MNL-133-97_ch8_11

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PCI BRIDGE DESIGN MANUAL CHAPTER 8

JUL 03

TABLE OF CONTENTDESIGN THEORY AND PROCEDU

NOTATION

8.0 AASHTO SPECIFICATION REFERENCES

8.1 PRINCIPLES AND ADVANTAGES OF PRESTRESSING

8.1.1 History 8.1.2 High Strength Steel 8.1.3 Prestressing Versus Conventional Reinforcing 8.1.4 Concrete to Steel Bond

8.2 FLEXURE

8.2.1 Allowable Stress Design (ASD) 8.2.1.1 Theory 8.2.1.1.1 Stage 1 Loading 8.2.1.1.2 Stage 2 Loading 8.2.1.1.3 Stage 3 Loading 8.2.1.1.4 Stage 4 Loading 8.2.1.1.5 Stage 5 Loading 8.2.1.1.5.1 Tensile Stresses - Normal Strength Concrete 8.2.1.1.5.2 Tensile Stresses - High Strength Concrete 8.2.1.1.5.3 Tensile Stresses -LRFD Specifications 8.2.1.2 Allowable Concrete Stresses 8.2.1.2.1 Standard Specifications

8.2.1.2.2 LRFD Specifications 8.2.1.3 Design Procedure 8.2.1.4 Composite Section Properties 8.2.1.4.1 Theory 8.2.1.4.2 Procedure 8.2.1.5 Harped Strand Considerations 8.2.1.6 Debonded Strand Considerations 8.2.1.7 Minimum Strand Cover and Spacing 8.2.1.8 Design Example 8.2.1.8.1 Design Requirement 1 8.2.1.8.2 Design Requirement 2 8.2.1.8.3 Design Requirement 3 8.2.1.8.3.1 Strand Debonding 8.2.1.8.3.2 Harped Strands 8.2.1.8.3.3 Other Methods to Control Stresses 8.2.1.8.4 Design Requirement 4 8.2.1.9 Fatigue


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8.2.2 Flexural Strength Design 8.2.2.1 Theory 8.2.2.2 Standard Specifications 8.2.2.2.1 Ultimate Moment Capacity

8.2.2.2.1.1 Required Parameters 8.2.2.2.1.2 Rectangular Section 8.2.2.2.1.3 Flanged Section 8.2.2.2.2 Maximum Reinforcement Limit 8.2.2.2.3 Minimum Reinforcement Limit 8.2.2.3 LRFD Specifications 8.2.2.3.1 Nominal Flexural Resistance 8.2.2.3.1.1 Required Parameters 8.2.2.3.1.2 Rectangular Sections 8.2.2.3.1.3 Flanged Sections 8.2.2.3.2 Maximum Reinforcement Limit 8.2.2.3.3 Minimum Reinforcement Limit 8.2.2.4 Flexural Strength Design Example 8.2.2.4.1 Design Requirement 1 8.2.2.4.1.1 Standard Specifications 8.2.2.4.1.2 LRFD Specifications 8.2.2.4.2 Design Requirement 2 8.2.2.5 Strain Compatibility Approach 8.2.2.6 Design Example - Strain Compatibility

8.2.2.6.1 Part l - Flexural Capacity 8.2.2.6.2 Part 2 - Comparative Results 8.2.3 Design of Negative Moment Regions for Members Made Continuous for Live

Loads 8.2.3.1 Strength Design 8.2.3.2 Reinforcement Limits - Standard Specifications 8.2.3.3 Reinforcement Limits - LRFD Specifications 8.2.3.4 Serviceability 8.2.3.5 Fatigue in Deck Reinforcement

8.3 STRAND TRANSFER AND DEVELOPMENT LENGTHS

8.3.1 Strand Transfer Length 8.3.1.1 Impact on Design 8.3.1.2 Specifications 8.3.1.3 Factors Affecting Transfer Length 8.3.1.4 Research Results 8 3.1.5 Recommendations 8.3.1.6 End Zone Reinforcement


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8.3.2 Strand Development Length 8.3.2.1 Impact on Design 8.3.2.2 Standard Specifications 8.3.2.3 LRFD Specifications

8.3.2.4 Factors Affecting Development Length 8.3.2.5 Bond Studies 8.3.2.6 Recommendations

8.4 SHEAR

8.4.1 Standard Specifications 8.4.1.1 Flexure-Shear Strength, V ci

8.4.1.2 Web-Shear Strength, V cw

8.4.1.3 Web Reinforcement Contribution, V s 8.4.1.3.1 Minimum Spacing Requirements 8.4.1.3.2 Minimum Shear Reinforcement 8.4.1.4 Application of Standard Specifications to Continuous Spans 8.4.2 1979 Interim Revisions 8.4.3 LRFD Specifications 8.4.3.1 Shear Design Provisions 8.4.3.1.1 Nominal Shear Resistance 8.4.3.1.2 Concrete Contribution, V c 8.4.3.1.3 Web Reinforcement Contribution, V s 8.4.3.1.4 Values ofβ and θ

8.4.3.2 Design Procedure 8.4.3.3 Longitudinal Reinforcement Requirement 8.4.4 Comparison of Shear Design Methods

8.5 HORIZONTAL INTERFACE SHEAR

8.5.1 Theory 8.5.2 Standard Specifications 8.5.3 LRFD Specifications 8.5.4 Comparison of Design Specifications

8.6 LOSS OF PRESTRESS

8.6.1 Introduction 8.6.2 Definition 8.6.3 Significance of Losses on Design 8.6.4 Effects of Estimation of Losses 8.6.4.1 Effects at Transfer 8.6.4.2 Effect on Production Costs 8.6.4.3 Effect on Camber 8.6.4.4 Effect of Underestimating Losses


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8.6.5 Prediction of Creep, Shrinkage and Relaxation Material Properties 8.6.5.1 Prediction of Creep Coefficient of Concrete 8.6.5.1.1 Creep Modification Factors 8.6.5.1.2 Modification Factors for Strength

8.6.5.1.3 Example 8.6.5.2 Prediction of Shrinkage Coefficient of Concrete 8.6.5.2.1 Shrinkage Modification Factors 8.6.5.2.2 Modification Factors for Strength 8.6.5.2.3 Example 8.6.5.3 Prediction of Relaxation of the Prestressing Steel 8.6.6 Methods for Estimating Losses 8.6.7 Elastic Shortening Loss 8.6.7.1 Computation of Elastic Shortening Loss 8.6.7.2 Elastic Shortening Example 8.6.8 Losses from theStandard Specifications 8.6.8.1 Shrinkage Loss 8.6.8.2 Elastic Shortening Loss 8.6.8.3 Creep Loss 8.6.8.4 Steel Relaxation Loss 8.6.8.5 Lump Sum Losses 8.6.9 Standard Specifications Example 8.6.10 Losses from theLRFD Specifications 8.6.10.1 Elastic Shortening Loss

8.6.10.2 Shrinkage and Creep Losses 8.6.10.3 Steel Relaxation Loss 8.6.10.4 Washington State Study 8.6.11 LRFD Specifications Example 8.6.12 Losses by the Tadros Method 8.6.12.1 Tadros Method Example

8.7 CAMBER AND DEFLECTION

8.7.1 Multiplier Method 8.7.2 Improved Multiplier Method 8.7.3 Examples 8.7.3.1 Multiplier Method Example 8.7.3.2 Improved Multiplier Method Example 8.7.4 Camber and Deflection Estimates Using Numerical Integration 8.7.4.1 Numerical Integration Example

8.8 DECK SLAB DESIGN

8.8.1 Introduction


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8.8.2 Design of Bridge Decks Using Precast Panels 8.8.2.1 Determining Prestress Force 8.8.2.2 Service Load Stresses and Flexural Strength 8.8.2.3 Standard Specifications

8.8.2.3.1 Minimum Thickness 8.8.2.3.2 Live Load 8.8.2.3.3 Reinforcement Requirements 8.8.2.3.4 Shear Design 8.8.2.3.5 Crack Control 8.8.2.4 LRFD Specifications 8.8.2.4.1 LRFD Specifications Refined Analysis 8.8.2.4.2 LRFD Specifications Strip Method 8.8.2.4.2.1 Minimum Thickness 8.8.2.4.2.2 Minimum Concrete Cover 8.8.2.4.2.3 Live Load 8.8.2.4.2.4 Location of Critical Sections 8.8.2.4.2.5 Design Criteria 8.8.2.4.2.6 Reinforcement Requirements 8.8.2.4.2.7 Shear Design 8.8.2.4.2.8 Crack Control 8.8.3 Other Precast Bridge Deck Systems 8.8.3.1 Continuous Precast Concrete SIP Panel System, NUDECK 8.8.3.1.1 Description of NUDECK

8.8.3.2 Full-Depth Precast Concrete Panels 8.8.4 LRFD Specifications Empirical Design Method

8.9 TRANSVERSE DESIGN OF ADJACENT BOX BEAM BRIDGES

8.9.1 Background 8.9.1.1 Current Practice 8.9.1.2 Ontario Bridge Design Code Procedure 8.9.2 Empirical Design 8.9.2.1 Tie System 8.9.2.2 Production 8.9.2.3 Installation 8.9.3 Suggested Design Procedure 8.9.3.1 Transverse Diaphragms 8.9.3.2 Longitudinal Joints Between Beams 8.9.3.3 Tendons 8.9.3.4 Modeling and Loads for Analysis 8.9.3.5 Post-Tensioning Design Chart 8.9.3.6 Design Method 8.9.3.7 Design Example


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8.9.4 Lateral Post-Tensioning Detailing for Skewed Bridges

8.10 LATERAL STABILITY OF SLENDER MEMBERS

8.10.1 Introduction

8.10.1.1 Hanging Beams 8.10.1.2 Beams Supported from Beneath 8.10.2 Suggested Factors of Safety 8.10.2.1 Conditions Affecting FSc 8.10.2.2 Effects of Creep and Impact 8.10.2.3 Effects of Overhangs 8.10.2.4 Increasing the Factor of Safety 8.10.3 Measuring Roll Stiffness of Vehicles 8.10.4 Bearing Pads 8.10.5 Wind Loads 8.10.6 Temporary King-Post Bracing 8.10.7 Lateral Stability Examples 8.10.7.1 Hanging Beam Example 8.10.7.2 Supported Beam Example

8.11 BENDING MOMENTS AND SHEAR FORCES DUE TOVEHICULAR LIVE LOADS

8.11.1 HS20 Truck Loading 8.11.2 Lane Loading, 0.640 kip/ft

8.11.3 Fatigue Truck Loading

8.12 STRUT-AND-TIE MODELING OF DISTURBED REGIONS

8.12.1 Introduction 8.12.2 Strut-and-Tie Models 8.12.2.1 Truss Geometry Layout 8.12.2.2 Nodal Zone and Member Dimensions 8.12.2.3 Strengths of Members 8.12.3 LRFD Specifications Provisions for Strut-and-Tie Models 8.12.3.1 Compression Struts

8.12.3.1.1 Unreinforced Concrete Struts 8.12.3.1.2 Reinforced Concrete Struts 8.12.3.2 Tension Ties 8.12.3.2.1 Tie Anchorage 8.12.3.3 Proportioning Node Regions 8.12.3.4 Crack Control Reinforcement 8.12.4 Steps for Developing Strut-and-Tie Models 8.12.4.1 Design Criteria 8.12.4.2 Summary of Steps

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8.12.5 Pier Cap Example 8.12.5.1 Flow of Forces and Truss Geometry 8.12.5.2 Forces in Assumed Truss 8.12.5.3 Bearing Stresses

8.12.5.4 Reinforcement for Tension Tie DE 8.12.5.5 Strut Capacities 8.12.5.6 Nodal Zone at Pier 8.12.5.7 Minimum Reinforcement for Crack Control

8.13 DETAILED METHODS OF TIME-DEPENDENT ANALYSIS

8.13.1 Introduction 8.13.1.1 Properties of Concrete 8.13.1.1.1 Stress-Strain-Time Relationship 8.13.1.2 Effective Modulus 8.13.1.3 Age-Adjusted Effective Modulus 8.13.1.4 Properties of Prestressing Steel 8.13.1.5 Reduced Relaxation under Variable Strain 8.13.2 Analysis of Composite Cross-Sections 8.13.2.1 Initial Strains 8.13.2.2 Methods for Time-Dependent Cross-Section Analysis 8.13.2.2.1 Steps for Analysis 8.13.2.2.2 Example Calculations 8.13.3 Analysis of Composite Simple-Span Members

8.13.3.1 Relaxation of Strands Prior to Transfer 8.13.3.2 Transfer of Prestress Force 8.13.3.2.1 Example Calculation (at Transfer) 8.13.3.3 Creep, Shrinkage and Relaxation after Transfer 8.13.3.3.1 Example Calculation (after Transfer) 8.13.3.4 Placement of Cast-in-Place Deck 8.13.3.5 Creep, Shrinkage and Relaxation 8.13.3.6 Application of Superimposed Dead Load 8.13.3.7 Long-Term Behavior 8.13.4 Continuous Bridges 8.13.4.1 Effectiveness of Continuity 8.13.4.2 Applying Time-Dependent Effects 8.13.4.3 Methods of Analysis 8.13.4.3.1 General Method 8.13.4.3.2 Approximate Method 8.13.4.3.2.1 Restraint Moment Due to Creep 8.13.4.3.2.2 Restraint Moment Due to Differential Shrinkage

8.14 REFERENCES

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NOTATIONDESIGN THEORY AND PROCEDU

JUL 03

A = area of cross-section of the precast beam [STD], [LRFD] A = distance to pickup points from each end of the beam — A c = area of concrete on the flexural tension side of the member [LRFD] A c = area of beam cross-section —

A cv = area of concrete section resisting shear transfer [LRFD] A cs = cross-sectional area of a concrete strut [LRFD] A g = gross area of section [LRFD] A k = area of cross-section of element k — A o = area enclosed by centerlines of the elements of the beam [LRFD] A ps = area of pretensioning steel [LRFD] A s = area of non-pretensioning tension reinforcement [STD], [LRFD] A s = total area of vertical reinforcement located within a distance

(h/5) from the end of the beam [LRFD] A sf = area of steel required to develop the ultimate compressive

strength of the overhanging portions of the flange [STD] A sr = area of steel required to develop the compressive strength of the

web of a flanged section [STD] A ss = area of reinforcement in strut [LRFD] A st = area of longitudinal mild steel reinforcement in tie [LRFD] A *s = area of pretensioning steel [STD] A s = area of compression reinforcement [LRFD] A v = area of web reinforcement [STD] A v = area of transverse reinforcement within a distance s [LRFD] A vf = area of shear-friction reinforcement [LRFD] A vh = area of web reinforcement required for horizontal shear — A v-min = minimum area of web reinforcement — a = depth of the compression block [STD] a = depth of the equivalent rectangular stress block [LRFD] a = length of overhang — b = effective flange width — b = width of beam [STD] b = width of top flange of beam — b = width of the compression face of a member [LRFD]

b´ = width of web of a flanged member [STD] bb = width of bottom flange of beam — bv = width of cross-section at the contact surface being investigated

for horizontal shear [STD] bv = effective web width [LRFD] bv = width of interface [LRFD] b w = web width [LRFD] Ca = creep coefficient for deflection at time of erection due to loads

applied at release —

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CR c = loss of pretension due to creep of concrete [STD] CR s = loss of pretension due to relaxation of pretensioning steel [STD] C(t,t 0) = creep coefficient of the concrete member at a certain age — C(t,t j) = creep coefficient at time t j (j = 0,1,2,…) —

Cb(t,t3) = creep at time t for beam concrete loaded at time t3 —Cd(t,t3) = creep at time t for deck concrete loaded at time t3 — Cu = ultimate creep coefficient for concrete at time of release of

prestressing — C´

u = ultimate creep coefficient for concrete at time of application ofsuperimposed dead loads —

c = distance from the extreme compression fiber to the neutral axis [LRFD] c = cohesion factor [LRFD] D = dead load [STD] D = nominal diameter of the strand [STD]

DC = dead load of structural components and non-structuralattachments [LRFD]

DW = load of wearing surfaces and utilities [LRFD] d = distance from extreme compression fiber to centroid of the

pretensioning force [STD] db = nominal strand diameter [STD], [LRFD] de = effective depth from the extreme compression fiber to the

centroid of the tensile force in the tension reinforcement [LRFD] dext = depth of the extreme steel layer from extreme compression fiber — di = depth of steel layer from extreme compression fiber —

dp = distance from extreme compression fiber to the centroid of thepretensioning tendons [LRFD] ds = distance from extreme compression fiber to the centroid of

nonprestressed tensile reinforcement [LRFD] dv = effective shear depth [LRFD] d´ = distance from extreme compression fiber to the centroid of

nonprestressed compression reinforcement [LRFD] E = modulus of elasticity — Ec = modulus of elasticity of concrete [STD], [LRFD] Ecb(t3) = age-adjusted modulus of elasticity for beam concrete at time t3 — E

cd(t

3) = age-adjusted modulus of elasticity for deck concrete at time t

3 —

Ec(t j) = modulus of elasticity at time t j (j = 0,1,2,…) — Ec(t0) = initial modulus of elasticity —Ec(t,t0) = modulus of elasticity at a certain time — Eci = modulus of elasticity of the beam concrete at transfer — Ep = modulus of elasticity of pretensioning tendons [LRFD] ES = loss of pretension due to elastic shortening [STD] Es = modulus of elasticity of pretensioning reinforcement [STD] Es = modulus of elasticity of reinforcing bars [LRFD]

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E*c = age-adjusted, effective modulus of elasticity of concrete for a

gradually applied load at the time of transfer of prestressing — E*

cb = age-adjusted, effective modulus of elasticity of the beam — E*

cd = age-adjusted, effective modulus of elasticity of the deck —

E*c(t,t0) = effective modulus of elasticity at certain time — E*

ck = age-adjusted, effective modulus of element k — e = eccentricity of prestressing strands — ec = eccentricity of the strand at midspan — eg = distance between the centers of gravity of the beam and the slab [LRFD] ei = initial lateral eccentricity of the center of gravity with respect to

the roll axis — em = average accentricity at midspan [LRFD] ep = eccentricity of the prestressing strands with respect to the

centroid of the section —

FSc = factor of safety against cracking — FSf = factor of safety against failure — Fb = allowable tensile stress in the precompressed tension zone at

service loads — Fcj = force in concrete for the j th component — Fpi = total force in strands before release — f = stress — f b = concrete stress at the bottom fiber of the beam — f c = specified concrete strength at 28 days [STD] f c = specified compressive strength at 28 days [LRFD]

f cds = concrete stress at the center of gravity of the pretensioning steeldue to all dead loads except the dead load present at the time thepretensioning force is applied [STD]

f cir = average concrete stress at the center of gravity of thepretensioning steel due to pretensioning force and dead loadof beam immediately after transfer [STD]

f ci = concrete strength at transfer [STD] f ci = specified compressive strength of concrete at time of initial

loading or pretensioning (transfer) [LRFD] f cgp = concrete stress at the center of gravity of pretensioning tendons,

due to pretensioning force at transfer and the self-weight of the

member at the section of maximum positive moment [LRFD] f cu = the limiting concrete compressive stress for designing

by strut-and-tie model [LRFD] f f = stress range [STD] f min = algebraic minimum stress level [STD] f pbt = stress in prestressing steel immediately prior to transfer [LRFD] f pc = compressive stress in concrete (after allowance for all pretensioning

losses) at centroid of cross-section resisting externally applied loads [STD]

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f pc = compressive stress in concrete after all prestress losses have occurredeither at the centroid of the cross-section resisting live load or atthe junction of the web and flange when the centroid lies in theflange. In a composite section, f pc is the resultant compressivestress at the centroid of the composite section, or at the junction

of the web and flange when the centroid lies within the flange,due to both prestress and to the bending moments resisted by theprecast member acting alone. [LRFD]

f pe = compressive stress in concrete due to effective pretensionforces only (after allowance for all pretension losses) atextreme fiber of section where tensile stress is caused byexternally applied loads [STD]

f pe = effective stress in the pretensioning steel after losses [LRFD] f pi = initial stress immediately before transfer — f pj = stress in the pretensioning steel at jacking [LRFD] f po = stress in the pretensioning steel when the stress in the

surrounding concrete is zero [LRFD] f ps = average stress in pretensioning steel at the time for which the

nominal resistance of member is required [LRFD] f pu = specified tensile strength of pretensioning steel [LRFD] f py = yield strength of pretensioning steel [LRFD] f r = modulus of rupture of concrete [STD], [LRFD] f s = allowable stress in steel under service loads — f s = ultimate stress of pretensioning reinforcement [STD] f se = effective final pretension stress — f si = effective initial pretension stress —

f *su = average stress in pretensioning steel at ultimate load [STD] f(t j) = stress at time t j — f r(t,t0) = relaxation stress at a certain time — f(t0) = tensile stress at the beginning of the interval — f y = yield strength of reinforcing bars [STD] f y = specified minimum yield strength of reinforcing bars [LRFD] f y = yield stress of pretensioning reinforcement [STD] f y = specified minimum yield strength of compression reinforcement [LRFD] f yh = specified yield strength of transverse reinforcement [LRFD] H = average annual ambient mean relative humidity [LRFD] h = length of a single segment — h = overall depth of precast beam [STD] h = overall depth of a member [LRFD] hcg = height of center of gravity of beam above road — hd = deck thickness — hf = compression flange depth [LRFD] hr = height of roll center above road — I = moment of inertia about the centroid of the non-composite

precast beam, major axis moment of inertia of beam [STD], [LRFD]

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I = impact fraction [STD] Ik = moment of inertia of element k — IM = dynamic load allowance [LRFD] Ieff = effective cracked section lateral (minor axis) moment of inertia —

Ig = gross lateral (minor axis) moment of inertia — K = factor used for calculating time-dependent losses — K r = factor used for calculating relaxation loss in strand that occurs prior

to transfer — K θ = sum of rotational spring constants of supports — k = factor used in calculation of average stress in pretensioning steel

for strength limit state; factor related to type of strand[LRFD]

k c = product of applicable correction factors for creep= k la k h k s — k cp = correction factor for curing period —

k la = correction factor for loading age — k h = correction factor for relative humidity — k s = correction factor for size of member — k sh = product of applicable correction factors for shrinkage= k cp k h k s — k st = correction factor for concrete strength — L = live load [STD] L = length in feet of the span under consideration for positive

moment and the average of two adjacent loaded spans fornegative moment [STD]

L = overall beam length or design span —

L = span length measured parallel to longitudinal beams [STD] L = span length [LRFD] LL = vehicular live load [LRFD] Lr = intrinsic relaxation of the strand — Lx = distance from end of prestressing strand to center of the panel [STD] l = overall length of beam — l d = development length — l t = transfer length — Mc = moment in concrete beam section — Mcr = cracking moment [LRFD] Mcr(t) = restraint moment due to creep at time t — M*

cr = cracking moment [STD] Md/nc = moment due to non-composite dead loads [STD] Mel = fictious elastic restraint moment at the supports — Mg = unfactored bending moment due to beam self-weight — Mg = self-weight bending moment of beam at harp point — Mgmsp = self-weight bending moment at midspan — Mk = element moment —

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Mlat = lateral bending moment at cracking — MLL = unfactored bending moment due to lane load per beam — Mmax = maximum factored moment at section due to externally applied loads [STD] Mn = nominal moment strength of a section [STD]

Mn = nominal flexural resistance [LRFD] Mn/dc = non-composite dead load moment at the section — Mr = factored flexural resistance of a section in bending [LRFD] Msh = shrinkage moment — Msr(t) = restraint moment due to differential shrinkage at time t — Msw = moment at section of interest due to self-weight of the member

plus any permanent loads acting on the member at time of release — Mu = factored bending moment at section [STD], [LRFD] Mx = bending moment at a distance x from the support — M0 = theoretical total moment in sections — M0k = theoretical moment in section of element k — m = stress ratio — N = number of segments between nodes (must be even number) — Nk = element normal force — Nc = internal element force in concrete — Ns = internal element force in steel — Nu = applied factored axial force taken as positive if tensile [LRFD] N0k = theoretical normal force in section of element k, positive when tensile — N0 = theoretical total normal force in sections —

n = modular ratio between slab and beam materials [STD], [LRFD] nk = modular ratio of element k — ns = modular ratio of steel element — PPR = partial prestress ratio [LRFD] Pc = permanent net compression force [LRFD] Pn = nominal axial resistance of strut or tie [LRFD] Pr = factored axial resistance of strut or tie [LRFD] Pse = effective pretension force after allowing for all losses — Psi = effective pretension force after allowing for the initial losses — Q = first moment of inertia of the area above the fiber being considered — R = radius of curvature — RH = relative humidity [STD] R n = strength design factor — R u = flexural resistance factor — r = radius of gyration of the gross cross-section — r = radius of stability

S = width of precast beam [STD] S = spacing of beams [STD], [LRFD]

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S = slab span [LRFD] S = span between the inside faces of the beam webs [LRFD] Sb = section modulus for the extreme bottom fiber of the

non-composite precast beam —

Sbc = composite section modulus for extreme bottom fiber of theprecast beam — SH = loss of pretension due to concrete shrinkage [STD] SN = the value of the integral — S(t,t0) = shrinkage coefficient at a certain age — St = section modulus for the extreme top fiber of the non-composite

precast beam — Su = ultimate free shrinkage strain in the concrete adjusted for

member size and relative humidity — s = longitudinal spacing of the web reinforcement [STD] s = length of a side element [LRFD] s = spacing of rows of ties [LRFD] t = time, days; age of concrete at the time of determination of creep

effects, days; age of concrete at time of determination of shrinkageeffects, days; time after loading, days —

t = thickness of web — t = thickness of an element of the beam — tf = thickness of flange — t0 = age of concrete when curing ends; age of concrete when load is

initially applied, days — ts = cast-in-place concrete slab thickness — ts = depth of concrete slab [LRFD] V c = nominal shear strength provided by concrete [STD] V c = nominal shear resistance provided by tensile stresses in the

concrete [LRFD] V ci = nominal shear strength provided by concrete when diagonal

cracking results from combined shear and moment [STD] V cw = nominal shear strength provided by concrete when diagonal

cracking results from excessive principal tensile stress in web [STD] V d = shear force at section due to unfactored dead load [STD] V i = factored shear force at section due to externally applied loads

occurring simultaneously with Mmax [STD] V n = nominal shear resistance of the section considered [LRFD] V nh = nominal horizontal shear strength [STD] V p = vertical component of effective pretension force at section [STD] V p = component of the effective pretensioning force, in the

direction of the applied shear, positive if resisting the applied shear [LRFD] V s = nominal shear strength provided by web reinforcement [STD] V s = shear resistance provided by shear reinforcement [LRFD] V u = factored shear force at the section [STD], [LRFD]

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V uh = factored horizontal shear force per unit length of the beam [LRFD] v u = average factored shear stress [LRFD] W = total weight of beam — w = a uniformly distributed load [LRFD]

w = width of clear roadway [LRFD] w = weight per unit length of beam — w c = unit weight of concrete [STD], [LRFD] x = distance from the support to the section under question — y = height of center of gravity of beam above roll axis

(beam supported from below) — y b = distance from centroid to the extreme bottom fiber of the

non-composite beam — y bc = distance from centroid to the bottom of beam of the composite section — y bs = distance from the center of gravity of strands to the bottom

fiber of the beam — y k = distance of the centroid of element k from edge — y r = height of roll axis above center of gravity of beam (hanging beam) — y s = height above soffit of centroid of prestressing force — y t = distance from centroid to the extreme top fiber of the

non-composite beam — y tc = distance from centroid to the top of deck of the composite section — z = lateral deflection of center of gravity of beam — zmax = distance from centerline of vehicle to center of dual tires — zo = theoretical lateral deflection of center of gravity of beam with the

full dead weight applied laterally — z o = theoretical lateral deflection of center of gravity of beam with the

full dead weight applied laterally, computed using Ieff for tilt angleθ under consideration —

α = super-elevation angle or tilt angle of support in radians — α = factor used in calculating elastic shortening loss — α = coefficient defined by (Eq. 8.6.2.5.1-3) to account for interaction

between steel and concrete in pretensioning loss calculations — αs = angle between compressive strut and adjoining tension tie [LRFD] β = factor indicating ability of diagonally cracked concrete to

transmit tension (a value indicating concrete contribution) [LRFD] β1 = factor for concrete strength [STD] β1 = ratio of the depth of the equivalent uniformly stressed compression

zone assumed in the strength limit state to the depth of the actualcompression zone [LRFD]

δc = time-dependent multiplier — ∆ = deflection — ∆ = camber measured with respect to the beam-ends —

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∆f cdp = change in concrete stress at center of gravity of pretensioningsteel due to dead loads except the dead load acting at the timethe pretensioning force is applied [LRFD]

∆f pCR = loss in pretensioning steel stress due to creep [LRFD] ∆f

pES = loss in pretensioning steel stress due to elastic shortening [LRFD]

∆f pR = loss in pretensioning steel stress due to relaxation of steel [LRFD] ∆f pR1 = loss in pretensioning steel stress due to relaxation of steel at

transfer [LRFD] ∆f pR2 = loss in pretensioning steel stress due to relaxation of steel after

transfer [LRFD] ∆f pSR = loss in pretensioning steel stress due to shrinkage [LRFD] ∆f pT = total loss in pretensioning steel stress [LRFD] ∆f s = total loss of prestress — ε = strain — ε

c = strain in concrete beam —

εcr = the time dependent creep strain — εf = the immediate strain due to the applied stress f — εfc = elastic strain in concrete — εfk = element strain — εfs = elastic strain in steel — εk = strain in element k — εp = strain in prestressing steel — εs = strain in mild steel — εs = tensile strain in cracked concrete in direction of tensile tie [LRFD]

εsh = free shrinkage strain —εshb(t,t2) = shrinkage strain of the beam from time t2 to time t —εshb(t3,t2) = shrinkage strain of the beam from time t2 to time t3 —εshd(t,t3) = shrinkage strain of the deck from time t3 to time t — εshu = ultimate free shrinkage strain in the concrete, adjusted for member

size and relative humidity — εsi = strain in tendons corresponding to initial effective pretension

stress — εx = longitudinal strain in the web reinforcement on the flexural

tension side of the member [LRFD]

ε0c = initial strain in concrete — ε1 = principal tensile strain in cracked concrete due to factored loads [LRFD] γ * = factor for type of pretensioning reinforcement [STD] φ = strength reduction factor [STD] φ = resistance factor [LRFD] φ = curvature — φc = curvature at midspan — φcr = curvature due to creep — φfk = element curvature —

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φk = curvature of element k — φ0 = curvature at support — λ = parameter used to determine friction coefficientµ [LRFD] µ = Poisson’s ratio for beams [STD]

µ = coefficient of friction [LRFD] θ = angle of inclination of diagonal compressive stresses [LRFD] θ = roll angle of major axis of beam with respect to vertical — θL = left end rotation of beam due to simple span loads — θR = right end rotation of beam due to simple span loads — θi = initial roll angle of a rigid beam — θmax = tilt angle at which cracking begins, based on tension at the top corner

equal to the modulus of rupture — θmax = tilt angle at maximum factor of safety against failure — ρb = reinforcement ratio producing balanced strain condition [STD] ρ* = ratio of pretensioning reinforcement [STD] ψ = a factor that reflects the fact that the actual relaxation is less than

the intrinsic relaxation — χ = aging coefficient — χ(t,t0) = aging coefficient at certain time —

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DESIGN THEORY AND PROCED8.11 Bending Moments and Shear Forces Due to Vehicular Live Loads/8.11.2 Lane Loading, 0.640 kip/ft

Table 8.11.1-1 Maximum Bending Moment

per Lane for HS20 TruckLoad

Load x/L Formula for maximum Minimumtype bending moment, ft-kips x,*ft L, ft

HS20Truck

0 - 0.333 72 9.33( )[( ) ]x L x L

− −0 28

0.333 - 0.500 72 4.67 112( )[( ) ]x L x L

− −− 14 28

* x is the distance from left support to the section being considered, ft

Table 8.11.1-2 Maximum Shear Force perLane for HS20 Truck Load

Load x/L Formula for maximum Minimum Maximumtype shear force, kips x,*ft L, ft L, ft

HS20Truck

0 - 0.50072 4.67

8[( ) ]L x

L

− −− 14 28 42

0 - 0.50072 9.33[( ) ]L x

L

− −0 42 -


8.11BENDING MOMENTS

AND SHEAR FORCESDUE TO VEHICULAR

LIVE LOADS

8.11.1HS20 Truck Loading

8.11.2Lane Loading, 0.640 kip/ft

In designing longitudinal members of bridges, the maximum bending moment andshear force at each section along the span, are computed for live loads. The loadposition must be determined to give the maximum values of shears and moments.The Standard Specifications use the HS20 design truck while theLRFD Specifications use the HL-93 loading which is a combination of the HS20 design truck and a lane

loading of 0.640 kip/ft. Design for the fatigue limit state in theLRFD Specifications ,requires that a special design truck be used. This section gives formulas which may becombined to get the maximum bending moments and shear forces due to the aboveloading cases.

Readers are referred to theStandard Specifications for details about the effects of theequivalent lane loading which must also be considered in design. It can be shown thatthis equivalent lane loading may govern the design of spans longer than 144.5 ft forbending moment and 120 ft for shear force.

The following formulas may be used to calculate the maximum bending momentand maximum shear force per lane at any point on a span for the HS20 design

truck. Certain limitations apply, as noted in the tables. The computed values shouldbe multiplied by a factor of 1/2 to obtain forces per line of wheels. The formulas arevalid only for simple spans and impact is not included (see AASHTO Manual forCondition Evaluation of Bridges, AASHTO, 1994).

The following formulas may be used to calculate the maximum bending momentand the maximum shear force per lane at any point on a span for a lane load of 0.640kip/ft. The formulas are valid only for simple spans and impact is not included.

Maximum bending moment = 0.64(x)(L x)2

−

, ft-kips

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DESIGN THEORY AND PROCED8.11.2 Lane Loading, 0.640 kip/ft/8.11.3 Fatigue Truck Loading

8.11.3Fatigue Truck Loading

Table 8.11.3-1 Maximum Bending Moment per Lane for HL-93 Fatigue

Truck Loading

Load x/L Formula for maximum Minimumtype bending moment, ft-kips x,*ft L, ft

FatigueTruck

0 - 0.24172 18.22( )[( ) ]x L x

L

− −0 44

Loading (LRFD) 0.241 - 0.500 72 11.78

112( )[( ) ]x L x

L

− −− 14 28


Maximum shear force = 0.642L

(L x)2− , kips

where x = the distance from left support to the section being considered, ft L = span, ft

When designing using the LRFD Specifications , consideration of the fatigue limitstate may be required (see LRFD Article 5.5.3.1). A special fatigue truck load isdefined in LRFD Article 3.6.1.4.1. This loading consists of a single design truck which has the same axle weights used in all other limit states, but with a constantspacing of 30.0 ft between the 32.0-kip axles. The following equations may be usedto calculate the maximum bending moment per lane at any point on the span for thefatigue truck loading. These values should be multiplied by a factor of 1/2 to obtainvalues per line of wheels. These formulas are valid only for simple spans and impactis not included.

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