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After reading this article you will learn about the design of curved bridges.

Curved bridges are normally provided for viaducts and interchanges where divergent traffic lanes are converted into a multilane bridge or over-bridge and vice versa. One such example is the Second Hooghly Bridge at Calcutta with six-lane divided carriageway on the main bridge over the river and on the approach viaducts on both Calcutta and Howrah side.

The interchanges on both Calcutta and Howrah side consist of a number of single or dual lane arms. A part of the Calcutta end viaduct and some of the arms of Calcutta and Howrah side interchanges are situated on curves as shown in Fig. 9.12.

Curved bridges over channels are sometimes required to be constructed when constraint of land inside a town or a city is such that construction of such a bridge is the only possibility.

**Type of Piers****:**

Selection of type of Piers for viaduct and interchange curved bridges is not a problem except in cases where traffic lanes are situated below. When the traffic lanes are located below the viaduct or interchange structures or where the bridge is constructed over a channel, the normal rectangular pier affects flow of traffic in case of the former and flow of water in case of the later (Fig. 9.13a).

Therefore, under such circumstances, circular pier either solid or hollow, with pier cap above at right angles to the axis of the bridge is the right solution (Fig. 9.13b) in which case the flow will be smooth.

**Layout of Bearings****:**

The axis of a bridge deck for a curved bridge is not a straight line and changes direction at every point and for this reason, the pier or abutment caps supporting the deck through the bearings are not parallel to each other although these are at right angle to the axis of the bridge at these locations.

But since the axis of the bridge changes direction from one pier cap to the other, it requires careful consideration in respect of fixing the axis of the metallic bearings, whether roller, rocker, hinged or sliding, although no such problem would normally arise in respect of elastomeric bearings or rubber pot bearings which are free to move in any direction and allow free horizontal movement and rotation of the superstructure.

The orientation of the free metallic bearings should be such that the direction of translation of the bearings shall coincide with the direction of movement of the bridge deck. The axis of a curved bridge changes direction at every point and hence the axis of the bridge at two adjacent piers is not the same.

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Therefore, it is to be decided in which manner the axis of the bearings shall be placed, whether at right angle to the bridge axis at such location or whether parallel to the pier-cap axis or in any other direction such that free movement of the deck due to temperature variation is allowed without any obstruction. The direction of movement of a curved bridge deck at the free bearings can be found theoretically from Fig. 9.14.

The curved bridge deck AG is divided into six equal segments, AB, BC, CD etc. and these lengths may be considered as equal to the chord lengths AB, BC, CD etc. specially when the number of division are large. Let length of these chords be equal to “1” and change in length due to temperature increase be “δ1”. Therefore, all the chords AB, BC, CD etc. get increased by 81 tangentially.

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These increased lengths may be resolved into two perpendicular directions viz. along AG and perpendicular to AG. Increase in length of AB, BC, CD along AG direction is δ1cosθ_{A}, δ1cosθ_{B}, δ1cosθ_{c} respectively and increase of AB, BC, CD along perpendicular direction (outwards) is δ1sinθ_{A}, δ1sinθB, δ1sinθc respectively.

Similarly, increase in length of DE, EF, FG along AG is δ1cosθ_{E}, δ1cosθ_{F}, δ1cosθ_{G} and along perpendicular direction (inwards) is δ1sinθ_{E}, δ1sinθF, δ1sinθ_{G} respectively. But since θ_{A} = θ_{G}, θ_{B} = θ_{F} and θc = θ_{E} and summation of the 8 δ1sinθ of the left half is outwards and summation of the δ1sinθ of the right half is inwards, these outward and inward movements balance and the nett movement in the perpendicular direction is zero. .

Therefore, the movement of the curved bridge deck AG due to temperature variation will be along AG i.e. the chord line joining the axis of the bridge from one pier to the other and the nett movement will be ∑δ1cosθ.

Hence the bearing axis shall be at right angles to the chord line AG as shown in Fig. 9.14d. However, when elastomeric bearings are used, no such consideration need be made since these bearings are free to move in any direction.

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**Reactions at Piers****:**

Fig. 9.15 shows the plan of a curved bridge deck. Both the dead load of the deck and the live load (specially when it is eccentric outwards) produce torsion in the deck thereby causing additional reaction over the normal reaction at outer edge or outer bearings at B and D but relief of some reaction at A and C. These aspects should be duly considered in the design of bearings, substructure and foundations.

Another factor which induces additional reaction at B and D is the centrifugal force of the moving vehicles. The centrifugal force acting at a height of 1.2 m above the bridge deck will cause moment which is equal to the centrifugal force multiplied by the depth of deck or girder plus 1.2 m and this will induce additional reaction at B and D.

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**Design of Superstructure****:**

Both the dead load and the live load will induce torsion in the deck. This A-ill not much affect the design of solid slab deck since the span is less and as such the torsional moment is less. However, the torsional stress may be checked and additional steel provided if the stress exceeds the permissible value.

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In addition, the inner corners A and C (where warping might take place due to deflection of the deck) shall be provided with some top reinforcement as in acute angle corners of a skew bridge. In girder bridges, the torsion due to dead and live load will thrust more load on the outer girder and give relief to inner girder in addition to the normal distribution of load.

The bending of the bridge deck in plan due to lateral centrifugal force has also to be duly considered,

The centrifugal force will also cause torsion of deck which may be taken as equal to the centrifugal force multiplied by the distance from the c g. of the deck to 1.2 m above the deck. This torsional moment will again thrust more load on the outer girder and give relief to the inner girder. Therefore, the outer girder for a curved bridge has to carry more load than the outer girder for a normal straight bridge.

To prevent the overturning of the moving vehicles due to centrifugal force, super elevation in the bridge deck as given by the following equation shall be provided.

Superelevation, e = V^{2}/225R (9.1)

Where, e = Super elevation in meter per meter

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V = Speed in Km. per hour

R = Radius in meter.

Super elevation obtained from equation 9.1 shall be limited to 7 per cent. On urban sections with frequent intersections it will, however, be desirable to limit the super elevation to 4 per cent. The super elevation may be provided in the deck slab by raising the deck slab towards the outer curve as shown in Fig. 9.16.

The required super elevation may be achieved by increasing the height of the pedestals towards outer curve (keeping the depth of the girder same for all) as shown in Fig. 9.16a or by increasing depth of the girders towards outer curve (keeping the pedestal height same for all) as in Fig. 9.16b but the former is preferable to the latter from economic and constructional point of view.

**Design of Bearings****:**

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In addition to the usual considerations for the design of the bearings, the effect of the centrifugal force and the torsional moment shall be duly considered and the design of the bearings shall be made accordingly.

The detailing of the bearings shall be such that the deck supported on the bearings is restrained from horizontal movement in the transverse direction due to the effect of centrifugal force in addition to the seismic force owing to dead and live loads.

**Design of Substructure and Foundations****:**

While preparing the design of substructure as well as the foundations, additional reaction on one side of the pier due to torsion and additional horizontal force at the top of the pier due to centrifugal force shall be given due consideration.

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