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    Department of Civil Engineering Page 1

    1. SYNOPSIS

    Since inception of Indian Railways in 1853 the Railway Engineershas history of more than 250 years of construction and maintenance of

    railway bridges. During the long journey they had achieved several

    heights and continuing to excellence. Recently Indian Railways has

    either constructed several worldclass bridges or they are in the

     process of construction. A technical review of design and construction

    of recent bridges (viz. Bogibeel, Chenab and New Jubilee Bridges)

    through light on the recent technological advancements attained bythe Indian Railways in the field of bridge engineering.

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    2. ABSTRACT

    The joints of Riveted Steel Truss Railway Bridge consist of gusset

     plates which lose their rigidity due to repeated passages of train loads;

    therefore the loss of rotational rigidity is to be taken into account inanalysis of Bridge. This joint flexibility tends to alter the vibration

    characteristics of the Bridge system and each component of the bridge

    responds dynamically to the rapidly varying loads and thus the time

    history obtained is a function of load variation and dynamics of the

    structure, which consequently affects fatigue life of the bridge

    components. In past, effect of semirigid joints has been studied in

    case of building frames.

    So here the knowledge of semirigid joints on building framehas been extended to Steel Truss Railway Bridge. This present

    article tries to study the influence of joint flexibility on the fatigue

    life of 76.2 m Truss bridge due to moving load at different

    speeds. The joint rotational stiffness is reduced by 5%, 10%, 25% and

    50%. The result of preliminary studies conducted on Steel Truss

    Bridge is presented. It is prime facia that upto 50% reduction in

    rotational Stiffness of the joints does not affect the stability of the

     bridge.

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    3. INTRODUCTION

    In India, Economic progress mainly depends on the railway and is

    considered as the Life line of the Nation. India has the second largest

    rail network in the world, transporting over four billion people

    annually and the total figure of existing railway bridges are approx.

    1,20,000. Out of these, 731 are long span open girders, 19014 are

    rolled steel joist or plate girders. So it can be seen that more than

    20% are Steel girder bridges. Due to continuous movement trains, the

    members and their connections are subjected to repeated loadings due

    to which the stiffness of the joint gets reduced, which are more prone

    to fatigue damage. The conventional static, dynamic or stability

    analysis of Steel Trusses bridges assumes that their members areconnected at rigid or hinged joints. However in reality Steel Trusses

    are reinforced at their joints by Gusset plates, which possess rotational

    flexibility. The presence of this gusset plates has an appreciable effect

    on the stiffness of the members of the Bridge and consequently on its

     behavior to Static and Dynamic loading. However, the behavior of

    connections is neither rigid nor pinned. Structures having such

    flexible Joints in which Joint flexibility becomes important are called

    as semirigid frame members. In fatigue assessment of the bridgecomponents the joints are assumed to be rigid as per RDSO, where

     joint flexibility is neglected which may affect the dynamic behavior

    of the bridge component, consequently its fatigue life. Therefore it

    is  necessary to evaluate the bridge components for semirigidconnections.

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    4. HISTORYBasic bridge designs are developed from natural bridges- a tree trunk

    has fallen across a stream, vines hanging over a river, or stones that

    make a stepping-stone path across a shallow stream. These natural bridges were probably built upon by ancient bridge builders. For

    example, someone may have built up the steeping stones, placed flat

    stone slabs or logs on top of them, and connected the stones to make a

    low bridge. This type of bridge was called a “clapper bridge.” It is one

    of the earliest bridge constructions. Such simple bridges are probably

    still built today in many places. In general, though, bridge

    construction has changed greatly.

    The ancient Romans refined bridge building with two importantcontributions. Nearly all of their bridges used the arch design- a

    structure that can support more weight than a flat surface can. Also,

    the Roman’s discovery of natural cement allowed them to build

    strong, long-standing bridges. Many of these ancient Roman bridges

    are still standing today.

    There were excellent bridge builders in Asia, too. Some early

     bridges in Asia used a cantilever design. This design enabled the

     builder to make simple, long-span bridges across fairly wide rivers.

    One famous bridge in China, built about 1300 years ago, is the Great

    Stone Bridge. Its graceful arch shape is not the same type of arch

    shape used by the Romans. Instead, this bridge is quite low, and the

    arch is very shallow.

    The Renaissance brought new scientific ideas to bridge building.

    Leonardo da Vinci and Galileo developed theories about the strength

    of building materials. Their theories have helped architects understandhow to make strong structures from lightweight materials. Bridge

     building became more exact as people began to use more

    mathematical theories about it. Another new development that

    changed bridge building was the development of metal.

    About 200 years ago, the first cast-iron bridge was built. This

    was the Iron Bridge at Coal brookdale in England. Before that time,

     bridges were made of stone, brick, clay, or timber. Eventually,

    wrought iron was used instead of cast iron. Much later, steel was

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    used. Many new bridges were created and tested during this time. The

    Britannia Tubular Bridge, completed in 1850, showed one such new

    development. It was built from rectangular tubes of wrought iron.

    Similar bridges are often used today.

    Other important developments came with the truss bridge and

    the suspension bridge designs. The truss is an old design, but it was

    improved when scientists and engineers knew enough about science

    and mathematics to work out the mechanics of the design. Covered

     bridges were usually built on the truss design. Truss bridges were

    improved even more when metal was used. The suspension bridge

    was another basic design that was changed by the use of metal. The

    Brooklyn Bridge is one famous suspension bridge built during thattime. It uses steel wires for the suspension cables.

    About a hundred years ago, engineers began using concrete for

     bridges. A new method called “prestressing” helps prevent concrete

    from cracking after a structure is built. Today, most new bridges are

    made of prestressed concrete and steel.

    5. GENERAL INFORMATION ABOUT TRUSSES.

     A truss is a structure comprising one or more triangular unitsconstructed with straight members whose ends are connected at

     joints or nodes.

     If all the bars lie in a plane, the structure is a planar truss The main parts of a planar truss. In other words, Trusses are designed to form a stable structure.

    6. TYPES OF TRUSSES.

     Kingpost and Queenpost Howe truss bridge Pratt truss bridge Warren Truss Bridges

     K-truss bridges Continuous truss bridges

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    6.1 Kingpost and Queenpost Kingpost:

    • It is used for simple short-span bridges: 40 feet. (probably, it wasthe first used with small open-work bridges).

    • Fewest number off truss members.- two diagonal members,

    kingpost braces, that meet at the apex of the truss, one horizontal

     beam and the king post which connect the apex to the horizontal

     beam below.

    • Kingpost braces are in compression, and the Kingpost, in

    compression. CHECAR

    Figure1 KINGPOST Queenpost:• It has two vertical post.• Very strong and stable. 

    • It´s more stable and can support a wider  span than a kingpost

    Figure2 QUEENPOST 

    6.2Howe Truss Bridge • William Howe, 1840.

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    • It became very popular and was considered one of the best designs

    for railroad bridges back in the day.

    •Wooden beams for the diagonal members, which were in

    compression. It used iron (and later steel) for the vertical members,

    which were in tension.

    Figure3 HOWE TRUSS

    6.3 Pratt Truss Bridge (1844)• Very common type but has many variations (Baltimore, 

    Pennsylvania, and the Parker)

    • The basic identifying features are the diagonal web members which

    form a V-shape. (Howe truss bridge has a A-shape).• Maximum length of the this bridge can be 250 feet. 

    • Commonly used for supporting railways. 

    •The Pratt truss’s verticals functioned as compression members and

    diagonals functioned as tension members.

    • The Pratt truss required more iron than a Howe truss, and due to the

    increased cost and less rigid construction, builders did not extensively

    use it for wooden trusses

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    Figure 4 PRATT TRUSS

    6.4 Warren Truss Bridge (1848)

    • It uses equilateral triangles to spread out the loads on the bridges.The equilateral triangles minimize the forces to only compression and

    tension. The forces for a member switch form compression to tension,

    especially to the members near the center of the bridge.

    • This bridges are often used with verticals to reduce the panel

    size.Without vertical present aesthetically pleasing appearance.

    6.5 K-truss Bridges• The length of members undergoing com pression is reduced.

    •This reduction in length enables components of bridges to endure thecompressional force.

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    • The design is complicated and it is considered to be one of the

    hardest bridges to build.

    Howrah Bridge, Kolkata, India. 

    Figure5 K-TRUSS 6.6 Continuous Truss Bridges.• Comparatively, it is more rigid and statically indeterminatestructure.

    • Only suitable in case where the differential settlements of  abutments

    and piers are not significant.

    • A continuous truss bridge may use less material  than a series of

    simple trusses, because a continuous truss distributes live loads across

    all the spans; in series of simple trusses• This have been used in span ranges of 150m to 400m. 

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    Figure6 CONTINUOUS TRUSS 

    7. TRUSS BRIDGES

    Trusses are used in bridges to transfer the gravity load of moving

    vehicles to supporting piers. Depending upon the site conditions and

    the span length of the bridge, the truss may be either through type or

    deck type. In the through type, the carriage way is supported at the

     bottom chord of trusses. In the deck type bridge, the carriage way is

    supported at the top chord of trusses. Usually, the structural framing

    supporting the carriage way is designed such that the loads from the

    carriage way are transferred to the nodal points of the vertical bridgetrusses.

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    Figure 7 Some of the trusses that are used in steel bridges

    Truss Girders, lattice girders or open web girders are efficient and

    economical structural systems, since the members experience

    essentially axial forces and hence the material is fully utilised.

    Members of the truss girder bridges can be classified as chord

    members and web members. Generally, the chord members resistoverall bending moment in the form of direct tension and

    compression and web members carry the shear force in the form of

    direct tension or compression. Due to their efficiency, truss bridges

    are built over wide range of spans. Truss bridges compete against

     plate girders for shorter spans, against box girders for medium spans

    and cable-stayed bridges for long spans. Some of the most commonly

    used trusses suitable for both road and rail bridges.

    For short and medium spans it is economical to use parallel chordtrusses such as Warren truss, Pratt truss, Howe truss, etc. to minimize

    fabrication and erection costs. Especially for shorter spans the warren

    truss is more economical as it requires less material than either the

    Pratt or Howe trusses. However, for longer spans, a greater depth is

    required at the centre and variable depth trusses are adopted for

    economy. In case of truss bridges that are continuous over many

    supports, the depth of the truss is usually larger at the supports and

    smaller at midspan.

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    As far as configuration of trusses is concerned, an even number of

     bays should be chosen in Pratt and modified Warren trusses to avoid a

    central bay with crossed diagonals. The diagonals should be at an

    angle between 50° and 60° to the horizontal. Secondary stresses can

     be avoided by ensuring that the centroidal axes of all intersecting

    members meet at a single point, in both vertical and horizontal planes.

    However, this is not always possible, for example when cross girders

    are deeper than the bottom chord then bracing members can be

    attached to only one flange of the chords.

    8 GENERAL DESIGN PRINCIPLES

    8.1 Optimum depth of truss girderThe optimum value for span to depth ratio depends on the magnitude

    of the live load that has to be carried. The span to depth ratio of a

    truss girder bridge producing the greatest economy of material is that

    which makes the weight of chord members nearly equal to the weight

    of web members of truss. It will be in the region of 10, being greater

    for road traffic than for rail traffic. IS: 1915-1961, also prescribes

    same value for highway and railway bridges. As per bridge rules published by Railway board, the depth should not be greater than

    three times width between centres of main girders. The spacing

     between main truss depends upon the railway or road way clearances

    required.

    8.2 Design of compression chord membersGenerally, the effective length for the buckling of compression chord

    member in the plane of truss is not same as that for buckling out-of- plane of the truss i.e. the member is weak in one plane compared to

    the other. The ideal compression chord will be one that has a section

    with radii of gyration such that the slenderness value is same in both

     planes. In other words, the member is just likely to buckle in plane or

    out of plane. These members should be kept as short as possible and

    consideration is given to additional bracing, if economical. The

    effective length factors for truss members in compression may be

    determined by stability analysis. In the absence of detailed analysisone can follow the recommendations given in respective codes. The

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    depth of the member needs to be chosen so that the plate dimensions

    are reasonable. If they are too thick, the radius of gyration will be

    smaller than it would be if the same area of steel is used to form a

    larger member using thinner plates. The plates should be as thin as

     possible without losing too much area when the effective section is

    derived and without becoming vulnerable to local buckling. Common

    cross sections used for chord members are shown in Fig. 7.

    Trusses with spans up to 100 m often have open section compression

    chords. In such cases it is desirable to arrange for the vertical posts

    and struts to enter inside the top chord member, thereby providing a

    natural diaphragm and also achieving direct connection between

    member thus minimizing or avoiding the need for gussets. However,

     packing may be needed in this case. For trusses with spans greaterthan about 100 m, the chords will be usually the box shaped such that

    the ideal disposition of material to be made from both economic and

    maintenance view points. For shorter spans, rolled sections or rolled

    hollow sections may be used. For detailed design of compression

    chord members the reader is referred to the chapter on Design of

    axially compressed columns.

    8.3 Design of tension chord membersTension members should be as compact as possible, but depths have

    to be large enough to provide adequate space for bolts at the gusset

     positions and easily attach cross beam. The width out-of-plane of the

    truss should be the same as that of the verticals and diagonals so that

    simple lapping gussets can be provided without the need for packing.

    It should be possible to achieve a net section about 85% of the gross

    section by careful arrangement of the bolts in the splices. This means

    that fracture at the net section will not govern for common steel

    grades.

    In this case also, box sections are preferable for ease of maintenance

     but open sections may well prove cheaper. For detailed design reader

    is referred to the chapter on Design of Tension members.

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    Figure 8 Typical cross-section for truss members

    9. DESIGN OF VERTICAL AND DIAGONAL

    MEMBERS

    Diagonal and vertical members are often rolled sections, particularly

    for the lightly loaded members, but packing may be required for

    making up the rolling margins. This fact can make welded members

    more economical, particularly on the longer trusses where the packingoperation might add significantly to the erection cost. Aesthetically, it

    is desirable to keep all diagonals at the same angle, even if the chords

    are not parallel. This arrangement prevents the truss looking over

    complex when viewed from an angle. In practice, however, this is  usually overruled by the economies of the deck structure where a

    constant panel length is to be preferred. Typical cross sections used

    for members of the truss bridges

    are shown in Fig. 8

    9.1 Lateral bracing for truss bridgesLateral bracing in truss bridges is provided for transmitting the

    longitudinal live loads and lateral loads to the bearings and also to

     prevent the compression chords from buckling. This is done by

     providing stringer bracing, braking girders and chord lateral bracing.

    In case of highway truss bridges, concrete deck, if provided, also acts

    as lateral bracing support system.

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    Figure 9 Lateral bracing systems 

    The nodes of the lateral system coincide with the nodes of the main

    trusses. Due to interaction between them the lateral system may cause

    as much as 6% of the total axial load in the chords. This should be

    taken into account. Fig.8 shows the two lateral systems in its original

    form and its distorted form after axial compressive loads are applied

    in the chords due to gravity loads. The rectangular panels deform asindicated by the dotted lines, causing compressive stresses in the

    diagonals and tensile stresses in the transverse members. The

    transverse bracing members are indispensable for the good

     performance of St. Andrew’s cross bracing system. 

    In diamond type of lateral bracing system the nodes of the lateral

    system occur midway between the nodes of the main trusses [Fig.8].

    They also significantly reduce the interaction with main trusses. With

    this arrangement, “scissors-action” occurs when the chords arestressed, and the chords deflect slightly laterally at the nodes of the

    lateral system. Hence, diamond system is more efficient than the St.

    Andrew’s cross bracing system. It is assumed that wind loading on

    diagonals and verticals of the trusses is equally shared between top

    and bottom lateral bracing systems. The end portals (either diagonals

    or verticals) will carry the load applied to the top chord down to the

     bottom chord. In cases, where only one lateral system exists (as in

    Semi through trusses), then the single bracing system must carry the

    entire wind load.

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    10. HOW BRIDGES WORK?

    Every passing vehicle shakes the bridge up and down, making waves

    that can travel at hundreds of kilometers per hour. Luckily the bridgeis designed to damp them out, just as it is designed to ignore the

    efforts of the wind to turn it into a giant harp. A bridge is not a dead

    mass of metal and concrete: it has a life of its own, and understanding

    its movements is as important as understanding the static forces.

    11. CONNECTIONS

    Members of trusses can be joined by riveting, bolting or welding. Dueto involved procedure and highly skilled labour requirement, riveting

    is not common these days, except in some railway bridges in India. In

    railway bridges riveting may be used due to fatigue considerations.

    Even in such bridges, due to recent developments, high strength

    friction grip (HSFG) bolting and welding have become more

    common. Shorter span trusses are usually fabricated in shops and can

     be completely welded and transported to site as one unit. Longer span

    trusses can be prefabricated in segments by welding in shop. Thesesegments can be assembled by bolting or welding at site. This results

    in a much better quality of the fabricated structure. However, the

    higher cost of shop fabrication due to excise duty in contrast to lower

    field labour cost frequently favour field fabrication in India.

    If the rafter and tie members are T sections, angle diagonals can be

    directly connected to the web of T by welding or bolting. Frequently,

    the connections between the members of the truss cannot be made

    directly, due to inadequate space to accommodate the joint length. Insuch cases, gusset plates are used to accomplish such connections

    (Fig. 9). The size, shape and the thickness of the gusset plate depend

    upon the size of the member being joined, number and size of bolt or

    length of weld required, and the force to be transmitted. The thickness

    of the gusset is in the range of 8 mm to 12 mm in the case of roof

    trusses and it can be as high as 22 mm in the case of bridge trusses.

    The design of gussets is usually by rule of thumb. In short span (8  –  

    12 m) roof trusses, the member forces are smaller, hence the thickness

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    of gussets are lesser (6 or 8 mm) and for longer span lengths (> 30 m)

    the thickness of gussets are larger (12 mm).

    Figure 9 

    12. LOADS ON BRIDGES

    The following are the various loads to be considered for the purposeof computing stresses, wherever they are applicable.

     Dead load Live load Impact load Longitudinal force Thermal force Wind load

     Seismic load Racking force Forces due to curvature Forces on parapets Frictional resistance of expansion bearings Erection forces

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    12.1 DEAD LOADThe dead load on a bridge consists of the weight of all its structural

     parts and all the fixtures and services like deck surfacing, kerbs,

     parapets, lighting and signing devices, gas and water mains,

    electricity and telephone cables. The weight of the structural parts has

    to be guessed at the first instance and subsequently confirmed after

    the structural design is complete.

    12.2 Live loads Bridge design standards of different countries specify the design loads

    which are meant to reflect the worst loading that can be caused on the

     bridge by traffic permitted and expected to pass over it. The

    relationship between bridge design loads and the regulationsgoverning the weights and sizes of vehicles is thus obvious, but other

    factors like traffic volume and mixture of heavy and light vehicles are

    also relevant. Short spans, say up to 10m for bending moment and 6m

    for shear   force, are governed by single axles or bogies with closelyspaced multiple axles. The worst loading for spans over about 20m is

    often caused by more than three vehicles.

    The worst vehicles are often the medium-weight compact vehicles

    with two axles, and not the heaviest vehicles with four, five or sixaxles. The criteria thus change from axle loads to worst vehicles as

    the span increases, with the mixture of vehicles in the traffic being an

    important factor for the longer spans. When axles or single vehicles

    are the worst case, the effect of impact has to be allowed for, but

    several closely spaced vehicles represent a jam situation without

    significant impact. The adjacent lanes of short span bridges may all be

    loaded simultaneously with the worst axles or vehicles, but this is less

    likely for long spans. Apart from the design loading for normal traffic,many countries specify special bridge design loading for the passage

    of abnormal vehicles of the military type or carrying heavy indivisible

    industrial equipment like generators. The passage of such heavy

    vehicles on public roads usually involves special permits from the

    highway authorities and often supervision by the police. In addition to

    these legal heavy loads, there is the growing problem of illegal

    overweight vehicles weighing as much as 40% over their legal limits.

    In each country traffic regulations limit the wheel and axle loads and

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    gross vehicle weights, and impose dimensional limits on axle spacing

    and size of vehicles. Goods vehicles may be of the following types:

     vehicles with two axles rigid vehicles with three or more axles articulated vehicles with two or three axles under the tractor and

    one or more axles under the trailer

     road trains comprising a vehicle and trailer.

    12.3 Longitudinal forces

    Longitudinal forces are set up between vehicles and the bridge deck

    when the former accelerate or brake. The magnitude of the force is

    given by

     

    where W is the weight of the vehicle, g is the acceleration due to

    gravity (¼9.81 m/s2) and V is the change in speed in time t.Usually the change in speed is faster during braking than while

    accelerating.

    Railway bridges: Railway bridges including combined rail and road

     bridges are to be designed for railway standard loading given in

     bridge rules. The standards of loading are given for:

    · Broad gauge - Main line and branch line

    · Metre gauge - Main line, branch line and Standard C

    · Narrow gauge - H class, A class main line and B class branch line

    The actual loads consist of axle load from engine and bogies. The

    actual standard loads have been expressed in bridge rules as

    equivalent uniformly distributed loads (EUDL) in tables to simplify

    the analysis. These equivalent UDL values depend upon the span

    length. However, in case of rigid frame, cantilever and suspension

     bridges, it is necessary for the designer to proceed from the basic

    wheel loads. In order to have a uniform gauge throughout the country,

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    it is advantageous to design railway bridges to Broad gauge main line

    standard loading. The EUDLs for bending moment and shear force for

     broad gauge main line loading can be obtained by the following

    formulae, which have been obtained

    from regression analysis:

    For bending moment:

    EUDL in kN = 317.97 + 70.83l + 0.0188l2 ≥ 449.2 kN (7.1)

    For shear force:

    EUDL in kN = 435.58 + 75.15l + 0.0002l2 ≥ 449.2 kN (7.2)

     Note that, l is the effective span for bending moment and the loaded

    length for the maximum effect in the member under consideration for

    shear. 'l ' should be in metres. The formulae given here are not

    applicable for spans less than or equal to 8 m with ballast cushion. For

    the other standard design loading the reader can refer to Bridge rules.

    12.4  Impact load

    Figure 10 Impact percentage curve for highway bridges for IRC class A and IRC class B

    loadings

    The dynamic effect caused due to vertical oscillation and periodical

    shifting of the live load from one wheel to another when the

    locomotive is moving is known as impact load. The impact load is

    determined as a product of impact factor, I, and the live load. The

    impact factors are specified by different authorities for different typesof bridges. The impact factors for different bridge for different types

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    coefficient of thermal expansion or contraction for steel is 11.7 x 10-6

    /.

    12.6  Wind load –  Wind load on a bridge may act

    · Horizontally, transverse to the direction of span

    · Horizontally, along the direction of span

    · Vertically upwards, causing uplift

    · Wind load on vehicles

    Wind load effect is not generally significant in short-span bridges; formedium spans, the design of sub-structure is affected by wind

    loading; the super structure design is affected by wind only in long

    spans. For the purpose of the design, wind loadings are adopted from

    the maps and tables given in IS: 875 (Part III). A wind load of 2.40

    kN/m2 is adopted for the unloaded span of the railway, highway and

    footbridges. In case of structures with opening the effect of drag

    around edges of members has to be considered.

    12.7  Racking force  –   This is a lateral force produced due to thelateral movement of rolling stocks in railway bridges. Lateral

     bracing of the loaded deck of railway spans shall be designed to

    resist, in addition to the wind and centrifugal loads, a lateral

    load due to racking force of 6.0 kN/m treated as moving load.

    This lateral load need not be taken into account when

    calculating stresses in chords or flanges of main girders.

    12.8  Forces on parapets - Railings or parapets shall have aminimum height above the adjacent roadway or footway surface

    of 1.0 m less one half the horizontal width of the top rail or top

    of the parapet. They shall be designed to resist a lateral

    horizontal force and a vertical force each of 1.50 kN/m applied

    simultaneously at the top of the railing or parapet.

    12.9  Seismic load  –   If a bridge is situated in an earthquake proneregion, the earthquake or seismic forces are given due

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    consideration in structural design. Earthquakes cause vertical

    and horizontal forces in the structure that will be proportional to

    the weight of the structure. Both horizontal and vertical

    components have to be taken into account for design of bridge

    structures. IS:1893  –   1984 may be referred to for the actual

    design loads.

    12.10 Forces due to curvature - When a track or traffic lane on a

     bridge is curved allowance for centrifugal action of the moving

    load should be made in designing the members of the bridge. All

    the tracks and lanes on the structure being considered are

    assumed as occupied by the moving load.

    12.11 Erection forces  –  There are different techniques that are used

    for construction of railway bridges, such as launching, pushing,

    cantilever method, lift and place. In composite construction the

    composite action is mobilised only after concrete hardens and

     prior to that steel section has to carry dead and construction live

    loads. Depending upon the technique adopted the stresses in the

    members of the bridge structure would vary. Such erectionstresses should be accounted for in design. This may be critical,

    especially in the case of erection technologies used in large span

     bridges.

    13. ECONOMY OF TRUSSES

    Trusses consume a lot less material compared to beams to span the

    same length and transfer moderate to heavy loads. However, the

    labour requirement for fabrication and erection of trusses is higher

    and hence the relative economy is dictated by different factors. In

    India these considerations are likely to favour the trusses even more

     because of the lower labour cost. In order to fully utilize the economy

    of the trusses the designers should ascertain the following:

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     Method of fabrication and erection to be followed, facility for shopfabrication available, transportation restrictions, field assembly

    facilities.

     Preferred practices and past experience. Availability of materials and sections to be used in fabrication. Erection technique to be followed and erection stresses. Method of connection preferred by the contractor and client

    (bolting, welding or riveting).

     Choice of as rolled or fabricated sections. Simple design with maximum repetition and minimum inventory

    of material.

    14. ADVANTAGE OF STEEL IN BRIDGE

    CONSTRUCTRION

    The steel is a very versatile material having many advantages over the

    other material. Presently, the mega bridge projects being undertaken

     by the Railways involves steel super structures. For longer spans, the

    railway has shown more confidence in steel compared with PSC. This

    is basically due to the fact that in case of steel bridges, rehabilitation

     procedures are easier and involve lesser delays, inspections are easier

    as it allows the deformations to be seen and easily

    evaluated/measured besides the basic fact that Railway engineers feel

    comfortable in constructing and maintaining steel bridges. Generally

    speaking, steel bridges may have the following advantages when

    compared to concrete/PSC bridges:

    -_Reduced dead loads.

    -_More economic foundations.

    -_Simpler erection procedures.

    -_Shorter execution time.

    -_Faster and easier rehabilitation.

    When constructed in insurgency affected areas like North-East and

    J&K and in high seismicity areas where damage to the bridges is

    more likely, steel bridges provides easier and faster options for

    rehabilitation. More over, structural redundancies can be easily inbuilt

    in steel bridges. A disadvantage of steel when compared to concrete is

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    the maintenance cost for the prevention of corrosion. However, it is

    now recognized that concrete bridges also have problems relating to

    maintenance i.e. relating to the effects of corrosion of steel

    reinforcement on the durability of the structure. In addition to the

    various points cited above, structural steel as the basic bridge

    construction material involves following advantages which have also

     played an important part in this shift of Railway engineer’s ideology

    from concrete/PSC bridge construction to steel bridge construction: -

    14.1 HIGH STRENGTH TO WEIGHT RATIO

    High strength to weight ratio of steel minimizes substructure costs,

    which is particularly beneficial in poor ground conditions. Minimum

    self weight is also an important factor in transporting and handling of bridge components specially in hilly areas like North-East and Jammu

    & Kashmir. In addition, it facilitates very shallow construction depth

    for girders, which over come problems with headroom and flood

    clearances and minimizing the length of approach ramps. The low self

    weight also minimizes foundation work in case of bridges being

    constructed near existing rail lines.

    14.2  HIGH QUALITY MATERIALSteel is a high quality material, which is readily available world wide

    in various certified grades, shapes and sizes. The testing regime

    carried out at the steel mills imparts confidence to the engineers for

    the bridge projects. Prefabrication in controlled shop condition leads

    to high quality work at minimum cost. The quality control extends

    from the material itself and follows on through the processes of

    cutting, drilling, welding and fit-up. The total weight of steel

    constructions is a fraction of the total weight of concrete bridges.Therefore steel bridges can be used with long spans, even in

    earthquake-prone areas.

    14.3  SPEED OF CONSTRUCTION

    The prefabrication of the components means that construction time on

    site in hostile environment is minimized. The light weight nature of

    steel permits the erection of large components. Besides this, resource,

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    such as water, aggregates etc may sometimes not be easily available

    at sites on this project, for the purpose of production of concrete.

    14.4 VERSATILITY

    The prefabrication of the components means that construction time on

    site in hostile environment is minimized. The light weight nature of

    steel permits the erection of large components. Besides this, resource,

    such as water, aggregates etc may sometimes not be easily available

    at sites on this project, for the purpose of production of concrete.

    14.5 RECYCLING

    Steel suits a wide range of construction methods and sequences.

    Installation may be by cranes, launching, slide-in-techniques ortransporters. For example, in Jammu & Kashmir area on Katra-

    Quazigund section, the erection of the main steel arch of Chenab and

    Anjikhad bridge is being planned by mechanized rope way. Steel

    gives the engineer flexibility in terms of erection sequence and

     programme. Components can be sized to suit access restriction at site,

    and once erected the steel girders provide a platform for subsequent

    operations.

    14.6  REPAIR & REHABILITATIONSteel is a ‘sustainable’ material. When a steel bridge reaches the end

    of its useful life, the girders can be cut into manageable sizes to

    facilitate demolition, and returned to steelworks for recycling. The

    increased emphasis of the green techniques for construction, steel is

    lot ‘Greener’ than concrete for bridges. 

    14.7  AESTHETICSSteel bridges can readily be repaired after accidental damages. In case

    of damage to the bridge due to derailment/accident, damage due to a

    terrorist activity or damage due to natural causes such as earthquakes,

    floods etc. complete steel spans can be replaced without much delay

    which is not the case with PSC super structures. This aspect is very

    important in the case of Railways where longer disruption to rail

    traffic can not be afforded.

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    14.8  DURABILITYSteel bridges now have a proven life span extending to well over 100

    years. In fact, old steel girders of vintage 1854 etc are also in use on

     branch lines. Steel has a predictable life, as the structural elements are

    visible and accessible. Any signs of deterioration are readily apparent,

    without the need for extensive investigations. Direct measurements of

    stresses are possible as all the parts / members are accessible and

    thickness of members is less. Corrosion is a problem, which can,

    however, be addressed by timely painting. In addition, the latest

    coatings are anticipated to last well beyond 30 years before requiring

    major maintenance.

    The potential durability of steel may be summarized in the following

    quote by a Mr. J.A.Waddell in 1921:“The life of a steel bridge that is scientifically designed, honestly and

    carefully built, and not seriously overloaded, if properly maintained,

    is indefinitely long.” 

    15. COMPONENTS OF SUPERSTRUCTURE OF

    A STEEL TRUSS BRIDGE

    Figure 11 15.1 Top chords:

    • The most highly stressed compression members.• Need special attention while proportioning and detailing. 

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    Figure 12 Common cross-sections of top chords

    15.2 Bottom chords:

    • The most highly stressed tension members.• Connections may be welded, bolted or riveted. 

    Figure 13 Common cross-sections of bottom chords

    15.3 Web members.

    • These members could be diagonals and verticals and may be

    subjected to tension and some to compression. ( it depends on the typeof the truss)

    • Vertical members working at compression are termed “post” and

    those in tension are called “hangers” 

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    Figure 14 Typical crosssections of web members .

    15.4 End posts or rakers:

    • These are located at the ends of a truss to carry the lateral forces

    from the top chord level to the bridge bearings.• For this purpose portal bracings are fixed onto them at the upper

    level.

    Figure15 15.5 Bracing system:

    • Considered as secondary members, but in fact, vital for thesuccessful performance of the primary members.

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    Bracings are designed to resist two types of forces:• Lateral forces: those acting transverse to the axis of the bridge.• Longitudinal forces: Those acting along the axis of the bridge. 

    Lateral bracing:• Placed between the top chords and bottom chords of a pair oftrusses.

    • In a road bridge, the deck slab can act as a stiffening member

     between the trusses.

    Figure 16 Lateral bracing systems

    15.6 Sway bracing or cross bracing:

    • Placed between trusses.• Are provided for distributing the transverse loads to the lateral

    system.• Also for providing torsional rigidity to the truss frame 

    Figure 17 Sway bracing

    15.7 Portal bracings :

    • Located at the end posts or rakers.• Provide end supports to the top lateral bracing system. 

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    Figure 18 Portal bracing

    16. ANALYSIS AND DESIGN OF TRUSSES

     Stability and Determinacy• A stable and statically determinate plane truss should have atleast three members, three joints and three reaction components.

    • To form a stable and determinate plane truss of “n” joints: 

    • The number of members (m) = 2n-3

    • If the stable, simple, plane truss has more than three reactions

    components, the structure is externally indeterminate.

    • If it has more than (m>2n-3) members, the structure is internally

    indeterminate and hence all of the member forces cannot be

    determined form the 2n available equations of static method of joints.

    • Truss analysis gives the bar forces in a truss, for a statically

    determined truss, these bar forces can be found by employing

    the laws of statics to assure internal equilibrium of the

    structure.

    • Method of Joints. 

    • Method of Sections. 

    For indeterminate truss:

    • Virtual work. 

    Another methods:

    • Finite Element Method• Computer Analysis. 

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    17. METHOD OF JOINTS.

    • Chose a node whose number of unknown forces doesn’t exceed two,

    then study its equilibrium using static equilibrium equations todetermine these forces:

    • ΣFx = 0, ΣFy = 0 and ΣM = 0 

    • Go to next node and study its equilibrium using the evaluated forces

    from the previous node then go to the next node and so one.

    Figure 19 Method of Joints.

    18 METHOD OF SECTIONS

    • If only a few member forces of  a truss are needed, the quickest way

    to find these forces is by this method.

    • In this method, an imaginary cut (section) is drawn through a stableand determinate truss. Thus, a section subdivides the truss into two

    separates parts. Since the entire truss is in equilibrium, any part of it

    must also be in equilibrium.

    •ΣFx = 0, ΣFy = 0 and ΣM = 0. 

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    Figure 20

    19. DEFLECTION OF A TRUSS

    • The virtual work method can be used to determine thedeflection of trusses.

    • with n equal to the virtual force in the member andequal to the change in length of the member.

    • A deflection is caused by three ways: 

    • Applied loads acting on each member. • Temperature changes 

    • Fabrication errors. 

    • Axial deformation: 

    • When the force of all the members are k nown we can determine the

    axial deformation of each member by using the equation:

    • If we modified the value of  into the equation for the deflection.

    m= number of members.

    n= the force in the member due to the virtual load.

     N= the force in the member due to applied load.

    L= length.

    A= area.E= Young´s Modulus of Elasticity.

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    • Temperatures Changes= is the coefficient of thermal expansionL = length of the member

    AT= Change in temperature.

    • Fabrication Errors: 

    • The original equation for deflection of a truss can be modified

    K= number of members undergoing fabrication errors.

    n= force in the member due to the virtual load= the change in length of the member due to fabrication errors.

    Total deflection of a Truss (Hibbeler, Structural Analysis) 

    20. DETERMINACY OF COPLANAR TRUSSES

     Since all the elements of a truss are two-force members, themoment equilibrium is automatically satisfied.

     Therefore there are two equations of equilibrium for each joint, j ,in a truss. If r is the number of reactions and b is the number of bar

    members in the truss, determinacy is obtained by

    b + r = 2j Determinate

    b + r > 2j Indeterminate

    21. STABILITY OF COPLANAR TRUSSES

     If b + r < 2j , a truss will be unstable , which means the structurewill collapse since there are not enough reactions to constrain all

    the joints. A truss may also be unstable if b + r ≥2j. In this case, stability will

     be determined by inspection

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     b + r < 2j Unstable

    b + r  2j Unstable if reactions are concurrent, parallel, or collapsiblemechanics

    21.1 External stability - a structure (truss) is externally unstable if its

    reactions are concurrent or parallel

    Figure21

    21.2 Internal stability - may be determined by inspection of thearrangement of the truss members.

     A simple truss will always be internally stable

     The stability of a compound truss is determined by examining howthe simple trusses are connected

     The stability of a complex truss can often be difficult to determine by inspection.

     In general, the stability of any truss may be checked by performinga complete analysis of the structure. If a unique solution can be

    found for the set of equilibrium equations, then the truss is stable.

     Internal stability

    Externally stable Internally stable

    Figure22 

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    Collapsible mechanism 

    Externally stable Internally unstableFigure23 

    Collapsible mechanism 

    Figure24 Externally stable Internally unstable

    22. TRUSS MEMBERS ARE CONNECTED BY

    SMOOTH PINS

     The stress produced in these elements is called the primary stress. The pin assumption is valid for bolted or welded connections if the

    members are concurrent.

     However, since the connection does provide some rigidity, the bending introduced in the members is called secondary stress.

     Secondary stress analysis is not commonly performed.

    Figure25 

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    23. ALL LOADING IS APPLIED AT THE JOINTS

    OF THE TRUSS

      Since the weight of each members is small compared to themember force, the member weight is often neglected.

     However, when the member weight is considered, it is applied atthe end of each member.

     Because of these two assumptions, each truss member is a two-force member with either a compressive (C) or a tensile (T) axial

    force.

     In general, compression members are bigger to help with instability

    due to buckling.

    24. CLASSIFICATION OF COPLANAR

    TRUSSES

    24.1Simple Truss

     The simplest framework that is rigid or stable is a triangle. Therefore, a simple truss is constructed starting with a basic

    triangular element and connecting two members to form additional

    elements.

     As each additional element of two members is placed on a truss,the number of joints is increased by one.

    Figure26 SIMPLE TRUSS 24.2 Compound Truss

     This truss is formed by connecting two or more simple trussestogether.

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     This type of truss is often used for large spans. There are three ways in which simple trusses may be connected to

    form a compound truss:

    1. Trusses may be connected by a common joint and bar.

    2. Trusses may be joined by three bars.

    3. Trusses may be joined where bars of a large simple truss, called themain truss , have been substituted by simple trusses, called secondary

    trusses 

    Figure27 COMPOUND TRUSS

    25. FORCE ANALYSIS (TRUSS)

    •  Loads members in tension and compression.•  Members are pinned at joints (Moment = 0).•  Triangles provide stability and strength.•  Top members in Compression.•  Bottom members in Tension.

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    26. APPLICABLE SPAN OF TRUSS BRIDGE

    Truss brides are generally applied within the following range.

    1. Simple truss bridge is in the range of 55 meter to 85 meter span.2. Continuous truss bridge is in the range of 60 meter to 300 meter

    span.

    3. Cantilever truss bridge is in the range of 300 meter to 510 meterspan (in Japan, only one bridge has longer span than 200 meter,

    and that is Minato Ohashi, with 510 meter span.)

    Among the 3 types of bridges, the simple truss bridge or the

    continuous truss bridge, either with approximate 60 meter to 100

    meter span is usually applied.

    27. ADVANTAGES OF TRUSS BRIDGE

    27.1 Mountain Region

    1. When the members are difficult to be transported to the site andwhen the conditions of construction is restricted.2. When a bridge in a curve alignment is required, a horizontal bent

    continuous bridge or a deck truss bridge with brackets can selected.

    27.2 Open Field Region

    1. Assurance of space below the bridge soffit due to adoption of

    through bridge2. Long span bridge over the mouth of a river or the coast.3. Double-deck bridge having upper and lower 2-layer road surface.

    28. DISADVANTAGES OF TRUSS BRIDGE

    1. As it is composed by many members, maintenance requirements

    such as repainting is needed.2. The sight view from the driver in case of through bridges.

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    29. OVER ALL HEIGHT OF TRUSS

    AND LENGTH BETWEEN PANELS

    1. For simple truss, the truss height is 1/7-1/8 time the span lengthand the length between panels is 1/6-1/8 time the span length.

    2. For continuous truss, the truss height is 1/9- 1/10 time the spanlength and the length between panels is 1/8-1/10 time the span

    length

    Figure28

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    30. CONCLUSIONS1. Bridge components are having substantial fatigue life even after

    considering the Joint Flexibility.

    2. Joint flexibility tends to alter the vibration characteristics ofeach component to loading environment in presence of realistic

    damping of 2%, thus the damage

     potential of each member which depends upon the stress range and

    cycle counts is also got affected ,however the change was only

    40%(max).

    3. In most of members fatigue life got increased, however life of

    some component got decreased, the maximum decrease observed is

    about 40% in one of Bottom Chords

    and Verticals.

    4. The variation in passage to failure exhibited by each component

    with the different speeds makes it difficult to find a particular

    trend, however the trend is similar at

    different flexibilities with change only in magnitudes.

    5. It can be concluded that the reduction of joint rotational stiffness

    up to 50% has less effect on structural stability of Steel Truss Railway

    Bridge.

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    31. SUGGESTIONIn comparison to the developed countries, the steel being used in

    Indian railways is of inferior quality. Following suggestions

    recommendations are given for early adoption of High Performancesteel over Indian Railways:-

    1. Furthermore studies should be conducted for the adoption of HPS

    or any other type of steel which suits Indian conditions and economy.

    2. The Railway Board in consultation with RDSO may jointly discuss

    the issue for convincing the steel industry including SAIL for

     producing the special type of steel for Indian bridges.

    3. A pilot project should be given to each railways for applying the

    High Performance steel in at least one bridge so that experience in the

    same can be gained.

    4. The major supplier of HPS in US and Europe is Arcelor-Mittal

    steel and Corus Group (recently tied up with Tata steel) and they have

    a very good Indian connection. These groups may be approached for

    their help and guidance.

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    32.Marvels of truss bridge in India:

    Howrah Bridge  –   The Bridge without Nuts & Bolts!: 

    How about visiting a vintage bridge which has no nuts & bolts

    in its construction but still standing tall for the last 66 years?

    Hard to believe? The Bridge in concern - one of the busiest in

    the world - is located at Howrah in West Bengal. The Howrah

     bridge, the sixth longest of its type, has been an emblem of the

    city of  Kolkata from its inception. So much so that the worldknows Kolkata  by its trams, the Victoria Memorial, and of

    course the Howrah Bridge. Opened to traffic in 1943, the

    construction of the bridge was started in 1937. The bridge has

    remained one of the most renowned landmarks of Kolkata. 

    More than 150,000 vehicles and 4,000,000 pedestrians cross

    over the bridge every day. Technically speaking, Howrah

    Bridge is a "Cantilever Truss" bridge, constructed entirely byriveting, without nuts or bolts!

    http://www.holidayiq.com/destinations/Howrah-Overview.htmlhttp://www.holidayiq.com/destinations/Howrah-Overview.htmlhttp://www.holidayiq.com/destinations/Howrah-Overview.htmlhttp://www.holidayiq.com/states/West-Bengal-Overview.htmlhttp://www.holidayiq.com/states/West-Bengal-Overview.htmlhttp://www.holidayiq.com/states/West-Bengal-Overview.htmlhttp://www.holidayiq.com/destinations/Kolkata-Overview.htmlhttp://www.holidayiq.com/destinations/Kolkata-Overview.htmlhttp://www.holidayiq.com/destinations/Kolkata-Overview.htmlhttp://www.holidayiq.com/destinations/Kolkata-Know.htmlhttp://www.holidayiq.com/destinations/Kolkata-Know.htmlhttp://www.holidayiq.com/destinations/Kolkata-Know.htmlhttp://www.holidayiq.com/Kolkata-Sightseeing.htmlhttp://www.holidayiq.com/Kolkata-Sightseeing.htmlhttp://www.holidayiq.com/Kolkata-Sightseeing.htmlhttp://www.holidayiq.com/destinations/Kolkata-Know.htmlhttp://www.holidayiq.com/destinations/Kolkata-Overview.htmlhttp://www.holidayiq.com/states/West-Bengal-Overview.htmlhttp://www.holidayiq.com/destinations/Howrah-Overview.html

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    The Godavari Bridge or Kovvur-Rajahmundry Bridge 

    The godavri bridge is a truss bridge spanning Godavari

    river in Rajahmundry, India.  It is Asia's second longest road-cum-railbridge crossing a water body, after the Sky Gate Bridge in Kansai

    International Airport, Osaka.  It is second of the three bridges that

    span the Godavari River at Rajahmundry. The Havelock Bridge being

    the earliest, was built in 1897, and having served its full utility, was

    decommissioned in 1997. The latest bridge is the Godavari Arch

    Bridge,  a bowstring-girder bridge,  built in 1997 and presently in

    service.

    The bridge is 2.7 kilometers long consisting of 27 spans of 91.4 m and

    7 spans of 45.72 m of which 6 spans of 45.72m are in 6 deg. curve at

    long Rajahmundry end to make up for the built up area. The bridge

    has a road deck over the single track rail deck, similar to the Grafton

    Bridge in New South Wales, Australia.  This bridge, in addition

    to Godavari Arch Bridge,  has been widely used to

    represent Rajahmundry in arts, media,  andculture.  It is one of the

    recognised symbols of  Rajahmundry. 

    http://en.wikipedia.org/wiki/Truss_bridgehttp://en.wikipedia.org/wiki/Godavari_riverhttp://en.wikipedia.org/wiki/Godavari_riverhttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Indiahttp://en.wikipedia.org/wiki/Kansai_International_Airporthttp://en.wikipedia.org/wiki/Kansai_International_Airporthttp://en.wikipedia.org/wiki/Osakahttp://en.wikipedia.org/wiki/Godavari_Riverhttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Tied-arch_bridgehttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Single_track_(rail)http://en.wikipedia.org/wiki/Grafton_Bridge,_New_South_Waleshttp://en.wikipedia.org/wiki/Grafton_Bridge,_New_South_Waleshttp://en.wikipedia.org/wiki/New_South_Waleshttp://en.wikipedia.org/wiki/Australiahttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Artshttp://en.wikipedia.org/wiki/Mass_mediahttp://en.wikipedia.org/wiki/Culturehttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/File:Godavari_bridge_panorama.jpghttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Culturehttp://en.wikipedia.org/wiki/Mass_mediahttp://en.wikipedia.org/wiki/Artshttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Australiahttp://en.wikipedia.org/wiki/New_South_Waleshttp://en.wikipedia.org/wiki/Grafton_Bridge,_New_South_Waleshttp://en.wikipedia.org/wiki/Grafton_Bridge,_New_South_Waleshttp://en.wikipedia.org/wiki/Single_track_(rail)http://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Tied-arch_bridgehttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Godavari_Arch_Bridgehttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Godavari_Riverhttp://en.wikipedia.org/wiki/Osakahttp://en.wikipedia.org/wiki/Kansai_International_Airporthttp://en.wikipedia.org/wiki/Kansai_International_Airporthttp://en.wikipedia.org/wiki/Indiahttp://en.wikipedia.org/wiki/Rajahmundryhttp://en.wikipedia.org/wiki/Godavari_riverhttp://en.wikipedia.org/wiki/Godavari_riverhttp://en.wikipedia.org/wiki/Truss_bridge

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    33. IMAGES

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    33. REFERENCE1. “Use and Application of High-Performance steels for steel

    structures” Structural Engineering documents No 8 Published by

    IABSE Oct 2005.2. NBSA White paper, “Advances in High performance steels for

    Highway bridges” by Alexander D Wilson, manager customer

    technical service, Mittal Steel USA

    3. “Prospects of High-Performance welded steel bridge” , Advances

    in bridge Engineering, Mar 24-25, 2006 by

    P.K.Ghosh,Professor,Department of Metallurgical and Material

    Engineering, IIT , Roorkee.

    4. “Improvements to High Performance steel “ by A. D. Wilson ,

    Mittal Steel USA

    5. IS 2062:1999 “Steel for general structural purposes”- specification,

    BIS ,N.Delhi

    6. IS 8500 1977 “Specification for Weld able structural steel” 

    ( Medium and High strength qualities ) , BIS, N. Delhi

    7. “Steel Structures, Design and Behaviour” by Charles G. Salmon &

    John E.Johnson , Harper & Row

    8. www.mittalsteel.com

    9. www.nsba.com

    10. www.iitr.ac.in/departments/CE/abe/413-419.pdf

    11. www.fhwa.dot.gov/bridge

    12. www.corusgroup.com

    13. Anon, “Design of Composite Trusses”, Steel Construction

    Institute”, Ascot, 1992. 

    14. Anon “Constructional Steel Design: An International Guide”,

    Elsevier, London, 1993. 

    15. S.Venkateswara Rao, and M.Prabhakar, Prestressingfor

    economical Bridge construction”, Indian Highways (IRC), 1990. 

    16. Vallenilla, C.R., and Bjorhovde, R., “Effective Width Criteria for

    Composite Beams”, Engineering Journal, AISC, Fourth Quarter,

    1985, pp. 169-175.