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PCI BRIDGE DESIGN MANUAL CHAPTER 8

JUL 03

TABLE OF CONTENTSDESIGN THEORY AND PROCEDURE

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





8.2.1.1.5.1 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



8.2.1.8.3.1 Strand Debonding

8.2.1.8.3.2 Harped Strands

8.2.1.8.3.3 Other Methods to Control Stresses


8.2.1.9 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


8.2.2.4.1.1 Standard Specifications

8.2.2.4.1.2 LRFD Specifications


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 LiveLoads

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



8.3.2.4 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 of β 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 the Standard 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 the LRFD 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


8.8.2.3.1 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


8.8.2.4.1 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 TO

VEHICULAR 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 PROCEDURE

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 compressivestrength 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 investigatedfor 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 loadsapplied 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,t0) = 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 ofprestressing —

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 thepretensioning force [STD]

db = nominal strand diameter [STD], [LRFD]

de = effective depth from the extreme compression fiber to thecentroid 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 ofnonprestressed tensile reinforcement [LRFD]

dv = effective shear depth [LRFD]

d ́ = distance from extreme compression fiber to the centroid ofnonprestressed 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 t 3 —

Ecd

(t3

) = age-adjusted modulus of elasticity for deck concrete at time t3

—

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 agradually 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 tothe roll axis —

em = average accentricity at midspan [LRFD]

ep = eccentricity of the prestressing strands with respect to thecentroid of the section —

FSc = factor of safety against cracking — FSf = factor of safety against failure —

Fb = allowable tensile stress in the precompressed tension zone atservice 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 initialloading 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 pretensioninglosses) 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-compositeprecast 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 priorto transfer —

K θ = sum of rotational spring constants of supports —

k = factor used in calculation of average stress in pretensioning steelfor 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 positivemoment 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 —

ld = development length —

lt = 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 memberplus 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 thenon-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-compositeprecast beam —

Su = ultimate free shrinkage strain in the concrete adjusted formember 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 creepeffects, 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 isinitially 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 theconcrete [LRFD]

V ci = nominal shear strength provided by concrete when diagonalcracking results from combined shear and moment [STD]

V cw = nominal shear strength provided by concrete when diagonalcracking 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 thedirection 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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∆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 attransfer [LRFD]

∆f pR2 = loss in pretensioning steel stress due to relaxation of steel aftertransfer [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 —

εc

= 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 membersize and relative humidity —

εsi = strain in tendons corresponding to initial effective pretensionstress —

εx = longitudinal strain in the web reinforcement on the flexuraltension 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 coefficient µ [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 cornerequal to the modulus of rupture —

θ ḿax = 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 thanthe intrinsic relaxation —

χ = aging coefficient —

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

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8.12.1Introduction

8.12

STRUT-AND-TIE

MODELING OF

DISTURBED REGIONS

Traditionally, models used in the analysis and design of concrete structures have beenbased on elastic theory and the basic assumption that plane sections remain plane,regardless of the loading. However, it is well known that disturbances do occur inregions near discontinuities, for example, at concentrated loads and abrupt changesin member dimensions. Such regions are referred to as “disturbed regions.”

Methods used to analyze and design disturbed regions must include procedures thatreflect the actual flow of stresses in such regions. In considering stress distributionbefore cracking, it is customary to apply elastic methods of analysis, especially whenpredicting where significant cracking will occur. Since significant stress redistributiontakes place after concrete cracks, elastic methods cannot adequately predict stressessubsequent to cracking.

A rational method for dealing with disturbed regions subsequent to cracking is theuse of strut-and-tie models. These models can give an excellent representation of theflow of forces in disturbed regions of cracked concrete systems.

The theoretical basis of the method prescribed in the LRFD Specifications [Article5.8.3.4.2] for the design of a section subjected to combined shear, axial load and flex-ure, is the modified compression field theory. This method considers the equilibriumof forces acting on the idealized, variable-angle truss, the compatibility of strains, andthe effects of cracking of concrete. Simplifications in the method include the use ofaverage values of stresses and strains over a length greater than the crack spacing.

In a typical calculation for shear reinforcement, the sectional dimensions, prestress-ing and material strengths have been chosen and the shear design involves selectionof adequate shear reinforcement and, if necessary, additional longitudinal reinforce-ment.

Figure 8.12.1-1 shows that there are three types of regions that need to be consideredin general shear design of a beam as follows:

1) Regions that can be appropriately treated as a system of struts and ties. Thisapproach is discussed in this section.

2) Disturbed regions of fanning compressive stresses characterized by radiatingcompressive stresses near supports and regions where the shear changes sign.

In such regions the value of θ varies. 3) Regions of uniform compressive stress fields where the value of θ is constant.

The second and third types of regions are discussed in Section 8.4.3 using the gen-eral modified compression field theory and the corresponding LRFD Specifications procedure.

DESIGN THEORY AND PROCEDURE8.12 Strut-and-Tie Modeling of Disturbed Regions/8.12.1 Introduction

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8.12.2Strut-and-Tie Models

The LRFD Specifications encourage the use of strut-and-tie models in design whereappropriate. It has been determined through sophisticated analysis and laboratorytesting, that cracked reinforced concrete carries load mainly by development of atruss system represented by compressive stresses in the concrete and tensile stressesin the reinforcement. Furthermore, upon the occurrence of significant cracking, the

originally curved principal stress trajectories in concrete tend toward straight lines,and it is appropriate to regard the resulting compressive forces as being carried bystraight compressive struts. Examples of strut-and-tie modeling of a simply supportedand a continuous deep beam are shown in Figures. 8.12.2-1a-1b and 8.12.2-2.

Figure 8.12.1-1Disturbed Regions and

Regions of UniformCompression Fields

Uniform compressionfield (θ constant)

Fanning compression field(θ varies)

Compression strut

Tension tie

θ

Figure 8.12.2-1a-1bStrut-and-Tie Model for a

Simple Deep Beam Nodal zone

Tension tie

a) Flow of Forces

Anchor tension tiebeyond this point to

develop necessary force

P P

A

B C

DP

P

0.85 φ max.

Effectiveanchorage

area

0.75 φ

ε1

f cu

Truss node

Compression strut

Tension-tieforce

b) Truss Model

PP

PP

A

B C

D

φs

f c′

f c′

DESIGN THEORY AND PROCEDURE8.12.1 Introduction/8.12.2 Strut-and-Tie Models

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8.12.2.1

Truss Geometry Layout

Important considerations in strut-and-tie modeling include the geometry of the trusssystem, the nodal zone and member dimensions, and the strengths of the compres-sion and tension members.

The significance of using correct geometry in defining a truss should be obvious inthe necessity to have a triangularized system of struts and ties. At first glance, theuse of a strut-and-tie truss system to resist loads seems like an easy solution thatany engineer should be readily able to accomplish. Since the real structure is a con-tinuum, however, there are an infinite variety of trusses that could be designed insidea concrete member. The best or most efficient truss layout will be one that mostclosely fits the applied load and reaction conditions while resisting forces throughthe shortest load paths.

Identification of the existing boundary conditions is the first step in selecting a trusslayout for the strut-and-tie system. In the hammerhead pier cap of Figure 8.12.2.1-1a-1b, two different sets of boundary conditions are shown depending on the loca-tions of the design lanes and loading on the roadway above. In Figure 8.12.2.1-1a ,the two 12 ft. design lanes are placed symmetric about the pier centerline and thegirder reactions on the pier cap, representing the top boundary condition, are allidentical. In Figure 8.12.2.1-1b, the two design lanes are shifted to the left side ofthe roadway and the reactions vary across the top of the pier cap, giving a second topboundary condition.

Regardless of the truss layout that might be selected within the pier cap, the forces inthe pier column can be directly calculated: with pure axial compression in the first caseand compression plus bending in the second case as shown in Figure 8.12.2.1-1a-1b.In the first case, the bottom boundary condition is simply an axial force acting at themiddle of the pier. The boundary condition in the second case, however, must be cal-culated and includes a column compression block and tension component as shown inFigure 8.12.2.1-1b. The forces shown in the pier of Figure 8.12.2.1-1b are assumedto exist at a distance “d” from the bottom of the pier cap – away from the disturbedregion and in the portion of the column assumed to have sectional model behavior.

Figure 8.12.2-2Strut-and-Tie Model for a

Continuous Deep Beam

Tension tie

Compression strut

PP

DESIGN THEORY AND PROCEDURE8.12.2 Strut-and-Tie Models/8.12.2.1 Truss Geometry Layout

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In the first case of Figure 8.12.2.1-1a , the truss layout in the pier cap need onlymeet the condition of developing a compression thrust at the bottom of the cap. Inthe second case of Figure 8.12.2.1-1b, the truss must develop both the compressionand the tension force in the pier column. Clearly two different truss layouts could bedesigned depending on which set of loads/boundary conditions was being consideredas shown in Figure 8.12.2.1-2a-2b. The truss in the Figure 8.12.2.1-2b would beinverted if the trucks were at the other side of the roadway.

Figure 8.12.2.1-1a-1bPier Cap under Symmetric

and Unsymmetric LaneLoading

uniform axia stress

axial force resultant

compression stress block rebar tension

resultantforce couple

b) Unsymmetric Loads

a) Symmetric Loads

DESIGN THEORY AND PROCEDURE8.12.2.1 Truss Geometry Layout

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It is essential in the development of a truss layout for strut-and-tie design that:

(1) all of the possible load combinations be identified.

(2) boundary forces, including internal forces from portions of the structure havingsectional type behavior, be calculated for each controlling load condition.

(3) appropriate strut-and-tie models be laid out and designed for each set of boundaryconditions.

The nodal zones are regions where the struts and ties of the truss join. While thetruss diagrams of Figure 8.12.2.1-2a-2b idealize the truss members as connecting atpoints, the actual structure has struts and ties with finite dimensions. The nodal zonesizes are related to both the effective tie member sizes and the mechanism by whichexterior loads are transferred into the structure. As shown in Figure 8.12.2.2-1a-1c

Figure 8.12.2.1-2a-2bTruss Layouts for theDifferent Load Cases

strut

tie

192k 672k

52k 84k 132k 212k

b) Unsymmetric Truss

strut

tie

480k

120k 120k 120k 120k

a) Symmetric Truss

8.12.2.2

Nodal Zone and

Member Dimensions

DESIGN THEORY AND PROCEDURE8.12.2.1 Truss Geometry Layout/8.12.2.2 Nodal Zone and Member Dimensions

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[LRFD Specifications Commentary Figure 5.6.3.3.2-1] the dimensions of the nodalzone and adjoining struts are controlled by the anchorage conditions of reinforcingtie bars or bearing areas of applied loads.

The strength of tension ties depends directly on the type and strength of reinforcingused in the ties. Strengths of the individual truss strut members are normally con-trolled by the limits on stresses within the nodal zones. The nodal zone compressivestresses are defined by the relation between compressive stress capacity and perpen-

dicular tension strains invoked by compression stress field theory. Figure 8.12.2-1a shows the principal tension strain, ε1, which may exist perpendicular to the compres-sion strut, BP. The strain, ε1, is dependent on the truss geometry and the tensile strainin adjoining truss members. The adverse effect of this tensile strain in the crackedconcrete must be considered in calculating the capacity of a strut. In such struts thelimiting compressive stress, f cu, is a function of f ́c and ε1. The value of ε1 is, in turn, afunction of the tension strain, εs, in the cracked concrete in the direction of the ten-sion tie, and the angle between strut and tie.

8.12.2.3

Strength of Members

Figure 8.12.2.2-1a-1cEffects of Anchorage

Conditions on Cross-Sectional Area of Strut

a) Strut Anchored by Reinforcement

x

x

dba

θ s

s

la Section x-x

dba

b) Strut Anchored byBearing and Reinforcement

ha

lb

θs

lb

θs hs

c) Strut Anchored by Bearing and Strut

l sinθb s+h cosθs s

l sinθb s + h cosθa s

6db

0.5ha

6dba 6dba

l sinθa s ≤ 6d ba ≤ 6d ba

DESIGN THEORY AND PROCEDURE8.12.2.2 Nodal Zone and Member Dimensions/8.12.2.3 Strength of Members

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LRFD Article 5.6.3.1 states that “strut-and-tie models may be used to determineinternal force effects near supports and the points of application of concentratedloads at strength and extreme event limit states.” The statement appearing in the sec-ond paragraph of this article is stronger, and more specific, namely, “the strut-and-tiemodel should be considered for the design of deep footings and pile caps or other

situations in which the distance between the centers of applied load and the support-ing reactions is less than about twice the member thickness.”

LRFD Article 5.6.3 provides the following specifications for strut-and-tie model-ing.

The factored resistance of strut, Pr , may be calculated as:

Pr = φPn [LRFD Eq. 5.6.3.2-1]

where

φ = 0.7 for bearing in concrete and for strut-and-tie models [LRFD Art. 5.5.4.2-1]

Pn = nominal resistance of a compressive strut

The nominal axial resistance of unreinforced struts is calculated as:

Pn = f cu A cs [LRFD Eq. 5.6.3.3.1-1]

where

f cu = limiting compressive stress in strut and is calculated from:

f cu = f ́c/(0.8 + 170ε1) ≤ 0.85f ́c [LRFD Eq. 5.6.3.3.3-1]

where

ε1 is the principal tensile strain in cracked concrete, and is taken as:

ε1 = (εs + 0.002)cot2αs [LRFD Eq. 5.6.3.3.3-2]

where

αs = smallest angle between the compressive strut and adjoining tensionties, deg

εs = tensile strain in the concrete in the direction of the tension tie, in./in.

A cs = effective cross-sectional area of the strut determined from a considerationof the available concrete area and the anchoring or bearing conditions at theends of the strut [LRFD Art. 5.6.3.3.2.]

For an individual strut, the more basic expression for ε1 includes an additional term,εs, outside the bracket of LRFD Eq. 5.6.3.3.3-2.

For a value of principal tensile strain, ε1 = 0.002, the concrete in the compression

strut can resist a compressive stress of 0.85f ́c, i.e., the limit for regions of the strutnot crossed by or joined to tension ties. It is thus conservatively assumed that theprincipal compressive strain, ε2, in the direction of the strut is equal to 0.002.

In the presence of a tension tie at a node, if the reinforcing bars are to yield intension, there must exist significant tensile strains in the concrete. In LRFD Eq.5.6.3.3.3-2, as εs increases, ε1 increases, and f cu in LRFD Eq. 5.6.3.3.3-1 decreases.From LRFD Eq. 5.6.3.3.3-2, it is seen that as αs decreases, cot

2αs and ε1 increase,and therefore f cu decreases. In the limit when αs = 0, the compressive strut directioncoincides with that of the tension tie (i.e., incompatibility occurs, and f cu = 0 whichis an impractical case).

8.12.3LRFD Specifications

Provisions forStrut-and-Tie Models

8.12.3.1

Compression Struts

8.12.3.1.1

Unreinforced Concrete Struts

DESIGN THEORY AND PROCEDURE8.12.3 LRFD Specifications Provisions for Strut-and-Tie Models/

8.12.3.1.1 Unreinforced Concrete Struts

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• 0.75φ f ́c for node regions anchoring only one tension tie

• 0.65φ f ́c for node regions anchoring tension ties in more than one direction

where φ = resistance factor for bearing on concrete = 0.7 [LRFD Art. 5.5.4.2.1]

Stress limits at a nodal zone are controlled by the type of truss members meeting at thenode. At nodes B and C (Figure 8.12.2-1a ) where compression members meet, and atbearing areas at these locations, a higher compressive stress (0.85φ f ́c) is allowed than at A and D where it is necessary to anchor the tension tie, AD. In the latter case, the allow-able maximum compressive stress is reduced to 0.75φ f ́c. This limit is reduced even fur-ther to 0.65φ f ́c when tension ties converge from more than one direction at a node.

The above reductions in the presence of tension ties reflect the detrimental effect oftensile strain in nodes in which tensile reinforcement is anchored. It can be seen thatstresses in nodal zones can be reduced by increasing the size of bearing plates, or byincreasing the dimensions of struts and tension ties.

LRFD Commentary Article C5.6.3.5 states that if ties consist of post-tensionedtendons, and if the stress in the concrete does not exceed f pc (at centroid of the tie’scross-section), there is no tensile strain in the nodal zone and the limit for concretecompressive stress may be taken as 0.85 φ f ́c.

In order to control crack widths in members designed with the strut-and-tie model(except slabs and footings), and to ensure minimum ductility so that significant redis-tribution of internal stresses is possible, LRFD Article 5.6.3.6 states that an orthogonalreinforcing grid must be provided near each face. The spacing of bars in such a gridshould not exceed 12.0 in., and the ratio of reinforcement area to gross concrete areashould exceed 0.003 in each direction.

In general, these crack control requirements lead to a substantial amount of well-distrib-uted reinforcement throughout the member. Accordingly, the LRFD Specifications allowfor crack control reinforcement located within the region of a tension tie to be includedin calculating the resistance of the tie.

The use of strut-and-tie models typically involves a trial-and-error procedure. Thefollowing steps, if followed, should help reduce the effort required:

1) Use strut-and-tie modeling for disturbed regions of the structural member. Solvefor internal forces, and their resultants outside of the disturbed regions using sec-tional analysis with all of the controlling load combinations. These forces from

sectional analysis may be considered as boundary forces for the disturbed regionmodel. Apply the resultant forces to the disturbed region along with any externalloads that fall on that part of the member.

2) Assume initial models for each of the appropriate controlling load cases and bound-ary condition force sets. Estimate likely member widths. Elastic stress distributionmay be used as a guide. Static equilibrium is then used to determine forces in mem-bers due to factored loads. These forces are used in checking member dimensions.It may be necessary to modify the assumed model if the members are determined tobe inadequate. A number of appropriate models for different applications are avail-able in the literature [Guyon, (1960); Gergely and Sozen, (1967); Schlaich, et. al.,(1987); Collins and Mitchell, (1991); Breen, et. al. (1994)].

8.12.3.4

Crack Control Reinforcement

8.12.4Steps for DevelopingStrut-and-Tie Models

DESIGN THEORY AND PROCEDURE8.12.3.3 Proportioning Node Regions/8.12.4 Steps for Developing Strut-and-Tie Models

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3) Draw the strut-and-tie model to a reasonably large scale. This will help avoiderrors and give a better appreciation of the proportions of the structure.

4) There is no single strut-and-tie model for a particular system. Generally, the forces will flow in accordance with the pattern of reinforcement. Well-distributed rein-

forcement should be provided to ensure the redistribution of internal forces in thecracked concrete.

5) Good detailing of the structure is essential to ensure that the assumed flow offorces can be achieved in the cracked structure. Accordingly, reinforcement intension ties must be effectively anchored to develop the strength of the member.Nodal zones must be checked to ensure satisfactory load transfer between strutsand ties.

6) Complicated stress fields such as fans, arches and bands can usually be replaced bysimple line struts. Unnecessary complication of the model is not warranted.

Regardless of the strut-and-tie model adopted, the following design criteria must be

met:

1. Limits on bearing stresses and on compressive stresses in struts

2. Satisfactory anchorage and careful detailing of tension tie reinforcement

3. Critical examination of nodal zones to determine their maximum capacities

4. Provision of adequate crack control reinforcement throughout, to ensure theredistribution of internal stresses after cracking of concrete

Step 1 Determine bearing areas [LRFD Arts. 5.6.3.5 and 5.7.5]

Step 2 Assume appropriate truss geometry (draw a large-scale diagram)

Step 3 Select tension-tie reinforcement [LRFD Art. 5.6.3.4]

Select reinforcement distribution [LRFD Art. 5.6.3.5]

Step 4 Check development of tension-tie reinforcement [LRFD Arts. 5.6.3.4.2 and 5.11]

Step 5 Check strength of compression struts [LRFD Art. 5.6.3.3.1]

Step 6 Select crack control reinforcement [LRFD Art. 5.6.3.6]

Step 7 Detail structure carefully

8.12.4.1

Design Criteria

8.12.4.2

Summary of Steps

DESIGN THEORY AND PROCEDURE8.12.4 Steps for Developing Strut-and-Tie Models/8.12.4.2 Summary of Steps

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Design the pier cap shown in the figure below.8.12.5Pier Cap Example

Figure 8.12.5-1Pier Cap Dimensions

��

� �

� ��

� �

�

�

� �

�

�

�

�

′

DESIGN THEORY AND PROCEDURE8.12.5 Pier Cap Example

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Node C:

FCD = 500

tan (26.5 )° = 1,002.8 kips

F AC = 500

sin (26.5 )° = −1,120.6 kips

Node D:

F AD = 500

sin (62.8 )° = −562.1 kips

FDE = FCD + F ADcos62.8° = 1,002.8 + 562.1cos62.8° = 1,259.7 kips

8.12.5.2

Forces in Assumed Truss

8.12.5.1

Flow of Forces andTruss Geometry

Figure 8.12.5.1-1a-1d Assumed Truss Geometry

Pu = 500 kips

a) Strut-and-Tie Model

Pu

1'-0"

3'-0" 8'-0" 8'-0"

Tension Tie

8'-0" 3'-0"

C D E F

4'-0"

2'-6"1'-4"

A 2'-6" 2'-6" B

Pu Pu Pu

b) Nodal Zone A, B

1'-4"

5'-0"

27.4°

c) Nodal Zones: C,D,E,F

l = 2'-0"b

h = 1'-0"a 26.5°

500k 500k 500k 500k 8.0'8.0' 4.0'4.0'

A B

C D FE

5.35'

1000 k 1000 k

1.25' 1.25'

F B F

θBE = 62.8°

d) Member Force Calculation

26.5°

FCD = 1,002.8 k

F A C = 1 ,1 2 0 .6 k

F A D =

5 6 2 . 1 k

FDE = 1,259.7 k FEF

θBF = 26.5°

F = 1,259.7 k AB

tan θBF = 5.35/10.75

θBE = 26.5°

tan θBE = 5.35/2.75

θBE = 62.8°

DESIGN THEORY AND PROCEDURE8.12.5.1 Flow of Forces and Truss Geometry/8.12.5.2 Forces in Assumed Truss

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Node A:

F AB = F ADcos62.8° + F ACcos26.5° = 562.1cos62.8° + 1,120.6cos26.5° = −1,259.7 kips= −FDE

Bearing Stresses at C and D:

F < Pr = φPn = φ(0.85f ́c A 1m), assume m = 1 [LRFD Eq.5.7.5-2]

Allowable F/A 1 = 0.7(0.85 x 5) = 2.98 ksi

Actual F/A 1 = 500 k

24 (10) = 2.08 ksi < maximum allowable O.K.

Bearing Stresses at A and B:

F/A 1 = 2,000k

60 (48)

= 0.69 ksi < maximum allowable O.K.

Therefore, bearing stresses are acceptable.

FDE = 1,259.7 kips

φf y A st ≥ 1,259.7

A st ≥ 1,259.7/0.9(60) = 23.3 in.2

Because 3’-9” is available for development at C (at inner edge), choose a bar thatcan be developed in this distance, i.e., choose #10, ldb = 43.1 in. < 45 in. available[LRFD Art. 5.11.2.1].

No. of bars required = 23.3/1.27 = 18.34 bars

Use (20) #10 bars (25.4 in.2) in 2 layers

FCD = 1,002.8 kips

A st > 1,002.8/0.9(60) = 18.6 in.2

If (20) #10 bars are used as in DE, A required

A provided

18.6

25.4s

s

= = 0.73

Top bars:

Required development length= 1.3(0.73)(43.1) = 40.9 in. < 45 in. available O.K.

Note that each strut is traversed by a tie at one end.

Strut AC:

Strut AC is critical due to the small angle it makes with the tension tie, CD.

F AC = 1,120.6 kips (compression)

End C: (Anchored by bearing and reinforcement)

Tensile strain in tie:

FCD/A sEs = 1,002.8/[(20)(1.27)(29,000)] = 1.36 x 10-3 in./in.

The tension-tie reinforcing bars are developed in the nodal zone. Therefore, thestrain in these bars will increase from zero at the ends to 1.36 x 10 -3.

Strain at center of strut, εs = 1/2(1.36x10-3) = 0.68 x 10-3

ε1 = (εs + 0.002)cot2 αs

8.12.5.5

Strut Capacities

8.12.5.3

Bearing Stresses

8.12.5.4

Reinforcement for

Tension Tie DE

DESIGN THEORY AND PROCEDURE8.12.5.2 Forces in Assumed Truss/8.12.5.5 Strut Capacities

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where

αs = 26.5° [LRFD Art. 5.6.3.3.3]

ε1 = (0.68 x 10-3 + 2.0 x 10-3)cot2 26.5° = 10.8 x 10-3

f cu = f ́c/(0.8 + 170ε1) ≤ 0.85f ́c [LRFD Art. 5.6.3.3.3]

= 5.0/[0.8 + 170(10.8 x 10-3)] = 1.90 ksi < (0.85f ́c = 4.25 ksi) Capacity of strut, AC at C = φf cu A cs

where

A cs = (lbsin θs + ha cos θs)(48) (Figure 8.12.2.2-1c)

= (24sin26.5° + 12cos26.5°) x 48 = (21.4 x 48) in.2

Capacity of AC at C = (0.70)(1.90)(48 x 21.4)

= 1,366.2 kips > F AC (1,120.6 kips) O.K.

End A of AC is obviously not critical (not crossed by tension tie and wider dimensionsthan at C).

Strut AD:

Strut AD is anchored by bearing and reinforcement at end A, and crossed by tie atend D.

F AD = 562.1 kips

End D: Tensile strain in tie, DE = 1,259.7/[(20)(1.27)(29,000)] = 1.71 x 10-3

Strain at center of strut, εs = 1/2(1.71 x 10-3) = 0.86 x 10-3

ε1 = (εs + 0.002)cot2 αs

where

αs = 62.8°

ε1 = (0.86 + 2.0)10-3cot2 62.8° = 0.755 x 10-3

f cu = f ́c/(0.8 + 170ε1) ≤ 0.85f ́c

f cu = 5.0/[0.8 + 170(0.755 x 10-3)] = 5.39 ksi > 0.85f ́c = 4.25 ksi

Capacity of strut AD at D: φf cu A cs

A cs = (lbsinθs + ha cos θs)(48) = (24sin62.8° + 12cos62.8°) x 48 = 1,288 in.2

Capacity of Strut = (0.7)(4.25)(1,288) = 3,831 kips > F AD = 562.1 k O.K.

End A is obviously not critical (not crossed by tie).

Figure 8.12.5.6-1Nodal Zone at Pier

1.3'H = 1,259.7 k

5 6 2

. 1 k

1, 1 2 0. 6 k

V = 1,000 k

2.5'

26.5°b =

6 2. 8 °

8.12.5.6

Nodal Zone at Pier

DESIGN THEORY AND PROCEDURE8.12.5.5 Strut Capacities/8.12.5.6 Nodal Zone at Pier

8/16/2019 MNL-133-97_ch8_12

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Horizontal component, H = 562.1cos62.8° + 1,120.6cos26.5° = 1,259.7 kips = F ABO.K.

This nodal zone is bounded only by compressive struts and bearing areas:

Allowable compressive stress, f = 0.85f ́c = 2.98 ksi [LRFD Art. 5.6.3.5]

Due to H: F/A = 1,259.7/[(1.3)(12)(48)] = 1.68 ksi < 2.98 O.K.

Due to V: F/A = 1,000/[(2.5)(12)(48)] = 0.69 ksi O.K.

Check at throat and provide this reinforcement throughout (assuming 12 in. spacing).

Minimum A s required = 0.003(12)(48) = 1.73 in.2/ft [LRFD Art. 5.6.3.6]

Use (2) #9 = 2.00 in.2, #9 each face at 12 in. on center = 2.00 in.2/ft, or #6 bars eachface at 6 in. spacing, 4 x 0.44 = 1.76 in.2/ft

Use #6 bars @ 6 in. on center vertically and horizontally.

8.12.5.7

Minimum Reinforcement

for Crack Control

Figure 8.12.5.7-1

Reinforcement Details

# 6 Horizontal bars@ 6 in. on centereach face

# 6 Vertical bars @ 6 in. on center throughout each face

(20)# 10 bars

A

A

3"

Orthogonal grid of # 6 bars@ 6 in. on center oneach face

(20) # 10 bars

4'-0"

2'-6"

4'-0"

6"

View A-A

DESIGN THEORY AND PROCEDURE8.12.5.6 Nodal Zone at Pier/8.12.5.7 Minimum Reinforcement for Crack Control

MNL-133-97_ch8_12

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