This paper describes the application of time-dependent analyses for developing progressively erected hybrid structures that combine precast/prestressed concrete with other structural building materials. Several examples of time-dependent analysis through the use of a specialized computer software program are discussed. These structures demonstrate how modern methods of design and construction can be applied to enhance the design of moderate to long span bridges. The featured structures combine the inherent advantages of precast/prestressed concrete with steel and cast-in-place concrete. The designs demonstrate how the strength and durability qualities of concrete can be integrated to produce structures that are light, elegant and economical while also having a minimum impact on the environment.

Modern bridge structures can achieve considerable economy in construction through the design of hybrid systems that combine the advantages of

precast/prestressed concrete with steel and cast-in-place concrete.1-4 Hybrid structures take advantage of specific material properties of the individual components. Structural steel, for example, has the particular advantage of high ten-sile strength, precision in its manufacture, and fast erection. Precast concrete has an intrinsic high quality and can be erected simply and quickly, while cast-in-place concrete has an inherent continuity and behaves monolithically.

The structures (see Fig. 1a) and also the individual struc-tural members, e.g., decks (see Fig. 1b), pylons, piers, struts,

Applications of Time-Dependent Analysis in the Design of Hybrid Bridge Structures

56 PCI JOURNAL

Stanislav Susky Partner

Strasky, Husty and Partners Consulting Engineers

Brno, Czech Republic

Jiri Strasky, Ph.D., P.E. Professor of Concrete Structures

Technical University of Brno Brno, Czech Republic

andPartner

Strasky, Husty and Partners Consulting Engineers

Brno, Czech Republic and Mill Valley, California

Jaroslav Navratil, Ph.D. Assistant Professor Technical University of Brno Brno, Czech Republic

July-August 2001 57

arches (see Fig. 1c), and even the ties and tendons (see Fig. 1d) can be as-sembled of steel, precast concrete or cast-in-place concrete.

Hybrid structures can not only take advantage of the material properties, but can also maximize the economy and speed of construction. In bridge construction, for example, the main loadbearing members formed by sus-pension or stay cables, hangers, beams, or arches are very often constructed in advance to reduce overall construction time and costs (see Fig. 2).

Combining precast and cast-in-place construction techniques adds to the economy in construction and main-tains a high standard of quality while reducing the time needed to complete construction. Precast members are often used as self-supporting perma-nent formwork for casting in place other parts of the structure.

Precast concrete members and light-weight steel can be erected with very

little disturbance to the existing en-vironment. The components can also be transported and installed by read-ily available and inexpensive trans-port trucks and erection cranes. These projects give opportunities for smaller contractors to successfully compete for and participate in the construction of new bridges that previously had been impossible because of their lack of ac-cess to highly specialized equipment previously required for large bridge projects.

During construction these structures are subjected to a wide range of load-ing conditions. Boundary conditions change, new structural members are assembled or cast, post-tensioning is applied, and temporary support ele-ments are removed. Concrete elements of various ages are combined and the concrete is gradually loaded.

Because of these widely varying loading conditions, the designer must take into account the redistribution of

internal forces caused by creep and shrinkage both during construction and throughout the service life of con-crete structures. Sophisticated meth-ods, therefore, are needed to carry out the structural analysis.

TIME-DEPENDENT ANALYSISSpecialized time-dependent analysis

(TDA) software has been developed to facilitate the volume change analysis of hybrid structures.

The creep and shrinkage of the struc-tural members are predicted through the mean properties of a given cross section, taking into account the average relative humidity and member size. The development of the modulus of elastic-ity over time is also considered.

The method is based on a step-by-step computer procedure in which the time domain is subdivided by discrete times into time intervals.5 A finite element analysis is performed

Fig. 1. Examples of hybrid structures and individual structural members.

(a) Hybrid structures(b) Hybrid deck

(c) Hybrid piers, struts, and arches(d) Hybrid ties and tendons

58 PCI JOURNAL

Fig. 2. Progressive erection of hybrid structures.

in each time node. The element stiff-ness matrix and load vector terms include the effect of axial, bending, and shear deformations.

The centroidal axis of the element can be placed eccentrically, relating to the reference axis connecting the nodes. Six external and two internal degrees of freedom are used. The static condensation of the internal node pa-rameters is used, thus ensuring the full compatibility between eccentric ele-ments. The elements represent concrete webs, deck layers, prestressing ten-dons, or reinforcement (see Fig. 3).

The various elements are installed or removed according to the method of construction, and they can express the influence of different ages of con-crete members in both the longitudinal and transverse directions of the struc-ture. Various construction operations, including the addition or removal of segments, changes of boundary condi-tions, loads, and prescribed displace-

ments, may be modeled. The stress-produced strain consists

of an elastic instantaneous strain εe(t) and creep strain εc(t). The development of the modulus of elasticity over time is considered. The creep prediction model is based on a linear stress-strain relationship to ensure the applicability of linear superposition. The numerical solution is based on the replacement of the Stieltjes hereditary integral by a fi-nite sum. The general creep problem is thus converted to a series of elasticity problems [see Eq. (1)].

ε ϕσ

m jj

jj

n

t t tt

E t( ) = +[ ]

=∑ 1

1

0

( , )( )

( )

( )

∆

The method used for the time-de-pendent analysis is based on a step-by-step computer procedure in which the time domain is subdivided by dis-

crete time nodes ti (i = 1, 2, ... n) into time intervals. The solution in the time node i is as follows:

1. The increments of strains, cur-vatures, and shear strains caused by creep during the interval < ti-1, ti > are calculated using the second summa-tion of Eq. (1). Correspondingly, the shrinkage strains are also calculated.

2. The load vector dFp is assembled as equivalent to the effects of general-ized strains calculated in Step 1.

3. The stiffness matrices K of the elements are calculated for the time ti, and the stiffness matrix of the whole structure Kg is assembled.

4. The system of equations Kgd∆g = dFp is analyzed. The vector of incre-ments of nodal displacements d∆g is added to the vector of total nodal dis-placements ∆g.

5. The elements are analyzed in the central coordinate system. The incre-ments of internal forces and incre-ments of elastic strains are calculated

July-August 2001 59

from the increments of displacements of the element nodes.

6. The changes of the structural con-figuration carried on in the time node ti are introduced.

7. The increments of generalized strains of the elements that are pre-stressed (or loaded by changes of temperature) in the time node ti are calculated. The prestressing loss due to the deformation of the structure is automatically included in the analysis through the increments of internal ele-ment forces.

8. The load vector dFZ is assembled as equivalent to the effects of general-ized strains calculated in Step 7. The increments of other types of long-term loading applied in the time node ti are added to the load vector dFZ.

9. The system of equations Kgd∆g = dFZ is analyzed. The vector of in-crements of nodal displacements d∆g is added to the vector of total nodal displacements ∆g.

10. The increments of internal forces and increments of elastic strains are calculated from the increments of dis-placements of the element nodes.

11. The increments of internal forces calculated in Steps 5 and 10 are added to the total internal forces. The incre-ments of elastic strains calculated in

Steps 5 and 10 are added together and saved to reflect the history of elastic strains as the increments in the time node ti.

12. Go to the first step of the time node i+l.

Because of the large quantity of input and output data, it was neces-sary to develop a user-friendly graphic interface for pre- and post-processing. From the user’s viewpoint, these en-hancements contribute significantly to the effectiveness of the system. A pre-processor guides the user through the preparation and input of the data. Post-processors allow the user to select and view only the data needed.

Because of the graphical environ-ment, it is possible to filter output data, switch individual parameters (internal forces, stresses, deformations) on and off, zoom in and out of the structure, and choose a time node to view the data. This makes checking the structure quick and easy and allows back-check-ing of the process. The TDA program has recently been included in the pro-gram system SCIA.6

The modified numerical procedure was also used for the time-dependent, geometrically nonlinear structural analysis. The computer procedure for the creep and shrinkage analysis

described previously was adapted so it could be integrated with the FEM software package ANSYS used at the Technical University of Brno.

The effects of creep and shrinkage calculated in Steps 1 and 2, and the external load applied in Steps 6 and 7, are converted to an equivalent load for the ANSYS structural model. After a full geometrical analysis is performed, the increments of displacements and internal forces are processed in the same way as described in Steps 5, 10, and 11, and the analysis moves on to a new time node.

BRIDGE APPLIcATIoNSReferences 7 and 8 describe a meth-

odology used in all designs performed by Strasky, Husty and Partners and in all research work done at the Techni-cal University of Brno. The program was also used in the design or check-ing of structures in which the first au-thor was involved in while working in the United States. The application of the computer program is illustrated with examples of relatively small con-crete bridges in which economy was achieved by combining precast/pre-stressed concrete with other structural materials.

Fig. 3. Modeling of a typical structure.

(a) Partial elevation(b) Cross section(c) Detail A(d) Finite element

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Wisconsin Avenue Viaduct, Milwaukee, Wisconsin

The computer program was first used in the United States for check-ing the design of the Wisconsin Av-enue Viaduct, completed in 1993.9,10

Fig. 4. Wisconsin Avenue Viaduct: Longitudinal section of a typical span.

Fig. 5. Wisconsin Avenue Viaduct: Cross section of a typical span.

The design of this bridge received the Honor Award in ACEC’s 1994 Excellence Awards Competition, and PCI’s Bridge Design Award and Harry H. Edwards Industry Advancement Award in 1994.

The bridge was designed to replace an existing concrete arch structure that had become severely deteriorated as a result of time, weather, traffic, and salt. Its structural solution addressed two issues: (1) the public acknowl-

July-August 2001 61

edged the need to replace the existing structure, but expressed the desire to retain a nostalgic design appropriate to its surroundings, and (2) in keeping with their rigid maintenance policies and budget, the state of Wisconsin ad-vocated a simple girder structure with a replaceable deck. The design of the new bridge fulfilled both requirements by creating a simple, economical, and aesthetically pleasing structure.

The viaduct has 11 spans with lengths of 22.86 + (8 x 48.16) + 12.80 + 22.86 = 443.80 m (1456 ft). The 48.16 m (158 ft) spans comprise two parallel precast arches that support the deck formed by precast girders and a composite slab (see Figs. 4 and 5). Each arch was assembled from two precast segments connected at the crown by a post-tensioned con-crete joint.

The precast segments have a trough cross section and were concentrically post-tensioned for handling before erection. Once erected, the troughs were filled with concrete and combined with the precast arch segment to form the completed arch ribs. The precast girders are supported by cast-in-place diaphragms located above the spandrel walls and at the midspan crown.

The bridge was designed to allow progressive erection from one abut-ment to the other, without the need to temporarily strengthen the piers against excessive unbalanced lateral thrust (see Figs. 6 and 7). The se-quence of construction generally fol-lowed distinct, overlapping stages that were arranged to keep the unbalanced thrust within acceptable limits. The construction sequence of a typical arch span was as follows:

1. Erect the precast arch segments. 2. Connect the segments and pre-

stress them at the crown. 3. Fill the arch with concrete (see

Fig. 8). 4. Erect the precast deck girders (see

Fig. 9). 5. Place the deck slab and dia-

phragms. Temporary falsework was needed

for erecting the arch elements at the crown, but no further falsework was needed because the arch ribs served as the platform for constructing the deck. The design life span of the deck slab is 30 years.

The time-dependent analysis was done for a typical arch span. Due to its symmetry, only one-half of the struc-ture was analyzed with corresponding boundary conditions on the axis of symmetry (see Fig. 10).

Fig. 6. Wisconsin Avenue Viaduct: Progressive erection of the bridge (Steps 1 to 9) and replacement of the deck (Steps 10 to 13).

1. Construction of the substructure, casting and post-tensioning of the arch segment in storage. 2. Erection of the arch segments. 3. Casting of the joints between the arch segments at midspan, casting of the girders. 4. Post-tensioning of the short continuity crown tendons, removal of the temporary tower at midspan, prestressing of the girders. 5. Casting of the arch fill and the spandrel wall. 6. Erection of the girders.

7. Casting of the diaphragms and composite slab. 8. Removal of temporary girder support at midspan and above spandrel walls. 9. Placement of sidewalks and railing. 10. Design service life of 30 years. 11. Removal of the composite slab, sidewalks, and railing. 12. Placement of the composite slab and loading of the structure by the new sidewalks and railing. 13. Design service life of 60 years.

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Fig. 7. Wisconsin

Avenue Viaduct: Progressive erection

of the bridge. (Photograph by

CH2M Hill.)

Fig. 8. Wisconsin Avenue Viaduct: Casting of the arch fill.

Fig. 9. Wisconsin Avenue Viaduct: Erection of precast girders.

The deck was modeled as a series of parallel elements that represented precast girders (PGc), diaphragms, composite slabs (CSc), girder rein-forcing bars (PGs), slab reinforcing bars (CSs), and formwork. The girder prestressing steel (PGp) was modeled by elements with a linear course of the eccentricity. The reference axis for all deck elements was the centroid of the girders.

The arch was modeled as a series of parallel elements representing precast arch segments (ASc), cast-in-place arch fill concrete (AFc), reinforcing bars for the arch segments (ASs), rein-forcing bars for the concrete fill (AFs), prestressing steel for the arch seg-ments (ASp), and continuity tendons (CT). The reference axis for all arch elements was the centroid of the arch segments.

The spandrel wall was modeled by vertical elements with the refer-ence axis situated in the centroid of the wall. The connections of the deck with the spandrel wall or the arch were modeled by very stiff members.

The time-dependent analysis followed the design of the placement of all mem-bers. The structure was analyzed in 70 time nodes. Changes to the structural system or loading were done in ten con-struction steps. After 30 years, the com-posite deck slab was replaced in three additional steps (see Fig. 6).

July-August 2001 63

Deformations and internal forces in all elements and in all time nodes were obtained through the analysis. The analysis allowed checking of the stresses and designing of the connec-tions between the precast and cast-in-place members. Fig. 11 shows the bending moments in the structure dur-

ing construction and replacement of the deck slab.

Arch Bridge crossing the Brno-Vienna Expressway

A new 67.50 m (221 ft) span steel-tube arch bridge carries local road traf-fic across the new Brno-Wien Freeway

Fig. 10. Wisconsin Avenue Viaduct: Modeling of the structure.

(a) Elevation (b) Cross section

Fig. 11. Wisconsin Avenue Viaduct: Bending moments in the structure at 13 erection and replacement steps.

close to Rajhrad, a small city in the Czech Republic.11 The height of the crossing, 13 m (43 ft) above the free-way, called for an arch structure. In evaluating the skew angle of the cross-ing, we determined that using only one arch as the loadbearing member would be the best architectural solution, and would prove to be very economical as well (see Fig. 12).

The 74.75 m (245 ft) radius circu-lar arch is formed by a single steel tube with a diameter of 900 mm (35 in.) and a thickness of 30 mm (1.2 in.) concrete infill. Internally, the steel tube was stiffened by diaphragms at 2.0 m (6.6 ft) on center. The arch was fixed in concrete foundations on each side of the freeway (see Fig. 13). The arch supports a slender trough-shaped cast-in-place concrete deck using edge girders in the shape of New Jersey bar-riers, which serve as stiffening girders as well as safety barriers.

The deck is post-tensioned by ten-dons situated at the edge girders. Ad-ditional straight tendons are situated in the deck slab. Four tendons make a loop above each of the outermost

64 PCI JOURNAL

steel support struts. The deck was con-nected to the arch by steel struts at 6.0 m (19.3 ft) on center located perpen-dicular to the axis of the arch.

These steel struts, which have a small box cross section and are con-nected to the stiffening diaphragms of the steel arch tube, are filled with concrete. These struts were triangular in shape to maintain the stability of the arch in both the vertical and transverse directions. The width of the triangle is kept constant, but the length varies. In the middle of the bridge, the arch has a fixed connection at the deck.

The first and last side spans, which are relatively long, are supported by inclined cast-in-place concrete struts that are pin connected with the deck and with the concrete foundation of the arch. These concrete struts, which are arranged directly under the edge girders, transfer the load directly from the edge girders to the foundations and thus ensure the stability of the struc-ture in the transverse direction.

The steel arch was erected from 12 m (39 ft) steel segments (see Fig. 14a). First, heavy spring segments with base plates and stiffeners were erected. Then, typical segments were erected from the footing to midspan. Erection towers supported the arch segments.

The position of the segments was carefully adjusted and the arch seg-ments were temporarily bolted. The segments were connected by full pen-etration welds. The arches were tem-porarily fixed into the footings and erection towers were removed. The two remaining towers did not support

Fig. 12. Arch Bridge Crossing the Brno-Vienna Expressway: Completed structure.

Fig. 13. Arch Bridge Crossing the Brno-Vienna Expressway: Structural arrangement.

the structure; they only stabilized the arch in the transverse direction.

After bracing the triangular-shaped steel struts with temporary bracing (see Fig. 14b), they were welded to the arch. After the erection of the steel members, the concrete fill of the struts and arch was cast (see Fig. 14c). Con-crete was pumped from the bottom of the arch to the top. To prevent air voids in the concrete, three openings with tubes were provided at the top of the arch.

The short tube in the center had a closure that was opened during the casting; the side tubes had a height of 3.0 m (9.8 ft). After the concrete reached the top of the central tube, the arch was closed. The concrete was pumped until it reached the top of the side tubes. The hydraulic pressure of

3.0 m (9.8 ft) high columns of con-crete guaranteed the compactness of the concrete.

After seven days, the top tubes were removed and falsework was installed for casting of the deck. The deck was placed as a single unit from one end cross beam to another (see Fig. 14d). After the concrete reached sufficient strength, the deck and end struts were post-tensioned. Hydraulic jacks were used to press the arch against the foun-dation to reduce the short-term defor-mation of the foundation and to main-tain prescribed forces and moments in the arch springs (see Fig. 14e).

This operation was repeated after one week. The joint between the con-crete foundation and the steel base plate was filled with epoxy concrete. Then, reinforcing steel and concrete

July-August 2001 65

were placed around the arch springs to protect the steel in the soil. The align-ment of the freeway was finished and the behavior of the bridge was verified by static and dynamic loading tests (see Fig. 14f).

The structure forms a system in which all structural members con-tribute to the resistance of the exter-nal load. It was analyzed as a three- dimensional structure assembled from brick, shell, tube, and beam elements.12 The time-dependent analysis was done on the model formed by a two-dimen-sional frame (see Fig. 15). The steel arch with its concrete fill and the steel struts with their concrete fill were rep-resented by parallel elements.

Parallel members modeled the con-crete deck and reinforcing steel of the deck and struts. Prestressing was rep-resented by straight members with the arrangement that corresponded to the layout of the tendons. A time-depen-dent analysis was performed to simu-late the erection procedure.

The analysis proved the advantage of using a static function of a steel tube filled with concrete. The concrete inside the tube cannot dry because it is encased in the steel tube. Therefore, the effects of creep and shrinkage are reduced and the portion of the stresses Fig. 14. Arch Bridge Crossing the Brno-Vienna Expressway: Construction sequence.

Fig. 15. Arch Bridge Crossing the Brno-Vienna Expressway: Calculation model for the time-dependent analysis.

(a) Elevation(b) Partial elevation(c) Cross section

66 PCI JOURNAL

that are resisted by steel or concrete is nearly constant over time. Usually, the redistribution of forces between concrete and steel is expressed by a change in the coefficient of the modu-lar ratio of elasticity n = ES/EC. In our case, n varied from 5.45 at time t0 to 8.20 at time t∞.

Detailed static and dynamic load-ing tests verified the static function and quality of the workmanship. The bridge was well accepted by the owner, engineers and by the public. The design of the bridge has since received awards from the Czech Steel Society, Czech Concrete Society, and Czech Ministry of Transportation.

cable-Stayed Bridge Across the Svratka River

A relatively small, 50 m (164 ft) span cable-stayed bridge recently re-placed an old steel truss bridge cross-ing the Svratka River at Zidlochovice, Czech Republic. Adjacent roads left little space on the banks of the river for the new bridge; therefore, the deck is suspended on one side by an in-clined pylon (see Fig. 16).

The deck of the new bridge, 18.60 m (61.0 ft) wide and only 700 mm (28 in.) deep, is formed by two longitudi-nal precast edge girders and transverse solid slab members connected by lon-gitudinal and transverse post-tension-ing (see Fig. 17). The stays are formed by cables of 18 - 0.6 in. (15 mm) diam-eter strands grouted in the steel tubes.

The cables are arranged in two planes and anchored in the longitu-dinal edge beams. In the main span, the cables are arranged as a semi-fan; in the side span they are parallel and anchored in the abutment. The pylon is formed by two steel columns filled with concrete and is fixed into the abutment.

The longitudinal girders, 1.40 m (4.6 ft) wide and 900 mm (35 in.) deep, were assembled from 5.00 m (16.4 ft) long match-cast segments. Contact joints were filled with epoxy resin. At the midspan of the girders, the stay an-chors are located in the pockets.

The girders were prestressed by four post-tensioning bars positioned in the corners of the cross section and by longitudinal tendons comprising 12 - 0.6 in. (15 mm) diameter strands.

Fig. 16. Cable-stayed bridge across the Svratka River: Completed structure.

Fig. 17. Cable-stayed bridge across the Svratka River.

(a) Cross section of the deck(b) Elevation(c) Cross section of the bridge

Fig. 18. Cable-stayed bridge across the Svratka River.

(a) Partial cross section(b) Normal stresses in the deck due to live load(c) Prestressing of the deck(d) Resultant stresses

July-August 2001 67

The cable layout corresponds to the course of the bending moments due to the dead load that originated in a con-tinuous beam with supports situated at stay anchors.

The transverse solid members are 1.00 m (3.3 ft) wide and have variable depths from 400 mm (16 in.) at the ends to 500 mm (20 in.) at their midspan. The transverse members were post-ten-sioned by 6 - 0.6 in. (15 mm) diameter strands. The concrete joint between the transverse members was post-tensioned by post-tensioning bars. The transverse members were connected by reinforcing bar loops in the concrete joints and by longitudinal post-tensioning.

The longitudinal girders were de-signed as fully prestressed members with a minimum compression of 1.0 MPa (145 psi) in the joints between them. The deck between the girders was designed as a partially prestressed structure in both the longitudinal and transverse directions. The behavior of the partially prestressed joints between the transverse members was verified by tests of a full-scale model of the deck.

The design of the bridge addressed two special problems: eliminating tor-sion in the longitudinal girders dur-ing the erection of the structure, and determining the level of the post-ten-sioning of the longitudinal girders and transverse members in such a way that

a redistribution of stresses between them is minimal. Erecting eccentric transverse post-tensioning of the joints between longitudinal and transverse members solved the first problem (see Fig. 18a).

The transverse members were pro-vided with steel brackets with nuts and screws located on the surface close to their ends. After a transverse member was erected, the screws were drawn until their heads touched the longi-tudinal girders. Then, the post-ten-sioning bars were partially tensioned. The force couple acting on the girder (under the screw’s head and the bar’s anchor) created a moment that bal-anced the torsion.

The geometry of the deck caused larger live load stresses in the longi-tudinal edge girders than in the deck between them (see Fig. 18b). There-fore, the erection procedure and post-tensioning of the deck was designed in such a way that the prestress of the longitudinal girders is larger then the prestress of the transverse members.

Since the joints between the trans-verse members were cast after the erection of the deck, the main com-pression caused by stay cables loads only the edge girders. The deck was successively erected in 5 m (16 ft) long parts corresponding to the length of the girder’s segments.

The assembly process was carried out as follows (see Fig. 19):

1. The edge longitudinal girders were erected; each new girder was post-tensioned to the same force as the previously erected ones (see Fig. 19a).

2. Stay cables were installed and tensioned to the prescribed level (see Fig. 19b).

3. Transverse members were erected and the joints between the transverse members and the longitudinal girders were post-tensioned (see Fig. 19c).

4. The forces in the stay cables were adjusted (see Fig. 19d).

After the erection of all members, the longitudinal and transverse joints between precast members were placed and the deck was post-tensioned by straight tendons. Due to the different levels of the prestressing, correspond-ing long-term creep is different. The shrinkage of concrete of the precast members and cast-in-place joints be-tween the transverse members is also different.

To eliminate the redistribution of stresses between the edge longitudinal girders and the deck between them, a parametric study was performed to analyze the effects of creep and shrinkage (see Fig. 20). The bridge deck, pylons, and stays were modeled as parallel members that represented longitudinal girders, transverse mem-

Fig. 19. Cable-stayed bridge across the Svratka River: Erection of the deck.

68 PCI JOURNAL

bers and joint between them, steel col-umns, and their concrete fill and stay cables. Prestressing was represented by straight members with the arrange-ment that corresponded to the layout of the tendons.

Based on the results of the analysis, the erection sequences were designed and executed. Since its completion in 1992, the bridge has performed ex-tremely well, and neither cracking nor any other types of damage has been observed.

Precast Segmental Structure with Replaceable cIP Deck Slab

A precast segmental structure with a replaceable cast-in-place deck slab was used in the design of a new via-duct in Plzen, Czech Republic. The design combines both precast and cast-in-place techniques. Precast seg-ments allow efficient erection without falsework (see Fig. 21), while a cast-in-place deck ensures the integrity of the bridge and creates an additional barrier against corrosion of the pre-stressing steel.

The viaduct is a part of a complex structure that includes an arch bridge

Fig. 20. Cable-stayed bridge across the Svratka River: Time-dependent analysis.

(a) Calculation model(b) Modeling of the deck

(c) Longitudinal section at the bridge axis(d) Detail “A”

Fig. 21. Precast segmental structure with replaceable CIP deck slab: Erected structure.

across the Radbuza River, a parking structure, and ramps connecting the arch bridge with the roof of the garage. The viaduct carries through traffic from the arch bridge over the building to the junction with ‘U Trati’ Street.

The continuous structure is formed by seven spans having lengths of 34.0 + (3 x 45.0) + (2 x 42.0) + 34.0 = 290.6

m (953 ft). The deck is formed by a three-cell box girder without traditional overhangs (see Fig. 22). The girder is supported by two pot bearings situated on single supports formed by elliptical piers with a hammerhead.

The 2.20 m (7.2 ft) deep box girder was assembled of precast match-cast segments of an open cross section and

July-August 2001 69

a cast-in-place deck slab. Typical seg-ments were formed by a curved bot-tom slab and two webs. Transverse ribs at midspan stiffen the segments during production and erection.

To protect the prestressing steel, no ducts were located in the top deck slab. The arrangement of the prestress-ing tendons was developed following the method for erecting the deck.

The open cross section with a wide bottom slab was well suited for the cantilever construction because the compression flange and prestressing force arm were designed to resist the negative bending moments. The bal-anced cantilevers originated above the piers (see Fig. 23).

During erection, the segments were post-tensioned by tendons inside the webs. After erecting the neighboring cantilevers, a midspan joint was cast, and the erected structure was post-ten-sioned by eight external tendons (see Fig. 24).

External tendons were anchored at the pier diaphragms and were deviated at blisters in the bottom corners of the box girder. Each tendon was situated in two spans and post-tensioned from only one side. After post-tensioning of the midspan joint, the top slab was cast. The slab was placed and post-tensioned in two steps.

First, the whole span without the

Fig. 22. Precast segmental structure with replaceable CIP deck slab: Cross section of the bridge.

Fig. 23. Precast segmental structure with replaceable CIP deck slab: Erection of segments.

part situated above the piers was placed and post-tensioned by two ex-ternal tendons (see Fig. 25a). After the remaining part of the slab was placed, the structure was post-tensioned (see Figs. 25b and 26). This procedure lim-ited the amount of work done inside the shallow box girder to checking of

the dead anchors. The structural solution of this bridge

was based on a very detailed static analysis of the space structure that was assembled from solid brick ele-ments. A time-dependent analysis was performed as well. The structure was modeled by a series of parallel mem-

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Fig. 24. Precast segmental structure with replaceable CIP deck slab: External cables.

Fig. 25. Precast segmental structure with replaceable CIP deck slab: Casting of the deck slab and post-tensioning of the external tendons.

Fig. 26. Precast segmental structure with replaceable CIP deck slab: Casting of the deck slab.

July-August 2001 71

bers that represented precast segments, a cast-in-place deck slab, reinforcing bars, and internal and external ten-dons. Analyses were done for a design schedule of the erection and checked for the actual procedure.

The CEB-FIP 90 Model Code was used for the creep and shrinkage analysis. Results are presented in Fig. 27, which shows contours of normal stresses at different erection and ser-

vice stages at sections above the pier and at midspan. Since the deck slab was cast and post-tensioned after the erection of the segments, it can be eas-ily replaced in the future.13

The viaduct was erected within one year without any significant technical problems. At present a cable-stayed structure using a similar cross section and technology is being designed.

Fig. 27. Precast segmental structure with replaceable CIP deck slab: Redistribution of normal stresses.

(a) Section above pier(b) Section at midspan

1 - At completion of the cantilever2 - After casting of the deck3 - Open to service4 - After three years5 - After 50 years

Fig. 28. Stress-ribbon bridge across the Sacramento River in Redding, California: Completed structure.

STRESS-RIBBoN BRIDGES The term “stress-ribbon” bridge de-

scribes a structure formed by a very slender concrete deck with the shape of a catenary. It can be designed with one or more spans and is character-ized by successive and complemen-tary smooth curves (see Fig. 28). Such structures can be constructed using precast concrete units or can be cast-

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in-place. Precast construction is usu-ally more economical.

For precast concrete structures, the deck is assembled from precast seg-ments that are suspended on bearing cables and shifted along them to their

final position (see Fig. 29). Prestress-ing is applied after casting the joints between the segments to ensure suf-ficient rigidity of the structure.

The main advantage of these struc-tures is that they have minimal effect

Fig. 29. Stress-ribbon bridge across the Sacramento River in Redding, California: Erection of a segment.

Fig. 30. Stress-ribbon bridges: Cross sections.

(a) Bridge Across the Sacramento River in Redding, California(b) Bridge Across the Rough River in Grants Pass, Oregon

on the environment because they use very little material and can be erected independently from the existing ter-rain. Moreover, because they do not need bearings or expansion joints, they need little long-term mainte-nance.

With their slender concrete decks, their stiffness and stability come from their geometry. They are able to resist not only uniformly distributed loads, but also large concentrated loads im-posed by heavy trucks.

Although stress-ribbon structures have low natural frequencies, our ex-perience has shown that the speed of motion of the deck created by walk-ing is within acceptable limits. Also, our detailed dynamic testing has con-firmed that these structures are safe from vandalism.

The first author has designed seven stress-ribbon bridges that have been built in the Czech Republic.14 He also participated in the design of the first stress-ribbon structure with a span of 127.00 m (417 ft) that was built across the Sacramento River in Redding, Cal-ifornia.15 Recently, he participated in the design of five stress-ribbon bridges that are now under construction in the United States and United Kingdom.16,17

The structural arrangement of the stress-ribbon bridge is determined by its static function and the construction process. During erection, the structure acts as a perfectly flexible cable; in service it acts as a stress ribbon that is stressed by normal forces and bending moments. The shape and the stress of the structure at the end of the erection determine the stresses that originate in the structure when in use.

Because significant bending mo-ments originate only at the supports, the deck of the stress ribbon can be formed by a very slender solid section that can be further reduced by waffles. The deck of stress-ribbon structures can be placed in formwork suspended on bearing cables or can be assembled from precast segments.

In our early structures, the segments were hung on bearing cables located in the troughs. After erection, the deck was post-tensioned by the second group of cables located in the ducts within the segments or in the troughs themselves (see Fig. 30a).

July-August 2001 73

The bearing cables were protected by cast-in-place concrete that was placed simultaneously with the joints between the segments. The longitu-dinal joint between the cast-in-place and precast concrete, where shrinkage cracks may have occurred, was cov-ered by an overlay.

Note that a similar approach was used in the design of the East/West Golf Cart Stressed-Ribbon Bridges at Rancho Santa Fe, California, which received a 1999 PCI Design Award for Best Non-Highway Bridge.17

In two of our latest designs – the Bridge Across the Blue River in Colo-rado and Bridge Across the Rough River in Grants Pass, Oregon – precast segments with a composite slab form the deck (see Fig. 30b). The segments were suspended on bearing cables and served as falsework and formwork for placing the composite slab, which oc-curred simultaneously with the joints between the segments.

Both the precast segments and the composite slab were post-tensioned by prestressing tendons that were situated together with the bearing cables within the cast-in-place slab. A continuous jointless deck slab provides protection to prestressing steel and requires mini-mum maintenance.

A typical section of the stress rib-bon is not able to resist the bending moments that originate at the supports (see Fig. 31a). There are two options for solving this problem. The deck can be supported by saddles (pier tables) from which it can rise after its post-tensioning or when the temperature drops, or short cast-in-place haunches have to be designed at the supports (see Fig. 32).

Although in the second option the bending moments are larger than in the structure of the constant depth, the resulting stresses are smaller. The haunches that are designed as partially post-tensioned members have to be sufficiently reinforced by reinforcing bars located close to their bottom fi-bers.

Today’s computer programs for nonlinear analysis allow expression of the static function of the stress-ribbon structures. The deck can be modeled by mutually connected parallel mem-bers that can express the function of

Fig. 31. Stress-ribbon bridges: Analysis of the structure.

(a) Geometry of the structure and bending moments in the deck due to the dead load and prestress at time to and t∞(b) Modeling of the deck

Fig. 32. Stress-ribbon bridges: Arrangement of the deck at supports and bending moment at the deck.

REFERENcES

74 PCI JOURNAL

the prestressing cable bearing, precast deck, and cast-in-place concrete (see Fig. 31b). In the initial stage, the forces in the bearing cables are in equilibrium with the self-weight of the deck.

Due to creep and shrinkage of con-crete, the sag of the structure decreases over time and the tensile forces in the deck increase. Also, due to the redistribution of forces between the cables and the concrete (post-tension-ing losses), tensile forces in the deck increase with time.

Since the ratio of the area of the pre-stressing steel to the concrete area is

larger than in conventional prestressed concrete structures, these losses have greater value and, therefore, have to be carefully analyzed. The time-de-pendent analysis has confirmed that bending moments in the deck due to dead load and prestress do not change significantly over time (see Fig. 32).

coNcLUDING REMARKS

Time-dependent analysis has proved to be a very useful tool for developing new types of progressively erected hy-

brid structures. The examples provided in this paper demonstrate how modern methods of construction can be used in a large variety of bridge projects.

The featured structures combine the inherent high quality of precast/pre-stressed concrete with steel and cast-in-place concrete, and incorporate both stay and external cable technology. The designs demonstrate how the strength and endurance qualities of concrete en-able the construction of beautiful light structures, with economy and minimal environmental impact.

cREDITS

1. Virlogeux, M., Association of Prestressed Concrete and Steel for Bridge Construction, Contribution of French Group, IABSE Symposium, St. Petersburg, Russia, 1991.

2. Mantreola, J., Fernades, Trojano, L., Astiz, M. A., and Mar-tinez, Cutillas, A., “Concrete as a Means for Innovation in Bridge Design,” Structural Concrete: The Bridge Between People, fib Symposium, Prague, Czech Republic, 1999.

3. Arenas, J. J., “Urban Bridges: A Civilised Civil Engineering,” Structural Concrete: The Bridge Between People, fib Sympo-sium, Prague, Czech Republic, 1999.

4. Development of Technology for Expressway Bridges, Japan Highway Public Corporation, Tokyo, Japan, 1998.

5. Navratil, J., “Time-Dependent Analysis of Concrete Frame Structures” (in Czech), Building Research Journal (Staveb-nicky Casopis), V. 40, No. 7, 1992.

6. ESA PrimaWin, Reference Manual, SCIA Software, Scientific Application Group, Belgium, 2000.

7. Strasky, J., “Progressive Assembly of the Deck of Cable-Stayed Bridges,” Symposium on Modern Prestressing Tech-niques and their Application, Kyoto, Japan, 1993.

8. Strasky, J., “Design and Construction of Cable-Stayed Bridges in the Czech Republic,” PCI JOURNAL, V. 38, No. 6, Novem-ber-December 1993, pp. 24-43.

9. Wanders, S. P., Maday, M. A., Redfield, C., and Strasky, J., “Wis-consin Avenue Viaduct – Design-Construction Highlights,” PCI JOURNAL, V. 39, No. 5, September-October 1994, pp. 20-34.

10. Wanders, S. P., Maday, M. A., Redfield, C., and Strasky, J., “Wisconsin Avenue Viaduct in Milwaukee, USA,” L’ Industria Italiana del Cemento, No. 707, Rome, Italy, 1996.

11. Strasky, J., “Arch Bridge Crossing the Brno-Vienna Express-way,” Proceedings of the Institution of Civil Engineers, Civil Engineering, London, England, November 1999.

12. Strasky, J., “Arch Bridge Crossing the Brno-Vienna Express-way,” International Bridge Conference, Pittsburgh, PA, 1998.

13. Strasky, J., “Segmental Structure with Replaceable CIP Deck Slab,” International Bridge Conference, Pittsburgh, PA, 2000.

14. Strasky, J., “Precast Stress-Ribbon Pedestrian Bridges in Czecho-slovakia,” PCI JOURNAL, V. 32, No. 3, May-June 1987, pp. 52-73.

15. Redfield, C., and Strasky, J., “Stressed Ribbon Pedestrian Bridge Across the Sacramento River in Redding, California, USA,” L’ Industria Italiana del Cemento, No. 663, Rome, Italy, 1992.

16. Strasky, J., “Stress-Ribbon Pedestrian Bridges,” International Bridge Conference, Pittsburgh, PA, 1999.

17. “Bridge Winners Showcase Design, Cost Efficiency,” Ascent, Precast/Prestressed Concrete Institute, Fall 1999, pp. 29-35.

• The Wisconsin Avenue Viaduct was designed by CH2M Hill, Milwaukee, Wisconsin, in collaboration with Charles Redfield and Jiri Strasky.

• The arch bridge crossing the Brno-Vienna Expressway, the cable-stayed bridge across the Svratka River, and the precast segmental structure with replaceable CIP deck slab were designed by Strasky, Husty and Partners, Brno, Czech Republic.

• East/West Golf Cart Stressed-Ribbon Bridges, Rancho Santa Fe, California, were designed by T.Y. Lin Inter-national-McDaniel, San Diego, in collaboration with Jiri Strasky.

• The Bridge Across the Blue River, Colorado, was de-signed by Jim Nolen, Aurora, Colorado, in collaboration with Charles Redfield and Jiri Strasky.

• The Bridge Across the Rough River, Grants Pass, Or-egon, was designed by OBEC, Consulting Engineers, Eu-gene, Oregon, and by Jiri Strasky.

The development of the segmental structure with replace-able CIP deck slab was carried out with the support of a grant from the Czech Ministry of Industry, FB-CV/69/98. The theoretical analyses of the stress-ribbon structures were supported by the Grant Agency of the Czech Republic, GACR 103/96/1635.