38

23.57

Limits

-2.07

19.60

Status

OK

Not OK

Msw /Z

The detailing of debonded prestressing strands should be carefully considered and recommendations from international codes should be sought. In general, consideration should be given to debonding no greater than 25% of the total number of strands in a beam. No more than 40% of these debonded strands (or 4No. whichever is the lesser) should then be stopped off at any one location. In addition the exterior strands in any horizontal row should ideally not be debonded and any strands anchoring a link (i.e. at a bend in a link) should never be debonded.

Table 14.1 Therefore debonding required at beams ends

Moment due to self weight at 748.62 mm from beam ends = 228.12 kNm P = 12051777.64 N

Precast concrete specialists

25


New Strand Layout Row

No. strand used & H/O soffit

10

4 @ 1430 mm

9

4 @ 1230 mm

8

2 @ 1115 mm

7

2 @ 880 mm

6

2 @ 730 mm

5

2 @ 480 mm

4

2 @ 340 mm

3

4 @ 210 mm

2

12 @ 110 mm

1

16 @ 60 mm

Initial Prestressing Force, Pi = 10462500 N Eccentricity = 214.3 mm Moment = 2242113750 Nmm

50 @ 417 mm

Table 14.2 14.2 Revised Stresses with Debonding 12nr strands at 60mm above the beam soffit have been chosen to be debonded with the check re calculated, the section passes with the revised strand layout.

prevents it transferring its force to the concrete along the debonded length. This allows the point of prestress transfer for debonded strand to be moved further into the span, to a point which tension in the top flange is no longer an issue.

Debonding involves passing a plastic sheath around the strand to prevent the strand bonding to the surrounding concrete and thereby

See Appendix B for images of this.

14.2.1. At End of Transmission Length

14.2.2. At End of Debonded Length

P = 9719175.516 N

Moment due to self weight at 6498.62mm from beam ends = 1597.54kNm P = 12051777.64 N Beam Top (MPa)

Beam Btm (MPa)

P/A

11.48

11.48

Pe /Z

-8.40

Msw /Z

Beam Top (MPa)

Beam Btm (MPa)

P/A

14.23

14.23

6.10

Pe /Z

-13.77

10.01

0.92

-0.67

Msw /Z

6.44

-4.68

Σ

4.00

16.91

Σ

6.90

19.56

Limits

-2.07

19.60

Limits

-2.07

19.60

Status

OK

OK

Status

OK

OK

Beam Top (MPa)

Beam Btm (MPa)

P/A

13.61

13.61

Pe /Z

-13.17

7.24

17.82

-12.95

Table 14.3

Table 14.4

14.3 Stress Check at Construction Stage

P = 11527894.8 N

The construction stage involves the beam alone supporting the weight of wet slab and any applicable construction loading in a simply supported condition. The limits here are 30MPa, (0.6fck)

MDL /Z

The prestressing force of 11527.8kN comes from the already worked out effective prestressing force after all losses have been accounted for after 30nr days.

Σ

18.26

7.90

Limits

30.00

30.00

Status

OK

OK

Table 14.5

26

Precast concrete specialists


14.4 Decompression and Stress Check at Service (midspan max) The stress check at service includes the live moment in addition to stresses resulting from differential shrinkage and temperature effects. The check is repeated for heating and cooling conditions (cooling is more critical for tensile stresses at the extreme bottom fibre).

P = 10207201 N

Heating Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MDL /Z

17.82

-12.95

0.00

0.00

17.82

-12.95

0.00

0.00

MSDL /Zi

0.41

-0.98

0.41

0.60

0.41

-0.98

0.41

0.60

MLive /Zi

2.15

-5.09

2.15

3.11

3.33

-7.89

3.33

4.82

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.10

0.27

0.10

2.06

0.12

0.33

0.12

2.47

Σ

21.97

1.42

2.25

5.55

23.17

-1.32

3.45

7.67

Limits

30.00

0.00

24.00

24.00

30.00

-4.06

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.6 Cooling Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MDL /Z

17.82

-12.95

0.00

0.00

17.82

-12.95

0.00

0.00

MSDL /Zi

0.41

-0.98

0.41

0.60

0.41

-0.98

0.41

0.60

MLive /Zi

2.15

-5.09

2.15

3.11

3.33

-7.89

3.33

4.82

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.21

-0.94

0.21

-1.27

0.25

-1.13

0.25

-1.52

Σ

22.08

0.21

2.36

2.22

23.30

-2.78

3.58

3.68

Limits

30.00

0.00

24.00

24.00

30.00

-4.06

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.7

Precast concrete specialists

27


14.5 Stress Check at Service (midspan min) This check is as per 14.4 only now minimum moments at midspan are being checked, which are taken from the MIDAS Civil model and shown below.

Minimum live characteristic moment at midspan = -623 kNm (MIDAS) Minimum live frequent moment at midspan = -423 kNm (MIDAS)

Heating Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MDL /Z

17.82

-12.95

0.00

0.00

17.82

-12.95

0.00

0.00

MSDL /Zi

0.41

-0.98

0.41

0.60

0.41

-0.98

0.41

0.60

MLive /Zi

-0.34

0.79

-0.34

-0.49

-0.49

1.17

-0.49

-0.72

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.10

0.27

0.10

2.06

0.12

0.33

0.12

2.47

Σ

19.48

7.30

-0.24

1.95

19.35

7.74

-0.37

2.13

Limits

30.00

0.00

24.00

24.00

30.00

-4.06

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.8 Cooling Frequent combination Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MDL /Z

17.82

-12.95

0.00

0.00

17.82

-12.95

0.00

0.00

MSDL /Zi

0.41

-0.98

0.41

0.60

0.41

-0.98

0.41

0.60

MLive /Zi

-0.34

0.79

-0.34

-0.49

-0.49

1.17

-0.49

-0.72

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.21

-0.94

0.21

-1.27

0.25

-1.13

0.25

-1.52

Σ

19.59

6.09

-0.13

-1.38

19.48

6.28

-0.24

-1.86

Limits

30.00

0.00

24.00

24.00

30.00

-4.06

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.9

28

Characteristic combination

Precast concrete specialists


14.6 Decompression and Stress Check at Service (End Of Diaphragm): The bridge was modelled with a 500mm diaphragm and the check for decompression and stress at service is shown below. The moments due to self weight and dead load are worked out using an equation along the length of the beam and the moments due to surfacing, GR1a and GR5 are taken from MIDAS Civil.

Distance from end of beam to end of diaphragm = 500 mm Most unfavourable transmission length = 748.62 mm This point lies within the transmission zone Number of strands debonded at this point = 12 e = 214.3 mm Allowing for a reduced prestress value within the transmission region: Peff = 5497859.1 N Characteristic and Frequent moments at the end of the diaphragm: Characteristic = -4468 kNm (From Midas) Frequent = -2560 kNm (From Midas) Moment due to superimposed dead load = -967.6 kNm (From Midas) Moment due to self weight of the beam = 153.63 kNm Moment due to dead load = 131.39 kNm

Heating Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

6.49

6.49

0.00

0.00

6.49

6.49

0.00

0.00

Pe /Z

-4.75

3.45

0.00

0.00

-4.75

3.45

0.00

0.00

MSW /Z

0.62

-0.45

0.00

0.00

0.62

-0.45

0.00

0.00

MDL /Z

0.53

-0.38

0.00

0.00

0.53

-0.38

0.00

0.00

MSDL /Zi

-0.77

1.82

-0.77

-1.11

-0.77

1.82

-0.77

-1.11

MLive/Zc

-2.03

4.81

-2.03

-2.94

-3.54

8.40

-3.54

-5.13

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.10

0.27

0.10

2.06

0.12

0.33

0.12

2.47

ÎŁ

1.29

15.66

-3.11

-2.21

-0.20

19.31

-4.60

-3.99

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.10

Precast concrete specialists

29


Cooling Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

6.49

6.49

0.00

0.00

6.49

6.49

0.00

0.00

Pe /Z

-4.75

3.45

0.00

0.00

-4.75

3.45

0.00

0.00

MSW /Z

0.62

-0.45

0.00

0.00

0.62

-0.45

0.00

0.00

MDL /Z

0.53

-0.38

0.00

0.00

0.53

-0.38

0.00

0.00

MSDL /Zi

-0.77

1.82

-0.77

-1.11

-0.77

1.82

-0.77

-1.11

MLive /Zc

-2.03

4.81

-2.03

-2.94

-3.54

8.40

-3.54

-5.13

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.21

-0.94

0.21

-1.27

0.25

-1.13

0.25

-1.52

ÎŁ

1.40

14.45

-3.00

-5.54

-0.07

17.85

-4.47

-7.98

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.11

14.7 Decompression and Stress Check at Service (end of transmission length) The most unfavourable transmission length calculated was 748.62mm and a check for decompression and stress at service is calculated across. The moments due to self weight and dead load are worked out using an equation along the length of the beam and the moments due to surfacing, GR1a and GR5 are taken from MIDAS Civil. Distance from end of beam to end of the transmission length (most unfavourable) = 748.62 mm e = 214.3 mm Peff = 8231613.5 N Characteristic and Frequent moments at the end of the diaphragm: Characteristic = -3965 kNm (From Midas) Frequent = -2290 kNm (From Midas) Moment due to superimposed dead load = -875 kNm (From Midas) Moment due to self weight of the beam = 228.12 kNm Moment due to dead load = 195.09 kNm

30

Precast concrete specialists


Heating Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

9.72

9.72

0.00

0.00

9.72

9.72

0.00

0.00

Pe /Z

-7.11

5.17

0.00

0.00

-7.11

5.17

0.00

0.00

MSW /Z

0.92

-0.67

0.00

0.00

0.92

-0.67

0.00

0.00

MDL /Z

0.79

-0.57

0.00

0.00

0.79

-0.57

0.00

0.00

MSDL /Zi

-0.69

1.64

-0.69

-1.00

-0.69

1.64

-0.69

-1.00

MLive/Zc

-1.81

4.30

-1.81

-2.63

-3.14

7.45

-3.14

-4.55

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.10

0.27

0.10

2.06

0.12

0.33

0.12

2.47

Σ

3.02

19.51

-2.81

-1.79

1.71

22.72

-4.12

-3.30

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.12 Cooling Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

9.72

9.72

0.00

0.00

9.72

9.72

0.00

0.00

Pe /Z

-7.11

5.17

0.00

0.00

-7.11

5.17

0.00

0.00

MSW /Z

0.92

-0.67

0.00

0.00

0.92

-0.67

0.00

0.00

MDL /Z

0.79

-0.57

0.00

0.00

0.79

-0.57

0.00

0.00

MSDL /Zi

-0.69

1.64

-0.69

-1.00

-0.69

1.64

-0.69

-1.00

MLive/Zc

-1.81

4.30

-1.81

-2.63

-3.14

7.45

-3.14

-4.55

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.21

-0.94

0.21

-1.27

0.25

-1.13

0.25

-1.52

Σ

3.13

18.30

-2.70

-5.12

1.84

21.26

-3.99

-7.29

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.13

14.8 Decompression and Stress Check at Service (@ end of different debonded lengths) The following calculation for debonding has an input for “X” and an input for “Distance from beam end to end of debonded length”, when input these locations are checked for decompression and stress at service to see where debonding is required and where it can stop. When a value is input into the distance from beam end box the effective prestressing force along with the eccentricity, moment due

to self weight, moment due to permanent slab weight will all change automatically, the rest of the moments are taken from MIDAS Civil. This check is required at the end of each debonding zone, (1m,1.5m, 2m, 3m, 4m & 6m in this case). The check has the same form at each location, as such for simplicity only one instance of the calulation is presented here.

Precast concrete specialists

31


Debonded length “X” = 6 Distance from beam end to end of debonded length = 6000 mm No. of remaining debonded strands after debonded length “X” = 0 nr Peff = 10207201 N e = 283.4 mm Moment due to self weight of the beam = 1505.61 kNm Moment due to permanent slab load = 1287.63 kNm Moment due to superimposed permanent load on composite section -57.7 kNm (From Midas) Characteristic and Frequent moments at distance chosen above Characteristic = -1256.3 kNm (From Midas) Frequent = -820.6 kNm (From Midas)

Checked debonded lengths: Dist. (mm)

No. remaining debonded strands

Result

1000

10

OK

1500

8

OK

2000

6

OK

3000

4

OK

4000

2

OK

6000

0

OK

Table 14.14

Heating Frequent combination

Characteristic combination

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MSW /Z

6.07

-4.41

0.00

0.00

6.07

-4.41

0.00

0.00

MDL /Z

5.19

-3.77

0.00

0.00

5.19

-3.77

0.00

0.00

MSDL /Zi

-0.05

0.11

-0.05

-0.07

-0.05

0.11

-0.05

-0.07

MLive /Zc

-0.65

2.40

-0.65

-0.94

-1.00

2.36

-1.00

-1.44

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.10

0.27

0.10

2.06

0.12

0.33

0.12

2.47

Σ

12.15

14.77

-1.01

0.83

11.82

14.79

-1.34

0.74

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.15 Cooling Frequent combination Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

Beam Top (MPa)

Beam Btm (MPa)

Interface (MPa)

Slab Top (MPa)

P/A

12.05

12.05

0.00

0.00

12.05

12.05

0.00

0.00

Pe /Z

-11.66

8.47

0.00

0.00

-11.66

8.47

0.00

0.00

MSW /Z

6.07

-4.41

0.00

0.00

6.07

-4.41

0.00

0.00

MDL /Z

5.19

-3.77

0.00

0.00

5.19

-3.77

0.00

0.00

MSDL /Zi

-0.05

0.11

-0.05

-0.07

-0.05

0.11

-0.05

-0.07

MLive /Zc

-0.65

2.40

-0.65

-0.94

-1.00

2.36

-1.00

-1.44

Diff Shrinkage

1.10

-0.35

-0.41

-0.22

1.10

-0.35

-0.41

-0.22

Temp Effects

0.21

-0.94

0.21

-1.27

0.25

-1.13

0.25

-1.52

Σ

12.26

13.56

-0.90

-2.50

11.95

13.33

-1.21

-3.25

Limits

0.00

30.00

24.00

24.00

-4.06

30.00

24.00

24.00

Status

OK

OK

OK

OK

OK

OK

OK

OK

Table 14.16

32

Characteristic combination

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14.9 SLS Stress Summary

The following tables and graphs take the worst case scenarioâ&#x20AC;&#x2122;s from heating and cooling for both frequent and characteristic combinations. Frequent Combination Beam Length (m)

Beam top (MPa)

Beam btm (MPa)

Compressive Limit 30.00

-4.06

0.50

1.30

15.66

30.00

-4.06

0.75

3.01

19.52

30.00

-4.06

1.00

3.26

21.26

30.00

-4.06

0.00

Decompression Limit

1.50

4.31

20.95

30.00

-4.06

2.00

5.24

20.33

30.00

-4.06

3.00

7.28

18.43

30.00

-4.06

4.00

8.98

17.28

30.00

-4.06

6.00

12.16

14.81

30.00

-4.06

15.25

22.07

0.21

30.00

-4.06

24.50

12.16

14.81

30.00

-4.06

26.50

8.98

17.28

30.00

-4.06

27.50

7.28

18.43

30.00

-4.06

28.50

5.24

20.33

30.00

-4.06

29.00

4.31

20.95

30.00

-4.06

29.50

3.26

21.26

30.00

-4.06

29.75

3.01

19.52

30.00

-4.06

30.00

1.30

15.66

30.00

-4.06

30.00

-4.06

30.50

Table 14.17

Figure 14.1

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Characteristic Combination Beam Length (m)

Beam top (Mpa)

Beam btm (Mpa)

Compressive Limit 30.00

-4.06

0.50

-0.19

19.30

30.00

-4.06

0.75

1.83

22.72

30.00

-4.06

1.00

2.12

22.32

30.00

-4.06

1.50

3.25

21.54

30.00

-4.06

2.00

4.34

20.80

30.00

-4.06

3.00

6.65

18.70

30.00

-4.06

0.00

4.00

8.55

17.29

30.00

-4.06

6.00

11.83

14.84

30.00

-4.06

15.25

23.29

-2.77

30.00

-4.06

24.50

11.83

14.84

30.00

-4.06

26.50

8.55

17.29

30.00

-4.06

27.50

6.65

18.70

30.00

-4.06

28.50

4.34

20.80

30.00

-4.06

29.00

3.25

21.54

30.00

-4.06

29.50

2.12

22.32

30.00

-4.06

29.75

1.83

22.72

30.00

-4.06

30.00

-0.19

19.30

30.00

-4.06

30.00

-4.06

30.50

Table 14.18

Figure 14.2

34

Decompression Limit

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15 Ultimate Limit State, ULS After the prestressed W beam has been designed to satisfy SLS criteria, a check must be carried out to ensure the ultimate moment of resistance and shear resistance are adequate to satisfy the requirements of the ultimate limit state, ULS.

15.1 Ultimate Limit State Flexure Check

Nuetral Axis Depth, X = 255mm Compression Force = 13827.44 kN Tension Force = 13850.59 kN

Moment Capacity of the Concrete Section: Compression Force = 13872 kN Lever Arm = 153mm Moment Capacity = 13872 x 153 = 2122.42 kNm

Moment of resistance due to prestress is calculated in the following table.

Row

No.

Strand @

Stress MPa

Strain

The calculation of the ultimate moment capacity of the composite section is carried out below. Seeing that the ultimate moment capacity is well in excess of the ultimate moment due to loads a more rigourous calculation need not be considered.

Force kN

Lever Arm mm

Moment kNm

10

4

1430

1038.29

5.19E-03

622.97

15.00

9.3

9

4

1230

1429.01

7.94E-03

857.41

215.00

184.3

8

2

1115

1440.00

9.51E-03

432.00

330.00

142.6

7

2

880

1462.45

1.27E-02

438.74

565.00

247.9

6

2

730

1476.78

1.48E-02

443.03

715.00

316.8

5

2

480

1500.67

1.82E-02

450.20

965.00

434.4

4

2

340

1514.04

2.02E-02

454.21

1105.00

501.9

3

4

210

1526.46

2.19E-02

915.88

1235.00

1131.1

2

12

110

1536.02

2.33E-02

2764.83

1335.00

3691.0

1

28

60

1540.79

2.40E-02

6471.33

1385.00

8962.8

Sum

13850.6

Sum

15622.2

Table 15.1 - Moment of resistance due to prestress Ultimate Moment of Resistance, MUR = 17744.60 kNm Ultimate Moment Due to Loads, MUL = 12335.70 kNm

15.2 Minimum Reinforcement Requirement Check According to BS EN 1992-1-1 clause 5.10.1(5(P)) Brittle Failure of prestressed members caused by failure of prestressing tendons should be avoided. The check for this is shown below. Minimum area of reinforcement = As,min = Mrep /zs.fyk where; Mrep = fctm I / y = 2162.64 kNm Zs = 1262.00 mm fyk = fp0.1K = 1637.00 MPa As, min = 1045.35 mm2 As, prov = 9300.00 mm2

EN 1992-2 CL.6.1(109) - (Exp 101a)

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15.3 Global Vertical Shear Design (Beam Ends) Shear in prestressed concrete is considered at the ULS. The design for shear involves the most severe loading conditions and most severe factors of safety all of which are outlined in the below calculation. The reaction of the prestressed beam in resisting shear is similar to that of an RC beam but with additional effects of the compression due to the prestressing force. This increases the shear capacity of the section considerably and is taken into account in the shear capacity without shear reinforcement calculation below.

The design procedure for vertical shear is as follows: 1. Calculate the ultimate design shear force VEd 2. Check if shear reinforcement is required using the equation for VRd,c & VRd,c min 3. Check the maximum shear resistance or crushing strength VRd,max and using this formula you can also rearrange it and work out the θ angle required, noting the requirement to fall within the 22° to 45° range. 4. Calculate the design shear links from Exp (6.8) of BS EN 1992-1-1 5. Finally calculate the nominal links required from Exp (9.5) of BS EN 1992-1-1

Ultimate Shear Force, VULT = 2619.26 kN Based on full web width of 345 mm The Ultimate shear force is made up of: 1.35 x (Beam DL + Slab DL + Surfacing DL + Superimposed DL + Characteristic LL) - (Table NA.9) VRd,c Shear capacity without shear reinforcement: VRd,c = [CRd,c.k(100ρl.fck)1/3 + k1.σcp]bwd = 862.10kN - Therefore Shear Links are Required - (Exp 6.2a) where; fck = 50 MPa k = 1+(200/d)1/2 ≤ 2.0 = 1.35 ρl = Asl /bwd ≤ 0.02 = 0.0075214 Asl is the area of tensile reinforcement, assuming 36nr strands in the tensile area = 4200 mm2

bw is the smallest width of the cross section in the tensile area = 345 mm σcp = NEd /Ac < 0.2fcd = 5.2 MPa Ned is the axial force in the cross section due to loading or prestressing = 4399811 N Ac is the area of concrete cross section = 846920 mm2 Minimum Shear Resistance, VRd,c = (vmin + k1σcp)bwd = 775.83 kN OK - Use the above VRd,c value where; Minimum shear strength, vmin = 0.035k3/2.fck1/2 = 0.39 MPa k = 1+(200/d)1/2 ≤ 2.0 = 1.35

- (Exp 6.2b)

- (Exp 6.3n)

Max Shear Loading allowed, VEd: VEd ≤ 0.5.bw.d.v.fcd = 4469.49kN - Ok, section can be used as shear force is less than the allowable - (Exp 6.5) where; v = 0.6[1-(fck/250)] = 0.48 v is a strength reduction factor for concrete cracked in shear. Note Max Fck for shear calc’s = 50MPa Maximum Shear Resistance(Crushing Strength), VRd,max: VRd,max = αcw.bw.z.v1.fcd/(cotθ + tanθ) = 3492.86kN where; z = 1618.57mm Strut angle, θ = 18.58 ° This value cannot be less than 22 °

- (Exp 6.9)

Design Links, VRd,s = (Asw/s).z.fywd.cotθ = 1.818 mm2/mm - (Exp 6.8) Therefore provide 4 legs of B12-225 c/c’s, As prov = 2.01 mm2/mm Nominal Links, Vnom, ρw = Asw/(s.bw.sinα) = 0.390 mm2/mm - (Exp 9.4) OK - Design links greater than minimum Maximum Links, Vmax Asw,maxfywd/bws ≤1/2.αcwv1fcd = 2.227 mm2/mm - (Exp 9.12) OK - Design links less than maximum Maximum spacing of shear links: sl

36

max

= 0.75d(1+Cotα) = 1080 mm

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- (Exp 9.6)


Shear vaules - taken from MIDAS X (m)

Shear due to SW (kN)

Shear due to surfacing (kN)

Shear due to parapet (kN)

Shear due to gr5 (kN)

0.6

447.2

161.2

13.9

1335.8

3

358.1

131

13.2

1172.4

5

287.8

166.8

11.4

1014.3

10

111

46.3

6.8

696.8

Table 15.2 Curtailment along length of beam X (m)

Shear Force per beam end (kN)

Chosen Link Dia (mm)

No. legs

Max Spacing (mm)

Asv /s req (mm2/mm)

Chosen spacing (mm)

Asv /s prov (mm2/mm)

0.6

2619.26

12

4

248.78

1.82

250

1.81

3

2241.20

12

4

316.58

1.43

300

1.51

5

1973.39

12

4

359.55

1.26

350

1.29

10 - Mid

1155.27

12

4

1159.01

0.39

550

0.82

Table 15.3 Slab reinforcement provisions - based on similar examples Bar Dia (mm)

Spacing (mm)

Top

16

150

Bottom

16

150

As prov (mm2/m) 1340.412866

1340.412866

Table 15.4

15.4 Shear at the interface between the beam and the slab Vertical shear forces in the beam always give rise to longitudinal (Interface) shear forces. The construction joint between the precast beam and the insitu slab is the area under consideration for this check. The following calculations will show if the shear reinforcement for vertical shear is adequate for interface shear or if extra links are required to deal with the interface shear. A solution for this is where a design for vertical shear is carried out, the design links are added to the beam and any additional interface shear links are added to make up the difference as U-bars being anchored back into the web. This will save a lot of additional vertical shear rebar as the interface shear links are required only at the interface. For the calculations a rough surface finish is used, which is very important as this has a large impact on the area of reinforcement required. A lot of design programmes are geared towards default values for c and Îź and again highlights the necessity to liaise with the precast beam manufacturer. In calculating the shear load at the interface an average shear force between 0 and 3m from the beam ends was taken. This value came to 1886kN, it was also checked at 3-6m and again from 6m to midspan. Please note the following calculations are for the full beam end using the 2nr webs. In summary you will see that interface links are far more severe than vertical shear links in this example. For the first 3m links are required at 175mm centres to satisfy interface shear whereas 250mm centres would have satisfied vertical shear.

Rebar crossing shear plane to resist longitudinal (interface) shear forces

Shear plane for interface shear check

Figure 15.1 - Interface Shear diagram

The value for minimum shear interface links is taken from FIB Bulletin 65: Model Code 2010 Section 6.3.5 as: Minimum reinforcement ratio = 0.2(fctm/fyk)>0.001 where fctm is that of the beam.

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37


Design value of interface shear stress: vEdi = βVEd/(z.bi) = 2.14 MPa - (Exp 6.24) where; β is the ratio of the longitudinal force in the new concrete area and the total longitudinal force either in the compression or tension zone, both calculated for the section considered = 1 VEd is the transverse shear force = 1886.72 kN z is the lever arm of the composite section = 1717.64 mm bi is the width of the interface = 514 mm z = I/AῩ = 1717.633917 mm where; I = 561.5109106 mm4 A = 600000 mm2 Ῡ = 544.8492304 mm Design shear resistance at the interface: - (6.2.5 (1)) Area of interface = 514000 mm Angle of reinforcement to interface: α = 90 deg Stresses resulting from forces normal to surface: σn = 0 MPa Surface roughness: - (6.2.5 (2)) Rough c = 0.4 μ = 0.7 Note: Do not set c=0 as shown in TG13 except when using specific load cases for fatigue Max value of interface stress: VRd,max = 0.5vfcd = 6.72 MPa - (Exp(6.25)) Asw/s = [VAῩ/I - cbfctd] / μfyd where; c = 0.40 μ = 0.70 fctd = 1.90 MPa for C50/60 fyd = 435.00 MPa b = 514.00 mm Note: Use fctd value appropriate to the concrete grade of the precast beam not the insitu concrete as advised in Cl.4.3.3.8 of EN 15050.

Shear vaules - taken from MIDAS X (m)

Shear due to surfacing (kN)

Shear due to parapet (kN)

Shear due to gr5 (kN)

0-3m

178.5

13.9

1225

3-6m

141.3

13.2

1006

6-Midspan

50

11.4

662

Table 15.5 Curtailment along length of beam using a stepped distribution X (m)

Average Shear Force per beam end (kN)

Chosen Link Dia (mm)

No. Legs

VEdi (MPa)

Asv/s min (mm2/mm)

Chosen Spacing (mm)

Asv/s prov (mm2/mm)

0-3m

1886.72

12

4

2.14

2.33

175

2.58

3-6m

1545.48

12

4

1.75

1.67

250

1.81

6-Midspan

969.09

12

4

1.1

0.84

500

0.90

Table 15.6

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15.5 Shear Check Between The Web and The Flange of The Composite Section This check is carried out looking specifically at the insitu slab. The rebar is that crossing the failure/shear plane to resist longitudinal shear forces as per fig 15.2 below.

Shear planes for shear check between web and flange

rebar crossing the failure/shear plane to resist longitudinal shear forces

Figure 15.2 - Shear between Web and Flange diagram

Design shear force per beam end = 1887 kN Thickness of web at junction with flange, bw = 514 mm Thickness of flange at junction with web, hf = 200 mm d = 1440 mm z = 1296 mm Total shear force transmitted from web to flange: VEd/z = 1456N/mm length Proportion of shear force transmitted to flange : (1-bw/b)/2 = 0.41 Longitudinal shear stress across junction: (VEd/z)*((1-bw/b)/2)*(1/hf) = 3.02 MPa Permissible shear stress = 0.4fctd = 0.76 MPa < 3.02 MPa Therefore transverse reinforcement is required Required amount of transverse reinforcement: θ = 0.5Sin-1 (2VEd/vfcd) = 14.06 ° Strut angle is limited to 26.5 ≤ θ ≤ 45 therefore θ = 26.5 ° Transverse reinforcement per unit length: Asf/sf = vEdhf /fydCotθ = 691.4 mm2/m As the slab is also subject to transverse bending, the area of transverse steel should equal the greater of 691.4 mm2/m or half this value plus the area required to resist transverse bending. Area of reinforcement required for transverse bending 1686.09 mm2/m Area of transverse reinforcement provided 2680.83 mm2/m Therefore transverse reinforcement provided is ok

(6.2.4(6))

(Exp 6.22)

(Exp 6.21) (6.2.4(5))

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39


16. Beam camber estimates Working out the camber of the prestressed beam due to the prestressing is shown below. This is an estimate as this is not an exact science but it is sufficent for this eaxmple. The final deflection due to the wet concrete being poured on the beam is also given.

Note: We have taken the elastic modulus at transfer and the full prestress force for working out the deflection due to prestress. For working out the deflection due to self weight and wet slab weight we have used an increased elastic modulus which has come from extensive factory testing and data. This modification factor is 1.2.

Formulae required: Upward deflection due to full prestressing moment, δ = Ml2/8EI Downwards deflections due to self weight and slab load, δ = 5Wl4/384EI

Beam

Length between supports (mm)

Ecm (transfer) (GPa)

Ecm (GPa)

I (mm4x109)

Mprestress (kNm)

Deflection due to prestress (mm)

Wself weight (kN/m)

Deflection due to selfweight (mm)

Wslab (kN/m)

Deflection due to wet slab (mm)

Ex works camber (mm)

Final camber after slab pour (mm)

Internal

30000

32.31

37.28

215.46

3676.65

59.42

21.17

-23.17

18.11

-19.81

36.25

16.44

Table 16.1

Figure 16.1

17 Creep Induced Sagging Moment at Supports Because of the delayed restraint between continuous spans, even in the absence of variable load a bending moment will arise in the connection, sagging if prestress is prevailing over permanent load, hogging if not. The evaluation of delayed bending moment can be performed by means of a suitable creep analysis method.

Net moment = M prestress - M Dead Load - M Superimposed Dead Load M prestress = 10210kN x 707.25mm = 7221kNm (The above is the long term force multiplied by the eccentricity from the centre of gravity of the composite section to the centre of gravity of the strands) M Dead Load = 4419kNm

40

Precast concrete specialists


M Superimposed Dead Load = 520kNm Net Moment = 2282kNm Prestress prevails - will cause sagging @ the support Sagging restraint moment = 2282(1-e-Ď&#x2020;) where; Ď&#x2020; = 1.47 MR = 2282(1-e-1.47) MR =1757kNm Place 6nr strands spaced at 160mm over the soffit. Fstrand = limit fypk to 0.75fypk (as per Mattock) = 1860x0.75 = 1395MPa Fstrand = 1395x150 = 209.25kN d = 1700-160 = 1540mm z = 0.95d = 1463mm M Restoring = 209.25 x 6 x 1.463 = 1836.8 > 1757 kNm - OK Adopt 6nr untensioned strand at 160mm above the beam soffit as per the below sketch.

Beam end

PROJECTING STRAND

TRANSVERSE HOLES

W11 Beam

Figure 16.2

Note: We feel that leaving strand projecting is not necessary with a precast downstand crosshead at the pier and when built into an abutment wall but we have carried out the check for completeness.

Precast concrete specialists

41


Appendix A

Appendix A1 - Bending Moment Diagram for gr5 Loading - internal beam - results taken from MIDAS

Appendix A2 - Bending Moment Diagram for gr1a Loading - internal beam - results taken from MIDAS

42

Precast concrete specialists


Appendix A3 - Bending Moment Diagram for surfacing Loading - internal beam - results taken from MIDAS

Appendix A4 - Shear Force Diagram for gr5 Loading - internal beam - results taken from MIDAS

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43


Appendix A5 - Shear Force Diagram for gr1a Loading - internal beam - results taken from MIDAS

Appendix A6 - Shear Force Diagram for surfacing Loading - internal beam - results taken from MIDAS

44

Precast concrete specialists


Appendix B

Appendix B1 - Photo showing W Beam end with transverse hole formers, lifting strand, prestressed strand, spacers, links, stopends and debonding all visible.

Appendix B2 - Photo showing W Beam being poured. The internal and external formers and fixed. The operative is vibrating the concrete with the other operative pouring the next beam on the line.

Precast concrete specialists

45


list of Symbols A Cross sectional area Cross sectional area of concrete Ac Area of a prestressing tendon or tendons Ap Cross sectional area of reinforcement As minimum cross sectional area of reinforcement As,min Cross sectional area of shear reinforcement Asw D Diameter Ec, Ec(28) Tangent modulus of elasticity of normal weight concrete at a stress of σc = 0 and at 28 days Effective modulus of elasticity of concrete Ec,eff Design value of modulus of elasticity of concrete Ecd Secant modulus of elasticity of concrete Ecm Tangent modulus of elasticity of normal weight concrete Ec(t) at a stress of σc = 0 and at time t Design value of modulus of elasticity of prestressing steel Ep Design value of modulus of elasticity of reinforcing steel Es Bending stiffness El EQU Static equilibrium F Action Design value of an action Fd Characteristic value of an action Fk Characteristic permanent action Gk l Second moment of area of concrete section L Length M Bending moment Design value of the applied internal bending moment MEd N Axial force Design value of the applied axial force NEd (tension or compression) P Prestressing force Initial force at the active end of the tendon immediately P0 after stressing Characteristic variable action Qk Characteristic fatigue load Qfat R Resistance SLS Serviceability limit state ULS Ultimate limit state V Shear force Design value of the applied shear force VEd b Overall width of a cross-section, or actual flange width in a T or L beam Width of the web on T, I or L beams bw d Diameter; Depth d Effective depth of a cross-section Largest nominal maximum aggregate size dg e Eccentricity Compressive strength of concrete fc Design value of concrete compressive strength fcd Characteristic compressive cylinder strength of concrete fck at 28 days Mean value of concrete cylinder compressive strength fcm Characteristic axial tensile strength of concrete fctk Mean value of axial tensile strength of concrete fctm Tensile strength of prestressing steel fp Characteristic tensile strength of prestressing steel fpk 0,1% proof-stress of prestressing steel fp0,1 Characteristic 0,1% proof-stress of prestressing steel fp0,1k Characteristic 0,2% proof-stress of reinforcement f0,2k Tensile strength of reinforcement ft

46

Precast concrete specialists

ftk Characteristic tensile strength of reinforcement fy Yield strength of reinforcement Design yield strength of reinforcement fyd Characteristic yield strength of reinforcement fyk Design yield of shear reinforcement fywd h Height h Overall depth of a cross-section i Radius of gyration k Coefficient; Factor l (or l or L) Length; Span r Radius t Thickness t Time being considered The age of concrete at the time of loading t0 u Perimeter of concrete cross-section, having area Ac x Neutral axis depth x,y,z Coordinates z Lever arm of internal forces α Angle; ratio β Angle; ratio; coefficient γ Partial factor Partial factor for concrete γC Partial factor for actions, F γF Partial factor for permanent actions, G γG Compressive strain in the concrete εc Compressive strain in the concrete at the peak stress fc εc1 εcu Ultimate compressive strain in the concrete Strain of reinforcement or prestressing steel at εu maximum load Characteristic strain of reinforcement or prestressing εuk steel at maximum load θ Angle λ Slenderness ratio μ Coefficient of friction between the tendons and their ducts ν Poisson’s ratio ν Strength reduction factor for concrete cracked in shear ξ Ratio of bond strength of prestressing and reinforcing steel ρ Oven-dry density of concrete in kg/m3 ρ1000 Value of relaxation loss (in %), at 1000 hours after tensioning and at a mean temperature of 20°C Reinforcement ratio for longitudinal reinforcement ρl Reinforcement ratio for shear reinforcement ρw Compressive stress in the concrete σc Compressive stress in the concrete from axial load σcp or prestressing Compressive stress in the concrete at the ultimate σcu compressive strain εcu τ Torsional shear stress Φ Diameter of a reinforcing bar or of a prestressing duct φ(t,t0) Creep coefficient, defining creep between times t and t0, related to elastic deformation at 28 days φ(∞,t0) Final value of creep coefficient Ψ Factors defining representative values of variable actions for combination values Ψ0 for frequent values Ψ1 for quasi-permanent values Ψ2


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