curved bridges

7/29/2019 Curved Bridges

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DESIGN OF HORIZONTALLYCURVED GIRDERS


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CURVED BRIDGES ARE PROVIDED

BECAUSE OF LIMITED RIGHTS OF WAY

TO MITIGATED TRAFFIC PROBLEM

TO SIMPLIFY COMPLICATED GEOMETRICES


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CURVED BRIDGE

CONFIGURATIONS


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USE OF CHORDS

Curved bridge beams are made as aseries of short segments/chords toapproximate the theoretical arc.

Max offset b/w Arc and its Chord iss = Lc2/8R

Where,

Lc = Chord Length/Arc Length

R = Radius of Curvature

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

CURVED BRIDGE CONFIGURATIONS


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BEAM CROSS-SECTION BOX BEAMS VERSUS I-BEAMS

For full-span-length curved beams made in plant (using post tensioning), box section ispreferred because of high torsional rigidity and handling considerations.

For segmental construction I-beams may be used. Straight segments are supported ontemporary shores and post-tensioned in the field after constructing diaphragm at the segment

joints.

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

CROSSBEAMSCrossbeams/Intermediated diaphragm are required to counteract

Effects of torsion

Lateral forces resulting from curvature

CURVED BRIDGE CONFIGURATIONS


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PRELIMINARY DESIGN

USEFUL GEOMETRIC APPROXIMATIONS

USEFUL STRUCTURAL APPROXIMATIONS


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USEFUL GEOMETRIC APPROXIMATIONS Arc to chord offset is

s = Lc2/8R

Excess of arc length over chord length

Arc length = (8s2

/3Lc) x Chord length

Center of gravity of an arcC.G. of an arc is offset from the chord by 2s/3, orLc

2/12R

Curved surfacesC.G. of the curved surface lies outside the center ofgravity of the centerline arc by an amount e.e = B2/12R

PRELIMINARY DESIGN


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USEFUL STRUCTURAL APPROXIMATIONSARE DISCUSSED IN THE CONTEXT

OFSTRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES

PRELIMINARY DESIGN


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STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES

Longitudinal Flexure

Torsion

Crossbeams

PRELIMINARY DESIGN


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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.

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 nextbeam to the outside. The outermost beam is the final resting place for this shifted load.

PRELIMINARY DESIGNSTRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES


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Torsion

It will be seen that torsional moment are related to the flexural moment M divided by the radius ofcurvature R.

Torsionin a Simple-Span Curved BeamConsider a short segment near mid-span



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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.



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Crossbeams 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 ofthe crossbeam.

The crossbeam must be deep enough to brace the bottom flange to resist this component.



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DETAILED DESIGN

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.


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Loading Stages-I-beams

1. Individual SegmentsThe segments are pre-tensioned in the plant to compensate for self-weight bending of theindividual segment.

2. Shoring LoadsThe individual segments are erected in the field, supported by final bearings and by shoresat 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 theshoring.

3. Non-composite GridworkPost-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 present in the shoresbecomes 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


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4. Composite GridworkThe 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

- Deflection and camber

- Prestress losses

- Ultimate strength

- Torsion (additional consideration)

DETAILED DESIGN


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DESIGN PROBLEM

Data- radius 600ft

- span length 120ft(measured along the arcat the centerline of the bridge)

- superelevation 6%

- design speed 40mph

- girder cross section bulb-tee-beam 6 in number spaced at 9ft

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


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Plan geometry- Arc-to-Chord offset La

2/8R = (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 2-8and 3-4 on the inside.


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Materials 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 12.9.1.2-1Concrete 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-relaxationArea of one strand =0.217 in.2

Ultimate strength, fpu =270.0 ksiYield 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


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Cross-Section Properties

Non-Composite SectionsA =area of cross-section of precast beam =911 in.2

h =overall depth of beam =72 in.

I =moment of inertia about the centroid of the precast beam =608,109 in.4

yb =distance from centroid to extreme bottom fiber of the precast beam =36.51 in.

yt =distance from centroid to extreme top fiber of the precast beam =35.49 in.

Sb =section modulus for the extreme bottom fiber of the precast beam =16,657 in.3

St =section modulus for the extreme top fiber of the precast beam =17,134 in.3

Ilat =lateral moment of inertia of precast beam =46,014 in.4wg =beam self-weight per unit length =0.949 kip/ft

Ec =modulus of elasticity, ksi =33,000(wc)1.5 fc [LRFD Eq. 5.4.2.4-1]

where

wc =unit weight of concrete =0.150 kcf

TheLRFD Specifications, Commentary C5.4.2.4, indicates that the unit weight of

normal weight concrete is 0.145 kcf. However, precast concrete mixes typically have a relatively low

water/cementitious materials ratio and high density. Therefore, a unit weight of 0.150 kcf is used in this

example. For high strength concrete, this value may need to be increased based on test results.


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Fc =specified strength of concrete, ksi

Therefore, the modulus of elasticity for:

cast-in-place slab, Ecs =33,000(0.150)1.5 =3,834 ksi

precast beam at transfer of post-tensioning (at 28 days minimum)

Eci =33,000(0.150)1.5 =4,888 ksiprecast beam at service loads, Ec =33,000(0.150)1.5 =4,888 ksi

The torsional constant, J, is estimated in accordance with LRFD Specifications, and Section 7.6.5.

J = A4 / (40.0 Ip)

The polar moment of inertia Ip is equal to the sum of I and Ilat. Ip =654,123 in.4

J = (911^4) / (40.0 x 654,123) = 26,324 in4

Properties of the 12- by 66-in. crossbeam:

A =792 in.2

I =287,496 in.4

Ilat =9,504 in.4 (for lateral bending)

J =33,120 in.4

wg =0.825 kip/ft


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Composite Sections Effective Flange Width

Effective flange width for interior beams shall be the lesser of: [LRFD Art. 4.6.2.6.1]

- (1/4) span =(120 x 12)/4 =360 in

- 12ts plus greater of web thickness or 1/2 beam top flange width

= (12 x 7.5) +(0.5 x 44) =112 in.

- average spacing between beams =(9 x 12) =108 in.

Therefore, the effective flange width is 108 in. for the beam.

For the interior crossbeams, the effective flange width is (12 x 7.5) +12 =102 in.

Modular Ratio

Modular ratio between slab and beam materials, n=Ecs/Ec=3,834/4,888=0.7845

Transformed Section Properties

Transformed flange width for interior beams =n (effective flange width)

=(0.7845)(108) =84.73 in.

Transformed flange area for interior beams =n(effective flange width)(structural thickness)

=(0.7845)(108)(7.5) =635.45 in.2

Ac =total area of the composite section =1,564 in.2

hc =overall depth of the composite section =80 in.

Ic =moment of inertia of the composite section =1,208,734 in.4

ybc =distance from the centroid of the composite section to the extreme bottom fiber of the precast beam

=53.05 in.

ytg =distance from the centroid of the composite section to the extreme top fiber of the precast beam

=18.95 in.


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ytc =distance from the centroid of the composite section to the extreme top fiber of the deck =26.95 in.

Sbc =composite section modulus for the extreme bottom fiber of the precast beam =22,784 in.3

Stg =composite section modulus for the top fiber of the precast beam =63,792 in.3

Stc =composite section modulus for the extreme top fiber of the deck slab =57,176 in.3

Iclat =moment of inertia of composite section for lateral bending =666,423 in.4

Composite properties of interior crossbeams:

Ac =1,397 in.2

Ic =765,432 in.4

Iclat =529,860 in.4 for lateral bending

Jc =54,204 in.4


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Loads

For a first approximation, all loads except the truck load will be assumed to be distributed

Dead Loads Dead Loads Acting on the Non-Composite Structure1. Beam and crossbeam weight:

Beams =(6)(120 ft)(0.949 kip/ft) =683 kips

Crossbeams =(4)(45 ft)(0.825 kip/ft) =149 kips

Total weight of beams and crossbeams =683 +149 =832 kips

2. Deck weight:

Gross area of deck =(120 ft)(51 ft) =6,120 ft 2

Actual thickness =8 in.

Deck weight =[8 in./(12 in./ft)](0.150 kcf )(6,120 ft 2 ) =612 kips

For a minimum haunch thickness of 0.5 in., the superelevation of 0.06 will cause the

average haunch thickness to be 0.5 in. +0.06(22 in.) =1.82 in., say 2 in. The haunch

weight is 0.150 kcf (2 in.)(44 in.)/(144 in.2 /ft 2 ) =0.092 kip/ft/beam

Haunch weight =(6)(120 ft)(0.092 kip/ft) =66 kips

Weight of deck, including haunch =678 kips


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Dead loads (contd)

Dead Loads Acting on the Composite StructureBarrier weight is given as 0.3 kip/ft/side

Barrier weight =(2)(120 ft)(0.3 kip/ft) =72 kips

Future wearing surface is 0.025 ksf

(0.025 ksf )(120 ft)(48 ft) =144 kips

Dead load on composite structure =72 +144 =216 kips

Total dead load =832 +678 +216 =1,726 kips


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Live Loads

Design live loading is HL-93 which consists of a combination of:

1. Design truck or design tandem with dynamic allowance.2. Design lane load of 0.64 kip/ft without dynamic allowance.

IM =33%

where IM =dynamic load allowance, applied to design truck or design tandem only

The number of design lanes is computed as:

Number of design lanes =the integer part of the ratio of w/12, where w is the clear roadway width, ft,between the curbs:

w =48 ft

Number of design lanes =integer part of (48/12) =4 lanes

Multiple presence factor, m:

For 4 lanes, m =0.65.

Stresses from truck and lane loads obtained from refined analysis will be multiplied by 0.65.


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Live Loads (contd)

Lane loadingTo maximize the effect of the live load, the 10-ft loaded width is shifted to the left within each design lane.

This causes the lane load to have an eccentricity of 1ft relative to the lane centerline, and the four lane loads

have an eccentricity of 1ft relative to the bridge centerline. The average arc length increases by the ratio of

601-ft radius/600-ft radius, to 120.2 ft.

The total lane loading for the four design lanes is (4)(120.2 ft)(0.64 kip/ft)(0.65)=200.0 kips.


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Correction FactorsThe bending moments in the exterior beam on the outside of the curve will be greater than in astraight bridge for two reasons:

Additional Span Length FactorThe outside beam is on a radius of 622.5 ft. This increases the span length by a factor of 622.5/600=1.0375.

Shift in Center of GravityCG of centerline arc is offset from a line through the centerline chord by an amount equal to 2/3 ofthe arc-to-chord offset.

2/3 x 3 = 2ft

The additional eccentricity caused by the extra area outside the centerline is equal toB2 /12R=(51 ft)2 / (12)(600) =0.36 ft,

Hence the eccentricity due to dead load is 2.36 ft.


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The next step is to find how much the load on the outside beam is increased because of this

eccentricity.

For six unit areas at 9-ft spacing,

The moment of inertia,I = 1,417.5 ft 4

The section modulus, S= 63 ft 3

For an arbitrary load of 1 kip per bearing,or 6 kips, at 2.36 ft eccentricity,

P/A +Pe/S =1 +6(2.36)/63 =1.2248.

This is the increase in load on the outsideexterior beam caused by the eccentricity ofthe load. The total correction factor forbending moment due to dead load is(1.0375)(1.2248) =1.271.


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For the lane loading, the LRFD requirement to place the load off-center of the lane adds 1 ft

to the eccentricity.

For a 6-kip load at 3.36-ft eccentricity, the load on the outside beam is1 +6(3.36)/63 =1.32.

The total correction factor for lane loading is (1.0375)(1.32) =1.370.


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Correction factor for the truck loading.

For the truck loading, LRFD Article 3.6.1.3.1 specifies that the center of the wheel load be

placed 2 ft from the curb. This causes the center of the vehicle to be 5 ft from the curb (also

the lane edge), so the eccentricity from the centerline of the lane is 1ft. The trucks are in the

center of the bridge, which has a 3-ft eccentricity with respect to the supports. Thus, the

vertical truck loading has an eccentricity of 4 ft


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Effects of Centrifugal Force

The total centrifugal force of 44.4 kips acts at a height of 6 ft. The vertical truck loading is249 kips. The horizontal force acting at 6 ft increases the eccentricity of the vertical load by

(44.4/249)(6 ft) =1.07 ft.

The total eccentricity of the vertical truck load is 5.07 ft, and the correction is1 +6(5.07)/63 =1.483

The total correction factor due to centrifugal force and truck loading is

(1.0375)(1.483) =1.538.


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Bending MomentsOutside Exterior Beam

For all loads, the bending moment may be estimated as that for a 120-ft straight beam multipliedby the correction factor.

For all loads except the truck loadings, the 120-ft straight beam bending moment is WL/8 divided by sixbeams in the bridge.

For the truck loading, the bending moment is scaled from that for a standard truck on a 120ft straight span.

Table shown below comprises summary of the estimated midspan bending moments for the outside exteriorbeam.


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The next step is to check stresses which is not cover here.


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Beam Gridwork

Computer Models

Model 1Beam Segments on Shores

Model 2Shore Loads Model 2 is the non-composite beam gridwork on

the nominal 120-ft span.

In model 2 shores are removed. The loads previously existing in the shores are

applied to Model 2.

The self-weight loads applied directly to Model 2.

The difference in total self-weight bending

moment in the out-side exterior beam is less than

0.1 percent.


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Beam Gridwork

Computer Models (contd)

Model 3Weight of Deck andHaunches

The total weight of the deck and the

haunches = 678 kips

This load is assumed to be applied as a

uniform load of 110.8 psf over the 6,120 sf

gross area of the deck.

The finite elements are only used as a means of

applying a uniform load.

The structural properties of the deck are

zeroed out, because this is a non-composite model.


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Beam Gridwork


Model 4Weight of Barriers and Future Wearing Surface Model 4 represents the composite structure. The general appearance of the model is the

same as Model 3

Composite section properties are used in the beam gridwork.

The 0.025 ksf uniform load is applied over the entire 51-ft width of the deck.

A net barrier load of 0.263 kip/ft (0.3 kip/ft less the 0.025 ksf acting over the 1.5-ft width occupied by the barrier) is applied as a line load along the longitudinal edges of the model.


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Beam Gridwork


Model 5Lane Loading The upper part of the below mention figureshows the specified location of the lane loads in a cross-section

through the bridge.

The lower part ofFigure shows the actual loads applied to the model.

The loads were chosen so that deck elements would be loaded uniformly and the total load would have thecorrect location of the resultant load.


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Beam Gridwork


Model 6Truck Loading with Centrifugal Force

For maximum positive moment, the minimum rear axle spacing of 14 ft controls. The maximum bendingmoment occurs with the middle axle load placed 2.33 ft from midspan.

The main axle wheel loads are 16 kips each, plus a 33 percent dynamic allowance, or 21.28 kips. For the

design speed of 40 mph, the centrifugal force is 0.2374 of the truck weight (without dynamic allowance).This force acts 6 ft above the roadway. The overturning moment per main axle is 0.2374 times 32 kipstimes 6 ft, or 45.58 ft-kips. Dividing by the 6-ft wheel spacing, the wheel loads due to centrifugal force are7.6 kip. The total main axle wheel loads, including the 0.65 factor, m, are 0.65 (21.28 7.6) =18.77 kipsand 8.89 kips. The front axle wheel loads are 1/4 of this, or 4.69 kips and 2.22 kips.


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Beam Gridwork


The wheel loads are placed on fictitious, pin-ended members in order to transfer the loads tothe main beams, as shown for the heavier axles.

The added pin-ended members and loads thatrepresent the truck loading for the conditionproducing maximum moment are shown inFigure.


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Summary of Bending Moments

The bending moments for each of the six beams from the six loading models are summarized in Tableshown below.

Pretensioning counteracts the moments from Model 1while post-tensioning is used to counteract themoments from Models 2 through 6.

TheLRFD Specifications [Article 3.4.1 and Table 3.4.1.1], states that for checking tension in prestressedmembers at service load, the Service III load combination may be used. This combination is 1.00 (DC+DW) +0.8(L +IM).

curved bridges

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