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DESIGN OF HORIZONTALLY CURVED GIRDERS

DESIGN OF HORIZONTALLY CURVED GIRDERS. CURVED BRIDGES ARE PROVIDED BECAUSE OF LIMITED RIGHTS OF WAY TO MITIGATED TRAFFIC PROBLEM TO SIMPLIFY COMPLICATED GEOMETRICES. OBJECTIVE.

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DESIGN OF HORIZONTALLY CURVED GIRDERS

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  1. DESIGN OF HORIZONTALLY CURVED GIRDERS

  2. CURVED BRIDGES ARE PROVIDED • BECAUSE OF LIMITED RIGHTS OF WAY • TO MITIGATED TRAFFIC PROBLEM • TO SIMPLIFY COMPLICATED GEOMETRICES

  3. 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.

  4. CURVED BRIDGE CONFIGURATIONS

  5. CURVED BRIDGE CONFIGURATIONS USE OF CHORDS • Curved bridge beams are made as a series of short segments/chords to approximate the theoretical arc. • Max offset b/w Arc and its Chord is s = Lc2/8R Where, Lc = Chord Length/Arc Length R = Radius of Curvature • Arc-to-Chord offset be limited to 1.5 ft.

  6. 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

  7. PRELIMINARY DESIGN USEFUL GEOMETRIC APPROXIMATIONS USEFUL STRUCTURAL APPROXIMATIONS

  8. PRELIMINARY DESIGN 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 arc C.G. of an arc is offset from the chord by 2s/3, or Lc2/12R • 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

  9. PRELIMINARY DESIGN USEFUL STRUCTURAL APPROXIMATIONSARE DISCUSSED IN THE CONTEXT OFSTRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES

  10. PRELIMINARY DESIGN STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES • Longitudinal Flexure • Torsion • Crossbeams

  11. 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. • 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.

  12. 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. • Torsionin a Simple-Span Curved Beam Consider a short segment near mid-span

  13. PRELIMINARY DESIGN STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES • 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.

  14. PRELIMINARY DESIGN STRUCTURAL BEHAVIOUR OF CURVED-BEAM BRIDGES 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 of the crossbeam. • The crossbeam must be deep enough to brace the bottom flange to resist this component.

  15. 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.

  16. DETAILED DESIGN Loading Stages-I-beams • Individual Segments The segments are pre-tensioned in the plant to compensate for self-weight bending of the individual segment. • 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. • 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 present 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.

  17. DETAILED DESIGN • 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. • Other Design Checks - Allowable stresses - Deflection and camber - Prestress losses - Ultimate strength - Torsion (additional consideration)

  18. DESIGN PROBLEM • Data - 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 - deck slab 8” - wearing surface ½” - design live load is HL-93

  19. 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.

  20. 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 12.9.1.2-1 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 <0.80fpy =194.4 ksi Modulus of elasticity, Ep =28,500 ksi [LRFD Art. 5.4.4.2] • Reinforcing bars: Yield strength, fy =60 ksi Modulus of elasticity, Es =29,000 ksi [LRFD Art. 5.4.3.2] • Future wearing surface: 2 in. additional concrete, unit weight =0.150 kcf • New Jersey-type barrier: Unit weight =0.300 kip/ft/side

  21. Cross-Section Properties • Non-Composite Sections A =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.4 wg =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 The LRFD 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.

  22. 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 ksi precast 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

  23. 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.

  24. 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

  25. 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 Structure • 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 • 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

  26. Dead loads (cont’d) • Dead Loads Acting on the Composite Structure Barrier 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

  27. Live Loads Design live loading is HL-93 which consists of a combination of: • Design truck or design tandem with dynamic allowance. • 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.

  28. Live Loads (cont’d) • Lane loading To 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.

  29. Live Loads (cont’d) • Truck Loading The total weight of the design truck is 8 +32 +32 =72 kips Including 33% impact, 1.33 x 72 =95.76 kips. For 4 trucks, including the multiple presence factor, m: 4(95.76)(0.65) =249.0 kips Total Live Load Total live load =200.0 +249.0 =449.0 kip Centrifugal Force The design speed is 40 mph. The centrifugal force coefficient is given by: C = [4/3] v2/gR where, C =coefficient to compute centrifugal force v =highway design speed, ft/sec g =gravitational acceleration, 32.2 ft/sec2 R =radius of curvature of traffic lane, ft The design speed in ft/sec =40 mph/0.682 =58.65 ft/sec C=0.2374 This is applied to the truck axle loads only, without the dynamic load allowance, and with the factor, m. The centrifugal force for four trucks is 4(72 kip)(0.2374)(0.65) =44.4 kips.

  30. Correction Factors The bending moments in the exterior beam on the outside of the curve will be greater than in a straight bridge for two reasons: • Additional Span Length Factor The 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 Gravity CG of centerline arc is offset from a line through the centerline chord by an amount equal to 2/3 of the arc-to-chord offset. 2/3 x 3 = 2ft The additional eccentricity caused by the extra area outside the centerline is equal to B2 /12R=(51 ft)2 / (12)(600) =0.36 ft, Hence the eccentricity due to dead load is 2.36 ft.

  31. 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 outside exterior beam caused by the eccentricity of the load. The total correction factor for bending moment due to dead load is (1.0375)(1.2248) =1.271.

  32. 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 is 1 +6(3.36)/63 =1.32. The total correction factor for lane loading is (1.0375)(1.32) =1.370.

  33. 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

  34. Effects of Centrifugal Force The total centrifugal force of 44.4 kips acts at a height of 6 ft. The vertical truck loading is 249 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 is 1 +6(5.07)/63 =1.483 The total correction factor due to centrifugal force and truck loading is (1.0375)(1.483) =1.538.

  35. Bending Moments – Outside 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 six beams 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 exterior beam.

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

  37. Beam GridworkComputer Models • Model 1 – Beam Segments on Shores • Model 2 – Shore 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.

  38. Beam GridworkComputer Models (cont’d) • Model 3 – Weight of Deck and Haunches • 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.

  39. Beam GridworkComputer Models (cont’d) • Model 4 –Weight 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.

  40. Beam GridworkComputer Models (cont’d) • Model 5 – Lane 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 of Figure 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 the correct location of the resultant load.

  41. Beam GridworkComputer Models (cont’d) • Model 6 – Truck Loading with Centrifugal Force • For maximum positive moment, the minimum rear axle spacing of 14 ft controls. The maximum bending moment 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 kips times 6 ft, or 45.58 ft-kips. Dividing by the 6-ft wheel spacing, the wheel loads due to centrifugal force are ±7.6 kip. The total main axle wheel loads, including the 0.65 factor, m, are 0.65 (21.28 ±7.6) =18.77 kips and 8.89 kips. The front axle wheel loads are 1/4 of this, or 4.69 kips and 2.22 kips.

  42. Beam GridworkComputer Models (cont’d) • The wheel loads are placed on fictitious, pin-ended members in order to transfer the loads to the main beams, as shown for the heavier axles. • The added pin-ended members and loads that represent the truck loading for the condition producing maximum moment are shown in Figure.

  43. Summary of Bending Moments • The bending moments for each of the six beams from the six loading models are summarized in Table shown below. • Pretensioning counteracts the moments from Model 1while post-tensioning is used to counteract the moments from Models 2 through 6. • The LRFD Specifications [Article 3.4.1 and Table 3.4.1.1], states that for checking tension in prestressed members at service load, the Service III load combination may be used. This combination is 1.00 (DC +DW) +0.8(L +IM).

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