Innovative Bridge Designs for Rapid Renewal (2014) / Chapter Skim
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From page 435... ...
435 A p p e n d i x F ABC Sample Design Calculations Three design examples are presented in this appendix, as follows: • Sample Calculation 1: Decked Steel Girder Design for ABC • Sample Calculation 2: Decked Precast Prestressed Concrete Girder Design for ABC • Sample Calculation 3: Precast Pier Design for ABC The design examples illustrate the design steps involved in the ABC design process as given in the breakdown below. The ABC design philosophy and design criteria have been described.
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From page 436... ...
436 21. Load Results 22.
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From page 437... ...
437 ABC SAMPLE CALCULATION – 1 Decked Steel Girder Design for ABC
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438 This document shows the procedure for the design of a steel girder bridge with precast deck elements for use in a rapid bridge replacement design. The procedure in this document employs Accelerated Bridge Construction (ABC)
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439 General: 1. Introduction 2.
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From page 440... ...
440 The design of this superstructure system follows AASHTO LRFD and is based on a bridge of three even spans, with no skew. The bridge has two 14-foot lanes and two 8-foot shoulders, for a total roadway width of 44' from curb to curb.
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From page 441... ...
441 Operator offers discount per volume of panels Example 4: True or False Verification Statements are an important operator that allow for the user to verify a system criteria that has been manually input. They are marked by lighter shading to make a distinction between the user inputs.
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From page 442... ...
442 Structural Steel Yield Strength: STable 6.4.1-1 Structural Steel Tensile Strength: STable 6.4.1-1 Concrete 28-day Compressive Strength: S5.4.2.1 Reinforcement Strength: S5.4.3 & S6.10.3.7 Steel Density: STable 3.5.1-1 Concrete Density: STable 3.5.1-1 Modulus of Elasticity - Steel: Modulus of Elasticity - Concrete: Modular Ratio: Future Wearing Surface Density: STable 3.5.1-1 Future Wearing Surface Thickness: (Assumed) The following load combinations will be used in this design example, in accordance with Table 3.4.1-1.
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From page 443... ...
443 Check Flange Proportion Requirements Met: S 6.10.2.2 Properties for use when analyzing under beam self weight (steel only) : Total steel area.
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444 Composite Section Properties (Uncracked Section - used for barrier dead load and live load positive bending) : Determine composite slab and reinforcing properties Slab thickness assumes some sacrificial thickness; use: Total section depth Effective width.
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From page 445... ...
445 Find composite section centroid: Centroid of steel from top of slab. Centroid of transformed composite section from top of slab.
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From page 446... ...
446 Composite Section Properties (Cracked Section - used for live load negative bending) : Find cracked section area and centroid: Find cracked section moments of inertia and section moduli: Phase 1: Steel girders are simply supported, and support their self-weight plus the weight of the slab.
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From page 447... ...
447 Moments due to Phase 1 DL: Shear due to Phase 1 DL: Phase 2: Steel girders are simply supported and composite with the deck slab, and support their self-weight plus the weight of the slab in addition to barriers on exterior modules. Barriers are assumed to be evenly distributed between the two exterior module girders.
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From page 448... ...
448 This section addresses the construction loads for lifting the module into place. The module is lifted from four points, at some distance, Dlift from each end of each girder.
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From page 449... ...
449 These factors represent the distribution of live load from the deck to the girders in accordance with AASHTO Section 4, and assumes the deck is fully continuous across the joints. Girder Section Modulus: Girder Area: Girder Depth: Distance between centroid of deck and centroid of beam: Modular Ratio: S3.6.1.1.2-1 Multiple Presence Factors: Interior Stringers for Moment: S4.6.2.2.1-1 One Lane Loaded: Two Lanes Loaded: Governing Factor: Interior Stringers for Shear: One Lane Loaded: Two Lanes Loaded: Governing Factor: Exterior Stringers for Moment: One Lane Loaded: Use Lever Rule.
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From page 450... ...
450 Governing Factor: Exterior Stringers for Shear: One Lane Loaded: Use Lever Rule. Two Lanes Loaded: Governing Factor: Factor for Use for Shear: Factor for Use for Moment: Case 1: Dead Load on Steel Only (calculated in Section 7)
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From page 451... ...
451 Case 3: Composite girders are lifted into place from lifting points located distance Dlift from the girder edges. Maximum moments and shears were calculated in Section 8.
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452 The flexural resistance shall be determined as specified in LRFD Design Article 6.10.6.2. Determine Stringer Plastic Moment Capacity First.
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From page 454... ...
454 Dp = distance from the top of slab of composite section to the neutral axis at the plastic moment (neglect positive moment reinforcement in the slab)
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From page 455... ...
455 Positive Flexural Compression Check: From LRFD Article 6.10.2 Check for compactness: Web Proportions: Web slenderness Limit: S 6.10.6.2.2 Therefore Section is considered compact and shall satisfy the requirements of Article 6.10.7.1. Negative Moment Capacity Check (Appendix A6)
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From page 456... ...
456 λ Web Plastification: Flexure Factor: ϕ Tensile Limit: ϕ Compressive Limit: Local Buckling Resistance: λ λ λ λ λ λ λ λ λ Lateral Torsional Buckling Resistance: Inflection point assumed to be at 1/6 span π
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From page 457... ...
457 ϕ Governing negative moment capacity: Phase 1: First, check the stress due to the dead load on the steel section only. (LRFD 6.10.3 - Constructability Requirements Reduction factor for construction ϕ Load Combination for construction Max Moment applied, Phase 1: (at midspan)
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From page 458... ...
458 Phase 4: Check flexural capacity under dead load and live load for fully installed continuous composite girders. Strength I Load Combination ϕ ϕ Strength III Load Combination ϕ Strength V Load Combination ϕ Check service load combinations for the fully continuous beam with live load (Phase 4)
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From page 459... ...
459 Bottom Flange: Check LL Deflection: ∆ from independent Analysis - includes 100% design truck (w/impact) , or 25% design truck (w/impact)
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From page 460... ...
460 Fatigue Moment Ranges at Detail Locations (from analysis) : γ γ Constants to use for detail checks: Category B Check: Stress at Bottom Flange, Fatigue I γ ∆ γ γ ∆ Category C' Check: Stress at base of transverse stiffener (top of bottom flange)
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From page 461... ...
461 γ γ ∆ FATIGUE CHECK: Ensure that single lane ADTT is less than If not, then the beam requires redesign. bp x tp Using LRFD Article 6.10.11 for stiffeners: 9tw x tw 9tw x tw ϕ *
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From page 462... ...
462 Axial Resistance of Bearing Stiffeners: ϕ for bearing stiffeners π ϕ Shear Connector design to follow LRFD 6.10.10. Stud Properties: Diameter Height of Stud Studs per row π Fati Resistance:gue
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From page 463... ...
463 Strength Resistance: ϕ ϕ Find required stud spacing along the girder (varies as applied shear varies)
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464 This section details the geometric and material properties of the deck. Because the equivalent strip method is used in accordance with AASHTO LRFD Section 4, different loads are used for positive and negative bending.
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465 This section calculates the dead loads on the slab. These are used later for analysis to determine the design moments.
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466 Longitudinal reinforcement ϕ ϕ π ϕ π ϕ Distribution Reinforcement (AASHTO 9.7.3.2) This section will conduct design checks on the reinforcing according to various sections in AASHTO LRFD.
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467 Unfactored Dead Load S 5.7.3.3.2Cracking Moment Minimum Factored Flexural Resistance CHECK CRACK CONTROL (AASHTO LRFD 5.7.3.4)
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From page 469... ...
469 Deck Properties: Deck Overhang Length Parapet Properties: Note: Parapet properties are per unit length. Compression reinforcement is ignored.
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From page 470... ...
470 Average width of section Cover ϕ Depth Factored Moment Resistance ϕ ϕ Parapet Base Moment Resistance (about longitudinal axis) : ϕ Development in tension ϕ ϕ ϕ ϕ ϕ ϕ ϕ Hooked bar developed in tension ϕ ϕ Lap splice in tension Distance from NA to Compressive Face β S 5.7.3.1.2-4
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From page 471... ...
471 Depth of Equivalent Stress Block β S 5.7.3.2.3 Nominal Moment Resistance S 5.7.3.2.2-1 Factored Moment Resistance ϕ S 5.7.3.2 Average Moment Capacity of Barrier (about longitudinal axis) : Factored Moment Resistance about Horizontal Axis ϕ ϕ Parapet Moment Resistance (about vertical axis)
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From page 472... ...
472 ϕ ϕ ϕ ϕ ϕ Nominal Moment Resistance - Tension on Outside Face ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ Vertical Nominal Moment Resistance of Parapet Parapet Design Factors: Crash Level Transverse Design Force Longitudinal Design Force Vertical Design Force (Down) Critical Length of Yield Line Failure Pattern:
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From page 473... ...
473 S A13.3.1-2 S A13.3.1-1 S A13.4.2-1 The parapet design must consider three design cases. Design Case 1 is for longitudinal and transverse collision loads under Extreme Event Load Combination II.
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From page 474... ...
474 DC - 1B: Design Section in Overhang Notes: Distribution length is assumed to increase based on a 30 degree angle from the face of parapet. Moment of collision loads is distributed over the length Lc + 30 degree spread from face of parapet to location of overhang design section.
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From page 475... ...
475 ϕ ϕ ϕ Maximum crash load moment at theoretical cut- ff point: ϕ π ϕ ϕ ϕ ϕ
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From page 476... ...
476 ϕ ϕ ϕ ϕ extension past second interior girder Does not govern - no live load on overhang. See sheet S7 from drawing set titled: "STANDARD CONCEPTS FOR ABC MODULAR SYSTEMS" Ensure compression splice and connection can handle the compressive force in the force couple due to the negative moment over the pier.
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From page 477... ...
477 Force in Flange: Calculate Bottom Flange Stress, Ignoring Concrete: Moment of inertia: Distance to center of bottom flange: Stress in bottom flange: Bottom Flange Force for design: Design Stress: Design Force: Compression Splice Plate Dimensions: Bottom Splice Plate: Built-Up Angle Splice Plate Horizontal Leg: Built-Up Angle Splice Plate Vertical Leg: Total Area: Average Stress: Proportion Load into each plate based on area: Check Plates Compression Capacity: Bottom Splice Plate: for bolted connection π
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From page 478... ...
478 Horizontal Angle Leg: for bolted connection π
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From page 479... ...
479 Vertical Angle Leg: for bolted connection π Additional Checks: Design Bolted Connections of the splice plates to the girders, checking for shear, bearing, and slip critical connections. See sheet S2 from drawing set titled: "STANDARD CONCEPTS FOR ABC MODULAR SYSTEMS" for closure pour drawing.
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From page 480... ...
480 Overall CG: Moment of Inertia: Section Moduli: Concrete Properties: Steel Properties: Negative Flexural Capacity: Slenderness ratio for compressive flange: λ Limiting ratio for compactness: λ Limiting ratio for noncompact λ Hybrid Factor: Flange compression resistance: λ λ λ λ λ λ
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From page 481... ...
481 Lateral Torsional Buckling Resistance: π Compressive Resistance: Tensile Flexural Resistance: For Strength For Service Ultimate Moment Resistance: from external FE analysis For additional design, one may calculate the force couple at the section over the pier to find the force in the UHPC closure joint. This force can be used to design any additional reinforcement used in the joint.
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From page 482... ...
482 ABC SAMPLE CALCULATION – 2 Decked Precast Prestressed Concrete girder Design for ABC
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From page 483... ...
483 DECKED PRECAST PRESTRESSED CONCRETE GIRDER DESIGN FOR ABC Unit Definition: kcf kip ft 3 This example is for the design of a superstructure system that can be used for rapid bridge replacement in an Accelerated Bridge Construction (ABC) application.
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484 ORDER OF CALCULATIONS Introduction1. Design Philosophy2.
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From page 485... ...
485 2. DESIGN PHILOSOPHY Geometry of the section is selected based on availability of standard formwork across many geographic regions, as evidenced by sections commonly used by many state transportation agencies.
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From page 486... ...
486 Concrete Limits: Allowable concrete stresses are generally in line with AASHTO LRFD requirements, with one exception. Allowable tension in the bottom of the section at final, Service III, is limited to 0 ksi, based on the research of NCHRP Project No.
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From page 487... ...
487 b1 b2 b3 bn+1 bn bn-1 bn-2 dn dn-1 dn-2 d1 d2 TYPICALGIRDER SECTION COMPRISED OF n TRAPEZOIDAL REGIONS y x 4. BEAM SECTION Use trapezoidal areas to define the cross-section.
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From page 488... ...
488 5. MATERIAL PROPERTIES Concrete: f'c 8 ksi Minimum 28-day compressive strength of concrete fci fc.rel f'c 6.4 ksi Minimum strength of concrete at release γc .150 kcf Unit weight of concrete K1 1.0 Correction factor for standard aggregate (5.4.2.4)
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From page 489... ...
489 6. PERMANENT LOADS Permanent loads to be considered in the design of this girder are self-weight, diaphragms, barrier, and future wearing surface.
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From page 490... ...
490 Load at Service: pfws 25 psf Assumed weight of future wearing surface wfws pfws S 0.199 klf Uniform load due to future wearing surface Mfws x( ) wfws x 2 L x( )
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From page 491... ...
491 Lsa 2.167 ft Length of semi-integral abutment Lgsa 4 in Length of girder embedded in semi-integral abutment Bsa S Wj 7.444 ft Width of semi-integral abutment cast with girder Dsa h 16 in 58 in Depth of semi-integral abutment Vsa Vcorb Lsa Bsa Dsa Ag tflange bf Lgsa 88.32 ft 3 Volume of semi-integral abutment cast with girder Wsa Vsa γc 13 kip Weight of semi-integral abutment cast with girder Semi-Integral Abutment Backwall Integral Abutment Backwall
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From page 492... ...
492 8. LIVE LOAD Vehicular loading conforms to the HL-93 design load prescribed by AASHTO.
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From page 493... ...
493 For a single loaded lane, use the Lever Rule. gmext1 S 0.5 bf Wb 5 ft S 0.65 Single loaded lane em 0.77 de 9.1 ft 1.042 gmext2 em gmint 0.659 Two or more loaded lanes gmext max gmext1 gmext2 0.659 Distribution factor for moment at exterior beams Distribution Factors for Shear: From Table 4.6.2.2.3a-1 for shear in interior girders, Verify this girder design is within the range of applicability for Table 4.6.2.2.3a-1.
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From page 494... ...
494 From Table 4.6.2.2.3c-1 for skewed bridges, θ skew 0 deg CheckSkew if θ 60 deg( )
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From page 495... ...
495 9. PRESTRESS PROPERTIES Because allowable tension at the service limit state is reduced to account for camber leveling forces, the prestress force required at midspan is expected to be excessive in the ends at release without measures to reduce the prestress moment.
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From page 496... ...
496 Nh 2 Nps 12if 4 12 Nps 24if 6 24 Nps 30if 6 Nps 30 Nps 30if Nh 14 Assumes all flange rows are filled prior to filling rows in web above the flange, which maximized efficiency. Use override below to shift strands from flange to web if needed to satisfy end stresses.
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From page 497... ...
497 Compute transformed section properties based on prestress layout. Initial Transformed Section (release)
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From page 498... ...
498 Minimum distance to harped prestress centroid from bottom of beam at centerline of bearingyh.brg.req ycgp.req ybti Ns Nh ys Ns Nh 16.488 in Minimum distance between uppermost strand and top of beamytop.min 18 in αhd 0.4 Hold-down point, fraction of the design span length Maximum slope of an individual strand to limit hold-down force to 4 kips/strandslopemax if dps 0.6 in= 1 12 1 8 0.125 Set centroid of harped strands as high as possible to minimize release and handling stressesyh.brg h ytop.min 0.5 Nh 1 2 2 in( ) 17 in yh.brg min yh.brg yhb slopemax αhd L 17 in Verify that slope requirement is satisfied at uppermost strand CheckEndPrestress if yh.brg yh.brg.req "OK" "Verify release stresses." "OK" yp.brg Ns ys Nh yh.brg Ns Nh 9.632 in Centroid of prestress from bottom at bearing slopecgp yp.brg yp αhd L 0.015 Slope of prestress centroid within the harping length ypx x( )
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From page 499... ...
499 Assumed time sequence in the life of the girder for loss calculations: ti 1 Time (days) between casting and release of prestress tb 20 Time (days)
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From page 500... ...
500 Kid 1 1 npi Aps Ag 1 Ag epg 2 Ixg 1 0.7 ψbif 0.809 Age-adjusted transformed section coefficient (5.9.5.4.2a-2) ∆ fpSR εbid Ep Kid 3.435 ksi Loss due to beam shrinkage(5.9.5.4.2a-1)
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From page 501... ...
501 Prestress Loss Summary ∆ fpES 15.978 ksi ∆ fpES fpj 7.9 % ∆ fpLT ∆ fpid ∆ fpdf 35.087 ksi ∆ fpLT fpj 17.3 % ∆ fpTotal ∆ fpES ∆ fpLT 51.065 ksi ∆ fpTotal fpj 25.2 % ∆ fp.est 25 % fpe fpj ∆ fpTotal 151.4 ksi Final effective prestress CheckFinalPrestress if fpe fpe.max "OK" "No Good" "OK" 11. CONCRETE STRESSES Stresses in the concrete section at release, during handling, and at final service are computed and checked against allowable values appropriate for the stage being considered.
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From page 502... ...
502 0 4 8 12 16 20 24 28 32 36 40 1 0 1 2 3 4 Stresses in Concrete at Release (Half Beam) Distance along Beam (ft)
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From page 503... ...
503 Pdia max Wia Wsa 13.2 kip Approximate abutment weight Pm Pj 1 ∆ fpES ∆ fpid fpj 1011.7 kip Effective prestress during lifting and shipping Define locations for which stresses are to be calculated: xe Lg 0 min Lt Lg Lend Lg max Lt Lg Lend Lg a' Lg αhd 0.5 T ie 1 last xe Compute moment in the girder during lifting with supports at the lift points. Mlift x( )
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From page 504... ...
504 BotLift3ie fbot.DIM.dec xeie BotLift3T 0.000 2.609 3.274 3.410 3.297 3.267( ) ksi Allowable stresses during handling: fcm fc.erec f'c 7.2 ksi Assumed concrete strength when handling operations begin fc.all.erec 0.6 fcm 4.32 ksi Allowable compression during lifting and shipping ft.all.erec ft.erec fcm 0.429 ksi Allowable tension during lifting and shipping 0 4 8 12 16 20 24 28 32 36 40 0 2 4 Stresses in Concrete During Lifting (Half Beam)
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From page 505... ...
505 BotLiftMaxie max BotLift1ie BotLift2ie BotLift3ie BotLiftMax T 0 2.637 3.31 3.502 3.297 3.267( ) ksi BotLiftMinie min BotLift1ie BotLift2ie BotLift3ie BotLiftMin T 0 2.609 3.274 3.41 2.808 2.744( )
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From page 506... ...
506 0 4 8 12 16 20 24 28 32 36 40 0 2 4 6 Stresses in Concrete at Service (Half Beam) Distance along Beam (ft)
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From page 507... ...
507 12. FLEXURAL STRENGTH Verify flexural resistance at the Strength Limit State.
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508 fps fpu 1 k c X( )
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From page 509... ...
509 13. SHEAR STRENGTH Shear Resistance Compute the factored shear at the critical shear section and at tenth points along the span due to the Strength I load combination, then compare it to the factored resistance calculated in accordance with AASHTO LRFD 5.8.
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From page 510... ...
510 Transverse reinforcement area and spacing providedAv 1.02 0.62 0.62 0.62 0.31( )
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511 Longitudinal Reinforcement Al.req x( )
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From page 512... ...
512 DClong Al.req Al.prov max DClong 0.93 CheckLong if max DClong 1.0 "OK" "No Good" "OK" Longitudinal reinforcement check 14. SPLITTING RESISTANCE Splitting Resistance Checking splitting resistance provided by first zone of transverse reinforcement defined in the previous section for shear design.
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From page 513... ...
513 ∆ cr t( )
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From page 514... ...
514 16. NEGATIVE MOMENT FLEXURAL STRENGTH Compute the factored moment to be resisted across the interior pier and determine the required reinforcing steel to be fully developed in the top flange.
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From page 515... ...
515 Reinforcing Steel Requirement in the Top Flange for Strength Reduction factor for strength in tensioncontrolled reinforced concrete (5.5.4.2) φ f 0.90 bc b1 26 in Width of compression block at bottom flange Distance to centroid of negative moment steel, taken at mid-depth of top flangednms h tsac 0.5 tflange tsac 37 in Factored load, in terms of stress in concrete at depth of steel, for computing steel requirement Ru Mu.neg.StrI φ f bc dnms 2 1.019 ksi m fy 0.85 f'c 8.824 Steel-to-concrete strength ratio ρreq 1 m 1 1 2 m Ru fy 0.0185 Required negative moment steel ratio Anms.req ρreq bc dnms 17.787 in 2 Required negative moment steel in top flange Full-length longitudinal reinforcement to be made continuous across jointAs.long.t 2.0 in 2 As.long.b 2.0 in 2 Additional negative moment reinforcing bar areaAbar 0.44 in 2 Additional reinforcement area required in the top mat (2/3 of total)
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From page 516... ...
516 ABC SAMPLE CALCULATION – 3a Precast Pier Design for ABC (70' Span Straddle Bent)
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517 PRECAST PIER DESIGN FOR ABC (70' SPAN STRADDLE BENT) BNofBm Total Number of Beams in Backward Span= BSpan Backward Span Length= BDeckW Out to Out Backward Span Deck Width= BBmAg Backward Span Beam X Sectional Area= BBmFlange Backward Span Beam Top Flange Width= BHaunch Backward Span Haunch Thickness= BBmD Backward Span Beam Depth or Height= BBmIg Backward Span Beam Moment of Inertia= yBt Backward Span Beam Top Distance from cg= NofCol Number of Columns per Bents= NofDs Number of Drilled Shaft per Bents= wCol Width of Column Section= bCol Breadth of Column Section= DsDia Drilled Shaft Diameter= HCol Height of Column= wEarWall Width of Ear Wall= hEarWall Height of Ear Wall= tEarWall Thickness of Ear Wall= tSWalk Thickness of Side Walk= bSWalk Breadth of Side Walk= BmMat Beam Material either Steel or Concrete= hbS Bottom Solid Height at Foam= htS Top Solid Height at Foam= γst Unit Weight of Steel= γc wc Unit Weight of Concrete= FNofBm = Total Number of Beams in Forward Span FSpan = Forward Span Length FDeckW = Out to Out Forward Span Deck Width FBmAg = Forward Span Beam X Sectional Area FBmFlange = Forward Span Beam Top Flange Width FHaunch = Forward Span Haunch Thickness FBmD = Forward Span Beam Depth or Height FBmIg = Forward Span Beam Moment of Inertia yFt = Forward Span Beam Top Distance from cg SlabTh = Slab Thickness RailWt = Railing Weight RailH = Railing Height RailW = Rail Base Width LeftOH = Left Overhang Distance RightOH = Right Overhang Distance DeckW = Out to Out Deck Width at Bent RoadW = Roadway Width BrgTh = Bearing Pad Thickness Bearing Seat Thickness NofLane = Number of Lanes wCap = Cap Width hCap = Cap Depth CapL = Cap Length wFoam = Width of Foam for Blockout hFoam = height of Foam for Blockout LFoam = Length Foam for Blockout
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From page 518... ...
518 SlabDCInt Dead Load for Slab per Interior Beam= SlabDCExt Dead Load for Slab per Exterior Beam= BeamDC Self Weight of Beam= HaunchDC Dead Load of Haunch Concrete per Beam= RailDC Weight of Rail per Beam= FSuperDCInt Half of Forward Span Super Structure Dead Load Component per Interior Beam= FSuperDCExt Half of Forward Span Super Structure Dead Load Component per Exterior Beam= FSuperDW Half of Forward Span Overlay Dead Load Component per Beam= BSuperDCInt Half of Backward Span Super Structure Dead Load Component per Interior Beam= BSuperDCExt Half of Backward Span Super Structure Dead Load Component per Exterior Beam= BSuperDW Half of Backward Span Overlay Dead Load Component per Beam= TorsionDCInt DeadLoad Torsion in a Cap due to difference in Forward and Backward span length per Interior Beam= TorsionDCExt DeadLoad Torsion in a Cap due to difference in Forward and Backward span length per Exterior Beam= TorsionDW DW Torsion in a Cap due to difference in Forward and Backward span length per Beam= DiapWt Weight of Diaphragm= tBrgSeat Thickness of Bearing Seat= bBrgSeat Breadth of Bearing Seat=
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From page 519... ...
519 Note: Use of Light-Weight-Concrete (LWC) may be considered to reduce the weight of the pier cap instead of styrofoam blockouts.
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From page 520... ...
520 FORWARD SPAN PARAMETER INPUT: FNofBm 12 FSpan 70 ft FDeckW 283 6 ft FBmAg 29.1 in2 FBmFlange 10.5 in yFt 14.85 inFHaunch 0 in FBmD 29.7 in FBmIg 3990 in4 BACKWARD SPAN PARAMETER INPUT: BNofBm 12 BSpan 70 ft BDeckW 283 6 ft BBmAg 29.1 in2 BBmFlange 10.5 in yBt 14.85 inBHaunch 0 in BBmD 29.7 in BBmIg 3990 in4 COMMON BRIDGE PARAMETER INPUT: Intermediate Bent between Forward and Backward span Parameters SlabTh 9 in Overlay 25 psf θ 0 deg DeckOH 1.75 ft BrgTh 3.5 in RailWt 0.43 klf RailW 19 in RailH 34.0 in tBrgSeat 0 in bBrgSeat 0 ft DeckW 283 6 ft NofLane 3 m 0.85 wc 0.150 kcf f'c 5 ksi Cap( )
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From page 521... ...
521 1. BENT CAP LOADING DEAD LOAD FROM SUPERSTRUCTURE: The permanent dead load components (DC)
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From page 522... ...
522 Forward Span Superstructure DC & DW per Interior and Exterior Beam: FSuperDCInt RailDC BeamDC SlabDCInt HaunchDC DiapWt FSuperDCInt 21.596 kip beam FSuperDCExt RailDC BeamDC SlabDCExt HaunchDC 0.5 DiapWt FSuperDCExt 21.824 kip beam FSuperDW OverlayDW FSuperDW 3.208 kip beam BACKWARD SPAN SUPERSTRUCTURE DEAD LOAD: consists of 12 W30x99 beams 12 beams were spaced 4.5' and 3'-4" alternately in backward span. For beam spacing see Typical Section Details sheet BBmSpa1 4.5 ft BBmSpa2 10 3 ft BIntBmTriW BBmSpa1 2 BBmSpa2 2 BIntBmTriW 3.917 ft BExtBmTriW BBmSpa1 2 DeckOH BExtBmTriW 4 ft RoadW 0.25 BDeckW 3 DeckW( )
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From page 523... ...
523 Total Superstructure DC & DW per Beam on Bent Cap: SuperDCInt FSuperDCInt BSuperDCInt SuperDCInt 43.192 kip beam SuperDCExt FSuperDCExt BSuperDCExt SuperDCExt 43.648 kip beam SuperDW FSuperDW BSuperDW SuperDW 6.417 kip beam TorsionDCInt max FSuperDCInt BSuperDCInt min FSuperDCInt BSuperDCInt ebrg TorsionDCInt 0 kft beam TorsionDCExt max FSuperDCExt BSuperDCExt min FSuperDCExt BSuperDCExt ebrgTorsionDCExt 0 kft beam TorsionDW max FSuperDW BSuperDW( ) min FSuperDW BSuperDW( )
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From page 524... ...
524 LIVE LOAD FOR SIMPLY SUPPORTED BRIDGE: HL-93 Loading: According to AASHTO LRFD 3.6.1.2.1, HL-93 consists of Design Truck + Design Lane Load or Design Tandem + Design Lane Load. Design Truck rather than Design Tandem + Design Lane Load controls the maximum Live Load Reactions at an interior bent for a span longer than 26'.
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From page 525... ...
525 Torsion on Bent Cap per Beam and per Drilled Shaft: Torsional load about center line of bent cap occurs due to horizontal loads acting on the superstructure perpendicular to the bent length or along the bridge length. Braking force, Centrifugal force, WS on superstructure, and WL cause torsion on bent.
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From page 526... ...
526 Icap2 wCap hCap3 12 wCap hCap hCap 2 ycg2 2 2 wFoam hFoam3 12 wFoam hFoam hFoam 2 hbS ycg2 2 Icap2 902191.259 in 4 EIcap2 Ec Icap2 Applicable for 6 CapL 41 EIcap2 2.686 10 7 kip ft2 OUTPUT of BENT CAP LOADING PROGRAM: The maximum load effects from different applicable limit states: DEAD LOAD MdlPos 3309.6 kft MdlNeg 30.1 kft SERVICE I MsPos 5377.1 kft MsNeg 45.1 kft STRENGTH I MuPos 7830.6 kft MuNeg 64.6 kft FLEXURE DESIGN: MINIMUM FLEXURAL REINFORCEMENT AASHTO LRFD 5.7.3.3.2 Factored Flexural Resistance, Mr, must be greater than or equal to the lesser of 1.2Mcr or 1.33 Mu. Applicable to both positive and negative moment.
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From page 527... ...
527 nsPos 3 (No. of Bottom or Positive Steel Layers)
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From page 528... ...
528 or simply use, ϕm 0.9 (AASHTO LRFD 5.5.4.2)
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From page 529... ...
529 Compression depth under ultimate load cNeg AsNeg fy 0.85 f'c β1 b cNeg 3.667 in aNeg β1 cNeg aNeg 2.933 in Thus, nominal flexural resistance: MnNeg AsNeg fy dNeg aNeg 2 MnNeg 2934.97 kip ft Factored flexural resistance MrNeg ϕm MnNeg MrNeg 2641.473 kip ft MuNeg 64.6 kip ft MinReinChkNeg if MrNeg Mcr_min "OK" "NG" MinReinChkNeg "OK" UltimateMomChkNeg if MrNeg MuNeg "OK" "NG" UltimateMomChkNeg "OK" Control of Cracking at Service Limit State AASHTO LRFD 5.7.3.4 exposure_cond 1 (for exposure condition, input Class 1 = 1 and Class 2 = 2) γe if exposure_cond 1= 1 0.75( )
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From page 530... ...
530 b 2 wFoam( )
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From page 531... ...
531 Negative Moment (Top Bars A) ρNeg AsNeg b dNeg ρNeg 3.863 10 3 kNeg ρNeg n 1 2 1 ρNeg n (Applicable for Solid Rectangular Section)
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From page 532... ...
532 SKIN REINFORCEMENT (BARS T) AASHTO LRFD 5.7.3.4 SkBarNo 8 (Size of a skin bar)
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From page 533... ...
533 3. BENT CAP SHEAR AND TORSION DESIGN SHEAR DESIGN OF CAP: Effective Shear Depth, dv max de a 2 0.9 de 0.72 h = (AASHTO LRFD 5.8.2.9)
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From page 534... ...
534 hh h tcover bcover (Height of shear reinforcement) hh 56 in bh b 2 bcover (Width of shear reinforcement)
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From page 535... ...
535 After Interpolating the value of Θ Β( ) Θ 30.773 deg Β 2.572 Nominal Shear Resistance by Concrete, Vc 0.0316 Β f'c ksi bv dv AASHTO LRFD EQ (5.8.3.3-3)
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From page 536... ...
536 Maximum Spacing Check: AASHTO LRFD Article 5.8.2.7 Vu 665.4 kip 0.125 f'c bv dv 1594.69 kip Svmax if Vu 0.125 f'c bv dv min 0.8 dv 24 in min 0.4 dv 12 in Svmax 24 in Svmax_check if Sv Svmax "OK" "use lower spacing" Svmax_check "OK" Av Avt At (Shear Reinf. without Torsion Reinf.)
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From page 537... ...
537 M'u 2620.013 kip ft Vu 665.4 kip Nu 0 kip Vs 1268.855 kip Tu 964.6 kip ft ph 212 in Vp 0 kip As 42.12 in 2 V's min Vu ϕv Vs AASHTO LRFD 5.8.3.5 V's 739.333 kip F M'u ϕm dv 0.5 Nu ϕn cotΘ Vu ϕv Vp 0.5 V's 2 0.45 Tu ph 2 ϕv Ao 2 F 1496.141 kip Fcheck if Aps fps As fy F "OK" "NG" AASHTO LRFD EQ 5.8.3.6.3 1( ) Fcheck "OK" 4.
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From page 538... ...
538 ho BrgTh BmH th SlabTh (Top of cap to top of slab height) ho 3.683 ft h6 ho 6ft (Top of cap to top of slab height + 6 ft)
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From page 539... ...
539 FBmLLRxn FLLRxn( ) DFSFmax Only Forward Span is Loaded( )
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From page 540... ...
540 BRAKING FORCE: BR (AASHTO LRFD 3.6.4) The braking force shall be taken as maximum of 5% of the Resultant Truck plus lane load OR 5% of the Design Tandem plus Lane Load or 25% of the design truck.
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From page 541... ...
541 Longitudinal Drag Force Coefficient for Drilled Shaft, CD_ds 0.7 pT CD V2 2 g γwater= (Longitudinal stream pressure)
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From page 542... ...
542 WA on Bent Cap (input as a punctual load) Water force on bent cap parallel to bent (X-direction)
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From page 543... ...
543 Msuper_Z WSsuper_X hCap 2 BrgTh hsup 2 Msuper_Z 10.72 kft beam WIND ON SUBSTRUCTURE: WS (AASHTO LRFD 3.8.1.2.3) Base Wind pressure, psub 0.04 ksf will be applied on exposed substructure both transverse & longitudinal direction Wind on Columns Wind force on columns parallel to bent (X-direction)
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From page 544... ...
544 WLPar pWLl TribuLength NofBm WLPar 0.233 kip beam WLNor pWLt TribuLength NofBm WLNor 0.583 kip beam Wind force on live load parallel to bent (X-direction)
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From page 545... ...
545 All these loadings as computed above such as DC, DW, LL, WL, WA, WS etc. are placed on the bent frame composed of bent cap and columns and drilled shafts.
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From page 546... ...
546 Total Load, DL+LL per Drilled Shaft of Intermediate Bent: Load_on_DShaft DL_on_DShaft RLL NofDs Load_on_DShaft 276.6 ton 5. PRECAST COMPONENT DESIGN Precast Cap Construction and Handling: w1 b h γc applicable for 0 ft Lcap 6 ft w1 3.375 klf (Cap selfweight)
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From page 547... ...
547 Maximum Positive Stress (ftP) & Negative Stress (ftN)
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From page 548... ...
548 Scol wCol bCol2 6 (Column Section Modulus) Scol 18432 in 3 Maximum Positive Stress (ftP)
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From page 549... ...
549 ABC SAMPLE CALCULATION – 3b Precast Pier Design for ABC (70' Conventional Pier)
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From page 550... ...
550 PRECAST PIER DESIGN FOR ABC (70' SPAN CONVENTIONAL PIER) FNofBm Total Number of Beams in Forward Span= BNofBm Total Number of Beams in Backward Span= FSpan Forward Span Length= BSpan Backward Span Length= FDeckW Out to Out Forward Span Deck Width= BDeckW Out to Out Backward Span Deck Width= FBmAg Forward Span Beam X Sectional Area= BBmAg Backward Span Beam X Sectional Area= FBmFlange Forward Span Beam Top Flange Width= BBmFlange Backward Span Beam Top Flange Width= FHaunch Forward Span Haunch Thickness= BHaunch Backward Span Haunch Thickness= FBmD Forward Span Beam Depth or Height= BBmD Backward Span Beam Depth or Height= FBmIg Forward Span Beam Moment of Inertia= BBmIg Backward Span Beam Moment of Inertia= yFt Forward Span Beam Top Distance from cg= yBt Backward Span Beam Top Distance from cg= NofCol Number of Columns per Bent=SlabTh Slab Thickness= NofDs Number of Drilled Shaft per Bent=RailWt Railing Weight= wCol Width of Column Section=RailH Railing Height= bCol Breadth of Column Section=RailW Rail Base Width= DsDia Drilled Shaft Diameter=DeckOH Deck Overhang Distance= HCol Height of Column=DeckW Out to Out Deck Width at Bent= wEarWall Width of Ear Wall=RoadW Roadway Width= hEarWall Height of Ear Wall=BrgTh Bearing Pad Thickness Bearing Seat Thickness= tEarWall Thickness of Ear Wall=NofLane Number of Lanes= tSWalk Thickness of Side Walk= wCap Cap Width= bSWalk Breadth of Side Walk=hCap Cap Depth= CapL Cap Length= BmMat Beam Material either Steel or Concrete= γc Unit Weight of Concrete= DiapWt Weight of Diaphragm= wc Unit Weight of Concrete= γst Unit Weight of Steel= SlabDCInt Dead Load for Slab per Interior Beam=
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From page 551... ...
551 SlabDCExt Dead Load for Slab per Exterior Beam= BeamDC Self Weight of Beam= HaunchDC Dead Load of Haunch Concrete per Beam= RailDC Weight of Rail per Beam= FSuperDCInt Half of Forward Span Super Structure Dead Load Component per Interior Beam= FSuperDCExt Half of Forward Span Super Structure Dead Load Component per Exterior Beam= FSuperDW Half of Forward Span Overlay Dead Load Component per Beam= BSuperDCInt Half of Backward Span Super Structure Dead Load Component per Interior Beam= BSuperDCExt Half of Backward Span Super Structure Dead Load Component per Exterior Beam= BSuperDW Half of Backward Span Overlay Dead Load Component per Beam= TorsionDCInt DeadLoad Torsion in a Cap due to difference in Forward and Backward span length per Interior Beam= TorsionDCExt DeadLoad Torsion in a Cap due to difference in Forward and Backward span length per Exterior Beam= TorsionDW DW Torsion in a Cap due to difference in Forward and Backward span length per Beam= tBrgSeat Thickness of Bearing Seat= bBrgSeat Breadth of Bearing Seat=
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From page 552... ...
552 Note: Use of Light Weight Concrete (LWC) may be considered to reduce the weight of the pier cap instead of using styrofoam blockouts.
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From page 553... ...
553 FORWARD SPAN PARAMETER INPUT: FNofBm 12 FSpan 70 ft FDeckW 283 6 ft FBmAg 29.1 in2 FBmFlange 10.5 in yFt 14.85 inFHaunch 0 in FBmD 29.7 in FBmIg 3990 in4 BACKWARD SPAN PARAMETER INPUT: BNofBm 12 BSpan 70 ft BDeckW 283 6 ft BBmAg 29.1 in2 BBmFlange 10.5 in yBt 14.85 inBHaunch 0 in BBmD 29.7 in BBmIg 3990 in4 COMMON BRIDGE PARAMETER INPUT: Bent in Question Parameters SlabTh 9 in Overlay 25 psf θ 0 deg DeckOH 1.75 ft BrgTh 3.5 in RailWt 0.43 klf RailW 19 in RailH 34.0 in tBrgSeat 0 in bBrgSeat 0 ft DeckW 283 6 ft NofLane 3 m 0.85 wc 0.150 kcf f'c 5 ksi Cap( )
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From page 554... ...
554 1. BENT CAP LOADING DEAD LOAD FROM SUPERSTRUCTURE: The permanent dead load components (DC)
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From page 555... ...
555 Forward Span Superstructure DC & DW per Interior and Exterior Beam: FSuperDCInt RailDC BeamDC SlabDCInt HaunchDC DiapWt FSuperDCInt 21.596 kip beam FSuperDCExt RailDC BeamDC SlabDCExt HaunchDC 0.5 DiapWt FSuperDCExt 21.824 kip beam FSuperDW OverlayDW FSuperDW 3.208 kip beam BACKWARD SPAN SUPERSTRUCTURE DEAD LOAD: Consists of 12 W30x99 beams 12 beams were spaced 4.5' and 3'-4" alternately in Backward span. For beam spacing see Typical Section Details sheet BBmSpa1 4.5 ft BBmSpa2 10 3 ft BIntBmTriW BBmSpa1 2 BBmSpa2 2 BIntBmTriW 3.917 ft BExtBmTriW BBmSpa1 2 DeckOH BExtBmTriW 4 ft RoadW 0.25 BDeckW 3 DeckW( )
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From page 556... ...
556 Total Superstructure DC & DW Reactions per Beam on Bent Cap: SuperDCInt FSuperDCInt BSuperDCInt SuperDCInt 43.192 kip beam SuperDCExt FSuperDCExt BSuperDCExt SuperDCExt 43.648 kip beam SuperDW FSuperDW BSuperDW SuperDW 6.417 kip beam TorsionDCInt max FSuperDCInt BSuperDCInt min FSuperDCInt BSuperDCInt ebrg TorsionDCInt 0 kft beam TorsionDCExt max FSuperDCExt BSuperDCExt min FSuperDCExt BSuperDCExt ebrgTorsionDCExt 0 kft beam TorsionDW max FSuperDW BSuperDW( ) min FSuperDW BSuperDW( )
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From page 557... ...
557 LIVE LOAD FOR SIMPLY SUPPORTED BRIDGE: HL-93 Loading: According to AASHTO LRFD 3.6.1.2.1 HL-93 consists of Design Truck + Design Lane Load or Design Tandem + Design Lane Load. Design Truck rather than Design Tandem + Design Lane Load controls the maximum Live Load Reactions at an interior bent for a span longer than 26'.
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From page 558... ...
558 Torsion on Bent Cap per Beam and per Drilled Shaft: Torsional load about center line of bent cap occurs due to horizontal loads acting on the superstructure perpendicular to the bent length or along the bridge length. Braking force, Centrifugal force, WS on superstructure, and WL cause torsion on bent.
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From page 559... ...
559 FLEXURE DESIGN: Minimum Flexural Reinforcement AASHTO LRFD 5.7.3.3.2 Factored Flexural Resistance, Mr, must be greater than or equal to the lesser of 1.2Mcr or 1.33 Mu. Applicable to both positive and negative moment.
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From page 560... ...
560 Mcr2 1.33 max MuPos MuNeg Mcr2 3009.524 kip ft Mcr_min min Mcr1 Mcr2 Therefore Mr must be greater than Mcr_min 1524.963 kip ft Moment Capacity Design (Positive Moment, Bottom Bars B) AASHTO LRFD 5.7.3.2 Bottom Steel arrangement for the Cap: Input no.
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From page 561... ...
561 cPos AsPos fy 0.85 f'c β1 b (AASHTO LRFD EQ 5.7.3.1.1-4)
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From page 562... ...
562 The Amount of Negative As Required, AsReq 0.85 f'c b dNeg fy 1 1 2 MuNeg 0.85 ϕm f'c b dNeg 2 AsReq 11.757 in 2 The Amount of Negative As Provided, NofBarsNeg Nn NofBarsNeg 8 AsNeg Ayn0 1 in 2 AsNeg 12.48 in 2 Compression depth under ultimate load cNeg AsNeg fy 0.85 f'c β1 b cNeg 4.588 in aNeg β1 cNeg aNeg 3.671 in Thus, nominal flexural resistance: MnNeg AsNeg fy dNeg aNeg 2 MnNeg 2662.278 kip ft MrNeg ϕm MnNeg (Factored flexural resistance) MrNeg 2396.05 kip ft MinReinChkNeg if MrNeg Mcr_min "OK" "NG" MinReinChkNeg "OK" UltimateMomChkNeg if MrNeg MuNeg "OK" "NG" UltimateMomChkNeg "OK" Control of Cracking at Service Limit State AASHTO LRFD 5.7.3.4 exposure_cond 1 (for exposure condition, input Class 1 = 1 and Class 2 = 2)
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From page 563... ...
563 kPos ρPos n 1 2 1 ρPos n (Applicable for Solid Rectangular Section) kPos 0.278 jPos 1 kPos 3 jPos 0.907 fssPos MsPos AsPos jPos dPos fssPos 29.174 ksi(Tensile Stress at Service Limit State)
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From page 564... ...
564 smaxNeg 700 kip in γe βsNeg fssNeg 2 dc1Neg smaxNeg 13.587 in sActualNeg b 2 sidecTop Nn0 0 1 (Equal horizontal spacing of top first Rebar row closest to Tension Face) sActualNeg 5.5 in Actual Max Spacing Provided in Top first row closest to Tension Face, saNegProvided 11.125 in sActualNeg max saNegProvided sActualNeg sActualNeg 11.125 in SpacingCheckNeg if smaxNeg sActualNeg "OK" "NG" SpacingCheckNeg "OK" SUMMARY OF FLEXURE DESIGN: Bottom Rebar or B Bars: use 10~#11 bars @ 5 bars in each row of 2 rows Top Rebar or A Bars: use 8~#11 bars @ 8 bars in top row SKIN REINFORCEMENT (BARS T)
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From page 565... ...
565 NoAskbar1 R AskReq AskBar NoAskbar1 2 per Side Maximum Spacing of Skin Reinforcement: SskMax min de 6 12 in AASHTO LRFD 5.7.3.4 SskMax 7.417 in NoAskbar2 if dl 3ft R dskin SskMax 1 1 NoAskbar2 4 per Side NofSideBarsreq max NoAskbar1 NoAskbar2 NofSideBarsreq 4 SskRequired dskin 1 NofSideBarsreq SskRequired 7.4 in NofSideBars 4 (No. of Side Bars Provided)
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From page 566... ...
566 Aoh Area enclosed by centerline of exterior closed transverse torsion reinforcement including area of holes therein= Total Flexural Steel Area, As AsNeg As 12.48 in 2 Nominal Flexure, Mn MnNeg Mn 2662.278 kft Stress block Depth, a aNeg a 3.671 in Effective Depth, de dNeg de 44.5 in Effective web Width at critical Location, bv b bv 4 ft Input initial θ 35 deg cotθ cot θ( ) Shear Resistance Factor, ϕv 0.9 Cap Depth & Width, h 48 in b 48 in Moment Arm, de a 2 42.665 in 0.9 de 40.05 in 0.72 h 34.56 in Effective Shear Depth at Critical Location, dv max de a 2 0.9 de 0.72 h (AASHTO LRFD 5.8.2.9)
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From page 567... ...
567 M'u max Mu Vu Vp dv AASHTO LRFD B5.2 M'u 1647.569 kip ft V'u Vu 2 0.9 ph Tu 2 Ao 2 (Equivalent shear)
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From page 568... ...
568 Sv 9 in (Input Stirrup Spacing)
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From page 569... ...
569 ϕv Vn 808.645 kip Vu 463.4 kip ϕVn_check if ϕv Vn Vu "OK" "NG" ϕVn_check "OK" Torsional Resistance, Tn 2 Ao 0.5 Avt fy cotΘ Sv AASHTO LRFD EQ 5.8.3.6.2 1( ) ϕv Tn 1875.9 kip ft Longitudinal Reinforcement Requirements including Torsion: AASHTO LRFD 5.8.3.6.3 AASHTO LRFD EQ 5.8.3.6.3 1( )
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From page 570... ...
570 4. COLUMN/DRILLED SHAFT LOADING AND DESIGN Superstructure to substructure force: AASHTO LRFD SECTION 3 LOADS and LOAD COMBINATIONS Subscript: X = Parallel to the Bent cap Length and Z = Perpendicular to the bent Cap Length th 2.5 in (Haunch Thickness)
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From page 571... ...
571 LIVE LOAD REACTIONS: LL Live load Reaction LL on cap can be taken only the vertical Rxn occurs when HL93 is on both the forward and backward span or when HL93 Loading is on one span only which causes torsion too. To maximize the torsion, LL only acts on the longer span between forward and backward span.
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From page 572... ...
572 CENTRIFUGAL FORCE: CF (AASHTO LRFD 3.6.3) Skew Angle of Bridge, θ 0 deg Design Speed v 45 mph f g( )
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From page 573... ...
573 Braking force normal to bent (Z-direction)
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From page 574... ...
574 Lateral stream pressure, pL CL V2 2 g γwater pL 0 ksf Bent Cap: Longitudinal stream pressure CL 1.4 pTcap CL V2 2 g γwater pTcap 0 ksf WA on Columns Water force on column parallel to bent (X-direction) WAcol_X wCol pT_col WAcol_X 0 kip ft If angle between direction of flow and longitudinal axis of pile = 0 then apply load at one exterior column only otherwise apply it on all columns.
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From page 575... ...
575 AASHTO LRFD table 3.8.1.2.2-1 (modified) If the bridge is approximately 30' high and local wind velocities are known to be less than 100 mph, wind load for this bridge should be from AASHTO LRFD TABLE 3.8.2.2-1.
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From page 576... ...
576 Wind force on columns normal to bent (Z-direction) WScol_Z psub bCol sin θ( )
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From page 577... ...
577 Moments at cg of the Bent Cap due to Wind load on Live Load MWL_X WLZ h6 hCap 2 MWL_X 2.736 kft beam MWL_Z WLX h6 hCap 2 MWL_Z 6.84 kft beam Vertical Wind Pressure: (AASHTO LRFD 3.8.2) DeckWidth FDeckW Bridge deck width including parapet and sidewalk Puplift 0.02ksf( )
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From page 578... ...
578 Backward Span Superstructure DC (FBDC)
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From page 579... ...
579 5. PRECAST COMPONENT DESIGN Precast Cap Construction and Handling: w b h γc (Cap selfweight)
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From page 580... ...
580 Modulus of Rupture: According PCI hand book 6th edition modulus of rupture, fr = 7.5\/f'c is divided by a safety factor 1.5 in order to design a member without cracking f'c 5 ksi (Compressive Strength of Concrete) Unit weight factor, λ 1 fr 5 λ f'c psi (PCI EQ 5.3.3.2)
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From page 581... ...
581 Maximum Positive Stress (ftP) & Negative Stress (ftN)
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Key Terms
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