Curved Bridges

DESIGN OF HORIZONTALLY CURVED GIRDERS

CURVED BRIDGES ARE PROVIDED

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BECAUSE OF LIMITED RIGHTS OF WAY

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TO MITIGATED TRAFFIC PROBLEM

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TO SIMPLIFY COMPLICATED GEOMETRICES

CURVED BRIDGES ARE PROVIDED

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BECAUSE OF LIMITED RIGHTS OF WAY

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TO MITIGATED TRAFFIC PROBLEM

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TO SIMPLIFY COMPLICATED GEOMETRICES

OBJECTIVE The objective of this presentation is to furnish design of curved bridges with a basic understanding of the configuration, geometric approximations and structural behavior and their interrelationship in simple structural loading system.

CURVED BRIDGE CONFIGURATIONS

CURVED BRIDGE CONFIGURATIONS 

USE OF CHORDS 

Curved bridge beams are made as a series of short segments/chords to approximate the theoretical arc.

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Max offset b/w Arc and its Chord is s = Lc2/8R  Where, Lc = Chord Length/Arc Length R = Radius of Curvature

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Arc-to-Chord offset be limited to 1.5 ft.

CURVED BRIDGE CONFIGURATIONS 

BEAM CROSS-SECTION 

BOX BEAMS VERSUS I-BEAMS For full-span-length curved beams made in plant (using post tensioning), box section is  preferred because of high torsional rigidity and handling considerations. For segmental construction I-beams may be used. Straight segments are supported on temporary shores and post-tensioned in the field after constructing diaphragm at the segment  joints.

CONTINUITY Continuity is very desirable in curved bridges. It reduces the effects of torsion.

CROSSBEAMS Crossbeams/Intermediated diaphragm are required to counteract  Effects of torsion Lateral forces resulting from curvature 

PRELIMINARY DESIGN  

USEFUL GEOMETRIC APPROXIMATIONS USEFUL STRUCTURAL APPROXIMATIONS

PRELIMINARY DESIGN 

USEFUL GEOMETRIC APPROXIMATIONS 

Arc to chord offset is s = Lc2/8R 

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Excess of arc length over chord length Arc length = (8s2/3Lc) x Chord length

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Center of gravity of an arc C.G. of an arc is offset from the chord by 2s/3, or  Lc2/12R 

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Curved surfaces C.G. of the curved surface lies outside the center of 

gravity of the centerline arc by an amount “e”. e = B2/12R 

PRELIMINARY DESIGN 

USEFUL STRUCTURAL APPROXIMATIONS ARE DISCUSSED IN THE CONTEXT OF

STRUCTURAL BEHAVIOUR OF CURVEDBEAM BRIDGES

PRELIMINARY DESIGN 

STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES

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Longitudinal Flexure

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Torsion

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Crossbeams

PRELIMINARY DESIGN  STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES 

Longitudinal Flexure 

Analysis as a straight beam For vertical load the beam is analyzed as a straight beam with span length equal to the arc length.

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Loads on outside beam The shears and moments in the outside-exterior beam are larger than other beams. This is caused  by the following factors: -Arc length on the outside of the curve is longer than the nominal length at the centerline of the  bridge. -Other beams will shed some of the their torsional moment by shifting load toward the next  beam to the outside. The outermost beam is the final resting place for this shifted load.

PRELIMINARY DESIGN  STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES 

Torsion It will be seen that torsional moment are related to the flexural moment M divided by the radius of  curvature R. 

Torsion in a Simple-Span Curved Beam Consider a short segment near mid-span

PRELIMINARY DESIGN  STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES 

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Torsion in a Fixed-Ended Beam

Continuous span are intermediated between simple-span and fixed-ended beams. Interior spans resemble the fixed case more closely, and the free ends of the exterior spans may  be closer to the simple-span case.

PRELIMINARY DESIGN  STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES 

Crossbeams 

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Crossbeams must be designed for the shears and moments resulting from the change in direction of the primary bending moment at the location of the crossbeams. The longitudinal forces in the bottom flange have a transverse component at the location of  the crossbeam. The crossbeam must be deep enough to brace the bottom flange to resist this component.

DETAILED DESIGN  

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Detailed design is done by using a beam gridwork computer model. Computer model may be created in a horizontal plane, ignoring grade and super  elevation. The extra weight caused by super elevation should be taken into account.

DETAILED DESIGN 

Loading Stages-I-beams 1.

Individual Segments The segments are pre-tensioned in the plant to compensate for self-weight bending of the individual segment.

2.

Shoring Loads The individual segments are erected in the field, supported by final bearings and by shores at intermediate locations. Post-tensioning ducts are spliced and crossbeams are cast. During this loading stage, stresses in the beams do not change. Loads are added to the shoring.

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 Non-composite Gridwork  Post-tensioning is applied to the non-composite gridwork after the crossbeams have cured sufficiently. This lifts the beams from the shores. The load that was p resent in the shores  becomes a load applied to the non-composite beam gridwork. The weight of the deck and haunch is applied to the non-composite gridwork.

DETAILED DESIGN 

4.

Composite Gridwork  The weights of future wearing surface, barriers, live load plus impact, and centrifugal force are applied to the composite gridwork. The simplifying assumptions for distribution of these loads in straight bridges cannot be used for curved bridges.

5.

Other Design Checks -

Allowable stresses

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Deflection and camber 

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Prestress losses

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Ultimate strength

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Torsion (additional consideration)

DESIGN PROBLEM 

Data

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radius 600ft span length 120ft(measured along the arc at the centerline of the bridge) superelevation 6% design speed 40mph girder cross section → bulb-tee- beam → 6 in number spaced at 9ft

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deck slab 8” wearing surface ½” design live load is HL-93

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Plan geometry Arc-to-Chord offset La2/8R = (1202) / (8x600) = 3ft >1.5ft (min. recommended offset) Subdividing the beam into three segments max offset = (120/3)2 / (8x600)

= .3333ft or 4” → barely detectable visually and acceptable. Over hangs: from beam centerline on the outside will be 2’-8”and 3’-4” on the inside.

 Materials •

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Cast-in-place slab: Actual thickness, ts =8.0 in. Structural thickness =7.5 in.  Note that a 1/2-in. wearing surface is considered an integralpart of the 8-in. deck. Concrete strength at 28 days, f ´ c =4.0 ksi Precast beams: AASHTO-PCI Bulb-tee with 2-in.-added width as shown in Figure 1 2.9.1.2-1 Concrete strength of beam at post-tensioning, f ´ ci =6.5 ksi Concrete strength at 28 days, f ´ c =6.5 ksi Concrete unit weight, wc =0.150 kcf  Design span =120.0 ft (Arc length at centerline of bridge) Post-tensioning strands: 0.6-in. dia, seven-wire, low-relaxation Area of one strand =0.217 in.2 Ultimate strength, fpu =270.0 ksi Yield strength, fpy =0.9fpu =243.0 ksi [LRFD Table 5.4.4.1-1] Stress limits for post-tensioning strands: [LRFD Table 5.9.3-1] at jacking: fpj =0.80fpu =216.0 ksi at service limit state (after all losses): fpe