tendon modeled as load-resisting elements

The tendon is not considered to be removed from the concrete member.

Rather, it is modeled as a distinct element linked to the concrete mem-ber (Figure  5.10). The change in the prestressing force is automatically accounted for in the equilibrium equations set up for the analysis of the segment.

For tendons modeled as resisting elements, four post-tensioning analysis types are shown in Figure 5.11: (1) beam type, (2) tendon type, (3) plane stress type, and (4) solid type. The former two are used in routine bridge analyses, whereas the latter two with more detailed modeling technique are used more in research or forensic analysis (LUSAS 2012). For post-tensioning, the tendons can be either external or internal where internal tendons can be either bonded or unbonded (Figure 5.12).

5.2.3 2d and 3d modeling

Based on the discussion in Section 2.4.5, two-dimensional (2D) or three-dimensional (3D) models can be generated based on the project’s needs.

For a 2D model, only one beam is considered and section properties of that beam are based on the locations of their respective neutral axes. Two 2D beam models representing two different stages of noncomposite and

Continuity tendon

truss element Cantilever tendon

truss element Rigid link

Beam frame element

(c)

Tendon initial length (force)

(a)

Tendon truss element Rigid link Node J

Beam frame Node I

(b)

Figure 5.10 Tendon modeled as an element linked to the concrete member. (a) Tendon as element. (b) Tendon element geometry. (c) Finite element modeling of the segmentally erected bridge with post-tensioning tendons.

short-term composite models, respectively, are demonstrated in Figure 5.13.

Many customized 2D prestressed beam computer programs are available for analysis where customization is made by dividing beams into small segments of prismatic members with tendons modeled as applied loading within each segment, which was discussed in Section 5.2.1. For a 2D beam

Analysis type: beam

Analysis type: tendon

Analysis type: plane stress (a)

(b) Analysis type: solid

Figure 5.11 2D and 3D post-tensioning analysis types. (a) 2D model. (b) 3D model. (Data from LUSAS^®, “LUSAS Bridge/Bridge Plus Bridge Engineering Analysis,” 2012, http://www.lusas.com/products/information/eurocode_pedestrian_loading .html.)

Prestressed bridge

Cast-in-place

Unbonded External tendons

Precast

Post-tensioned Pretensioned

Internal tendons

Bonded/grouted

Figure 5.12 Types of prestressing analysis.

Composite NA Noncomposite NA

Figure 5.13 2D model with its associated neutral axis (NA) locations. (a) Framing plan.

(b) Cross section.

model, moments and shears are direct results from analysis, and there is no need to integrate stresses to get beam moments for strength limit state capacity check. No matter which code is adopted for design, stress limits for concrete and steel are always given.

On the other hand, the 3D modeling technique has become more sophis-ticated and more popular nowadays to understand the behavior of a bridge during different construction stages. Instead of modeling tendons as applied loading, they are modeled as resisting elements as described in Section 5.2.2. In routine bridge analyses, prestressed beams are usually modeled as beams while tendons are modeled as a series of truss elements with embedded pretensioning forces. For a complete 3D bridge model, in which deck are simulated by shell or solid elements with rigid connection to beam elements, tendons can be modeled by spatial truss elements sharing appropriate nodes with shell, solid or beam elements. An illustration of 2D modeling is described in Section 5.3, and a more detailed demonstration of 3D modeling is covered in Sections 5.4 through 5.7.

5.3 2d illustrated exaMPle of a PrototyPe Prestressed/Post-tensioned concrete Bridge in the united states

Based on AASHTO specifications (2013), a design case for a concrete alter-nate with a continuous prestressed and then post-tensioned precast I-beam bridge is analyzed as a single beam staged from simple to continuous beams.

The total length of the bridge is 198.86 m (652′-5″), with five continuous spans of 39.5  m (129′-7″) each (Figure 5.14a). The clear roadway width is 13.41 m (44′), and out-to-out distance is 17.98 m (59′) with 3–3.66 m (12′) lanes. Five 1880-mm (74″) deep precast bulb-T girders are used in the design with 3.81-m (12′-6″) girder spacing (Figure 5.14b). A 200-mm (8″) deck slab is used in the composite construction with another 13-mm (1/2″) wearing surface.

Precast girder is formed by the semi-light weight concrete with initial concrete strength (f_ci′) of 31  MPa (4500  psi) and final concrete strength (f_c′) of 48.3  MPa (7000  psi). Concrete strength of the cast-in-place con-crete is 34.5  MPa (5000  psi). All the prestressing tendons are 1862-MPa (270-ksi) stress-relieved seven-wire strands with modulus of elasticity of 1.9 × 10⁵ MPa (28 × 106 psi). The prestressing steel strand’s diameter is 13  mm (1/2″), and the post-tensioning steel strand’s diameter is 15  mm (0.6″). Figure 5.15 shows the profile of the post-tensioning conduits (pre-stressing strands are not shown) and three cross sections at the end spans.

Cross sections A–A and C–C (Figure 5.15b) show the thickened webs at the ends of the precast beam. The construction sequence is listed as follows:

128′ 6′′ Bearing to pier 128′ 6′′ Bearing to pier128′ 6′′ Bearing to pier 33′ 4′′

4′12′ Slope: 3/16′′ / FT.Slope: 3/16′′/FT.Slope: 1/4′′/FT. 4′ 6′′

6”

BearingsBearingsBearingsPier no. 2 (Expansion)Pier no. 3 (Semi-fixed) Beam erection plan

Pier no. 3 (Seul-fixed)

Pier no. 5 (Expansion)

Span EBearing Bent no: 6 (Expansion)

Bent no: 6Intermediate diaphragm See details drwg. c-29BearingsSee detail “C” Span ASpan BSpan C Span D 9′ 6′′

2 S PA @

52′ 4′′42′ 10′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′43′ 4′′42′ 4′′ 12′ 7′′4′′3 SPA @51′ 1 3/4′′34′ 6 1/4′′130′ Pier to pier130′ Pier to pier130′ Pier to pier647′ Bearing to bearing (a) (b)

Skew: 25°0′ 0′′ LT.Intermediate Diaphragm spacing

(T

4 Spaces @ 12′ 6′′-50′ 0′′ Figure 5.14 (a) Five-span precast and (b) prestressed concrete bridge made continuous with post-tensioning tendons.

1. Erect precast prestressed beams on early-made concrete abutments and supports

2. Install duct splices for post-tensioning tendons and pour beam splices and diaphragms at piers. At this stage, stress and grout tendons T1 3. Pour in-span diaphragms. At this stage, stress and grout post-tension

tendons T2

4. Pour deck. At this stage, stress and grout tendons T3 for full post-tensioning

5. Construct sidewalk and barrier/railing and complete the job

In the process three 2D beam models with different section properties are built. The first noncomposite sectional model with different levels of ten-don forces is used for stages 1, 2, and 3. The second short-term composite sectional model with full tendon forces is used for stage 4, whereas the third long-term composite sectional model with full tendon forces is used for stage 5. Note here that short-term and long-term composite sections are used by AASHTO to refer to the section properties of n and 3n, respectively, where n is the modulus ratio between steel and concrete materials. For the consideration of pretensioning/post-tensioning tendon modeling and its

Flat square anchor plate system

A B Midspan

T3 T3 T4

T1 (a) B

(b)

C C

A

T3 T2

T1

End view A–A Section B–B Section C–C

T1

T4 T1T2 T3

T4 T2 T3

Figure 5.15 Post-tensioning (a) layout and their (b) cross sections at the end span of a continuous precast prestressed/post-tensioned concrete bridge.

analysis, the “tendon modeling through primary moments” as discussed in Section 5.2.1 is used in the calculation by Merlin-DASH/PBEAM, a 2D line girder program. This tedious procedure of generating primary fixed-end moments can also be employed to a generic finite element analysis pack-age, but the process would be cumbersome. Results show that the program checks stress limits of the concrete (Figure 5.16) and the reinforcing steel under the serviceability limit states as well as ultimate moments and shears under the strength limit states.

1.0 Girder bottom stresses at service

Girder top stresses at service

260 325

Figure 5.16 (a) Top and (b) bottom stresses of a five-span precast, prestressed concrete bridge.

5.4 3d illustrated exaMPle of a

douBle- cell Post-tensioning concrete Bridge— Verzasca 2 Bridge, switzerland

In European practice, post-tensioning is more popular. A Swiss bridge with cast-in-place double-cell concrete beam is taking as an example in this section. The Bridge Verzasca 2, which locates on the main road between Bellinzona and Locarno, in the south of Switzerland, was built in 1990–

1991 and consists of six spans between 25.24 and 39.70  m (82.8′ and 130.3′), with a total length of 203.6 m (668′). The pier supports are skewed at an angle of 28.8°, whereas the abutments are placed perpendicular to the bridge axis. The superstructure is a post-tensioned continuous girder with a cast-in-place double-cell section (Schellenberg et al. 2005).

The cross section changes in the region over the piers where negative moments are expected. In this region the three webs of the double-cell sec-tion are widened. Also, the bottom flange is thickened continuously from 200 to 300 mm (8″ to 12″) in this region.

Diaphragms are placed over each pier, providing a higher torsional rigid-ity. Accounting for the diaphragms as well as a cross section of the beam, a total of three cross sections can be determined. The post-tensioning ten-dons are anchored approximately at the section of dead load point of con-traflexure, where the webs change their width, providing required spaces for the tensioning procedure.

Each tendon stretches over one span including both neighboring piers in such a way that the tendons overlap over a single pier. Their distribution over the cross section is shown in Figure 5.17.