This manual contains information related to the analysis and design of cast-in-situ concrete box girder bridges prestressed with post-tensioning tendons. The manual provides an overview of features of the construction of cast-in-place concrete box girder bridges, material.
Historical Overview
Chapter 1—Introduction 2 of 355 Today, cast-in post-tensioned box beam construction is used throughout the United States. The span range for cast-in box girder construction is shown to vary from 100 feet to 250 feet.
Typical Superstructure Cross Sections
The upper end of the span range represents continuous bridges, bridges without limitations on box girder depth, or bridges on a tangent span. Top slab - the entire width of the concrete deck, including the portions between the webs and overhangs outside the webs.
Longitudinal Post-Tensioning Layouts
The tendon spacing is increased at the ends of the span to properly locate the post-tensioning anchorages. Within the spans of the continuous unit, the chords overlap with a geometry similar to that shown in Figure 1.6.
Loss of Prestressing Force
Post-Tensioning System Hardware
- Basic Bearing Plates
- Special Bearing Plates or Anchorage Devices
- Wedge Plates
- Wedges and Strand-Wedge Connection
- Permanent Grout Caps
- Ducts
- Duct Size
- Corrugated Steel Duct
- Corrugated Plastic
- Plastic Fittings and Connections for Internal Tendons
- Grout Inlets, Outlets, Valves and Plugs
- Post-Tensioning Bars Anchor Systems
In this case, the inner channel area should be 2.5 times the net area of the strand tendon. These channels must be seamless and made of polyethylene or polypropylene and meet the requirements of section 4.3.5.2 of "Guide Specifications for Grouted Post-Tensioning, (PTI/ASBI M.
Overview of Construction
- Falsework
- Superstructure Formwork
- Reinforcing and Post-Tensioning Hardware Placement
- Placing and Consolidating Superstructure Concrete
- Superstructure Curing
- Post-Tensioning Operations
- Tendon Grouting and Anchor Protection
Longitudinal construction joints are usually located in the webs a few inches above the top of the bottom slab (three-stage molding) and a few inches above the top of the webs in the top slab fillets (two-stage and three-stage molding). The first phase of box girder construction is the pouring of the bottom slab concrete.
Concrete
- Compressive Strength
- Development of Compressive Strength with Time
- Tensile Strength
- Modulus of Elasticity
- Modulus of Elasticity Variation with Time
- Poisson’s Ratio
- Volumetric Changes
- Coefficient of Thermal Expansion
- Creep
- Shrinkage
Age of concrete under load - the younger the concrete under load, the more the concrete will creep. Relative humidity - the higher the relative humidity of the loading environment, the lower the concrete will creep.
Prestressing Strands
- Tensile Strength
- Modulus of Elasticity
- Relaxation of Steel
- Fatigue
The AASHTO LRFD specifications do not provide a stress-strain relationship other than within the elastic range as defined by Ep. The Precast Concrete Institute presents the stress-strain relationship up to the ultimate capacity of the strand.
Reinforcing Steel
Steel stress relaxation is a result of an increase in elongation over time while under an applied stress. Where: fst=steel stress level at start of time interval from (t1) to (t) (psi) fsu* = average stress in prestressed reinforcement at ultimate load (psi) fy*=yield point stress of prestressing steel (psi).
Introduction
Cross Section Properties and Sign Convention
Stress Summaries in a Prestressed Beam
The sum of these three load effects (i.e. self-weight with axial and eccentric prestress) as shown in figure 3.5 results in compressive stresses throughout the depth of the beam. The additional compression at the bottom of the beam is available to offset tensile stresses caused by other load effects such as superimposed dead loads, highway traffic loads and temperature induced stresses.
Selection of Prestressing Force for a Given Eccentricity
The sign of Ma is determined by the sign of the allowable stress on the cross-section. Determine the prestress limits for maximum and minimum stress at the top and bottom of the beam.
Permissible Eccentricities for a Given Prestressing Force
Chapter 3—Pretension with Posttension 46 of 355 Graphically, these limiting eccentricities of this example are shown in Figure 3.12. The cable profile can lie within the hatched area in the sketch, taking into account the minimum and maximum permissible bending stresses in the beam.
Equivalent Forces Due To Post-Tensioning and Load Balancing
Note that the equivalent horizontal force in the lower diagram has been made equal to the preload force. Example: For the beam of the previous examples, compare the prestressing force required for minimum bottom stress at mid-span and the prestressing force required to balance the total applied load of 4225 k/ft.
Post-Tensioning in Continuous Girders
The secondary moments are the result of the retention of the rotation of the end rotations of the primary moment beam on the intermediate beams. The total post-tensioning moment is the sum of the primary and secondary moments at each section along the length of the bridge. The effective eccentricity at mid-span is halved to Femax/2.
Tendon Profiles—Parabolic Segments
Although the tendon profile has no true eccentricity at the central support, the effective eccentricity is equal to the maximum eccentricity at the center of the spans. Chapter 3—Pre-tension with post-tensioning 54 of 355 Example: using the twin-span girder of the previous examples and tendon profile. Secondary moments for the two-span beam in this section could be easily calculated due to the symmetry of the structure.
Instantaneous Losses
- Friction and Wobble Losses
- Elongation
- Anchor Set
- Two-End Stressing
- Elastic Shortening
The force on the left side of the tendon is equal to a jacking force of 835 kips. Where, Δ1 = Shortening of the superstructure under the force of tension Tendon 1 (in) F1 = Tension force for Tendon 1 (kips). Where, Δ2 = Shortening of the superstructure under the force of tension Tendon 2 (in) F2 = Tension force for Tendon 2 (kips).
Time-Dependent Losses
- General
- Concrete Shrinkage
- Concrete Creep
- Steel
More generally, the center of gravity of the poststress is eccentric to the center of gravity of the concrete superstructure, and the stress is different from the axial value. SS is a gain in prestressing stress due to the shrinkage of the composite cover plate. AASHTO LRFD Article 5.9.5.4.5 provides guidance on the application of the general equation for post-tensioned non-segmental beams.
Introduction
Establish Bridge Layout
Project Design Criteria
Span Lengths and Layout
Cross Section Selection
Superstructure Depth
Superstructure Width
Cross Section Member Sizes
- Width and Thickness of Cantilever Wing
- Individual and Total Web Thickness
- Top Slab Thickness
- Bottom Slab Thickness
- Member Sizes for Example Problem
AASHTO LRFD Article 9.7.1.1 states that the minimum thickness of the top plate, exclusive of grinding and grooving, shall not be less than 7'' unless approved by the owner. The thickness of the top plate and the clear spacing between the webs or laps are shown in Figure 5.7. If we were to use the minimum thickness of each web, which is 12'', the above limits would suggest 5 or 4 webs.
Longitudinal Analysis
- Approach
- Analysis by Method of Joint Flexibilities
- Span Properties and Characteristic Flexibilities
- Analysis Left to Right
- Analysis Right to Left
- Carry-Over Factors
Note: For numerical convenience, the modulus of elasticity of concrete is factored by the coefficients of flexibility.
Bending Moments
- Effect of a Unit Uniform Load
- Dead Load—DC (Self Weight and Barrier Railing)
- Dead Load—DW (Future Wearing Surface)
- Live Load—LL
- Uniform Load Component
- Truck—Positive Moment in Span 1 or 3
- Truck—Positive Moment in Span 2
- Truck—Negative Moment over Piers
- Live Load Moment Totals
- Thermal Gradient (TG)
- Post-Tensioning Secondary Moments
DC moments are found as a linear scaling of the uniform unit load effects shown in Figure 5.11. The DW moments are found as a linear scaling of the uniform unit load effects shown in Figure 5.11. Chapter 5—Conceptual Design 92 of 355 Figure 5.16 shows the bending moment for one truck with the rear axle located 36' from the start of the bridge.
Required Prestressing Force After Loss
Chapter 5—Preliminary Design 100 of 355 Msum in this equation is the sum of all bending moments except the secondary moments of prestress. Assuming moderately corrosive conditions, the AASHTO LRFD table states that the allowable stress in the concrete after all losses is 0.19√f'c, where f'c has the units ksi. Using equations 5.25 and 5.26, the minimum prestressing force requirements at the three sections studied, after all losses, are:.
Prestressing Losses and Tendon Sizing for Final Design (Pjack)
- Losses from Friction, Wobble, and Anchor Set
- Losses from Elastic Shortening
- Losses from Concrete Shrinkage
- Losses from Concrete Creep
- Losses from Steel Relaxation
- Total of Losses and Tendon Sizing
The difficulty arises here because the value of fcgp, the stress of the concrete at the prestressing center of gravity, is calculated after bond and before any long-term loss. Subtracting this from the tendon stress at the center of the midspan (171.62 ksi) produces an estimate of the ultimate stress in the tendon at this location of 151.62 ksi (56 percent of fpu). For posttensioned structures with unbonded tendons, the value fcgp can be calculated as the stress at the center of gravity of the prestressing steel averaged along the length of the member.
Service Limit State Stress Verifications
- Service Flexure—Temporary Stresses (DC and PT Only)
- Service Limit State III Flexure Before Long-Term Losses
- Service Limit State III Flexure After Long-Term Losses
- Principal Tension in Webs after Losses
Allowable concrete stresses in concrete before losses are presented in AASHTO LRFD Article 5.9.4.1. Allowable concrete stresses in concrete before losses are presented in AASHTO LRFD Article 5.9.4.2. The stresses checked on piers 2 and 3 could have been checked on the pier face.).
Optimizing Post-Tensioning Layout
The maximum principal stress of -44 ksf is greater than would be allowable for a segmental box girder, but does reflect a level of stress that can be sufficiently strengthened during final design. In Span 2, however, the effect of reducing the maximum positive eccentricity in Spans 1 and 3 will reduce the secondary moments, and the poststress demand in that span. These power requirements are compared to those shown at the top of page 101. Increasing the eccentricity in the side spans reduces the overall posttensioning demand by 510 jumps, or about 6 percent.
Introduction
Bending Moments Caused by Unit Effects
Effect of a Unit Uniform Load
Effect of a Unit Lateral Displacement (Side-Sway Correction)
Effect of a Unit Contraction
Dead Load—DC (Self Weight and Barrier Railing)
Dead Load—DW (Future Wearing Surface)
Live Load—LL (Lane and Truck Components)
Envelope of Uniform Load Component
Truck—Positive Moment in Span 1 or 3
Truck—Positive Moment in Span 2
Truck—Negative Moment over Piers
Post-Tensioning Secondary Moments—Unit Prestressing Force
Chapter 6—Substructure Considerations 119 of 355 The top diagram in Figure 6.12 shows the secondary moments considering the torsional restraint provided by the columns. The secondary moment component resulting from axial shortening is slightly overestimated in this case as it does not take into account the shear forces drawn by the columns. For this example, the refinement reduces the secondary moments from the axial shortening by about 4 percent.
Thermal Gradient (TG)—20°F Linear
Moments Resulting from Temperature Rise and Fall
Temperature Rise—40°F Uniform Rise
Temperature Fall—40°F Uniform Fall
Moments Resulting from Concrete Shrinkage
The bending moment diagram in Figure 6.16 assumes that the same concrete mix is used for the superstructure and substructure. Also, these results assume that the superstructure and substructure concrete are poured at the same time (have the same age). For example, the bending moments will increase if the columns are cast in front of the superstructure, since they are made of the same concrete.
Moments Resulting from Concrete Creep
For the example problem under consideration, the creep moment is estimated by the linear scaling of the creep bending moment diagram. The bending moments due to creep of the concrete associated with shrinkage are shown in figure 6.17. For example, if the substructure of the example bridge is made of the same concrete but cast at an earlier date than the superstructure, the older substructure will resist the creep of the younger superstructure concrete.
Bending Moments Summaries
These creep moments occur when the relative creep characteristics of superstructure and substructure concrete are different, whether they are of different mix design or of different casting dates. The moments to be considered when calculating this creep moment will be the summation of permanent loads (DC and PT). The moments would be estimated by scaling the sum of the permanent loads by a relative ratio of residual creep coefficients for the two concretes.
Post-Tensioning Force Comparison (after all losses, with
Side Span Positive Bending
Middle Span Positive Bending
Consider the cross-section of the one-cell box beam with inclined webs shown in Figure 7.31. At = total area of transverse torsional reinforcement in the outer web of the box girder (in2). No guidelines are presented for reinforcement or forces in the remainder of the cross section.
Centerline of Middle Span
Introduction
Most of these computer programs also allow the input of post-tensioning tendons by geometric definition. The effects of post-tensioning are determined by applying equivalent loads to the stiffness-based solution. BD2, a two-dimensional analysis package, was used in the preparation of Design Example 1, which is in Appendix C of this manual.
Modeling Concepts
- Straight Bridges Supported on Bearings
- Nodes
- Elements
- Post-Tensioning
- Straight Bridges with Integral Piers
- Curved Bridges
- Other Three-Dimensional Analyses
The nodes shown in Figure 7.3 are defined vertically in the center of gravity of the cross-section of the superstructure. Their effect is usually small on the results of upgrade planning (see Chapter 6). Figure 7.6 shows a cross-section of the bridge used in design case 1 with built-in columns.
Strength Limit Verification—Flexure
- Factored Loads for Longitudinal Flexure
- Flexural Resistance
- Strain Compatibility
- Material Stresses and Internal Forces
- Internal Equilibrium
- Resistance Factors (ϕ)
- Limits of Reinforcing
- Procedure
Forces in the prestressing steel are found by multiplying the stress in the steel by the surface area of the reinforcing steel. However, this should not be inferred due to the initial strain level in the prestressing steel. Example: Determine the nominal bending capacity of the cross-section of the previous example taking into account light reinforcement in the bottom plate.
Strength Limit Verification—Shear
- LRFD Design Procedures for Shear and Torsion
- General Requirements
- Sectional Model Nominal Shear Resistance
- Effective Web Width
- Effective Shear Depth
- Shear Resistance from Concrete (Vc)
- Method 2 (Simplified MCFT)
- Method 3 (Historical Empirical)
- Shear Resistance from Transverse (Web) Reinforcing Steel (Vs)
- Shear Resistance from Vertical Component of Effective
- Longitudinal Reinforcing
- Torsion Reinforcing
Vp = shear resistance provided by the component of the effective prestressing force in the direction of the applied shear. Where: fpc = compressive stress at the center of gravity of the section after all losses (ksi). The general expression for the shear contribution of the transverse reinforcement is determined by LRFD equation.
Introduction
Methods of Analysis
Empirical Method—This method, presented in Section 9.7 of the LRFD, represents the required top plate reinforcement for bridges meeting the criteria of Section 9.7.2.4. Approximate method - Section 4.6.2 presents an approximate method of analysis where the deck is divided into strips perpendicular to the supporting elements (webs). Refined method—Article 4.6.3 allows the use of refined cross-sectional analyzes by methods listed in Article 4.4.
Applicable AASHTO LRFD Specifications
- Section 9—Deck and Deck Systems
- Section 3—Loads
- Section 4—Analysis
- Section 13—Railing
Where the plate spans mainly in the transverse direction, only the axles of the design truck of Article 3.6.1.2.2 or design tandem of Article 3.6.1.2.3 must be fitted to the deck sheet or the top plate of box divers. The live loads of the Design Truck and Design Tandem are shown in adjacent lanes in figure 8.2. For designing all other components—2.0 feet from the edge of the design track.
Strip Method Analysis for a Multi-Cell Box Girder Superstructure
The Transverse Model
Transverse Bending Moment Results
Transverse Design Moments
Top Slab Transverse Bending Moment Resultsfor a Single-Cell Box Girder
Introduction
Analysis for Uniformly Repeating Loads
Analysis for Concentrated Wheel Live Loads
Live Load Moments in Cantilever Wings
Negative Live Load Moments in the Top Slab
Positive Live Load Moments at Centerline of the Top Slab
Transverse Post-Tensioning
Transverse Post-Tensioning Tendon Layouts
Preliminary Estimate of Required Prestressing Force
Transverse Post-Tensioning Tendon Placement and Stressing
Effects of Curved Tendons
In-Plane and Out-of-Plane Forces
AASHTO LRFD Design Approach
Regional Effects—Transverse (Regional) Bending
Local Shear and Flexure in Webs
- Shear Resistance to Pull-out
- Cracking of Concrete Cover
Out-of-Plane Force Effects
End Anchorage Zones
Diaphragms at Supports
Single-Cell Box Girder Transfer of Vertical Shear Forces
Single-Cell Box Girder Transfer of Torsion to Bearings
Multi-Cell Box Girder Diaphragms
