REINFORCED CONCRETE BRIDGES · 2014-06-20 · REINFORCED CONCRETE BRIDGES ASS. PROF. PHD. DIMITAR...

DIMITAR DIMITROV

REINFORCED CONCRETE BRIDGES

REINFORCED CONCRETE BRIDGES ASS. PROF. PHD. DIMITAR DIMITROV

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Contents Notation Abreviations Glossary

CHAPTER 1:Basic for the RCB design 1.1 Bridge definition 1.2 Bridge classification 1.3 Historical review of the bridges 1.4 Main parts of the beam type RCB 1.5 Codes for RCB design 1.6 General background for RCB design.

CHAPTER 2: Actions on bridges. Traffic loads on road, railway and pedestrian bridges. Design groups and combinations.

2.1 Classification of actions 2.2 Some permanent actions on bridges 2.3 Traffic actions for road bridges according EC1 2.4 Traffic actions on railway bridges 2.5Design combinations of actions

CHAPTER 3:Carriageway, drainage, water proofing (insulation), expansion joints, safety barriers, sidewalks.

3.1 Pavement and water proof 3.2 Drainage 3.3 Transition structures at expansion joints 3.4 Safety barriers. Sidewalks

CHAPTER 4: Appropriate systems for the main RCB structures. Special features of the static analysis.

4.1 Statically determinate reinforced concrete bridge structures. Advantages and disadvantages. 4.1.1 Simple span bridges 4.1.2 Simple span with cantilever bridges

4.1.3 Gerber type bridges 4.2 Statically indeterminate reinforced concrete bridge structures. Advantages and disadvantages. Special features of the static analysis. 4.2.1 Continuous bridge beams

4.2.2 Frame bridges 4.3 Different methods for construction of the bridge systems. Division of the bridge structure into separate precast elements. Principles for division. Restoration to the continuity 4.4 Construction decisions for skew and curved bridge constructions


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CHAPTER 5: RCB slabs 5.1 One way RCB slabs.Static analysis and detailing. 5.1.1 Static scheme 5.1.2 Static calculation 5.1.3 Design for bending and shear at ULS 5.1.4 Detailing 5.2 Cantilever RCB slabs.Static analysis and detailing

5.2.1 Application of the cantilever RCB slabs 5.2.2 Actions 5.2.2.1 Permanent actions 5.2.2.2 Traffic actions

5.2.3 Static analysis 5.2.4 Design checks and detailing 5.3 Two way RCB slabs. Static analysis and detailing

5.3.1 Application of two way RCB slabs 5.3.2 Static analysis 5.3.3 Design checks and detailing

5.4 RCB slabs without beams. Static analysis and detailing 5.4.1 Application of RCB slabs w/o beams. Main dimensions

CHAPTER 6: Static analysis of the bridge superstructure

6.1 Methods for static analysis of RCB 6.2 Leonhardt’s method. Application. Influence lines

6.2.1 Transversal distribution of the traffic loads 6.2.2 Influence lines and internal forces at the main longitudinal beams

6.2.3 Internal forces at the transversal beams 6.2.4 Special cases 6.2.4.1 Case with more than 8 main beams

6.2.4.2 Application for continuous beams 6.3 Method of infinitely stiff transversal beam 6.4 Method of lever arm 6.5 Finite element method (FEM). 6.6 Finite strip method(FSM) 6.7 Analysis of RCB structures without transversal beams.

CHAPTER 7: RCB box section bridges. 7.1 Application of RC box section bridges 7.2 Types of torsion for RCB structures

7.2.1 Pure torsion 7.2.2 Warping torsion 7.2.3 Equilibrium torsion 7.2.4 Compatible torsion 7.3 Section properties for torsion calculations


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7.4 Static analysis of RC box section bridges 7.5 Design checks and detailing

CHAPTER 8: Design checks and detailing for RCB elements w/o prestressing

8.1 General assumptions and notations 8.2 Design at ULS of members with rectangular section subjected to bending 8.3 Design at ULS of members with T-sections subjected to bending 8.4 Design at ULS of members with rectangular section subjected to bending

with axial force 8.3 Design checks for RCB elements for shear and torsion

8.5 Design checks for RCB elements for shear 8.6 Design checks for RCB elements subjected to shear and torsion

8.7 Design check for shear stresses between web and flanges of T-sections 8.8 Detailing of RCB beams.

CHAPTER 9: Prestressed RCB structure 9.1 Application of the presstressing in the RCB 9.2 Application of pre-tensioning and post-tensioning in the RCB structures 9.3 Losses of prestress. 9.3.1 Immediate losses of prestress for pre-tensioning 9.3.2 Immediate losses of prestress for post-tensioning 9.3.3 Time dependent losses of prestress for pre- and post tensioning. 9.4 Design checks for prestressed RCB beams 9.4.1 Initial prestressing force determination for simple span RCB beams.

Check for decompression 9.4.2 Normal stress check 9.4.3 Principal stress check at SLS

9.4.4 Check for bending at ULS 9.4.5 Check for shear at ULS

CHAPTER 10: Modern systems for RCB construction 10.1 Cast-in-place construction of bridges using conventional scaffolding 10.2 Cast-in-place “span by span” construction of bridges 10.3 Precast bridge construction using conventional cranes 10.4 Precast bridge construction using “truss crane”(gantry) 10.5 Cast-in-place balanced cantilever bridges 10.6 Precast balanced cantilever bridges 10.7 Incrementally launched bridges

CHAPTER 11: Design checks for elastomeric bearings 11.1 General

11.2 Design checks 11.2.1 Stress check and determination of the bearing area 11.2.2 Check for shear strain


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11.2.3 Check for friction 11.2.4 Check for rotation

CHAPTER 12: Bridge substructure 12.1 Abutments 12.2 Piers

CHAPTER 13: Seismic design of RCB Notation Geometry A – Cross sectional area a(b) – the distributed contact surface area dimensions or elastomeric bearing dimensions in plan a0(b0) – the contact surface area dimensions at the top of the pavement a’(b’) – dimensions of the steel plats inside the elastomeric bearing aeff,span(supp) – effective width of a slab at span or support Ac – Cross sectional area of concrete Ap- Cross sectional area of prestressing reinforcement As – Cross sectional area of reinforcement As1 – Final required cross sectional area of reinforcement As,min – Minimum cross sectional area of reinforcement As,prov – Provided cross sectional area of reinforcement As,req – Required cross sectional area of reinforcement by design As1,Φ – Cross sectional area of one reinforcement bar Asw – Cross sectional area of shear reinforcement A1 –effective area of the elastomeric bearing Ar –reduced effective area of the elastomeric bearing CG – Center of gravity EI – Bending stiffness I – Second moment of area of concrete section L – length;span N – Total number of reinforcing bars S – first moment of the cross section n – Number of reinforcing bars for 1m d – Effective depth of a cross section; Distance between main bridge beams dg – Largest nominal maximum aggregate size e – Eccentricity h – Height;Overwall depth of a cross section hf – Overall depth of a slab i – Radius of gyration k – Coefficient l0 – Effective length of column


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lcl – Clear span leff – Effective span lnom – Span between axes of the supports lmax,cl – Maximum clear span of a slab lmin,cl – Minimum clear span of a slab 1/r – Curvature at a section s- Distance between reinforcing bars or stirrups u – Perimeter of concrete cross section, having area Ac

x – Neutral axis depth x,y,z – Coordinates z – Lever arm of internal forces; stiffness factor λ – Slenderness ratio φ – creep coefficient ρ – Reinforcement ratio for required longitudinal reinforcement ρl – Reinforcement ratio for provided longitudinal reinforcement ρw – Reinforcement ratio for shear reinforcement Φ – Diameter of a reinforcing bar Φs– Diameter of longitudinal reinforcing bar Φw– Diameter of transverse reinforcing bar Actions and Effects A – Accidental action (situation) E – Effect of action F – Action Fd – Design value of an action Fk– Characteristic value of an action Gd – Design permanentconcentrated action Gk – Characteristic permanentconcentrated action gd – Design permanentdistributed action gk – Characteristic permanentdistributed action M – Bending moment MEd – Design value of the applied internal bending moment N – Axial force NEd – Design value of the applied axial force (tension or compression) Qk–Characteristic variableconcentrated action Qd–Design variableconcentrated action qk–Characteristic variabledistributed action qd–Design variabledistributed action R – Resistance S – Seismic action (situation) T – Torsional moment


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TEd – Design value of the applied torsional moment TRd,max – Design value of the applied torsional moment V- Shear force VEd – Design value of the applied shear force VRd,c – Design shear resistance of the member without shear reinforcement VRd,max – Design value of the maximum shear force which can be sustained by the member, limited by crushing of the compression struts v – velocity vx,d –design horizontal displacement of the elastomeric bearing along axis x

vy,d –design horizontal displacement of the elastomeric bearing along axis y

σ – normal stress τ – shear stress γF – Partial factor for actions F γG – Partial factor for permanent actions G γQ – Partial factor for variable actions Q ψ – Factors defining representative values of variable actions ψ0 – for characteristic values ψ1 – for frequent values ψ1,inf – for infrequent values ψ2 – for quasi-permanent values Material Characteristics φ – Angle of internal friction fc – Compressive strength of concrete fcd – Design value of concrete compressive strength fck – Characteristic compressive cylinder strength of concrete at 28 days fcm – Mean value of concrete cylinder compressive strength fctb – Tensile strength prior to cracking in biaxial state of stress f ctk – Characteristic axial tensile strength of concrete f ctm – Mean value of axial tensile strength of concrete f ctx – Appropriate tensile strength for evaluation of cracking bending moment f p – Tensile strength of prestressing steel f pk – Characteristic tensile strength of prestressing steel fp,0.1 – 0.1% proof-stress of prestressing steel f p,0.1k – Characteristic 0.1 % proof-stress of prestressing steel f p,0.2k – Characteristic 0.2 % proof-stress of reinforcement f t – Tensile strength of reinforcement f tk – Characteristic tensile strength of reinforcement f y – Yield strength of reinforcement f yd – Design yield strength of reinforcement f yk – Characteristic yield strength of reinforcement


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f ywd – Design yield of shear reinforcement

Abreviations FEM=Finite Element Metod – Метод на крайните елементи FSM=Finite Strip Method – Метод на крайните ивици LM=LoadModel – Товарен модел RCB=ReinforcedConcreteBridge(s) – Стоманобетонен/нни мост/мостове ULS=UltimateLimitState(s) – Крайно/и гранично/и състояние/я SDOF=Single Degree of Freedom – Една степен ана свобода SLS=ServiceabilityLimitState – Експлоатационно/и гранично/и състояние/я SW=Special Wagons - Тежки ж.п.вагони TS=Tandem system –Двуосна група TDC = Transversal distribution coefficient –Коефициент на напречно разпределение UDL=UniformlyDistributedLoading – Равномерно разпределено въздействие Glossary Abutment = End support for the bridge beams or girders, placed where the roadway ends and the bridge begins (See Figure 1.11) – Устой Approach span = A span leading up to or away from the main span of a suspension or a cable – stayed bridge (See Figure 1.8 and 1.9). Also, the first or last span of a multispan, continuous bridge –Прилежащ отвор Approach slab.= Cast in placeslab that provides the connection “roadway-bridge” (See Figure 1.11) –Преходна плоча Arch – (See Figure 1.5) – Дъга (криволинейна греда), Свод (криволинейна повърхнина) Balanced cantilever erection = Method for erection of frame or continuous bridges – Уравновесено конзолно изграждане Bearings= Devices provided at the ends of the beams to transfer reactions to abutments or piers –Лагери Cable-stayed bridge(See Figure 1.8) –Вантов мост Cable stay = The element that connects the pylon and the beam or truss at cable-stayed bridge (See Figure 1.8) – Ванта Carriageway = The part of the road used only for the road traffic – Пътно платно Close-endabutment – Устой от тип “подпорна стена” Collarbeam=Thebeambelowthebearings (SeeFigure 1.11) – Подлагерна греда (Ригел при опора от тип “рамка” или кусинет при опора от тип “стена”) Culvert = A bridge with span less then 5m – Водосток Curb - Бордюр Cyclic loading (Also called repetitive loading) =Alternately applying and removing loads, causing member to endure cyclic stresses of some minimum value to some maximum value –Циклично въздействие Deck (or Slab) =Flooring that supports vehicular traffic. The deck may be made of reinforced concrete, steel or wood – Пътна плоча


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Drainage basin area= The area in square kilometers that provides the design water quantity and it passes under the bridge – Водосборна област Drainage=The device that provides the removal of the water from the bridge surface –Отводнител Expansion joint = The device that allows the differential movement between adjacent parts of the superstructure or superstructure and substructure – Деформационна фуга Fatigue =A fracture phenomenon resulting from a fluctuating stress cycle–Умора Guard wall = The element of the abutment that supports the longitudinal earth pressure at the level of the superstructure (See Figure q.1.11) – Гардбаластова стена Grade line= The line of the roadway in vertical direction at the longitudinal profile– Нивелета Incrementallaunching = Method for bridge erection using longitudinal movement of the superstructure – Тактово изтласкване Kerb –Бордюр Open-end abutment – Обсипан устой Overpass (Over crossing)= A bridge whose structure is above other roadway – Надлез Road alignment= The line of the roadway in horizontal direction (in plan) – Трасе на пътя Scaffolding = The temporary structure that supports the formwork and the cast in place concrete–Скеле Sidewalk (Footway)=The parts of the bridge for pedestrian access or the parts away from the carriageway–Тротоар Skew bridge =A bridge whose longitudinal axis is not perpendicular to the abutments or piers – Кос мост Skew =The angle between the bridge longitudinal axis and the axis of the abutments or piers – Косота Span = Distance between centers of bearings at supports –Отвор Substructure =Portion of the bridge that supports the superstructure. The substructure includes bridge bearings and every other bridge element below the bearings, such as abutments and piers.- Долно строене Superstructure =Generally,the portion of the bridge above the bridge bearings. The superstructure may include only a few components, such as a reinforced concrete slab in a slab bridge, or it may include several components, such as flooring, beams, trusses and bracing in a truss bridge. In suspension and cable-stayed bridges, components such as suspension cables, hangers, stays, towers(pylons), bridge deck, and the supporting structure comprise the superstructure – Връхна конструкция Suspension bridge (See Figure 1.9) – Висящ мост Tresle = A bridge usually situated at urban areas at the road to the major transportation object like airports, ports or others - Естакада Underpass (Under crossing) = A bridge built to carry a city street or a county road under a highwayл It may provide pedestrian access cross the highway- Подлез Viaduct = A bridge that allows highways or railroads to pass over a valley. The valley may contain streets, railroads, or other features. –Виадукт


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CHAPTER1: Basic for the RCB bridge design 1.1 Bridge definition. Bridge is a facility,whichcrosses an obstacle. The minimum span length of the bridge is

5m(in USAminimum 6 feet=6m). If the span length is less than 5m the facility is called “culvert”. There are typical projects for different culvert types – pipe type, slab type and others. Normally they are not designed.

1.2 Bridge classification. Criteria №1- bridge function - road bridge - railway bridge - pedestrian bridge - combined bridge Note 1:Combined is a bridge with roadway and railway traffic. There are not so

many examples of combined bridges in the world bridge practice. Two combined bridges above DanubeRiver have been executed –near the towns Russe and Vidin.

Note 2: The typical road bridge is provided with pedestrian sidewalks. The typical railway bridge is provided with paths for inspection. These bridges have not been considered as combined bridges.

Criteria №2- obstacle for the bridge - bridge over river - overpass– see Glossary - underpass– see Glossary - viaduct– see Glossary - trestle– see Glossary Criteria №3- bridge situation - in straight line w/o skew. The skew angle is 900. - in straight line with skew, shortly “skew bridges” – Figure 1.1 - curved –Figure 1.2 - unspecified – Figure 1.3


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Figure 1.1. Skew bridge

Figure 1.2. Curved bridge

Figure 1.3.

Abutment Abutment

skew

bridgeaxis

pier

axis

skew

Abutment

90°

Exeption (rare case)

Abutment

Pier Pier

Abutment Abutment

Pier

Pier


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Criteria №4- bridge structural system - Beam type – definition: bridges with only vertical reaction under vertical loadsFigure

1.4 - Arch type-Figure 1.5 - Frame (Integral) type-Figure 1.6 - Combined-Figure 1.7 - Cable stayed-Figure 1.8 - Suspension-Figure 1.9

Figure 1.4. “Beam type” bridge

Figure 1.5. Arch type bridge

Figure 1.6. “Frame” type bridges

Vertical reactionA

A

OR OR

A - A

Vertical reaction

OR SLAB

A

A

A - A

OR

OR

Abutment Pier Abutment

Integral bridge Semi Integral bridge

Bearing Bearing


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Figure 1.7. Example of combined bridge

Figure 1.8.

Main span Approach spanApproach span

"Harp" cable stays

"Fan" cable stays "Radial" cable stays


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Figure 1.9. Types of suspension bridge

Criteria №5- material for construction - reinforced concrete - prestressed reinforced concrete - steel - combined – steel and reinforced concrete - wood - stone - brick

Criteria №6- RCB construction - Fully cast in situ - Fully precast - Precast beams and cast in situ slab

Criteria №7 – carriageway location - Above the main bridge construction - Below the main bridge construction-see Figure 1.7 - In the middle of the main bridge construction-Figure 1.10

Figure 1.10. Carriageway in the middle of the main structure

Single - Span

Three - Span

Multi - Span

A - A


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1.3 Historical review of the bridges They say that the first “bridge” was a fallen tree above a river. The ancient bridges were built mainly by stone or wood. Roman bridges have become

emblems of civilization. The typical construction form was arch. Still exist in South France the unique aqueduct Pont-de-Guard with 3 rows of arches.

The first metal bridge was built above Severn River (now it is the border between England and Wales) in 1779. The used material is iron. Unfortunately, this material is brittle and is not appropriate for bridges which are subjected to repeated loads. The next bridge construction stage is the steelbridge design and construction.

At the second half of the 19-th century started the use of the reinforcedconcrete for building and bridge structures. The first application of prestressingwas at the end of the 19-th century. The first attempts were without success because the engineers and the researches did not know the nature of the prestress losses. At the first half of 20-th century started the extensive application of the prestressed concrete.

In the Middle Ages in Bulgaria mostly stone arch bridges were built. Several bridges still exist and have been used.

The first application of the prestressing in the Bulgarian bridge practice is at 1958. Almost all modern bridge construction systems have been applied in the Bulgarian

bridge engineering – balanced cantilever erection and concreting, incremental launching, “span by span” concreting and others. The span record is 180m – the spans at the Second Danube Bridge Vidin-Calafat.

1.4 Main parts of the RCB beam type There are two main parts of the beam type bridge structure:

- Superstructure – the part above the bearings; - Substructure – the part below the bearings.

The parts of the superstructureare: 1. Bridge slab 2. Longitudinal main beam 3. Transversal beam The parts 2 and 3 form the bridge beam grid. The functions of the transversal beams

are: A.To improve (with the role of the slab) the distribution of the traffic loads among the

main beams. B. To increase the lateral stability of the structure and the torsional (mainly) stiffness of

the structure. The modern tendency in the field of bridge engineering is to design transversal beams

only at the supports. It is difficult the construction of the middle transversal beams. In this case the role of the beams in the middle is carried out by the slab.

The parts of the substructureare: 1. Abutment. The main parts of the abutment are: 1.1. Abutment wall(columns) 1.2. Abutment guard wall 1.3. Wingwalls 1.4. Approach slab 1.5. Collar beam


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1.6. Abutment foundation – flat or pile 2. Pier. The main parts of the pier are: 2.1. Collar beam 2.2. Wall(columns) 2.3. Foundation – flat or pile 1.5 Codes for RCB design There are three groups of codes for RCB design:

- Codes for actions; - Codes for design; - Codes for seismic design.

The German code DIN 4227 was used in the past in Bulgariafor the design of prestressed reinforced concrete bridges.

From the beginning of 2011 the bridge construction design is according the system Eurocodes.

For seismic design of bridges is used “Regulation for seismic design of buildings and facilities” from 2012.

The most important Eurocodes for bridge design are as follows: 1. Actions БДС EN 1991-2 Part 2 – for traffic actions on bridges БДС EN 1991-1-3 Part 1-3 – for snow action БДС EN 1991-1-4 Part 1-4 – for wind actions БДС EN 1991-1-5 Part 1-5 – for temperature action БДС EN 1991-6 Part 1-6 – for actions during bridge execution 2. Design БДС EN 1992-2 Part 2 – for reinforced concrete bridges 3. Seismic design БДС EN 1998-2 Part 2 – seismic design of bridges

1.6 General background for RCB design 1.6.1 Grade line and road alignment, site plan and profiles.

The bridge engineer selects the place of the bridge according the grade line. The start and end of the bridge depend on the height of the construction fill of the road. As a rule, the bridge starts when the costs for the construction and maintenance of the fill are similar to the same costs for the bridge. It is assumed that 6m height of the fill is this border. The grade line determines the longitudinal slope of the bridge. The minimum slope for drainage is 0.5%.

The correct place of the bridge depends mainly on the road alignment and the obstacle. The relevant construction form could be selected after detailed analysis. The modern bridge theory and practice allow the design of large variety of bridges:

- Normal or skew; - Curved (situated in a horizontal or vertical curve) - Unspecified form.

The transversal profiles are required mainly for: - Curvedbridges;


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- Slope ground line. In this case it could be necessary to design the foundations at different levels – see Figure 1.11.

Figure 1.11. Bridge foundations at different levels

1.6.2 Geotechnical investigations and report A complete geotechnical study of a site will: - determine the subsurface stratigraphy; - define the physical properties of the earth materials; - evaluate the generated data and formulate solutions to the project. Geotechnical issues that can affect a project can be grouped as follows: - Foundation Issues – Including the determination of the strength, stability and

deformations of the subsurface materials under the loads imposed by the structure foundations;

- Earth Pressure Issues – Including the loads and pressure imposed by the earth materials on the foundations and against supporting structures;

- Construction Considerations- The characteristics of materials to be excavated and the conditions that affect deep foundation or ground improvement;

- Groundwater Issues – including occurrence, hydrostatic pressures, flow and erosion. Site and subsurface characteristics directly affect the choice of the foundation type,

foundation construction methods and bridge cost. Subsurface and foundation conditions also frequently directly affect the route alignment and bridge type selection.

For many projects, it is appropriate to conduct the geotechnical investigations in phases. For the preliminary phase only historical information and a limited field exploration may be adequate. The results or the first-phase study can then be used to develop a preliminary geological model of the site, which is used to determine the key foundation design issues and plan the design-phase site investigation.

1.6.3 Bridge hydraulics. Hydrological study for the bridge design mainly deals with the properties, distribution

and circulation of the water on the land surface. The primary objective is to determine the peak runoff discharges, water profiles and velocity distribution.

The base flood is the 100-year discharge (1% frequency). The design discharge Q2% [m3/sec] is the 50-year discharge (2% frequency) and is calculated according (1.1):

Q2%=42kA (1.1), where: - k is a coefficient that depends on the type and the length L[km]of the drainage basin

area;

Ground line

slope


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- A[km2] is the drainage basin area. Many times the historical flood is so large that a structure to handle the flow becomes

uneconomical. It is the engineer’s responsibility to determine the design discharge. The total required area that provides the water flood under the bridge ΣAj can be

determined according (1.2): ΣAj=Q2%/vmax (1.2), where vmax is the maximum stream velocity taken from the

hydrological report. Bridge scour (see Figure 1.12) is the result of the erosive action of flowing water,

excavating and carrying away material from the bed and banks of streams. Determining the magnitude of the scour is complicate by the cyclic nature of the scour process. Scour should be investigated closely in the field when designing a bridge. The engineer usually places the top of footings at or below the potential scour depth. Therefore, the determination of the depth of scour is very important.

The total potential scour comprises the following components: 1.Long-term degradation. The problem for the bridge engineer is to estimate the long-

term bed elevation changes that will occur during the lifetime of the bridge – as a rule 100 years.

Figure1.12. Bridge scour 2.Local scour.When upstream flow is obstructed by obstruction such as piers,

abutments and embankments, flow vortices are formed at their base – see Figure 1.13. The vortex action removes bed material from around the base of the obstruction.

Figure 1.13. Vortices caused by the pier as an obstacle

J

Pier

Long - term degradation

River bed

Local scour

Flow

VorticesPier


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3.In considering local scour, the bridge engineer needs to look into the following factors:

- flow velocity; - flow depth; - flow attack angle to the obstruction; - obstruction width and shape – see Figure 1.14.; - bed material characteristics.

Figure1.14. Preferred pier shapes for minimizing the scour

The following is a summary of the best solutions for minimizing scour damage: 1. Streamlining bridge elements to minimize obstructions to the flow; 2. Founding bridge pier foundations sufficiently deep to not require prevention

measures; 3. Founding abutment foundations below the estimated local scour depth 1.6.4 Technological background The bridge designer should be well acquainted with the following data for the design: 1. Materials for the bridge construction as follows:

- for high strength concrete – cement and aggregates; - steel for reinforcement and/or for prestressing(tendons); - for prestressing devices(depending on the used prestressing system in the relevant

country). The system which ismostly used in Bulgariais “Freyssinet”; - for road surface and insulation.

2. Equipment for: - concreting; - reinforcement; - prestressing (presses, anchorage); - lifting(cranes); - production of precast elements; - construction of deep foundations(e.g.piles – driven piles or drilled shafts).

4. Typical bridge construction systems used at the relevant country

River flow

Pier

NOT ALLOWED!


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The bridgeengineer must select the most relevant construction system for the designed bridge. Almost all known and applied bridge construction systems in the world have been used in Bulgaria.

CHAPTER 2: Actions on bridges. Traffic loads on road, railway and pedestrian

bridges. Design groups and combinations.

2.1 Classification of actions - Direct actions. They can be represented as concentrated forces or distributed loading.

For this action type one could use the words “load” and “force” simultaneously. Example – traffic actions on road or railway bridges

- Indirect actions - They cannot be represented as “loads” or “forces”, but they produce internal forces due to the restrain of the deformations. Example – temperature change, shrinkage, creep, settlement of the supports.

According the nature and lasting of the actions they are as follows: - Permanent actions (G) – self weight of the structural and non structural bridge

parts, earth pressure, hydrostatic pressure, prestressing, creep, shrinkage, settlement of the supports and others;

- Variable actions (Q) – traffic actions on bridges, wind, temperature changes, river flow and others;

- Accidental actions (A) – mainly collision forces on bridge supports, elastic barriers or explosion;

- Seismic action (Ae) – actions during an earthquake The actions are characteristic Fk and design actions Fd. The values of the (G)

characteristic actions are usually mean values. They are calculated using the geometry dimensions of the element and the volume density of the relevant material – see 2.2.

For the ULS checks are used design actions Fd=Fk*γf, where γf is a partial safety coefficient for actions.

2.2 Some permanent actions on bridges Example 1: Self weight of the structural elements of the bridge: - Slab characteristic self weight gf,k=hf*25=……… /kN/m2/ - Main beam characteristic self weight g1,k=Ac*25=……. /kN/m/ Example 2: Self weight of the non structural elements of the bridge: - Self weight of the pavement above the bridge slab gp=0.11*22=2.42kN/m2: - Self weight of the parapet and the elastic barrier – could be assumed 1kN/m each Example 3: Earth pressure For the bridges three earth pressure types are taken into account – active, passive and

in peace. The relevant coefficients are asfollows: Ka=tg2(450-φ/2) Kp=tg2(450+φ/2) K0=tg(450-φ/2) , where φ is the angle of the internal friction. This action is mostly important for bridge abutments.


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2.3 Traffic actions for road bridges according EC1 The road actions in EC1-3 should be applied only for the design of road bridges with: - individual span lengths less than 200m and with - carriageway widths not greater than 42m. For bridges having larger dimensions, traffic loads should be defined or agreed by the

client. Usually for span lengths exceeding 200m, the main models for characteristic values(ex.LM1) are considered to be conservative.

The load models for road bridges defined in the Eurocodes do not describe actual loads. They have been selected so that their effects (with dynamic amplification included) represent the effects on the actual traffic.

The carriageway has been divided into notional lanes. The typical national lane is with 3m width. The locations of the lanes are not necessarily related to their numbering. For each individual verification the load models on each notional lane should be applied on such a length that the most adverse effect is obtained.

The characteristic values for the vertical loads have been used. The load models for vertical loads represent the following traffic effects: Load Model 1(LM1):Concentrated and uniformly distributed loads, which cover most of

the effects of the traffic of lorries and cars. This model is intended for general and local verifications.

Load Model 2(LM2): Single axis load applied on specifictyre contact areas which cover the dynamic effects of the normal traffic on very short structural elements. This model should be separately considered and is only intended for local verifications.

Load Model 3(LM3): Special vehicles. This model is intended to be used only when is required by the client.

Load Model 4(LM4): A crowd loading. This load model should be considered only when required by the client. It is intended only for general verifications. The uniformly distributed load is qfk=5kN/m2for road bridges. For footbridges qfk depends on the span length.

The main load model is LM1 – see Figure 2.1. It consists of two partial systems: - Double-axle Concentrated Loads (TS – tandem system), each axle having a weight

αQQk[kN] - see Table 2.1.Only one TS should be considered per lane. The relative distances are 120cm at the direction of the traffic (longitudinal) and 200cm at the transversal direction.Each TS should be located in the most adverse position in its lane. The contact surface of each wheel is to be taken as square and of side 0.4m. For the assessment of general effects, the TS may be assumed to travel along the axes of the notional lanes.

For span lengths greater then 10m each TS in one lane could be replaced by a one-axle concentrated load of a weight equal to the total weight of the two axles.

- Uniformly Distributed loads. These loads are αq*9=0.8*9=7.2kN/m2 for notional lane №1 and 2.5kN/m2 for the rest. These loads should be applied only in the unfavorable parts of the influence lines longitudinally and transversally.


22

Figure 2.1. Load Model LM1

Table 2.1. Basic values for LM1

Location Axle loads Qik[kN] for TS qik[kN/m2] for UDL

Notional lane Number 1 αQ*Q1k=0.8*300=240 αQ*q1k=0.8*9=7.2 Notional lane Number 2 αQ*Q2k=0.8*200=160 2.5 Notional lane Number 3 αQ*Q3k=0.8*100=80 2.5

Other lanes 0 2.5 Sidewalks 0 qfk,red=3kN/m2

The load model LM2 consists of a single axle load αQ*Qk=0.8*400=320kN (dynamic

amplification included), which should be applied at any location on the carriageway. The contact surface of each wheel is a rectangle of sides 0.35m and 0.6m. The distance between the wheels is 200cm -see Figure 2.2

Figure 2.2. Load Model LM2

2.00Bridge longitudinal

axis direction

0.60

0.35

Q1k=160 kN

Q1k=160 kN

0.50*

2.00

0.50*

Lane №1Q1k=300 kN q1k=9 kN/m 2

Lane №2Q1k=200 kN q1k=2,5 kN/m 2

Lane №3 Q1k =100 kN q1k =2,5 kN/m 2

32.00

2.00

0.50*

2.00

0.50*

0.50*

2.00

0.50*

1.20

TandemQ1kQl/41kq1k

0.40

0.40

For w l =3,00m


23

The dispersal through the pavement and deck is taken as 1H:1V(450) down to the level of the middle plane of the slab – see Figure 2.3.

Figure 2.3. Dispersal of concentrated loads through pavement and slabs

2.4 Traffic actions on railway bridges

For railway bridges two models of railway loading are given; one to represent normal traffic on mainline railways(LM71) and one to represent abnormal heavy loads (LM SW). Dynamic amplification is not included.

The load model LM71 consists of 4 equal concentrated forces Qvk=250kN/axis and UDL qvk=80kN/m/axis – see Figure 2.4.

Figure 2.4. Load Model LM71 and characteristic value for vertical loads

Load model LM SW (see Figure 2.5) consists of UDL: - qvk=133kN/m for LM SW/O and a=15m and c=5.3m - qvk=150kN/m for LM SW/2 and a=25m and c=7m

Figure 2.5: Load Models SW

2.5 Design combinations of actions Three design situations are defined as follows:

- Persistent situation (P) - situation associated with the normal bridge use; - Transient situation (T) – short time situation (e.g. bridge repair plus snow action); - Accidental situation (A) – situation associated with collision action or explosion - Seismic situation(S). – situation during earthquake

Pavement

SLAB

45°

Qvk=250 kNqvk=80 kN/m qvk=80 kN/m

no limitation 0.80 m 1.60 m 1.60 m 1.60 m 0.80 m no limitation

250 kN 250 kN 250 kN

qvk kN/m

c aa

qvk kN/m


24

Partial factors for the different actions depend on the design situation and the relevant limit state checks. Below are given only the main partial factors used for the design of a simple span bridge.

The action values are as follows: - characteristic value qk– value associated with the bridge design life 100 years; - infrequent value - value associated with 1 year return period. –he value is qk*ψ1,inf; - frequent value – value associated with 1 week return period. The value is qk*ψ1; - quasi-permanent value – value with 50% probability for occurrence each day. The

value is qk*ψ2; Design combination for ULS: - for permanent actions: self weight of structural and non structural elements–

γf=1.35(unfavorable) and γf =1(favorable); - for traffic and other variable actions and P/T situations: γf=1.35(unfavorable) and

γf=1(favorable); Design combination for SLS check for decompression at the beam sections

(frequent values): - for permanent actions: self weight of structural and non structural elements – γf

=1.00; - for traffic actions and P/T situations -frequent values. ψ1=0.75 for TS and ψ1=0.4 for

UDL; - for the prestressed force - rinf=0.9 for post tensioning or 0.95 for pre tensioning; Design combination for SLS check for the beam sections for infrequent values: - for permanent actions: self weight of structural and non structural elements – γf

=1.00; - for traffic actions and P/T situations -frequent values. ψ1,inf=0.80 for TS and ψ1,in–

=0.80 for UDL; - for the prestressed force -rinf=0.9 for post tensioning or 0.95 for pre tensioning; Design combination for SLS check for the beam sections for characteristic values: - for permanent actions: self weight of structural and non structural elements – γf

=1.00; - for traffic actions and P/T situations -frequent values. Ψ0=1 for TS and ψ0=1 for

UDL– - for the prestressed force - γf =0.9 for post tensioning or 0.95 for pre tensioning; Design combination for SLS check for the beam sections for the quasi-permanent

values: - for permanent actions: self weight of structural and non structural elements – γf

=1.00; - for traffic actions and P/T situations -frequent values. Ψ2=0 for TS and ψ2=0 for

UDL– - for the prestressed force - rinf=0.9 for post tensioning or 0.95 for pre tensioning; It is obvious that the quasi-permanent design combinations include only permanent

actions(including prestressing considered as a permanent action).


25

CHAPTER 3: Carriageway, drainage, water proofing (insulation), expansion joints. Safety barriers.Sidewalks

3.1 Pavement and water proof Contemporary road and city bridges, in general, have asphalt concrete pavement. The asphalt concretepavementfor bridges works in unfavorable conditions. The asphalt

concrete serves as an additional protection from water penetration inside the reinforced concrete construction. For these reasons, asphalt concrete pavement must be designed in a suitable way and to be well-constructed.

In the past the thickness was taken to be at least 8 cm.Currently, the pavement must be at least 2 layers dense asphalt concrete with overall depth 10 cm and water proof at least 1 cm.

Previously, asphalt concrete on bridges was the same quality as the one on soil. This resulted in damages and destruction in the asphalt concrete pavement of many of the bridges.Lately, asphalt concrete supplements are put on bridges, which ensure better adhesion with the stone materials, increasedelasticity and durability of the pavement.

The function of water proof is to do not allow water to the reinforced concrete construction. Rainwater is soft and causes corrosion of the concrete, which brings about a change in the chemical composition of the solid concrete. The appearance of white spots is a certain sign of corrosion. Corroded concrete becomes more porous, its strength decreases and the reinforcement is subject to corrosion. During the maintenance of the roads in the winter, many substances that are aggressive towards the concrete (salt, etc) have been used. The water proof of thebridges is subject to dynamical vertical and horizontal loads from the vehicles. Because of all the above reasons, there are more requirements to the bridge materials and details in water proof than in buildings.

The slope of the bridge slab should follow the slope of the bridge pavement. Theflatness and the quality of the execution of the top surface of bridge slab should guarantee the direct placement of water proof on it. It is forbidden from the execution of the cement screed under the water proof and protection concrete above it.

The type of the water proof should be selected during the bridge execution.For new bridges, sheet type water proofs are preferred.

In the bridges, the following kinds of water proof are used: - Sheet type water proof - Plastered type water proof The sheet typewater proofs are normally with thickness 1-5 mm and are imported in a

roll.Only one or two layers are put. In our practice, sometimes three-layer isolations are used.

The materials for plasteredtype water proof are liquid.They are smeared on the surface that is to be isolated.

In the past, on the water proof is placed cement screed,which is reinforcedwith mesh ф5/15cm.Its function is toprotect the water proof during the laying of the asphalt concrete, as well as during the repair of the pavement. During the repair of some of the bridges, it was found out that the concrete screedhas been damagedand it causes damage on the pavement above it.

In the cases without protective screed, a thicker asphalt concrete pavement is recommended – 10-12 cm.


26

3.2 Drainage

Even the best designed and executed water proof does not offer enough guarantee to the protection of the reinforced concrete structure. The fallen water should be drained as quickly as possible. This could be achieved with the slopes of the carriageway and drainage equipment.

The transversal slopes could be two-way and they are assumed at least 2%. In horizontal curve, the slope is one-way and, depending on the radius, could reach 6%.

The bridge construction should have longitudinal slope. It is determined by the grade line.

The diameter of the pipe for typical drainage is 150 mm. In the past, it was considered enough for the lower end of the pipe to be 10 – 20 cm below the bridge slab. Nowadays, it is required of the pipes to end below the lower edge of the construction.

The longitudinal distance between drainage is assumed to be no more than 10 m if the slope is 0.5% and no more than 25 m, if the slope is above 1%. Another requirement of the drainage is to do now absorbwater from more than 400 m2 bridge pavement surface.

3.3 Transition structures at expansion joints The number and the place of the expansion joints depend on the static scheme of the

superstructure. The transition structures should guarantee: - water density; - flatness of the surface of the pavement; - relative displacement of the adjacent elements of the superstructure, or the elements of

the superstructure and the abutment. The displacement of the expansion joints could be determined using methods of

structural mechanics and they depend on the structural system, stiffness and the bearings of the superstructure.

In the past, expansion joint with copper sheet, was often used. Nowadays, it is forbidden of the design and the execution of the copper sheet. The expansion joint should be determined in the design as:

- closed type - opened type The designer should determine the dilatation (opening and closing) of the expansion

joints. For dilatation up to 75 mm (±37.5 mm) is used expansion joint with rubber profile,

attached with steel profiles, anchored in the reinforced concrete structure. The displacements in the expansion joints could be ensured with elastic asphalt concrete,

which allows big deformation.This execution allows displacements up to 50 mm (±25 mm). 3.4 Safety barriers. Sidewalks

The safety barriers could be: - stiff (made of concrete or reinforced concrete) - elastic (made of steel)


27

Safety barriers, made of concrete and reinforced concrete, undergo almost no deformation and the energy of the impact is absorbed at the expense ofdeformation of the vehicle. The simplest barrier is a kerb with 50 cm height. Its disadvantage is the low overturning security of the kerb.

The concrete and reinforced concrete barriers are more expensive than the elastic barriers. Maintenance costs for the stiff barriers are less then those for the elastic barriers.

Elastic barriers are easy repaired and removed. Thesteelparapet must be with a minimumheight 1.1 m. At the expansion joints must be

ensured the dilatation of the steel parapet.

CHAPTER 4: Appropriate systems for the main RCB structures. Special features of the static analysis.

4.1 Statically determinate reinforced concrete bridge structures. Advantages and

disadvantages. 4.1.1 Simple span bridges The main advantages of this bridge system are as follows: - internal forces caused by temperature action do not appear; - internal forces caused by differential movement action do not appear; - easy division into precast element and restoration to the unity of the bridge structure The main disadvantages of this bridge system are as follows: - this system is applied for relatively short spans; - presence of expansion joints at each support; - presence of many supports increases the price of the bridge; - big construction height, respectively decreased clearance below the bridge. The second disadvantage is eliminated using “temperature continuous” bridge slab – see

Figure 4.1.

Figure 4.1. Temperature continuous bridge slab

But in this case the bridge designer should calculate the bending moments caused by the

bridge beam rotation – see Figure 4.2.

Beam Beam

SlabEJ EJ

EJ - expansion joint


28

Figure 4.2. Bending moments at the transition slab caused by beam rotation

If necessary the slab span may be increased using isolation between the beam and the slab – see Figure 4.3.

Figure 4.3. Isolation between slab and beam

In this case the length of the transition slab has been increased. The adjacent bearings at the internal supports must be fixed and movable – see Figure 4.1. Otherwise, the behavior of the bridge structure will be similar to the continuous structure.

휑2 휑 1

4.islab. 휑2

2.islab .휑1

2.islab. 휑2

4.i slab .휑1

+

Slab

Beam Beam

Slab

Beam Beam

isolation


29

4.1.2 Simple span with cantilever bridges The main advantages of this bridge system are as follows: - advantages of the simple span bridges; - the bending moment at the span is less than the same moment for simple span(if the

spans are equal); - the possibility for regulation of the bending moments(with changing the cantilever

length or increasing the permanent actions at the cantilever span) – see Figure 4.4.

Figure 4.4. Simple span with cantilever

- the construction height is less than the same for the simple span bridge with the same

span; - abutments, which are one of the most expensive bridge elements, do not exist.

The main disadvantages of this bridge system are as follows: - the vertical downward and upward deflection at the end of the cantilever causes cracks

at the pavement; - consolidation of the fill at the end of the bridge. The influence of the first disadvantage might be decreased using expansion joints. The influence of the second disadvantage might be decreased using approach slabs. Unfortunately, the above mentioned undesired effects cannot be completely eliminated.

For this reason, the application of this bridge system is limited. The system is replaced with continuous beam with different spans and same bending moment’s distribution. The disadvantage of this design decision is that the end bearings must resist compression and tension – see Figure 4.5

Figure 4.5. Continuous span bridge with different span

Mc Mc

Mspan=Mc

b

b

a

a

a - a

balast

b - b

Q=>R ↓ Q=>R↑

R - reaction inboth direction


30

4.1.3 Gerber type bridges This system has the advantages of the statically determinate systems and the continuous

beams which are mentioned below. The places of the hinges must be carefully selected to prevent: - total collapse of all main and hanged beams; - tensile forces. Example: The possibilities for three span bridge are shown at Figure 4.6. Only the first

option meets both two requirements.

Figure 4.6. 3-span Gerber type bridge

The main disadvantage of this system is the presence of hinges and joints. This leads to

traffic discomfort. For this reason the application of this system now is very limited. Several Gerber bridges still exist in Sofia.

4.2 Statically indeterminate reinforced concrete bridge structures. Advantages and disadvantages. Special features of the static analysis.

4.2.1 Continuous bridge beams This bridge system is appropriate for founding at stiff soils. The main advantages are as

follows: - the bending moment at the span is less than the bending moment at the simple span

beam with same span length; - the construction height is less than the same for simple span beam with same span

length; - the clearance bellow the bridge is bigger than the same for simple span beam with

same span length; - expansion joints at the intermediate supports are missing;

1stoption

2doption

3doption

=

=

Q

Total collapse if part 1 is destroyed

21

3

RCtensile


31

- the possibility for bending moments regulation assuming the first span shorter than the next spans;

- the possibility for normal and shear stress regulation with increasing the depth of the bridge structure or the web of the beams at the supports.

The main disadvantages are as follows: - Internal forces caused by differential soil settlement appear;

- The division into separate elements is difficult; - The design and the construction are complicated.

4.2.2 Frame bridges This bridge system has the advantages and disadvantages of the continuous beams. Each

system, which is mentioned above, could be designed as frame system if the connection between the superstructure and the substructure is stiff.

4.3 Different methods for construction of the bridge systems. Division of the bridge structure into separate precast elements. Principles for division. Restoration to the continuity The all bridge structures, which are considered above, could be designed and constructed

as follows: - slab or slab and beam totally cast in place; - precast beams and cast in place slab; - totally precast elements;

The bridge structures are long structures. So, the construction is usually divided into separate cast in place or precast elements. The restoration of the unity and the continuity of the bridge structure using different construction methods should be provided.

The bridge construction can be divided (see Figure 4.7) using: - vertical longitudinal planes - see Figure 4.7a; - horizontal longitudinal planes - see Figure 4.7b; - vertical transversal planes see - Figure 4.7c.

Figure 4.7. Division of the bridge structure

The rules for the division are as follows: A. The division sections should be where M=0; B. The different elements should have approximately equal size, length and weight;

a) b) c)


32

C. The size, length and weight of the elements should guarantee convenient transport and assembly;

D. If possible, the number of the connections should be minimized. The simultaneous satisfaction of all these rules is almost impossible. Contemporary, one of the most widely applied bridge continuous beam structure is so

called “hidden collar beam structure”. The division sections are at the support. At construction stage the beams are simple span beams. The procedure for bridge construction is as follows:

A. The precast beams are temporarily supported at scaffolding; B. The abutments and the piers are completed; C. The bearings are placed at the supports. If the bridge structure is frame structure the

connection is stiff and the pier reinforcement is anchored at the joint; D. The formwork for the slab and for the “hidden collar beam” is placed; E. The reinforcement for longitudinal and transversal negative and positive bending

moments and for shear is placed; F. The slab and the “hidden collar beam” are cast-in-placed. At the serviceability stage the bridge structure is continuous or frame structure. 4.4 Construction decisions for skew and curved bridge constructions The skew bridges could be easily constructed with straight precast or cast in place beams

and cast in place “skew” bridge slab –see Figure 4.8.

Figure 4.8. Skew bridges

The curved bridges (see Figure 4.9) could be constructed with:

- parallel piers; - radial piers.

Straight beams

a) Simple span b) 2-span skew continuous bridge

pier

axis


33

Figure 4.9. Curved bridges

The advantage of the first design decision is that the length of all precast or cast in place

beams is equal. The advantage of the second decision is the aesthetics of the bridge. CHAPTER 5: RCB slabs

5.1 One way RCB slabs. Static analysis and detailing. 5.1.1 Static scheme

When the ratio between the longer and the shorter span of the slab exceeds 2 then the slab is assumed as aone way reinforced concrete slab.

We consider a strip in the short direction with unspecified width. It is assumed 1m for simplification. This strip is continuous and is supported by the beams. For one way building slabs, the beams have been considered as stiff beams. For bridge road slabs, the beams are assumed as deformed in the span.

For bending moment calculation “by hand”, it is assumed a basic simple span beam with a span L equal to the sum of the clear short span of the slab and the slab depth. The bending moment in the middle of the span has been calculated. The bending moments in the continuous one way slab are determined by multiplying this moment with relevant coefficients.

For shear force calculation, it is assumed a simple span beam with a span equal to the clear distance between the top flanges of the beams. The calculated shear forces are assumed as final design forces.

5.1.2 Static calculation I. Bending moments 1. Bending moment at the basic beam due to the permanent characteristic actions Mgk=gk*L2/8 kNm/m 2. Same due to the load model LM1 2.1 Bending moment caused by UDL at the first notional line q1k=9*0.8=7.2kN/m2 MUDL=7.2*L2/8 2.2 Bending moment caused by TS1 at the first notional line A part of the slab, which is called distribution length, is assumed to contribute for the

carrying of traffic loads. It depends on the position of the load. If it is in the middle of the span then:

a) Parallel piers b) Radial piers

pieraxes

"curved"slab

straightbeams


34

aeff,span=a+l/4 For TS1 check for interaction between the adjacent forces must be carried out. If

aeff,span<1.2m then interaction does not exist and the equivalent UDL for TS1 is qeq=(240/2)/(b*aeff,span).

Otherwise (see Figure 5.1) qeq=(2*240/2)/[b*(aeff,span+1.2)]. If it is at the face of the support thenaeff,sup=a. The maximum moment may be calculated using the influence line for the bending

moment in the midspan. The value MTS1 is equal to the area of the influence line below the load multiplied with

the qeq.

Figure 5.1. Interaction between adjacent wheels

2.3 Same due to the load model LM2 The procedure is the same with different lengths a and aeff,span. The equivalent UDL for

LM2 is qeq=(320/2)/(b*aeff,span). The value MLM2 is equal to the area of the influence line below the load multiplied with the qeq.

L

1,20 m

a eff,s

pan

b

interaction

b

aa


35

2.4 Combinations for the design actions for ULS design I- g+LM1 MI=γg*Mgk+γq*(MUDL+MTS1) II- g+LM2 MII=γg*Mgk+γq*MLM2 The partial safety coefficients γg=γq=1.35 Design basic bending moment is Md=max(MI;MII) 2.5 Bending moments at the continuous beam -see Figure 5.2

- bending moment at the middle of the span MEd,span=0.5Md; - bending moment at the support MEd,sup=0.7Md.

Figure 5.2. Bending moments at continuous bridge slab

II. Shear force The design shear force is at the section at distance d/2 from the support (face of the

beam). Here d is the effective slab depth d=hf-cnom-Ф/2≈hf-4 /cm/. The concrete cover cnom=cmin +Δcdev=20+10=30[mm] 1 Shear force caused by permanent characteristic actions Vgk=gk*(L-d)/2 2 Shear force caused by LM1 2.1 Shear force caused by UDL at the first notional line q1k=9*0.8=7.2kN/m2

VUDL=7.2*(L-d)/2 2.2 Shear force caused by TS1 The contact surface area has been placed at the most unfavorable position according

influence line, i.e. close to section “d/2”. The concentrated vertical force Qk/2=240/2kN is divided into several forces Qjk according the dimensions bj of the effective area Qjk=Qk*bj/b. The forces Qjk have been divided to the respective length a,eff,j from the effective area qjk=Qjk/aeff,j. The shear force VTS1is equal to the sum of the distributed forces qjk multiplied by the respective values vj from the influence line “Vd”.

+

+

-

Md

0,7Md

0,5Md

Menvelope

gq


36

2.3 Shear force caused by TS2 The procedure is the same as 3.2.2 with different dimension “b” with different contact

surface area and different concentrated vertical force Qk/2=320/2kN. 2.4 Combinations for the design actions for ULS design I- g+LM1 VI=γg*Vgk+γq*(VUDL+VTS1) II- g+LM2 VII=γg*Vgk+γq*VLM2 Design shear force is VEd=max(VI;VII)

5.1.3 Design for bending and shear at ULS I. Bending moment calculations The procedure for the calculation of the required area for the tensile reinforcement is

similar as the procedure at the project on “Reinforced concrete”. The differences are as follows:

-minimum concrete grade C30/37 -minimum slab depth is 160mm - design concrete strength is fcd=αcc*fck/γc=0.85*30/1.5=17MPa The number of the bars for 1 linear meter is equal to the calculated area As1 divided to

the area of one bar. The steel grade is B500B with fyd=500/1.15=435MPa. Minimum reinforcement As,min=0.26(fctm/fyk)bd=0.26*(2.9/500)bd=0.0015bd>

0.0013bd. The minimum number of bars is 6 and the minimum diameter is 10. The maximum

number of bars for 1m is 14. The calculation starts by determining of the reinforcement at the middle of the span.

The typical detailing is only with straight bars. The total number of the bars is equal to the number of the bars for 1 meter multiplied by the longer span of the slab.

For the support design the procedure is repeated. II. Shear force calculations The procedure is the same as at the project on “Reinforced concrete” The check is

“design for maximum shear without shear reinforcement”. VEd<VRd,c

Otherwise: - the slab depth should be increased. - tensile top reinforcement should be increased - the concrete class should be increased

5.1.4 Detailing The distribution reinforcement should be at least 20% from the bottom main

reinforcement and at least 4 bars with diameter 8mm. It is placed at the bottom and at the top of the slab.

At the short sides of the slab at the transversal beams are designed top bars without calculation with minimum area 30%As1. The minimum number of the bars is 6 and the minimum diameter is 8mm.

At the free edges of the slab w/o calculation “U” bars are placed with number and diameter as the top reinforcement. The overlapping with the top bars is 40 times bar diameter, but at least 2hf.


37

5.2 Cantilever RCB slabs. Static analysis and detailing 5.2.1 Application of the cantilever RCB slabs

These slabs could be: - a part of main bridge slab; - a part of main bridge slab and beam structure; - a part of main bridge box structure.

5.2.2 Actions 5.2.2.1 Permanent actions

The permanent distributed actions Gj=Aj*γj per 1 meter (see Figure 5.3) could be determined according the relevant areas Aj and volume densities γj of the materials (reinforced concrete, pavement or insulation). The volume density for the pavement and the insulation is assumed 22kN/m3.

It may be assumed G=1kN/m distributed actions for the parapet and for the safety barrier. 5.2.2.2 Traffic actions - The crowd action (LM4) is uniformly distributed 5kPa load according the EN 1991-

3 in case of missing traffic at the carriageway. At the top of the parapet uniformly distributed horizontal load 1kN/m is applied acting at both directions. This crowd action is applied at the sidewalk, at the space between the curb and the elastic barrier (dynamic clearance) and at the carriageway.

- Load model LM1. For local verifications (e.g. bridge slab design) it is allowed to place the contact surface area 40/40cm for the TS1 close to the curb – see Figure 5.3 . It is obviously the most unfavorable situation. For UDL q1k=7.2kPa is applied. The action applied at the sidewalk and at the space between curb and the elastic barrier (dynamic clearance) is qfk,red=3kPa. The dispersal of the traffic actions is similar to the one way slab design – see Figure 5.3. For the exam scale drawing for the dispersal is recommended.

Figure 5.3. Cantilever bridge slab

footing path

Fk

curb

wheel of the

TS

1aa0

central line Δ1Δ1

Δ2

Δ2

휓bot

휓top

pavement45°

45°

32%2%

2a

ljGj

beam

bL2c


38

- Load model LM2. For local verifications (e.g. bridge slab design) it is allowed to place the contact surface area 35/60cm close to the curb. This load model is not acting simultaneously with actions at the sidewalk.

5.2.3 Static analysis

I. Bending moments and shear forces caused by the permanent actions The minimum negative bending moment and the maximum shear force at the fixed end of

the cantilever should be calculated. Mgk=∑Gj*lj, where lj are the relevant distances from the action Gj and the fixed end. Vgk=∑Gj

II. Bending moments and shear forces caused by the trafficactions 2. Bending moments and shear forces caused by the crowdaction LM4 It could be assumed equivalent crowd force Fk=5kPa*lc. Again, the minimum negative

bending moment and the maximum shear force at the fixed end of the cantilever are easily calculated.

MFk=Fk*lk VFk=Fk 2. Bending moments and shear forces caused by the trafficaction The effective slab width for bending moment calculation aeff is equal to the distributed

contact surface area am width plus 1.5 times the distance X between the center of the area and the fixed end – see Figure 5.4.

Figure 5.4. One full contact surface area is placed at the cantilever slab

Δ2

Δ2

x

b

a 1

l c

Δ1

Δ1

Δ2

Δ2Δ1 Δ11

b 0

a 0 a2

Fixed end

a m= 12 (a1+a2)

Qk

Plan


39

The effective slab width for shear force calculation aeff is equal to the distributed contact surface area am width plus 0.5 times the distance X between the center of the area and the fixed end.

Two cases should be considered: A. One full contact surface area is placed at the cantilever (see Figure 5.4) MQk=Qk*X/aeff VQk=Qk/aeff

Where Qk=120kN for LM1 (TS1) and 160kN for LM2 B. One partial contact surface area is placed at the cantilever The acting force only at the cantilever is Qk,1=Qk*b1/b where b1 is the distance from the

fixed end to the end of the distributed contact surface area width and b is the total distributed contact surface area width – see Figure 5.5. The internal forces have been calculated in the same way as in Case A but instead Qk is used Qk1.

2. Design combinations The combinations for ULS check are as follows: MEd=1.35(Mgk+MFk+MQk) VEd=1.35(Vgk+VFk+VQk) 5.2.4 Design checks and detailing 1. Check for bending The rules for the design of rectangular section with only tension top reinforcement should

be applied. 2. Check for shear The check for resisting the shear stresses without shear reinforcement should be

completed. The favorable contribution of the inclined force at the compressive and tensile zone must be considered.

VEd,red=VEd-MEd(tgψtop+tgψbot)/z where ψtopandψbot are the angels of the inclination of the top and the bottom chord respectively – see Figure 5.3.

Figure 5.5. One partial contact surface area is placed at the cantilever slab

x

b

a m

lc

Fixedend

Q1

b

Plan


40

3. Detailing The general rules for bridge slab design are valid. In addition, the top reinforcement should

be well anchored at the adjacent span – at distance at least equal to the cantilever length. 5.3 Two way RCB slabs. Static analysis and detailing

5.3.1 Application of two way RCB slabs The two way RCB slabs are applied in the bridge structures with enough transversal

beams. The ratio long size/short size of the slab must be less than 2. The slab depth is assumed approximately 1/15 from the short slab span but not less than 16cm.

The internal adjacent two way slabs are usually with equal sizes and in this case the two way slabs could be assumed fixed each other.

Contemporary the application of this type RCB slab is very limited due to the design of the structures without transversal beams at the beam span.

The permanent action determination is typical for the bridge slabs. The surface contact area and its dispersal are similar to the one way bridge slabs. 5.3.2 Static analysis The bending moments caused by the permanent actions might be calculated using the

Marcus method. The slab will be case №1 (see below why). Other approach is described bellow.

The bending moments caused by the traffic actions could be calculated using base simplesupported slab (case№1) -see Figure 5.6a. The slab sizes are la and lb and la is the shorter span. The design spans la and lb are the clear dimensions of the slab plus the slab depth.

The limitations for the calculations are as follows: the contact surface area is only one; the contact surface area is situated at the middle of the slab; the size a is parallel to la and the size b is parallel to lb.

The bending moments Ma and Mb at the middle of the slab are calculated using the formulas:

Ma=αQk Mb=βQk

The coefficients α and β depend on the ratios lb/la, a/la and b/la. In this case UDL from LM1 (maximum at notional lane 1 – q1k=7.2kN/m2) could be

considered as one “contact surface area” with dimensions 3m and la (or lb) depending on the traffic direction – see Figure 5.6b.


41

Figure 5.6. Two way bridge slab

If the contact surface areas are more than one the following approach is applied: A) Equivalent UDL is calculated using the formula qeq=Qk/(ab) [kN/m2] Qk=120kN for LM1 (TS1) and 160kN for LM2 B) The individual contact surface areas are united into several sub areas Aj(the sign could

be plus or minus). The areas Aj cover the limitations above. C) For every single Aj is calculated notional action Qj=Aj*qeq The design bending moments Ma and Mb at the simple supported slab are calculated using

the superposition formulas: For TS1 and case with 4 contact surface areas – see Figure5.7: Ma,k=α1Q1k-α2Q2k-α3Q3k+α4Q4k Mb,k=β1Q1k-β2Q2k-β3Q3k+β4Q4k

Figure 5.7. TS1

For LM2 and case with 2 contact surface areas – see Figure 5.8: Ma,k=α1Q1k-α2Q2k Mb,k=β1Q1k-β2Q2k

/2b

a

la

l b

a

3 m

lb la

contactsurface

area

UDLq1k

a) b)

1.20 m

2.00 m

= - - +1 2 3 4

a1 a2 a3=a1 a4=a2

b 1 b 2=b

1

b 3

b 4=b

3


42

Figure 5.8. LM2

The bending moments Ma,g and Mb,g caused by the permanent uniformly distributed action

could be calculated using the following approach: - the total permanent load is G=g*la*lb; - the permanent load is considered as a “contact surface area” with dimensions la and

lb. The design combinations for the bending moments at the simple supported slab are similar

to the design combinations for the one way slabs. They are separately determined for each direction for the slab.

The design bending moments for every direction a and b for the simply supported slab are: MEd=1.35(Mgk+MQk) The design bending moments for the fixed two way slab are as follows:

- at the middle of the span Mspan=0.525MEd - at the support - Msup=0.75MEd

The shear force calculation “by hand” is very difficult. Conservatively, it might be assumed that the design shear force is the same as for the one way slab with design span equal to a clear short span of the slab.

5.3.3 Design checks and detailing

The reinforcement determination is the same as for the one way bridge slab. For each direction effective depth should be calculated. For the higher bending moment is required higher effective depth.

The detailing of this bridge slab type is similar to the one way slabs. The rules for the reinforcing at each direction apply.

5.4 RCB slabs without beams. Static analysis and detailing 5.4.1 Application of RCB slabs w/o beams. Main dimensions The main advantages of this bridge system are as follows:

2.00 m

= -1 2a1 a2

b 1 b 2=b

1


43

- the possibility for application at each road situation (skew, curve, special); - low construction height; - easily construction. The main disadvantages of this bridge system are as follows: - heavy self weight; - application for relatively short spans; - significant expenses for formwork, reinforcement and concrete; - difficult dispersal at segments. The typical construction of this bridge system is cast-in-place erection. The slab cross

sections are as follows (see figure 5.9): - solid slabs without voids; - voided slabs with round or rectangular voids.

Figure 5.9. Bridge slab cross sections

The ratio “slab height/span length” depends on: - type of the cross section of the slab (solid or voided); - structural system (simple span slab or continuous); - type of the slab (prestressed or not).

The typical limits for this ratio are 1/18-1/25. It is important that the self weight bending moment depends on the third degree of the

span length. So, the application of solid slab w/o voids is limited for relatively short spans. When the ratio “span length/slab width” exceeds 5 (long and relatively narrow bridge)

the bridge structure may be assumed as a “beam”, which resists longitudinal bending, shear and torsion. Otherwise, it may be assumed as a “slab”,which resists longitudinal and transversal bending, shear and torsion.

The voided slabs are very effective bridge structure, especially for prestressed slabs. In this case the slab self weight is significantly decreased without considerable stiffness reduction.

The voided slabs with round voids have been more easily constructed than the slabs with rectangular voids. The materials for the round voids are:

- pressed pasteboard pipes; - steel sheet pipes; - rubber pipes filled with air under pressure. The pipes should be connected to the shuttering (usually water proof plywood) with rings

for stability during concreting. The construction phases for voided slabs with rectangular voids are as follows: - shuttering, reinforcement and concreting of the bottom slab; - shuttering, reinforcement and concreting of the web;

a) Solid slab b) Voided slab


44

- permanent shuttering, reinforcement and concreting of the top slab. The slab cross section for continuous voided slabs is not sufficient to resist punching at

the supports. It is necessary to assume and design solid slab at the supports. See bellow for punching design and detailing.

The solid and voided slabs could be constructed with precast elements. The most typical sections are as follows:

- solid beam elements, connected with concrete dowels. This system is typical for culverts – see Figure 5.10

- hollow box beams with cast-in-place slab – see Figure 5.11; - flanged beams with connected top and bottom flanges see Figure 5.12, etc.

Figure 5.10. “”Precast” solid slab

Figure 5.11. “Precast” and cast in place hollow slab

Figure 5.12. “”Precast” voided slab

5.4.2 Static analysis The internal forces, reactions and the displacements for the solid bridge slabs could be

determined using different computer programs based on FEM. The “shell” type elements are used.

The following main recommendations could be applied: - the local coordinate axes for the elements should follow the direction of the main

reinforcement; - the finite elements set must be condensed at the supports;

concrete dowels

concrete dowels


45

- the concentrated loads should be distributed to the element joints; - the triangle finite elements should be avoided; - for the internal forces at the slab it is enough to restrain only the vertical deflection

at the supports; - the elastomeric bearings should be modeled with springs. . The internal forces caused by permanent actions for bridge slabs are usually 60-85%

from the maximum moment. So, it is possible to place the concentrated traffic actions at the nearest joint without significant mistake.

The voided slabs could be modeled with bridge beam grid. At transverse direction the top and bottom slab are exchanged with one equivalent slab with depth equal to the sum of the top and bottom slab depths – see Figure 5.13.

Figure 5.13.

5.4.3 Design checks The internal forces for the bridge slabs at longitudinal direction are higher than at

transversal direction. The slab behavior is similar to one very wide “beam”, which resists bending moments, shear and torsion.

At transversal direction the reinforcement is calculated to resist bending moments at the cantilever parts and small transverse bending moments.

The punching shear design at the supports is decisive for the slab depth. The punching shear resistance is considerably decreased if elastomeric bearings are used. In this case the following measures could be applied:

- increased slab depth; - increased concrete grade; - design heads; - design shear reinforcement (stirrups or bent bars). 5.4.4 Detailing The typical approach for detailing is as follows: - divide the slab width into several hidden “beams” – (see Figure 5.14); - place open stirrups; - place bottom reinforcement that resists torsion; - place longitudinal bottom reinforcement (bottom and top reinforcement for

continuous slabs); - place top transversal reinforcement;

x x

b'w h' w

b

bw

h

Element type 1

bw

h w+h

' w

h w

Element type 2

type 2

type 1


46

- place cover stirrups.

Figure 5.14.

For voided slabs the voids have been placed between the stirrups. For skew bridge slabs the typical approaches for detailing are as follows: A. Rectangular reinforcing for: - long and narrow bridges without significant skew – see Figure 5.15a; - very wide bridges without significant skew – see Figure 5.15b.

Figure 5.15. Skew bridge slabs

B. Skew reinforcing for other cases.


47

CHAPTER 6: Static analysis of the bridge superstructure

6.1 Methods for static analysis of RCB Methods for static analysis of RCB structures may be summarized in three main groups: A. Analytical methods These methods are usually used for calculations “by hand”. They have been applied for

structures that suit following requirements: - the elements of the structure are struts; - the number of the unknown quantities is small; - the connection “supporting member-structure” is simple (hinge supports or fixed

supports). The final results of the application of these methods are: - the functions of the internal forces or displacements; - the exact values of the internal forces or displacements at selected cross sections or

points. These following methods from the structural mechanic are included in this group: - Method of forces; - Method of displacements; - Mixed method;

The following analytical methods for RCB static analysis are used: - Leonhardt-Andrae’s method; - Method of infinitely stiff transversal beam (method of eccentric compression); - Method of lever arm.

B. Semi analytical methods These methods are used both for calculations “by hand” or by computer program. They

are applied for more complicated bridge structures. The final result of the application of these methods is the functions of the unknown

internal forces or displacements for the entire structure or for each structure element. The following semi analytical methods for RCB static analysis are used: - Semi analytical method of finite strips (FSM);

- Method of infinite Fourrier’s rows (used for RCB structures without transversal beams at the span of the main beam).

The FSM is suitable for bridge thin walled sections, eg. box sections. A. Numerical methods These methods are used only for calculations by computer program. They are effective

for very complicated bridge structures: - Skew or - Curved or - Continuous or frame or - Supported at elastomeric bearings. These methods are always used for dynamic analysis of bridge structures (impact

actions, seismic actions).


48

The final result of the application of these methods is the unknown internal forces or displacements at the joints of the grid for each structure element. The functions for the forces or displacements remain unknown.

These methods are approximate methods. The most widely applied method of this group is Finite Element method (FEM). 6.2 Leonhardt-Andrae’s method. Application. Influence lines The following assumptions for the bridge structure must be valid using Leonhardt-

Andrae’s method: - equal distance between beams; - the bridge must be situated in a straight line; - the bridge is without skew; - the second moment of cross section area for the end beam and for the internal beams

are approximately equal; - the beams are simply supported; - the base case is with one transversal beam at the middle of the span and two

transversal beams at the supports of the beams. The following assumptions for further simplification are assumed: - The main beams are spring supports of the central transversal beam. - The end transversal beams are continuous beams at stiff supports. - The slab is infinitely stiff at its own plane. - The slab is replaced by its contribution to the stiffness of the main and the transversal beams - The torsional stiffness of the main beam and of the transversal beam is neglected. In this case it is assumed hinge connection between the main beams and the transversal

beams and only one vertical unknown force at the hinge. The case with more than one transversal beam is reduced to a base case assuming (see

Figure 6.2): - i=1.6 for 3 and 4 internal (w/o end beams) transversal beams. For i see formula (6.1); - i=2 for 5,6….internal(w/o end beams) transversal beams The static scheme of the bridge grid is shown at Figure 6.1.


49

Figure 6.1. Static scheme for Leonhardt-Andrae’s method application

Figure 6.2.

The calculation is conducted through two stages: - At transversal direction – transversal distribution of the traffic action is determined; - At longitudinal direction – the internal forces are calculated.

The following scheme for the calculation must be applied: - Determination of the influence lines “Rj” for the vertical force at the hinges; -Loading of the influence lines with the nominal traffic action. For simplification the value

of the action is assumed equal to unity. The place of the actions must be at the most unfavorable condition. The calculated force is called “transversal distribution coefficient”.

The influence lines for the reactions at main beam “j” at the middle of the span “Rj,span”are straight or curved lines. The criteria is the ratio (“main beam static span length L”)/(”width of the bridge structure” - the distance between the end beams b1). It is assumed that for ratio L/b1≥2 (long and narrow bridges) the transversal beam could be assumed as infinitely stiff and the influence lines are straight lines. Only two points are required for the influence lines determination. The values depend on the number of the beams and the bridge geometry at transverse direction.

A

B

Jtr

JIIRj,span

Rj,support

1 i j n

d d db1

1/4

2/4

1/4L

supp

ort

span

supp

ort

"A"

Rj

"B"

Rj

L

h f

beff=L/2

Jtr

d d d

Iststage

IIdstage JII (or J)

i j

Jtr

2 transversalbeams

Jtr

3 transversal beams


50

Otherwise (ratio less than 2) the influence lines are curved lines – see Figure 6.3. The values at the beams depend on:

- Numbern of the main beams; - stiffness factor coefficient z, calculated using formula (6.1)

z=i*(Jtr/JII)*(0.5L/d)3 (6.1) Here Jtr is the second moment of the transversal beam cross section including the

effective slab depth beff=L/2. - number of the transversal beams. For the case of missing internal transversal beams

or 1 and 2 internal transversal beam i=1 at Formula (6.1). In case of missing internal beams Jtr=(L/2)*hf

3/12

Figure 6.3. Influence line for the internal forces at the span “푅 , ”

The sum of the values Rji for one beam must be equal to unity. At the cantilever parts of

the bridge it is assumed linear distribution of the influence line. The tangent line at the end might be used. The intermediate values may be calculated by linear interpolation.

The influence lines for the forces at the end of the span“Rj,support are influence lines for the reaction at the continuous beam -see Figure 6.4. For simplification it is assumed triangle distribution of the influence line.

Figure 6.4. Influence line for the internal forces at the support “푅 , ”

Rj

j1 nspan

"Rj,span"Rjn

RjjRj1

tangent linelinear distribution

+

-

j1 n

"Rj,support"+ 1

1+"Rj,support"


51

6.2.1 Transversal distribution coefficient

The influence lines are loaded using the following rules -:see Figure 6.5 - the nominal value is 1kN/m2 for UDL at first notional line and 2.5/7.2=0.35kN/m2

for the other notional lines. For the sidewalk the nominal value is 3/7.2=0.42kN/m2; - For UDL only positive part of the influence line has been loaded. The reaction for the

beam j (called notionally TDS – transversal distribution coefficient) is TDCj,UDL=∑(nominal values)*(area ωof the influence line below the UDL position);

Figure 6.5. TDC determination for UDL

- the nominal value is 1kN/axis for TS1 at first notional line (2*0.5kN for each wheel-

see Figure 6.6) and 160/240=0.666kN for the second notional line TS2(2*0.333kN for each wheel) and 80/240=0.333kN for the second notional line TS2(2*0.167kN for each wheel).

- The minimum distance between the curb and the first TS1 wheel is 50cm; - The distance between each TS wheel is 200cm; - The minimum distance between the adjacent TS wheels is 100cm; - The TS are applied since the reaction is increased. Otherwise, the TS is omitted!


52

Figure 6.6. TDC determination for TS

ThenTDC1,TS=∑(nominal value)*(value of the influence line ηbelow the wheel position).

6.2.2 Internal forces at the main longitudinal beams The schedule is as follows: 1. The influence lines for the internal forces should be determined and drawn – see

Figure 6.7; 2. The transversal distribution for the each traffic load should be considered. It is

assumed constant transversal distribution for the middle one half of the bridge L/2. The relevant TDC coefficients for the “span” and for the “support” are used;

3. The influence lines should be loaded at the most unfavorable condition. For the TS and for span lengths greater than 10m the internal forces caused by TS can be

calculated by replacing each tandem system by a one-axle concentrated load equal to the total weight of the two axes, i.e. 2*240=480kN.

j1 n

b1

1 kN/axis

휂1 휂2 "Rj,span"

50 200 100 200 3rd line (if possible)

2/3 kN/axis

0.5 0.5 1/3 1/3

h"R fj,support"


53

Figure 6.7.

For TS bending moment M (or shear force V)= 480*(value from the influence line

below the force 480kN)*(value from the TDS diagram line below the force 480kN). For UDL bending moment(or shear force)=7.2*∫(influence line diagram)*(TDC

diagram below the UDL position). 6.2.3 Internal forces at the transversal beams The following approach for approximate determination of the internal forces at the

transversal beams has been applied: - determination of the influence line for the internal force at the design section using

the method “general formula” – see Figure 6.8; “Mm”=”R1”*l1+”R2”*l2-“F*lf” “Vm”=”R1”+”R2”-“F=1”

m

Qk=480kN

L/4 L/2 L/4

Beam j

+m1"Mm"

TDCj,TSTDCspanj,TS

"Vm"+

-

V1

k1


54

Figure 6.8.

m

1 2 3 4 5

1 2

F=1 LF m

Qm

Mm

R1 R2

L1

L2

-

R11R12 R1m

R13 R14

R15

"R1"

R21 R22 R2m R23R24 R25

"R2"

"[1.LF]"L1 L2

"[1]"1 1

"Mm"

minM

maxM maxM

minM

Sc Sc

Sc Sc

?1Mm1

Mm2

Mmm

Mm3

Mm4 Mm5

"Qm"

Qm1

1

Qm2

Qmmt

QmmdQm3

Qm4

Qm5


55

- longitudinal distribution of the traffic loads between transversal beams – see Figure 6.9;

- It is possible to assume triangle or parabolic distribution at longitudinal direction.

Figure 6.9. Longitudinal distribution of TS loading

- loading of the influence line at the most unfavorable condition for positive and

negative values –see Figure 6.10.

Figure 6.10.

ScSc23Sc

200 100 200

1 2 jm

+ +- -

"Mm"

"Vm"

23Sc

ScSc23Sc

23Sc

200 100 200

240

A B C D E

240 120

휂1

휂1

휂2

휂2

OR S c= (240/2)*휂 j

"Sc"


56

6.2.4 Special cases 6.2.4.1 Case with more than 8 main beams The following approaches for simple calculations are applied: - using the values from the table for 8 beams and neglecting the values for distant

beams; - forming a coupled “beams” and reducing the overall number of the beams – see

Figure 6.11. The calculated internal forces are referred for two beams.

Figure 6.11. Reducing n=16 beams into n=8 beams

6.2.4.2 Application for continuous beams The above considered methods might be used for continuous beam type bridges as

follows: - determination of the influence lines for the required internal forces at the relevant

section see Figure 6.12. for “MB”; - determination of the transversal distribution as shown for simple span beam;

- loading of the influence lines. The distance between zero points from the bending moment diagram is assumed for each

span for stiffness coefficient calculation (see Figure 6.12.).

1 2

2 coupled beams

j

n=16

"Rspan1 "

R11 R1j

R1n


57

Figure 6.12.

6.3 Method of infinitely stiff transversal beam. This method belongs to the analytical group of methods. It is assumed that for long and

narrow bridges (L>2*b1) the local transversal bending of the transversal beam could be neglected. This method is applied if at least one transversal beam is constructed. Actually, the transversal beam is not infinitely stiff (Itr=∞). Conditionally stiffness factor z=∞.

The deformation at transversal dimension consists of two components (see Figure 6.13): - translation; - rotation.

Figure 6.13. Method of infinitely stiff beam

L1 L2 L3

L~0.8L1 L~0.6L2 L~0.8L3

g

A B C D

"M "B

b j

e

j1 nc

Q

QM=Q.e

b2 b1

Q

(translation)

Rj =Qn +

휑 (rotation)

j

Rj ~ bj


58

The forces for every main beam for the first component (translation) are Ri=Q/n, where n

is the number of the beam. The forces for the second (rotational) component are proportional to the distance between the axis of the beam and the axis of the structure.

The influence lines are straight lines. The values do not depend on the stiffness of the beams and on the bridge length. They depend only on the bridge geometry and on the number of beams.

The values Rij could be calculated as follows: Rij=1/n+[bibj/[2*(b1

2+b22…)]. (2)

For bj see Figure 6.13. 6.4 Method of lever arm

This method is applied for structures with small transversal stiffness: - bridge structures without transversal beams at the span; - bridge structures with big distance between main beams.

Conditionally the stiffness factor is assumed equal to z=0. In this case it is assumed hinge connection between the transversal and the main beams at the span – see Figure 6.14. Actually, the transversal continuous beam is divided into several simple span beams.

The influence lines are straight lines with Rii=1 for the beam “i” and Rij=0. In this case the transversal distribution at the span and at the supports is equal and the transversal distribution coefficient for the span and for the supports is the same.

Figure 6.14. Method of lever arm

The obtained results from the application of this method are as follows: - the internal forces for end main beams are lower than the actual internal forces; - the internal forces for internal main beams are higher than the actual internal forces;

6.5 Finite element method (FEM)

R j

j1

1 "R spanj "

"R suppj "


59

The main advantage of FEM is that it is universal for application. The FEM application is easy. Actually, FEM is a numerical method for solution of differential equations (ordinary or partial). It may be applied for every process that could be described with differential equation.

It may be applied for every simple or complicated RCB structure. The most widely applied computer programs for static and dynamic analysis of structures which use FEM are:

- SAP 2000; - ANSYS; - TOWER; - PLANET; - SOPHISTIC; - Others.

It is possible to make “exact” model of the structure with considering all features of the structure (geometry, stiffness, material properties, bearings, etc.).

The results of the application of this method are the internal forces and the displacements at the grid joints.

The main disadvantages of FEM are as follows: - the functions of the internal forces or displacements remain unknown; - the output results are difficult for treatment; - as a rule, at the joints where the regularity is broken the results are not reliable.

There are three main FEM models for RCB structures: - only with frame elements; - only with shell elements; - combination of frame and shell elements.

A. Model with frame elements The RCB structure is modeled as a beam grid – see Figure 6.15. The elements are:

- main beam with the effective slab width; - transversal beams with the effective slab width; - slab between the transversal beams.


60

Figure 6.15. The main disadvantages of this simple model are:

- the slab that connects the main and the transversal beams is replaced only by its stiffness;

- the eccentricity between centers of gravity of the slab, transversal beams and main beams is ignored;

h fh 1

1a 1 1a1

c a a a c

2 3 4 3 2

l/8+b2/2 l/4 l/4 l/4 l/8+b2/2

h 2

b2

b3

h 3l/4 l/4 l/4 l/4

l/2 l/2

l

b)

a)

1a

1a

1

1

2

2

3

34

1a

h fh 1

b1

h fh 1

b1

1

h 2

b2

2

h f

b3

h 3 h f

l/44

3

h f

l/4

d)

c)


61

- this model is appropriate only for vertical actions and not appropriate for dynamic analysis

The actions that are not situated on the frame elements should be distributed between the main beams using the rule “simple span beam reactions”.

B. Model with shell elements This model is applied for RCB slender opened or closed structures. It is appropriate for prestressed RCB. As a rule, this type RCB are with thin webs and the

web shape is similar to vertical “slab”. This model is appropriate for gravity and dynamic calculations.

C. Model with combination of frame and shell elements The elements of the model are:

- slab is modeled with shell elements; - beams are modeled with frame elements; - the slab and beam joints are connected with very stiff frame elements or “constrains”.

The axes of the frame elements connect the centers of gravity. This model is appropriate for gravity and dynamic calculations.

6.6 Finite strip method (FSM) This method belongs to the group of semi analytical methods. It is applied for RCB

structures with opened or closed cross section that meet the following requirements: - the axis of the structure is straight; - the stiffness in longitudinal direction is constant; - the depth of the cross section elements is constant; - at the end of the structure are placed transversal diaphragms.

The end diaphragms have stiffness, which is equal to infinity in its plane and zero outside its plane. FSM is very effective for box beam structures.

The RCB has been divided in longitudinal direction into “finite strips” – see Figure 6.16. The strips are simply supported at the end diaphragms and “fixed” at the longitudinal “joints” that connect the strips. The results of this method are the functions of the displacements and the internal forces for each finite strip.

Figure 6.16. The main advantage is that the number of the finite “elements” (strips) is small and the

calculation process is considerably simplified. But it is obvious that FSM is not universal for application as FEM.

6.7 Analysis of RCB structures without transversal beams.

X3 Z

X1 X

X2 Y

X 2

X1

X3


62

For this type RCB FEM or FSM could be easily applied. Another method is the method of infinite Fourier’s rows. The construction has been divided with longitudinal cross sections. The unknown internal forces at each section are bending moment, shear force, normal force and sliding force. – see Figure 6.17.

Figure 6.17.

The functions are searched into Fourier row form. The unknown four (one for each

unknown internal forces) Fourier coefficients are calculated using the following four equations:

- differential movement and rotation at the joints is zero; - longitudinal and transversal curvature for the joint sides is equal. CHAPTER 7: RC box section bridges

7.1 Application of RC box section bridges RC box section bridges are used mainly for long span continuous or frame bridges and

for bridges with significant torsional moments caused by permanent or traffic actions (e.g. curved bridges).

The main advantages of the RC box section bridges are as follows: - self weight of the structure is considerably reduced; - significant bending stiffness; - significant torsional stiffness; - aesthetics. The elements of the box section are top and bottom slab, cantilever slabs and web walls –

see Figure 7.1. As a rule, the top plate is with constant depth and the bottom web depth is increased for compression resistance at the intermediate supports – see Figure 7.1.

(j)

Mj(x)

(j-1) (j+1)

Vj(x)

=>+ Tj(x)

Nj(x)


63

The web walls could be vertical or inclined - see Figure 7.1. The inclined web walls have nice view. On the other hand, in this case the bottom slab width is decreased and its depth should be increased at the supports.

Figure 7.1. Box section types

Normally, the overall height of the RC box section has been increased at the support

direction. The most typical case is RC box section with one cell. It is easy for construction. But, in

case of very wide RCB the transversal bending is increased and the following measures have been applied:

- design additional cell(s); - transversal prestressing at the cantilever and top slabs Design of additional cells leads to significant construction difficulties. The typical bridge construction systems for RC box bridges are as follows: - cast-in-place construction; - balanced cantilever erection (segmental or concreting); - concreting “span by span”; - precast construction;

7.2 Types of torsion for RCB structures 7.2.1 Pure torsion It is the condition of the structure where torsional moment T≠0 and all other internal

forces are equal to zero. This is not typical case for RCB structures. 7.2.2 Warping torsion In this case at one or more sections the warping is limited. “For closed thin-walled

sections, warping torsion may normally be ignored” (EN 1992-1-1, Section 6.3.3 (1). “In open thin walled members it may be necessary to consider warping torsion. For very slender cross-sections the calculation should be carried out on the basis of a beam-grid model and for other cases on the basis of truss model. In all cases the design should be carried out according the design rules for bending and longitudinal normal force, and for shear” (EN 1992-1-1, Section 6.6.3 (2)).

7.2.3 Equilibrium torsion This is the case where the static equilibrium of a structure depends on the torsional

resistance of elements of the structure. The typical example is torsion at the collar beam of the abutment or pier.

a

a

a

a

top slab

web walls

bottom slab

constant

increased

cantilever slab


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7.2.4 Compatible torsion This is the case where, in statically indeterminate structures, torsion arises from

consideration of compatibility only. If it is assumed that the structure is not dependent on the torsional resistance for its stability, then it is normally to neglect this type of torsion. This approachis used in the Leonhard-Andrae method.

7.3 Section properties for torsion calculations The torsional resistance of a section may be calculated on the basis of a thin-walled

closed section, in which equilibrium is satisfied by a closed shear flow. Solid sections may be modeled by equivalent thin-walled sections. Complex shapes, such as T-sections, may be divided into a series of sub-sections, each of which is modeled as an equivalent thin-walled section, and the total resistance taken as sum of the capacities of the individual elements – see Figure 7.2.

Figure 7.2.

The distribution of the acting torsional moments over the sub-sections should be in proportion to their uncracked torsional stiffnesses. For non-solid sections the equivalent wall thickness should not exceed the actual wall stiffness.

7.4 Static analysis of RC box section bridges The simplest model of one RC box bridge structures is a single frame element subjected

to bending, shear and torsion. The disadvantages of this model are as follows: - in case of wide top slab the normal stress distribution is considerably different then

the actually calculated; - the possible transversal bending has been ignored; - the possible contour deformation has been ignored The FEM could be used. But in case of using shell elements the torsional stiffness of the

bridge structure has been considerably decreased then the actual overall torsional stiffness (up to 50%).

The FSM is the most appropriate method in case of straight RCB with constant stiffness. For local analysis and for simple consideration of the contour deformation the following

approach has been applied: - the cross section is assumed as outside statically determinate structure -see Figure

7.3; - the variable action has been applied and “frame” has been calculated.

Figure 7.3.

OR

JT JT2

= > JT=max( JT1; J T2)

=>

Q

M


65

7.5 Design checks and detailing The effects of bending, shear and torsion for both hollow and solid members may be

superimposed. The webs are constructed applying the rules for the walls.

CHAPTER 8: Design checks and detailing for RCB elements w/o prestressing

8.1 General assumptions and notations The general assumptions are:

- plane sections remain plane; - the tensile strength of the concrete is ignored; - the strain in bonded reinforcement, whether in tension or compression, is the same as

that in the surrounding concrete; - the stresses in the concrete in compression are given by the design stress-strain

relationship; - the stresses in the reinforcing steel are given by the design stress-strain relationship; - the maximum strains are 3.5‰for the concrete and 25‰ for the steel; - in case of centric compression the maximum strains is 2‰ for the concrete (1.75‰ in

case of bilinear stress-strain relation) The general notations are:

- εc is the maximum compressive strain at the concrete fiber; - εs1 is the maximum strain at the level of the center of gravity of the tensile

reinforcement; - εs2 is the maximum strain at the level of the center of gravity of the compressive

reinforcement; - d is the effective depth of the section; - x is the depth of the neutral axis of the section; - 0.8x is the depth of the equivalent rectangular block; - z is the lever arm of the internal forces (z=d-0.4x); - As1 and As2 is the reinforcement cross section area at the tensile or compressive zones

respectively The ultimate moment resistance of a section can be determined using so called “strain

compatibility method” by either algebraic or iterative approaches. An iterative approach is possible using the following steps:

1. Guess a neutral axis depth and calculate the strains in the tension and compression reinforcement by assuming a linear strain distribution and a strain of εcu2=3.5‰ at the extreme fiber of the concrete in compression;

2. Calculate from the stress-strain idealizations the steel stresses appropriate to the calculated steel strains;

3. Calculate from the stress-strain idealizations the concrete stresses appropriate to the strains associated with the assumed neutral axis depth;

4. Calculate the tensile and compressive forces at the section. If these are not equal, then adjust the neutral axis depth and return to step 1;

5. When the forces are equal then take moments about a common point in the section to determine the ultimate moment resistance.

An algebraic approach is described at the next point.


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8.2 Design at ULS of members with rectangular section subjected to bending As a rule the bridge reinforced concrete sections are detailed in several rows. In this case

for first assumption it might be assumed d=h-10cm. First step is to find the damage type – brittle (damage of the concrete) or plastic (damage

of the steel). The approach is conducted in several steps: 1. Assume simultaneously damage of the steel and the concrete - εcu=3.5‰(point B)

and εud=25‰(point A) . Using similar triangles depth of the neutral axis x=0.123d; 2. The depth of the equivalent rectangular block is 0.8x=0.098d. The lever arm is z=d-

0.098d/2=0.951d. 3. The compressive force is Fc=fcd*(0.098d)*b; 4. The moment of resistance at this case is MAB=Fc*z=fcd*(0.098d)*b*0.951d=0.093bd2. 5. If MEd is less then MABthen the case is “damage of the steel”. Because 0.951d≤z≤d then the assumption z=0.951d is conservative and the “mistake” will be less then 5%. 6. If MEd exceeds MABthen the case is “damage of the concrete”. The moment MEd cannot be resisted. It is necessary to calculate new x using the equation: MEd=MRd=Fc*z= 0.8x*fcd*b*(d-0.4x) The roots that are not real, negative and the root x≥d shall be neglected. The root x1≥0.123d is assumed.

Check for brittle failure x1≤xlim=0.45d. Otherwise compressive reinforcement is required. 7. In case of only tensile reinforcement strain of the reinforcement is εs1=(3.5‰)(d-x)/x

If εs1≥εyd=fyd/Es then σs1=fyd and the reinforcement area

As1=Fs/fyd Fc=Fs=0.8x*fcd*b If εs1<εyd=fyd/Es then σs1<=fyd but this is not practical because the tensile reinforcement is

working with low stress. If compressive reinforcement is required the schedule is as follows: 1. Determination of the moment of resistance if x=xlim=0.45d Ms1=0.8xlim*b*fcd(d-0.4xlim)=0.2952*b*fcd*d 2. Required compressive reinforcement shall be determined for the moment ΔM=MEd-Ms1

3. The compressive force at the reinforcement is Fs2=ΔM/(d-d2). Deformation εs2 at the level of the compressive reinforcement is εs2=(3.5‰)(d-xlim)/xlim.

4. See point 7 above. As2=Fs2/σs2 5. The force at the tensile reinforcement is Fs1=Fc+Fs2 and As1=Fs1/fyd 8.3 Design at ULS of members with T-sections subjected to bending There are two approaches for the design of T-sections subjected to bending: 1. Assume lever arm z=d-hf/2. The web contribution is neglected. Then Fs1=MEd/z

The compressive chord (slab) is working almost in uniform compression and the strain is limited to 2‰ (1.75‰ in case of bilinear stress-strain relation)at the center of the slab.

2. Using convenient Tables.


67

8.4 Design at ULS of members with rectangular section subjected to bending with axial force

For the RCB is actual the case “bending with axial compressive force”, e.g. bridge pier

walls or columns. Because there are a lot of combinations for the bridge actions the design is conducted using interaction diagrams.

In practice computer programs will be required to use “strain compatibility method”, e.g. program “GALA REINFORCEMENT”.

8.5 Design checks for RCB elements for shear The design of RCB members for shear is usually carried out as a check after the flexural

design and therefore basic section sizes and properties should already be chosen. For flanged beams with thin webs the maximum shear resistance achievable VRD,max, may, however, need to be considered at the initial sizing stage to ensure that the web thickness is great enough. It may also beneficial to increase the section sizes and reduce the shear reinforcement content for economical and build ability reasons.

In case of sections with changing depth EC2 allows to take account of the vertical components of the inclined tension and compression chord forces in the shear design of members with shear reinforcement. These components are added to the shear resistance based on the links.

Truss model has been applied at ULS for the shear reinforcement design and check for the compressive struts.

For members not requiring design shear reinforcement VEd≤ VRd,c

VRd,c=0.12k(100ρlfck+0.15σcp)1/3 bd, but minimum Vmin=(0.035k3/2 fck

½+0.15σcp)bd In this case minimum shear reinforcement is provided with ρmin==0.10(fck)½:/fyk ρ=Asw/(bs) For members requiring design shear reinforcement VEd≥VRd,c

Asw/bs=VEd/(0.9dfydcotθ) The angle θ=22-450 is assumed by the designer. The maximum resistance of the compressive struts is: VRd,max= αcwbwzν1fcd/(cotθ+tanθ) 8.6 Design checks for RCB elements subjected to shear and torsion

The torsional resistance of sections may be calculated on the basis of a thin-walled closed section even if the section is solid.

The crushing limit for combined shear and torsion is calculated assuming the same value of the compressive strut angle for both effects

TEd/TRd,max+VEd/VRd.max ≤1 Where TRd,max= αcwbw(2Akteff)zν1fcd/(cotθ+tanθ) The required longitudinal reinforcement for torsion is: Asl/uk=(TEdcotθ)/(2Akfyd) The required transversal reinforcement for torsion is: Ast/st=(TEdtanθ)/(2Akfyd)


68

This reinforcement has been added to the shear reinforcement. 8.7 Design check for shear stresses between web and flanges of T-sections Longitudinal shear in flanges is checked using a truss model. The check covers the

crushing resistance of the compressive struts and the tensile strength of the transverse reinforcement.

For the details see the course on “Reinforced concrete” . For the angle θ is assumed cotθ=2. If additional transverse reinforcement is required then it is added to the slab

reinforcement. 8.8 Detailing of RCB beams. Concrete cover and distance between longitudinal bars

The minimum concrete cover for slabs and beams is 20+10=30mm for slabs and 20+15=35mm for beams. The minimum clear distance between bars and rows of bars must be at least:

- 20mm; - maximum bar diameter; - dg+5, where dg is the largest nominal maximum aggregate size in milimeters. For wide beams with top reinforcement the distance between the bars at the middle of the

cross section is assumed at least 15 cm for easy concreting. Anchorage of the reinforcement

As given at the course on “Reinforced concrete” .See the formulas for the anchorage length.

Splicing of the reinforcement The lap length l0for tensile reinforcement depends on:

- anchorage length; - ratio “spliced reinforcement area/ overall reinforcement area”

According EC2 the distance between the adjacent lap lengths should be at least 0.3*l0. Minimum mandrel diameter to avoid damage to reinforcement

The minimum mandrel diameter depends on the bar diameter. The required values should be specified at the drawings.

Detailing of shear reinforcement The minimum diameter of the stirrups is 10mm. The minimum distance between stirrups is

10cm. The maximum distance is 3/4d where d is the effective depth of the beam.

CHAPTER 9: Prestressed RCB structure 9.1 Principle of the prestressing The main aim of the prestressing is to create compressive stresses at these parts of the

sections where tensile stresses appear under permanent and traffic actions. The results of the application of the prestressing in the RCB structures are as follows:


69

- the stiffness of the prestressed elements is considerably higher then the stiffness of the cracked reinforced elements with same cross section dimensions;

- the deflections of the prestressed elements are considerably lower then the same for the cracked reinforced elements with same cross section dimensions;

- the cross section dimensions for the prestressed elements could be decreased, respectively the clearance under the bridge would be increased;

- the durability of the prestressed concrete elements has been increased; - as a rule, the vertical component of the prestressing force is favourable for the

principal stress check and stirrups might not be necessary by calculation; - favourable vertical prestressing might be used for additional stiffness increasing and

decreasing principal stresses; - prestressing is favourable for fatigue checks when the stresses are only with negative

sign (compressive stresses) The disadvantages of the prestressing are as follows: - the construction is more difficult; - the requirements for the construction quality and control are considerably higher than

for reinforced concrete construction w/o prestressing; - the anchorage devices are expensive

9.2 Application of pre-tensioning and post-tensioning in the RCB structures Both ways of prestressing have been used in the Bulgarian bridge construction practice.

The systems “Stobet” and “Freissinet” are applied. Both systems are used for cast-in-place and for precast bridge elements and for different reinforced concrete bridge construction systems.

The prestressing tendon for “Stobet” system consists of group of ropes. The nominal diameter of the ropes is 12 or 15mm. Three types of this system have been used – “Stobet”100, 200 and 300, where these values are the prestressing forces in tones for onetendon. The prestressing force for one rope is maximum 180kN.

The prestressing tendon for “Freisssinet” system consists of group of : Type 1.Round smooth bars. Two types of tendons have been used – 24d5 or 12d7mm. Type2.Group of wires – for “C” type system NC15 with N wires with nominal diameter

16.0mm. – see Figure 9.1 with N=19.


70

Figure 9.1. The schedule of the construction of prestressed concrete precast elements is as follows: For pre-tensioned elements (see Figure 9.2) - tensioning and anchorage of the

tendons(ropes) at the end blocks, shuttering (at one side for easier construction), reinforcing, closing the formwork, concreting, release the tendons when enough concrete strength has been reached. The prestressing force is transferred to the concrete by bond and friction (effect of Hoyer). The main advantages of this system are the simplicity of the construction and lack of anchorage devices.

Figure 9.2.


71

Only straight prestressing tendons have been used in the bridge construction practice in our country. In the world practice straight and inclined tendons are used. The inclination has been constructed using deviators – see Figure 9.3.

Figure 9.3.

The main disadvantages for this system are the appearance of unbalanced tensilestresses

(see Figure 9.4) at the top of the beam near the supports and lack of vertical component of the prestressing force, respectively the admissible principal stresses might be increased.

Figure 9.4.

The effects of first disadvantage are minimized using top prestressing tendons and/or isolation of some tendons using rubber or plastic pipes – see Figure 9.5

Figure 9.5.


72

For post-tensioned elements: shuttering (at one side for easier construction), reinforcing, placing the ducts, closing the formwork, concreting, tensioning and anchorage of the tendons, cement mortar injection. The prestressing force is transferred to the concrete by anchorage devices. The main advantage of this system is the possibility for the construction of the tendons in most convenient way according internal force diagrams. The main disadvantages are the difficult construction and the use of expensive anchorage devices.

9.3 Losses of prestress 9.3.1 Immediate losses of prestress for pre-tensioning Losses due to wedge draw-in of the anchorage devices Losses due to relaxation of the pretensioning tendons during the period which

elapses between the tensioning of the tendons and prestressing of the concrete Losses due to thermal effects in case of heat curing – these losses could be neglected Losses at the transfer of prestress to concrete - loss due to elastic deformation of

concrete as the result of the action of pre-tensioned tendons when they are released from the anchorage

9.3.2 Immediate losses of prestress for post-tensioning Losses due to the instantaneous deformation of concrete

The effect of these losses is the different initial prestressing force at each tendon. The prestressing force for the last prestressed tendons is the highest. In this case, the first tendons should be prestressed with higher force value.

Losses due to friction The losses due to friction in post-tension may be estimated from the formula

∆ ( )xP ( )max 1 kx

eP (9.1)

where θ is the sum of the angular displacements over a distance x (irrespective of direction or sign), μ is the coefficient of friction between the tendon and its duct, k is the unintentional angular displacement for internal tendons (per unit length) and x is the distance along the tendon from the point where the prestressing force is equal to Pmax (the force at the active end during tensioning).

The value μ depends on the surface characteristics of the tendons and the duct, on the presence of rust, on the elongation of the tendon and on the tendon profile.

The value k depends on the quality of workmanship, on the type of duct or sheath employed and on the degree of vibration used in placing of concrete.

The effect of these losses may be decreased if two active ends have been used and if the influence of the above considered unfavorable factors is minimized.

Contemporary post-tensioningwithout bond between tendons and concrete (without cement grout injection) is used in the world bridge construction practice. In this case, tendons have been covered with corrosion protection layer. The advantage of this type of prestressing is the possibility for additional prestressing during the use of the bridge, if necessary.

Losses at anchorage due to the wedge draw-in of the anchorage devices during the operation of anchoring after tensioning and due to the deformation of the anchorage itself.


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9.3.3 Time dependent losses of prestress for pre- and post tensioning. They may be calculated by considering the following two reductions of the stress: - due to the reduction of the strain caused by the deformation of concrete due to creep

and shrinkage under the permanent loads; - due to the relaxation under tension. The losses could be calculated using formula (9.2)

0 ,

,2

0

0.8 ( , )

1 1 1 0.8 ( ,

ppcs pr c QP

cmp pc s r p c s r

p p cz cp

cm c c

EE t t

EA AP E A A t tE A I

(9.2)

These losses could be assumed 15%. 9.4 Design checks for prestressed RCB beams. 9.4.1 Initial prestressing force determination for simple span RCB beams. Check

for decompression There are two main ways for prestressing of simple span bridge beams: - only straight prestressing tendons at the bottom of the beam; - combination of straight, curved and straight and curved tendons. The required prestressing force is determined according the bending moments at the

middle of the span. The normal stresses at the cross section of the beam at each stage are as follows: First stage (construction stage), noted with (I)

Beam self weight (g1) -bending; Prestressing (P) – eccentric compression; Slab self weight (g2) – bending.

Second stage (serviceability stage), noted with (II) Pavement self weight (g3) – bending; Traffic loads (q) – bending.

There are three prestressing force values: - Pmax – prestressing force which the press transfers to the tendons; - Pm0 – prestressing force at the end of prestressing process (after immediate losses of

prestress); - P∞ - prestressing force after time t=∞ (after time depending losses – creep, shrinkage

and slow relaxation). The required prestressing force is calculated for SLS check, called “’decompression” -

only compressive stresses 100mm away from each tendon. As some of the tendons are within 100 mm of the concrete surface, the decompression must be checked at the extreme fiber – bottom of the beam. According EC2-2 decompression should be checked in the frequent load combination.

The stresses could be calculated for the overall section because tensile stresses do not appear for this combination. The sum of the stresses for the bottom fiber is:

σb=(Mg1+Mg2)/SbI- P∞/AI- (P∞*e) )/Sb

I+ (Mg3+ψ1Mq))/SbII ≤0


74

The smallest value P∞ is the solution of the equation. The first assumption for the point of application of the prestressing force is the center of the bottom flange.

e=ycI-Hbf/2

The prestressing force Pm0 is called “initial prestressing force”. This force is the transferred force to the concrete cross section. The first assumption for the time dependent losses might be 15%. In this case P∞=0.85Pmo.

The required cross section area of the prestressing tendons is Asp=Pmo/σmo. The prestessing system is “Freissinnet”. This is a system for post tensioning. Example: Steel grade is Y1770-S7. For this steel grade: - fp,0.1k=1520MPa is the steel yielding (conditional) stress limit; - fpk=1770 MPa is the characteristic steel strength.

The stress σm0 depends on these stresses. It is taken the minimum value between 0.85fp,0.1k=0.85*1520=1292MPa and 0.75fpk=0.75*1770=1327.5MPa Wires with a1=150 mm2 area are used for this prestressing system. There are two main possibilities for the cable arrangement: 1. 3C15 – three wires with total area A1=3a1=450 mm2 and d=45mm duct diameter

2. 4C15 – four wires with total area A1=4a1=600mm2and d=50mm duct diameter

The number of the cables is Nc=Asp/A1 The rules for the tendon arrangement at the bottom flange are: 1. Minimum concrete cover for the ducts – d+15mm; 2. Minimum clear distance between the ducts – d mm; 3. “Chess” shape arrangement is not allowed. The actual initial prestressing force is Pm0=Nc*A1*σm0. The exact center of gravity of the

pretressing tendons should be determined. The center of gravity is the application point for the prestressing forces.

9.4.2 Normal stress check Because of the decompression for frequent combination at SLS it is allowed to

calculate the stresses using the section properties for the overall concrete cross section using the formulas:

σi=±Mj/Si σi=-P/A±P*e/Si where i=b (bottom of the beam), t (top of the beam), ts (top of the slab); j=type of the action; The first moments of the concrete cross section area Si must be for the relevant stage (I -

construction or II – serviceability). The section properties for the construction stage (I) apply for the action g1,g2,Pmo and P∞ and the same for the serviceability stage (II) apply for the actions g3 and traffic actions.

Two limits for the prestressing force have been defined at EC2: 1. Upper limit Psup=rsup*P 2. Lower limit Psup=rinf*P, Where: - rsup=1.10 and rinf=0.9 for post tensioning. - rsup=1.05 and rinf=0.95 for pre tensioning.


75

The design stress combinations are: 1. Construction stage – beam self weight (g1) plus initial prestressing force Pm0*1.10(1.05), because prestressing force is unfavorable at this stage. 2. Serviceability stage At this stage prestressing action is favorable and P∞ is multiplied with 0.9(0.95).

Design combinations are as follows: Characteristic combination

g1+g2+P∞*0.9+g3+1*TS+1*UDL Non frequent combination

g1+g2+P∞*0.9+g3+0.8*TS+0.8*UDL Frequent combination

g1+g2+ P∞*0.9+g3+0.75*TS+0.40*UDL Quasi permanent combination

g1+g2+ P∞*0.9+g3+0*TS+0*UDL The stresses should not exceed the admissible stress limits as follows: 1. Construction stage Maximum compressive stress (at the bottom extreme fiber) – 0.6fck=0.6*35=21MPa Maximum tensile compressive stress (at the top extreme fiber) – fctm=3.2MPa 2. Serviceability stage 2.1 Decompression for the frequent combination – tension 100 mm away from each tendon is not allowed! 2.2 Maximum tensile stress at the non frequent combination (at the bottom extreme fiber) – fctm=3.2MPa 2.3 Maximum compressive stress at the characteristic combination (at the top extreme fiber) – 0.6fck=0.6*35=21MPa 2.4 Maximum compressive stress at the quasi static combinations (at the top extreme fiber) – 0.45fck=0.45*35=15.75MPa If some of the stresses do not meet these requirements the following measures apply:

- increasing the prestressing force (adding cable); - rearrangement of the cables; - changing the beam cross section (increasing beam height); - increasing the concrete grade.

9.4.3 Principal stress check at SLS This check is conducted for the frequent combination.Because of the decompression

the shear stresses and the principal stresses can be calculated using formulas for elastic materials and for overall cross section.

22

1 2 2

(9.3)

ii

VSdJ

(9.4)


76

The shear stresses are calculated for the relevant stages – construction stage (I) and serviceability stage (II). The respective section properties should be applied. The shear stresses at the following levels should be calculated:

- top end of the bottom flange; - bottom end of the web; - center of gravity of the beam section; - middle level between the two centers of gravity; - center of gravity of the slab and beam section; - top end of web; - bottom end of the top flange; - top end of the beam; - bottom end of the slab The second moments of area above or below each level about centroidal axis must be

calculated. The second moments for the level at centroidal axis and for the areas above and below it must be equal. Otherwise, the center of gravity of the beam section or center of gravity of the slab and beam section is not correctly calculated.

The shear force VI at first stage includes: - beam self weight shear force Vg1; - slab self weight shear forceVg2;

- vertical components of the prestressing forces for each prestressing tendons (with negative sign)

These components should be calculated using the formula for each tendon parabola at each section. The prestresssing force for each tendon is assumed to be equal and is calculated as follows:

P1=P∞/Nc, where Nc is the number of cables The total shear force at first stage VI could be positive or negative. The shear force VII at second stage includes: - pavement self weight shear force; - traffic action shear force The normal stress at each level is calculated using linear interpolation. The normal stress

at the bottom of the slab is σ = M(II)/Stb(II), where M(II)=Mg3+Mq. The maximum principal stress should be less than the design concrete tensile strength

fctk,0.05. For concrete grade C35/45 fct,0.05=2.20MPa. Otherwise the following measures are applied: - increasing normal stresses (adding cable); - changing cable layout; - increasing web depth or/and beam height 9.4.4 Check for bending at ULS This check shall be provided: -for section in the middle in case of simple span beam; -for the section with maximum moment at the span and the section with minimum

moment at the support for continuous beam.


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The check is MEd≤MRd. The bending moment MEd is taken from the bending moment calculations for ULS. The moment of resistanceMRd is calculated according the defined damage criteria for

damage of the concrete or damage of the steel – see general assumptions below. The following general equilibrium equations must be satisfied: MRd=Fc*z=Fsp*z Fc=Fsp The design assumptions for the calculation are as follows: 1. The cross sections remain plain during the application of the actions; 2. The appearance of the cracks at the tensile zone is possible. The concrete at the

tensile zone is ignored; 3. The strain in bonded reinforcement is the same as the strain in the concrete at the

same level; 4. The stress-strain relation for concrete is linear and parabolicwith εc2=2‰(point A)or

bilinear with εc3=1.75‰. For both diagrams εcu2= εcu3=3.5‰ (point B) 5. The stresses in the prestressed steel are given by the stress-strain relationship. Two

bilinear relationships are given. It is used the relationship with horizontal top branch; 6. The criterions for “damage” of the concrete are: 6.1“Maximum compressive concrete strain is 3.5‰” (maximum value from the stress-

strain relation); 6.2“Compressive concrete strain at the level of the center of the slabis εc2=2‰(or

εc3=1.75‰); 7. The criterion for “damage” of the steel is “the maximum tensile strain caused by all

permanent and traffic actions and prestressing after losses is 20‰”. The strain caused by prestressing force after time depending losses P∞ is determined using Hook law as follows

εsp=P∞/(Ap*Esp) . For wires Esp=195000MPa. The neutral axis depth x should be determined to satisfy the equilibrium equation Fc=Fsp Two methods have been used: 1. Strain compatibility method; 2. Iteration method – described bellow The first assumption is “damage” of the concrete (εc=3.5‰) at the extreme compressive

fiber (top of slab)and steel simultaneously - εsp=20‰-εsp. Using similar triangles the stress εcfat the center of gravity of the slab is calculated. If εcf≥εc2=2‰(or εc3=1.75‰) then it is assumed εcf= εc2=2‰ (or εc3=1.75‰) and the strain εc at the top of the slab is adjusted.

Then theneutral axis depth x1is x1=d*εc/( εc+ εsp) , where d is the effective depth. Without significant mistake equivalent rectangular block with 0.8x1 is determined. If

0.8x1 ≤hf then Fc=(0.8x1)*fcd*beff. Here fcd is for the slab concrete grade. If 0.8x1≥ hf then the beam top flange and the beam concrete grade should be taken into

account. The force at the steel is determined according εsp.The strain at yielding

isεpd=fpd/Esp=(1520/1.15)/195000=6.78‰Ifεsp≥εpd then σsp=fpd=1520/1.15=1321.7MPa. Fsp=σsp*Asp


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If Fc=Fsp then the moment resistance could be calculated taking moment according any point at the cross section. The most convenient point is the top of the slab. The moment resistance is

MRd=Fsp*d-∑(Fcj*acj) If Fc ≥ Fsp (typical case) then new neutral axis depth x2≤x1 is assumed and the design

steps from the first iteration are repeated until Fc=Fsp. Obviously this is the case “damage of the steel”.

If Fc≤ Fsp then new neutral axis depth x3≥x1 is assumed and the design steps from the first iteration are repeated until Fc=Fsp. Obviously this is the case “damage of the concrete”.

If MEd≥MRd then the following measures apply: 1. Cross section dimensions must be increased; 2. Prestressing force (if possible) must be increased; 3. The slab concrete grade must be increased. 9.4.5 Check for shear at ULS Truss model has been applied at ULS for the shear reinforcement design and check for

the compressive struts. The design shear forces for simple span bridge at the relevant stages at ULS are as

follows: - construction stage VEd, I=1.35(Vg1+Vg2)-0.9VP∞; - serviceability stage VEd,II=1.35(Vg3+VLM1) The relevant effective depths d1 and d2 should be used.The checks are as follows: 1. Maximum resistance of the compressive struts is VRd,max=

αcwbw,nomzjν1fcd/(cotθ+tanθ)where αcwtakes into account the favorable contribution of the prestressing and is assumed equal to

1+σcp/fcd if 0≤σcp ≤0.25fcd 1.25 if 0.25fcd ≤σcp ≤0.5fcd

2.5(1-σcp/fcd) if 0.5fcd≤σcp ≤fcd

bw,nom=bw-0.5∑ф for post tensioning with bonded prestresed reinforcement, where ф is the diameter of the duct

zj=0.9dj is the lever arm for the relevant section. ν1=0.6(1-fck/250) takes into account the reduced compressive strength of the compressive

struts subjected to tension Angle θ=300-450 for prestressed members. The minimum resistance is if the angle is 300. In this case cotθ + tanθ= 1.732+0.577=2.309 If VEd≥VRd,max then the following measures apply: - increasing concrete grade; - increasing cross section dimensions 2. For members not requiring design shear reinforcement VEd≤ VRd,cwhere VRd,c=0.12k(100ρlfck+0.15σcp)1/3 bwd The determination of VRd,c isdifficult because ρl depends on both reinforcements –

prestressed and not prestressed. So the check is VEd≤Vmin Vmin=(0.035k3/2 fck

½+0.15σcp)bwd In this case minimum shear reinforcement is provided with


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ρmin==0.10(fck) ½:fyk=0.00118 ρ=Asw/(bws) For members requiring design shear reinforcement VEd≥Vmin

Asw/bws=VEd/(0.9dfydcotθ) and angle θ=300 Asw=n*As1 where n=2 is the number of the stirrup legs and As1 is the cross section of one

leg. CHAPTER 10: Modern systems for RCB construction

10.1 Cast-in-place construction of bridges using conventional scaffolding Scaffolding (falsework,am.) may be defined as a temporary framework on which the

permanent structure is supported during its construction. As a rule the scaffolding is associated with the construction of bridge superstructures. It is also important to note the difference between “formwork” and “scaffolding”. Formwork is used to retain plastic concrete in its desired shape until it has hardened. It is designed to resist the fluid pressure of plastic concrete and additional pressure generated by vibrators. Because formwork does not carry the dead load of the concrete, it can be removed as soon as the concrete hardens. Scaffolding does carry the dead load of concrete, and therefore it has to remain in place until the concrete becomes self-supporting.

In case of conventional scaffolding system the various elements are erected individually to form the completed system.

10.2 Cast-in-place “span by span” construction of bridges In span-by-span bridge construction method, construction starts at one end and

proceeds continuously to the other end. Generally, this method is used where access to the ground level is restricted either by physical constrains or by environmental concerns. The construction joints are placed at the points of contra flexure. Conventional scaffolding supported at the ground level could be applied. Otherwise, form travelers are used. They are supported either on the bridge piers or on the edge of previously erected span and the next pier – see Figure 10.1.

Figure 10.1.

completed part concreting

completed part concreting


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The connection between adjacent segments is completed using prestressing -see Figure

10.2.

Figure 10.2.

10.3 Precast bridge construction using conventional cranes Conventional cranes have been used in case of relatively light precast elements and

easy access to the place for construction.

1 2

6

1 2

7 8

b)

a)

6 Connecting device

7 Tendons stessed after concreting of part 1

8 Tendons stessed after concreting of part 2


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The crane position for maximum hoisting capacity should be at the middle of the span – see Figure 10.3.

Figure 10.3.

10.4 Precast bridge construction using “truss crane”(gantry) “Truss crane” is used in cases of:

- long and heavy precast elements; - very high piers; - overwater bridges

Truss crane consists of two connected parallel space diagonal trusses. It is self-moved equipment. The truss can move longitudinally and transversally. For overturning stability the length of the truss crane should be at least equal to the sum of two bridge span lengths – see Figure 10.4.

1

1 2

1 Fill for 1st stage erection

2 Fill for 2nd stage erection


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Figure 10.4.

Truss cranes were used for the construction of the viaducts at “Hemus” and “Trakia”

motorways.

10.5 Cast-in-place balanced cantilever bridges Balanced cantilever segmental construction has been recognized as one of the most

efficient methods for building bridges without the need for scaffolding. This method has great advantages over other forms of construction in urban areas where temporary scaffolding would disrupt traffic and services below, in deep valleys and over waterways.

Construction commences from the permanent piers and proceeds in a “balanced” manner to midspan – see Figure 10.5. A final closure joint connects cantilevers (so called “birds”) from adjacent piers. The structure is hence self-supporting at all stages.

A

A 1 2

A 1 2

A 1 2

B

Ac)

b)

a)

L > l1 + l2 l1 l2


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Figure 10.5.

The most common methods are as follows:

- monolithic connection to the pier; - permanent or temporary double bearings and vertical temporary post-tensioning – see

Figure 10.6 The “birds” are usually constructed in 3- to 6-m-long segments. The adjacent segments

are connected using prestressing.

65432106 5 4 3 2 1

78

65432106 5 4 3 2 1

32103 2 1

0

d)

c)

b)

a)


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Figure 10.6.

The cast-in-place technique is preferred for long and irregular span lengths with few

repetitions.

10.6 Precast balanced cantilever bridges The first attempts to use balanced cantilever construction were made in 1929 in Brazil.

Precast segments were used by Eugene Freyssinet for construction of six bridges over Marne River in France (1946 to 1950).

The features of this method that provide significant advantages over the cast-in-place method are listed as follows:

- casting the superstructure segments may be started at the beginning of the project and at the same time as the construction of the substructure;

- rate of erection is usually 10 to 15 times the production achieved by the cast-in-place method. The time required for placing reinforcement and tendons, and most importantly, the waiting time for curing of the concrete is eliminated from the critical path;

- segments are produced in an assembly-line factory environment, providing consistent rates of production and allowing superior quality control. The concrete of segments is matured, and hence the effects of shrinkage and creep are minimized

The joints between segments are either left dry using dowels or made of a very thin layer of epoxy cement grout.

1

2

1 Temporary bearings

2 Temporary prestressed tendons


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10.7 Incrementally launched bridges The incremental launching technique has been used since its introduction by Fritz

Leonhardt in 1961 for the RioCaroniBridge in Venezuela. The method entails casting the superstructure, or a portion thereof, at a stationary location behind one of the abutments. The completed or partially completed structure is then jacked into place horizontally, i.e., pushed along bridge alignment – see Figure 10.7.

Figure 10.7.

Subsequent segments can then be cast onto the already completed portion and in turn

pushed onto the piers. Because all of the casting operations are concentrated at a location easily accessible from the ground, concrete quality of the same level expected from a precasting yard can be achieved. The procedure has the advantage that it obviates the need for scaffolding.

There are two peculiarities associated with the technique, which must be appreciatedby the designer. The first is that the alignment must be straight or, if it involves curves, the curvature must be constant. The second is that during launching, every section of the girder will be subjected to both the maximum and minimum moments and the leading cantilever portion will be subjected to higher moments.

The techniques for reducing launching bending moments are as follows: - using a light but stiff structural-steel launching nose attached to the leading

cantilever;

1 2

1 2

1 2 3

A A

S

S

S

S


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- for longer spans (longer than 100m), the steel nose is not as effective and temporary piers are the solution.

CHAPTER 11: Design checks for elastomeric bearings

11.1 General An elastomeric bearing is made of elastomer (either natural or synthetic rubber). It

accommodates both translational and rotational movements through the deformation of the elastomer. Elastomer is flexible in shear but very stiff against volumetric change. Under compressive load, the elastomer expands laterally. To sustain large load without excessive deflection, reinforcement is used to restrain lateral bulging of the elastomer. So, plain (w/o reinforcement) or reinforced elastomeric bearings are used – see Figure .11.1.

Rectangular or round elestomeric bearings have been used.

Figure 11.1.

11.2 Design checks 11.2.1 Stress check and determination of the bearing area The area of the bearing is determined according maximum bearing stress: σmax=Rmax/Ar<=σadm (11.1},where Rmax is the maximum bearing reaction caused by all permanent actions Rg and by

traffic actions Rq. Ar=A1(1-vx,d/a’-vy,d/b’) is the reduced effective area of the bearing; a’ and b’ are the dimensions of the steel plates and A1=a’*b’ is the effective area –

see Figure 11.1; vx,d and xy,d are the design displacements along axes x and y respectively. The transversal distribution and the most unfavorable location of the traffic actions

must be considered. The admissible stress for the rubber is given by the manufacturer and is appr.10MPa (1kN/cm2). The initial minimum stress against sliding of the bearing σmin=Rg/Ab must be at least 3MPa.

It is recommended for better stress distribution ratio b/a<1.5 between the bearing dimensions. 11.2.2 Check for shear strain

2.5t rt st b

a' (b') (D') 4a (b) (D)


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It is assumed that only elastomeric layers are flexible in shear – see Figure 11.2. The overall thickness of the layers is Hr.

Figure 11.2.

The shear strain is caused by: - lateral deformation of the bearing caused by the vertical reaction; - horizontal displacement caused by temperature, braking force, shrinkage or seismic

action; - rotation of the bearing caused by bending. The shear strain is controlled by the maximum allowable horizontal displacement. It is

provided by the manufacturer. The check is performed for unloaded and for loaded bridge. For unloaded bridge the

displacement Δ is caused by temperature difference and shrinkage.

11.2.3 Check for friction The check is H<T, where H is the horizontal force for the bearing and T is the friction

force at the contact surface (bearing area) between the bearing and the base. The friction force T depends on the compressive stresses at the surface and on the friction coefficientμ. The check is performed for unloaded and for loaded bridge.

For unloaded bridge the design combination is “permanent actions + temperature + shrinkage”. The friction force is Tmin=Rg*μ and horizontal force is Ht+s=tgθ*A*G, where

- tgθ =vd/Hr – see Figure 11.2.; - A=a*b is the total bearing area - G is the shear modulus given by the manufacturer (appr.0.9MPa). For loaded bridge H=Ht+s + Hb,1and T=Tmin + Tq, where Tq=Rq*μ. Here Hb,1 is the part of the braking force resisted by the bearing. Example: For simple

span bridge assuming equal stiffness of the bearings Hb,1=Hb/nB , where nBis the number of the bearings in the span.

If H>T then : - outer steel plates bonded to elastomer apply; - the bearing is anchored to the superstructure and to the substructure

=> 휃 Hr= trsteel plates

H Vd


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11.2.4 Check for rotation `The check is Δφ<Δg+q, where Δφ is the displacement of the outer face of the bearing

cause by maximum rotation and Δg+q is the vertical displacement of the bearing caused by the corresponding reactions – see Figure 11.3.

Figure 11.3.

The displacement Δφ=a/2*tgφ≈a/2*φ, where a is the dimension of the bearing at

longitudinal direction. The rotation φ is calculated using the methods of structural mechanics. The displacement Δg+qis calculated according the Hoock’s law using equivalent modulus for elasticity of the reinforced elastomer.

If Δφ>Δg+q then the number of the elastomeric layers must be increased.

CHAPTER 12: Bridge substructure

12.1 Abutments As a component of a bridge, the abutment provides the vertical support to the bridge

superstructure at the bridge ends, connects the bridge with the approach roadway and retains the roadway base materials from the bridge spans. According the design solution, the abutment may be a part from lateral force resisting system.

Unlike the bridge abutment, the earth-retaining structures are mainly designed for sustaining lateral earth pressures.

From the view of the relation between the bridge abutment and roadway or water flow that the bridge overcrosses, bridge abutments can be divided into two categories: open-end abutment, and close-end abutment.

For the open-end abutment, there are slopes between the bridge abutment face and the edge of the roadway or river canal that the bridge over crosses. These slopes provide a wide open area for the traffic flows or water flows under the bridge. It imposes much less impact on the environment and the traffic flows under the bridge than a closed-end abutment. Also, future widening of the roadway or water flow canal under the bridge by designing a new retaining wall is easier. However, the existence of slopes usually requires longer bridge spans and some extra earthwork. This may result in an increase in the bridge construction cost.

The closed-end abutment is usually constructed close to the edge of the roadways or water canals. Because of the vertical clearance requirements high abutment walls must be constructed. It is very difficult, or it is not possible, to do the future widening to the

superstructureRg+q

Δg+qΔ휑

C of the bearingL

휑


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roadways and water flow under the bridge. Also, the high abutment walls and larger backfill volume often result in higher abutment construction costs and more settlement of road approaches than for the open-end abutment.

Generally, the open-end abutments are more economical, adaptable, and attractive than the closed-end abutments. However, bridges with closed-end abutments have been widely constructed in urban areas and for railway bridges because of the right-of-way restriction and the large scale of the live load for trains, which usually results in shorter bridge spans.

Based on the connections between the abutment stem and the bridge superstructure, the abutments can be grouped in two categories:

- monolithic abutment (integral bridges); - seat-type abutment The monolithic abutment is monolithically constructed with the superstructure. There is

no relative displacement allowed between the bridge superstructure and abutment. All the superstructure forces at the bridge ends are transferred to the abutment stem and then to the abutment backfill soil and footings. The advantages of this type of abutment are its initial lower construction cost and its immediate engagement of backfill soil that absorbs the energy when the bridge is subjected to transitional movement. However, the passive soil pressure induced by the backfill soil could result in difficult-to-design abutment stem. In the practice this type of abutment is mainly constructed for short span bridges.

The seat-type abutment is constructed separately from the bridge superstructure. The superstructure seats on the abutment stem through bearings or other devices. This type of abutment allows the bridge designer to control the superstructure forces that are to be transferred to the abutment stem and backfill soil. By adjusting the devices between the bridge superstructure and abutment the bridge displacement can be controlled.

Abutment wingwalls act as a retaining structure to prevent the abutment backfill soil and roadway soil from sliding transversely. The wingwall types are as follows:

- cantilever wingwall; - continuous support wingwall A wingwall design is similar to the retaining wall design. However, live-load surcharge

needs to be considered in the design. 12.2 Piers Piers provide vertical supports for spans at intermediate points and perform two main

functions: - transferring superstructure vertical actions to the foundations; - resisting horizontal actions acting on the bridge Pieris usually used a general term for any type of substructure located between

horizontal spans and foundations. However, it is used also particularly for a solid wall in order to distinguish it from columns and bents. From a structural point of view, a column is a member that resists the lateral force mainly by flexure action whereas a pier is a member that resists the lateral force mainly by a shear mechanism. A pier that consists of multiple columns is called a bent.

There are several ways of defining pier types. One is by its connectivity to the superstructure: monolithic or cantilevered. Another is by its sectional shape: solid or hollow; round, rectangular, octagonal, etc. It can also be distinguished by its framing configuration: single or multiple column bent; hammerhead or pier wall. Selection of the type of piers for a bridge should be based on functional, structural and geometry requirements. Aesthetics is


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also a very important factor of selection since modern highway bridges are part of a city’s landscape.

Solid wall piers are designed at water crossings since they can be constructed to proportions that are both slender and streamlined. These features lend themselves well for providing minimal resistance to flood flows -see Figure 12.1a

Hammerhead piers are often used in urban areas where space limitation is a concern – see Figure 12.1b. They are aesthetically appealing.

A column bent pier consists of a cap beam and supporting columns forming a frame. The columns can be either circular or rectangular in cross section. They are the most popular forms of piers in the modern highway system.

Figure 12.1. Pier types

The distribution of the horizontal actions (seismic action, breaking action, etc.) depends

on the stiffness of the piers. For long bridges, it is preferred to design movable bearings at the abutments – see Figure 12.2. The superstructure is assumed infinitely stiff in horizontal direction. Stiffness is defined as a horizontal force for unit displacement.

For monolithic connection between pier and superstructure the stiffness of pier (i): Ki=12EJi/(Li

3) For hinge connection between pier and superstructure the stiffness of pier (i): Ki=3EJi/(Li

3) It is obvious that for short piers hinge connection should be designed.

Figure 12.2.

CHAPTER 13: Seismic design of RCB

Schedule of the topic: 1. General. Comparison between seismic behavior of buildings and bridges 2. Earthquake damages to RCB 3. Dynamic analysis

3.1 Division of the RCB on separate oscillation units 3.2 Model with single- degree –of- freedom (SDOF)

a) solid wall pier b) hammerhead pier

EJi = constL i


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3.3 Multimode spectral analysis 4. Capacity design approach 13.1 General. Comparison between seismic behavior of buildings and bridges The seismic design of RCB in our country is based on the aseismic code from 2012. For seisimic bridge design EC 8-2 has been applied simultaneously. The differences between the seismic behavior of buildings and bridges are as follows: A. Criterion: place of the elements of the lateral force resisting system The diaphragms and frames at the building structureare placed according the

architectural planes at the most convenient places. Usually the distance between them is small (less than 6m) and the behavior (vibration) of all elements is the same. As a rule, the behavior of the structure at the two main directions is the same.

The abutments and piers resist seismic actions for the bridges. They are placed according the obstacle, the type of the structure, soil conditions and other factors. The distance between them (span length) could be significant (e.g., in case of prestressed bridge structure).

B. Criterion: Structure sizes and shape The sizes of the buildings are relatively small. The distance between the elements of the

lateral force resisting system is much less then the length of the seismic wave. For long bridges (L>600m) and for long span bridges (L1>250m), it is possible

oscillation of the adjacent piers at opposite direction. This might cause unseating of the bridge superstructure.

C. Criterion: Places of the concentrated weight For the buildings these places are the slabs between floors. For the bridges these places

are at the level of the superstructure. In this case for simple bridges modes with SDOF could be applied.

D. Criterion : Foundation The footings for the building structure are isolated or continuous footings or foundation

mat. As a rule they are placed on same level and the soil conditions are constant. The footings for bridge structure are placed sometimes on different levels and the soil

conditions for the adjacent footings are different. Pile foundations are common design solution.

E. Criterion : Hinges and bearings As a rule hinges and bearings do not exist at the buildings. Hinges and bearings are

typical for bridges and they change the behavior of the structure during earthquake. F: Criterion : Prestressing Prestressing has been rarely applied for buildings. It is typical for bridges and in this

case the bridge structure behavior is almost elastic. The possibility for energy dissipation is reduced.

13.2 Earthquake damages to RCB Even when the cause of a particular collapse is well understood, it is difficult to

generalize about the causes of bridge damage. In past earthquakes, the nature and extent of damage that each bridge suffered have varied with the characteristics of the ground motion at


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the particular site and the construction details of the particular bridge. No two earthquakes or bridge sites are identical.

Despite this, one can learn from past earthquake damages, because many types of damage occur repeatedly. Be being aware of typical vulnerabilities that bridges have experienced, it is possible to gain insight into structural behavior and to identify potential weakness in existing or new bridges. Historically, observed damage has provided the impetus for many improvements in earthquake engineering codes and practice.

The typical damages are as follows: A. Unseating at expansion joints. Earthquake ground shaking, or transient or

permanent ground deformations resulting from the earthquake, can cause superstructure movements that cause the supported span to unseat. Unseating is especially a problem with the shorter seats that were common in older constructions.

B. Damage to superstructures. As a rule, superstructures tend to remain essentially elastic during earthquakes. So, superstructure damage is unlikely to be the primary cause of collapse of a span. The most commonform of damage to superstructures is due to pounding of adjacent segments at the expansion joints.

C. Damage to bearings. Failure of the bearings can cause redistribution of internal forces, which may overload either the superstructure or the substructure.

D. Damage to columns. Column failures are often the primary cause of bridge collapse. Most damage to columns can be attributed to inadequate detailing, which limits the ability of the column to deform in elastically. In reinforced concrete columns, the detailing inadequacies can provide flexural, shear, splice, or anchorage failures, or, as is often the case, a failure that combines several mechanisms.

E. Damage to abutments.The typical damage is occurred in seat abutments due to pounding of backwalls by the superstructure. Other cause is the increased passive earth pressure during the lateral displacement against soil.

13.3 Dynamic analysis 13.3.1 Division of the RCB into separate oscillation units The division of the RCB structures into separate units (see Figure 13.1) depends on: - type of the bridge; - expansion joints; - bearings

Figure 13.1. Division of the bridge into oscillation units

O R

I I I I I I I I I I I I


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In general, according EC-8 the bridges are two types – common or special. Common

bridges are straight, without significant differences in span length and stiffness of the piers. They are with less than 6 spans. Other bridges are special bridges. Common bridges could be divided into simple units and model with SDOF might be applied. Special bridges are analyzed with multimode spectra analysis.

13.3.2 Model with single- degree –of- freedom (SDOF) Single-mode spectral analysis is based on the assumption that earthquake design actions

for structures respond predominantly in the first mode of vibration. This method is most suitable to regular linear elastic analysis of the forces and displacements of the bridge. It is not applicable for irregular bridges because higher modes of vibration significantly affect the distribution of the forces and resulting displacements. This method can be applied to both continuous and non-continuous bridge superstructures in either the longitudinal or transverse direction.

The period of first mode of vibration is T=2π* K/M , where M=Q/g is the mass of the bridge superstructure and part of the substructure (usually 50%), g=9.81m/sec2and K is the sum of the stiffness of the piers.

13.3.3 Multimode spectral analysis This method is appropriate for structures with irregular geometry, mass, or stiffness.

These irregularities induce coupling in three orthogonal directions within each mode of vibration. Also, for these bridges, several modes of vibration contribute to the complete response of the structure. A multimode spectral analysis is done by modeling the bridge structure consisting of three-dimensional frame elements with structural mass lumped at various locations to represent the vibration modes of the components. Usually, five elements per span are enough to represent the first three modes of vibration.

The maximum response cannot be computed by adding the maximum response of each mode because different modes attain their maximum values at different times. Two commonly used methods are square root of sum of squares (SRSS) and the complete quadratic combination (CQC).

For structures with closely spaced dominant mode shapes, the CQC method is precise whereas SRSS estimates inaccurate results. The SRSS method is suitable for estimating the total maximum response for structures with well-spaced modes.

13.3.4 Capacity design approach The so-called “capacity design” has become a widely accepted approach in modern

structural bridge design. The main objective of this approach is to ensure the safety of the bridge during LARGE earthquake attack. For ordinary bridges, it is assumed that the performance for LOWER-LEVEL earthquake is automatically satisfied. The procedure of the capacity design involves the following steps:

1. Choose the desirable mechanisms that can dissipate the most energy and identify plastic hinge locations – commonly considered in columns;

2. Proportion bridge structure for design loads and detail plastic hinge for ductility; 3. Design and detail to prevent undesirable failure patterns, such as shear failure or joint

failure.

REINFORCED CONCRETE BRIDGES · 2014-06-20 · REINFORCED CONCRETE BRIDGES ASS. PROF. PHD. DIMITAR...

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