Bridge structures Kết cấu thép và bê tông

Toma, S.; Duan, L. and Chen, W.F. “Bridge Structures”
Structural Engineering Handbook
Ed. Chen Wai-Fah
Boca Raton: CRC Press LLC, 1999
BridgeStructures
ShoujiToma
DepartmentofCivilEngineering,
Hokkai-GakuenUniversity,Sapporo,Japan
LianDuan
DivisionofStructures,California
DepartmentofTransportation,Sacramento,
CA
Wai-FahChen
SchoolofCivilEngineering,
PurdueUniversity,
WestLafayette,IN
10.1General
10.2SteelBridges
10.3ConcreteBridges
10.4ConcreteSubstructures
10.5FloorSystem
10.6Bearings,ExpansionJoints,andRailings
10.7GirderBridges
10.8TrussBridges
10.9RigidFrameBridges(RahmenBridges)
10.10ArchBridges
10.11Cable-StayedBridges
10.12SuspensionBridges
10.13DefiningTerms
Acknowledgment
References

FurtherReading
Appendix:DesignExamples
10.1 General
10.1.1 Introduction
Abridgeisastructurethatcrossesoverariver,bay,orotherobstruction,permittingthesmoothand
safepassageofvehicles,trains,andpedestrians.Anelevationviewofatypicalbridgeisshownin
Figure10.1.Abridgestructureisdividedintoanupperpart(thesuperstructure),whichconsistsof
theslab,thefloorsystem,andthemaintrussorgirders,andalowerpart(thesubstructure),whichare
columns,piers,towers,footings,piles,andabutments.Thesuperstructureprovideshorizontalspans
suchasdeckandgirdersandcarriestrafficloadsdirectly.Thesubstructuresupportsthehorizontal
spans,elevatingabovethegroundsurface.Inthischapter,mainstructuralfeaturesofcommon
typesofsteelandconcretebridgesarediscussed.Twodesignexamples,atwo-spancontinuous,
cast-in-place,prestressedconcreteboxgirderbridgeandathree-spancontinuous,compositeplate
girderbridge,aregivenintheAppendix.
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FIGURE 10.1: Elevation view of a typical bridge.
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10.1.2 Classification
1. Classification by Materials
Steel bridges: A steel bridge may use a wide variet y of structural steel components
and systems: girders, frames, trusses, arches, and suspension cables.
Concrete bridges: There are two primary types of concrete bridges: reinforced and
prestressed.
Timber bridges: Wooden bridges are used when the span is relatively short.
Metal alloy bridges: Metal alloys such as aluminum alloy and stainless steel are also
used in bridge construction.

2. Classification by Objectives
Highway bridges: bridges on highways.
Railway bridges: bridges on railroads.
Combined bridges: bridges carrying vehicles and trains.
Pedestrian bridges: bridges carrying pedestrian traffic.
Aqueduct bridges: br idges supporting pipes with channeled waterflow.
Bridges can alternatively be classified into movable (for ships to pass the river) or fixed
and permanent or temporary categories.
3. Classification by Structural System (Superstructures)
Plate girder bridges: The main girders consist of a plate assemblage of upper and
lower flanges and a web. H- or I-cross-sections effectively resist bending and shear.
Box girder bridges: The single (or multiple) main girder consists of a box beam
fabricated from steel plates or formed from concrete, which resists notonly bending
and shear but also torsion effectively.
T-beam br idges: A number of reinforced concrete T-beams are placed side by side
to support the live load.
Composite girder bridges: Theconcrete deck slab works in conjunction with the steel
girders to support loads as a united beam. The steel girder takes mainly tension,
while the concrete slab takes the compression component of the bending moment.
Grillage girder bridges: The main girders are connected transversely by floor beams
to form a grid pattern which shares the loads with the main girders.
Truss bridges: Truss bar members are theoretically considered to be connected with
pins at their ends to form triangles. Each member resists an axial force, either
in compression or tension. Figure 10.1 shows a Warren truss bridge with vertical
members, which is a “trough bridge”, i.e., the deck slab passes through the lower
part of the bridge. Figure 10.2 shows a comparison of the four design alternatives
evaluated for Minato Oh-Hasshi in Osaka, Japan. The truss frame design was
selected.
Arch br idges: The arch is a structure that resists load mainly in axial compression.
In ancient times stone was the most common material used to construct magnif-

icent arch bridges. There is a wide variety of arch bridges as will be discussed in
Section 10.10
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FIGURE 10.2: Design comparison for Minato Oh-Hashi, Japan. (From Hanshin Expressway Public
Corporation, Construction Records of Minato Oh-Hashi, Japan Societ y of Civil Engineers, Tokyo [in
Japanese], 1975. With permission.)
Cable-stayed bridges: The girders are supported by highly strengthened cables (often
composed of tightly bound steel strands) which stem directly from the tower. These
are most suited to bridge long distances.
Suspension bridges: The girders are suspended by hangers tied to the main cables
which hang from the towers. The load is transmitted mainly by tension in cable.
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This design is suitable for very long span bridges.
Table 10.1 shows the span lengths appropriate to each type of bridge.
4. Classification by Support Condition
Figure 10.3 shows three different support conditions for girder bridges.
Simply supported bridges: The main girders or trusses are supported by a movable
hinge at one end and a fixed hinge at the other (simple support); thus they can be
analyzed using only the conditions of equilibrium.
Continuously supported bridges: Girders or trusses are supported continuously by
more than three supports, resulting in a structurally indeterminate system. These
tend to be more economical since fewer expansion joints, which have a common
cause of service and maintenance problems, are needed. Sinkage at the supports
must be avoided.
Gerber bridges (cantilever bridge): A continuous bridge is rendered determinate
by placing intermediate hinges between the supports. Minato Oh-Hashi’s bridge,

shown in Figure 10.2a, is an example of a Gerber truss bridge.
10.1.3 Plan
Before the structural design of a bridge is considered, a bridge project will start with planning the
fundamental design conditions. A bridge plan must consider the following factors:
1. Passing Line and Location
A bridge, being a continuation of a road, does best to follow the line of the road. A right
angle bridge is easy to design and construct but often forces the line to be bent. A skewed
bridge or a curved bridge is commonly required for expressways or railroads where the
road line must be kept straight or curved, even at the cost of a more difficult design (see
Figure 10.4).
2. Width
The width of a highway bridge is usually defined as the width of the roadway plus that of
the sidewalk, and often the same dimension as that of the approaching road.
3. Type of St ructure and Span Length
The types of substructures and superstructures are determined by factors such as the
surrounding geographical features, the soil foundation, the passing line and its width, the
length and span of the bridge, aesthetics, the requirement for clearance below the bridge,
transportation of the construction materials and erection procedures, construction cost,
period, and so forth.
4. Aesthetics
A bridge is required not only to fulfill its function as a thoroughfare, but also to use its
structure and form to blend, harmonize, and enhance its surroundings.
10.1.4 Design
The bridge design includes selection of a bridge type, structural analysis and member design, and
preparation of detailed plans and drawings. The size of members that satisfy the requirements
of design codes are chosen [1, 17]. They must sustain prescribed loads. Structural analyses are
performed on a model of the bridge to ensure safety as well as to judge the economy of the design.
The final design is committed to drawings and g iven to contractors.
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TABLE 10.1 Types of Bridges and Applicable Span Lengths
From JASBC, Manual Design Data Book, Japan Association of Steel Bridge Construction, Tokyo (in Japanese), 1981. With permission.
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FIGURE 10.3: Supporting conditions.
FIGURE 10.4: Bridge lines.
10.1.5 Loads
Designers should consider the following loads in bridge design:
1. Primary loads exert constantly or continuously on the bridge.
Dead load: weight of the bridge.
Live load: vehicles, trains, or pedestrians, includingthe effect of impact. A vehicular
load is classified into three parts by AASHTO [1]: the truck axle load, a tandem
load, and a uniformly distributed lane load.
Other primary loads may be generated by prestressing forces, the creep of concrete, the
shrinkage of concrete, soil pressure, water pressure, buoyancy, snow, and centrifugal
actionsorwaves.
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2. Secondary loads occur at infrequent intervals.
Wind load: a typhoon or hurricane.
Earthquake load: especially critical in its effect on the substructure.
Other secondary loads come about with changes in temperature, acceleration, or tempo-
rary loads during erection, collision forces, and so forth.
10.1.6 Influence Lines
Since the live loads by definition move, the worst case scenario along the bridge must be determined.
The maximum live load bending moment and shear envelopes are calculated conveniently using
influence lines. The influence line graphically illustrates the maximum forces (bending moment and

shear), reactions, and deflections over a section of girder as a load travels along its length. Influence
lines for the bending moment and shear force of a simply supported beam are shown in Figure 10.5.
For a concentrated load, the bending moment or shear at section A can be calculated by multiplying
the load and the influence line scalar. For a uniformly distributed load, it is the product of the load
intensity and the net area of the corresponding influence line diagram.
10.2 Steel Bridges
10.2.1 Introduction
The main part of a steel bridge is made up of steel plates which compose main girders or frames
to support a concrete deck. Gas flame cutting is generally used to cut steel plates to designated
dimensions. Fabrication by welding is conducted in the shop where the bridge components are
prepared before being assembled (usually bolted) on the construction site. Several members for two
typical steel bridges, plate girder and truss bridges, are given in Figure 10.6. The composite plate
girder bridge in Figure 10.6a is a deck type while the truss bridge in Figure 10.6b is a through-deck
type.
Steel has higher strength, ductility, and toughness than many other structural materials such as
concrete or wood, and thus makes an economical design. However, steel must be painted to prevent
rusting and also stiffened to prevent a local buckling of thin members and plates.
10.2.2 Welding
Welding is the most effective means of connecting steel plates. The properties of steel change when
heated and this change is usually for the worse. Molten steel must be shielded from the air to prevent
oxidization. Welding can be categorized by the method of heating and the shielding procedure.
Shielded metal arc welding (SMAW), submerged arc welding (SAW), CO
2
gas metal arc welding
(GMAW), tungsten arc inert gas welding (TIG), metal arc inert gas welding (MIG), electric beam
welding, laser beam welding, and friction welding are common methods.
The first two welding procedures mentioned above, SMAW and SAW, are used extensively in bridge
construction due to their high efficiency. Both use an electric arc, which is generally considered the
mostefficient methodofapplyingheat. SMAWisdone byhand andissuitable forweldingcomplicated
joints but is less efficient than SAW. SAW is generally automated and can be very effective for welding

simple parts such as the connection between the flange and web of plate girders. A typical placement
of these welding methods is shown in Figure 10.7. TIG and MIG use an electric arc for heat source
and inert gas for shielding.
An electric beam weld must notbe exposed to air, and therefore must be laidin avacuum chamber.
A laser beam weld can be placed in air but is less versatile than other types of welding . It cannot be
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FIGURE 10.5: Influence lines.
used on thick plates but is ideal for minute or artistic work. Since the welding equipment necessary
for heating and shielding is not easy to handle on a construction site, all welds are usually laid in the
fabrication shop.
The heating and cooling processes during welding induce residual stresses to the connected parts.
The steel surfaces or parts of the cross section at some distance from the hot weld, cool first. When
the area close to the weld then cools, it tries to shrink but is restrained by the more solidified and
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FIGURE10.6: Membernames ofsteel bridges. (FromTachibana, Y. andNakai,H., BridgeEngineering,
Kyoritsu Publishing Co., Tokyo, Japan [in Japanese], 1996. With permission.)
cooler parts. Thus, tensile residual stresses are trapped in the vicinity of the weld while the outer
parts are put into compression.
There are two types of welded joints: groove and fillet welds (Figure 10.8). The fillet weld is placed
at the junction of two plates, often between a web and flange. It is a relatively simple procedure
with no machining required. The groove weld, also called a butt weld, is suitable for joints requiring
greater strength. Depending on the thickness of adjoiningplates, theedges are beveled in preparation
for the weld to allow the metal to fill the joint. Various groove weld geometries for full penetration
welding are shown in Figure 10.8b.
Inspection of welding is an important task since an imperfect weld may well have catastrophic
consequences. It is difficult to find faults such as an interior crack or a blow hole by observing only

the surface of a weld. Many nondestructive testingprocedures are available which use various devices,
such as x-ray, ultrasonic waves, color paint, or magnetic particles. These all have their own advantages
and disadvantages. For example, the x-ray and the ultrasonic tests are suitable for interior faults but
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FIGURE 10.7: Welding methods. (From Nagai, N., Bridge Engineering, Kyoritsu Publishing Co.,
Tokyo, Japan [in Japanese], 1994. With permission.)
require expensive equipment. Use of color paint or magnetic particles, on the other hand, is a cheap
alternative but only detects surface flaws. The x-ray and ultrasonic tests are used in common bridge
construction, but ultrasonic testing is becoming increasingly popular for both its “high tech” and its
economical features.
10.2.3 Bolting
Bolting doesnot require theskilled workmanshipneeded for welding, andis thus asimpler alternative.
It is applied to the connections worked on construction site. Some disadvantages, however, are
incurred: (1) splice plates are needed and the force transfer is indirect; (2) screwing-in of the bolts
creates noise; and (3) aesthetically bolts are less appealing. In special cases that need to avoid these
disadvantages, the welding may be used even for site connections.
There are three types of high-tensile strength-bolted connections: the slip-critical connection, the
bearing-type connection (Figure 10.9), and the tensile connection (Figure 10.10). The slip-critical
(friction) bolt is most commonly used in bridge construction as well as other steel structures because
it is simpler than a bearing-type bolt and more reliable than a tension bolt. The force is transferred by
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FIGURE 10.8: Types of welding joints. (From Tachibana, Y. and Nakai, H., Bridge Engineer ing,
Kyoritsu Publishing Co., Tokyo, Japan [in Japanese], 1996. With permission.)
the friction generated between the base plates and the splice plates. The friction resistance is induced
by the axial compression force in the bolts.
The bearing-type b olt transfers the force by bearing against the plate as well as making some use

of friction. The bearing-type bolt can transfer larger force than the friction bolts but is less forgiving
with respect to the clearance space often existing between the bolt and the plate. These require that
precise holes be drilled and at exact spacings. The force transfer mechanism for these connections is
shown in Figure 10.9. In the beam-to-column connection shown in Figure 10.10, the bolts attached
to the column are tension bolts while the bolts on the beam are slip-critical bolts.
The tension bolt transfers forceinthe direction of the bolt axis. Thetension type of bolt connection
is easy to connect on site, but difficulties arise in distributing forces equally to each bolt, resulting
in reduced reliability. Tension bolts may also be used to connect box members of the towers of
suspension bridges where compression forces are larger than the tension forces. In this case, the
compression is shared with butting surfaces of the plates and the tension is carried by the bolts.
10.2.4 Fabrication in Shop
Steel bridges are fabricated into members in the shop yard and then transported to the construction
site for assembly. Ideally all constructional work would be completed in the shop to get the highest
quality in the minimum construction time. The larger and longer the members can be, the better,
within the restr ictions set by transportation limits and erection tolerances. When crane ships for
erection and barges for transportation can be used, one block can weigh as much as a thousand tons
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FIGURE 10.9: Slip-critical and bearing-type connections. (From Nagai, N., Bridge Engineering,
Kyoritsu Publishing Co., Tokyo, Japan [in Japanese], 1994. With permission.)
and be erected as a whole on the quay. In these cases the bridge is made of a single continuous block
and much of the hassle usually associated with assembly and erection is avoided.
10.2.5 Construction on Site
The designer must consider the loads that occur during construction, generally different from those
occurring after completion. Steel bridges are particularly prone to buckling during construction.
The erection planmust be made prior to the main design and must bechecked for e very possible load
case that may arise during erection, not only for strength but also for stability. Truck crane and bent
erection (or staging erection); launching erection; cable erection; cantilever erection; and large block
erection (or floating crane erection) are several techniques (see Figure 10.11). An example of the

large block erection is shown in Figure 10.43, in which a 186-m, 4500-ton center block is transported
by barge and lifted.
10.2.6 Painting
Steel must be painted to protect it from rusting. There is a wide variety of paints, and the life of a
steel structure is largely influenced by its quality. In areas near the sea, the salty air is particularly
harmful to exposed steel. The cost of painting is high but is essential to the continued good condition
of the bridge. The color of the paint is also an important consideration in terms of its public appeal
or aesthetic quality.
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FIGURE 10.10: Tension-type connection.
10.3 Concrete Bridges
10.3.1 Introduction
For modern bridges, both structural concrete and steel g ive satisfactory performance. The choice
between thetwo materialsdepends mainly uponthe costofconstruction andmaintenance. Generally,
concrete structures require less maintenance than steel structures, but since the relative cost of steel
and concrete is different from country to country, and may e ven vary throughout different parts of
the same country, it is impossible to put one definitively above the other in terms of “economy”.
In this section, the main features of common types of concrete bridge superstructures are briefly
discussed. Concrete bridge substructures will be discussed in Section 10.4. A design example of a
two-span continuous, cast-in-place, prestressed concrete box girder bridge is given in the Appendix.
For a more detailed look at design procedures for concrete bridges, reference should be made to the
recent books of Gerwick [7], Troitsky [24], Xanthakos [26, 27], and Tonias [23].
10.3.2 Reinforced Concrete Bridges
Figure 10.12 shows the t ypical reinforced concrete sections commonly used in highway bridge su-
perstructures.
1. Slab
A reinforced concrete slab (Figure 10.12a) is the most economical bridge superstructure
for spans of up to approximately 40 ft (12.2 m). The slab has simple details and standard

formwork and is neat, simple, and pleasing in appearance. Common spans range from
16 to 44 ft (4.9 to 13.4 m) w ith st ructural depth-to-span ratios of 0.06 for simple spans
and 0.045 for continuous spans.
2. T-Beam (Deck Girder)
The T-beams (Figure 10.12b) are generally economic for spans of 40 to 60 ft (12.2 to 18.3
m), but do require complicated formwork, particularly for skewed bridges. Structural
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FIGURE 10.11: Erections methods. (From Japan Construction Mechanization Association, Cost
Estimation of Bridge Erection, Tokyo, Japan [in Japanese], 1991. With permission.)
depth-to-span ratios are 0.07 for simple spans and 0.065 for continuous spans. The
spacing of girders in a T-beam bridge depends on the overall width of the bridge, the
slab thickness, and the cost of the formwork and may be taken as 1.5 times the structural
depth. The most commonly used spacings are between 6 and 10 ft (1.8 to 3.1 m).
3. Cast-in-Place Box Girder
Box girders like the one shown in Figure 10.12c, are often used for spans of 50 to 120 ft
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FIGURE 10.12: Typical reinforced concrete sections in bridge superstructures.
(15.2 to 36.6 m). Its for mwork for skewed structures is simpler than that required for the
T-beam. Dueto excessive dead load deflections, the useof reinforced concrete box girders
over simple spans of 100 ft (30.5 m) or more may not be economical. The depth-to-span
ratios are typically 0.06 for simple spans and 0.055 for continuous spans with the girders
spaced at 1.5 times the structural depth. The high torsional resistance of the box girder
makes it particularly suitable for curved alignments, such as the ramps onto freeways. Its
smooth flowing lines are appealing in metropolitan cities.
4. Design Consideration
A reinforced concrete highway bridge should be designed to satisfy the specification or

code requirements, such as theAASHTO-LRFD [1] requirements (American Association
of State Highway and Transportation Officials—Load and Resistance Factor Design) for
all appropriate service, fatigue, strength, and extreme event limit states. In the AASHTO-
LRFD [1], service limit states include cracking and deformation effects, and strength
limit states consider the strength and stability of a structure. A bridge structure is usually
designed for the strength limit states and is then checked against the appropriate service
and extreme event limit states.
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10.3.3 Prestressed Concrete Bridges
Prestressed concrete, using high-strength materials, makes an attractive alternative for long-span
bridges. It has been widely used in bridge structures since the 1950s.
1. Slab
Figure 10.13 shows Federal Highway Administration (FHWA) [6] standard types of pre-
cast, prestressed, voided slabs and their sectional properties. While cast-in-place, pre-
stressed slab is more expensive than reinforced concrete slab, precast, prestressed slab is
economical when many spans are involved. Common spans range from 20 to 50 ft (6.1 to
15.2 m). Structural depth-to-span ratios are 0.03 for both simple and continuous spans.
FIGURE 10.13: Federal Highway Administration (FHWA) precast, prestressed, voided slab sections.
(From Federal Highway Administration, Standard Plans for Highway Bridges, Vol. 1, Concrete Super-
structures, U.S. Department of Transportation, Washington, D.C., 1990. With permission.)
2. Precast I Girder
Figure 10.14 shows AASHTO [6] standard types of I-beams. These compete with steel
girders and generally cost more than reinforced concrete with the same depth-to-span
ratios. The formwork is complicated, particularly for skewed structures. These sections
are applicable to spans 30 to 120 ft (9.1 to 36.6 m). St ructural depth-to-span ratios are
0.055 for simple spans and 0.05 for continuous spans.
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FIGURE 10.14: Precast, prestressed AASHTO (American Association of State Highway and Trans-
portation Officials) I-beam sections. (From Federal Highway Administration, Standard Plans for
Highway Bridges, Vol. 1, Concrete Superstructures, U.S. Department of Transportation, Washington,
D.C., 1990. With permission.)
3. Box Girder
Figure 10.15 shows FHWA [6] standard types of precast box sections. The shape of a
cast-in-place, prestressed concrete box girder is similar to the conventional reinforced
concrete box girder (Figure 10.12c). The spacing of the girders can be taken as twice the
structural depth. It is used mostly for spans of 100 to 600 ft (30.5 to 182.9 m). Structural
depth-to-span ratios are 0.045 for simple spans and 0.04 for continuous spans. These
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sections are used frequently for simple spans of over 100 ft (30.5 m) and are particularly
suitable for widening in order to control deflections. About 70 to 80% of California’s
highway bridge system is composed of prestressed concrete box girder bridges.
FIGURE 10.15: Federal Highway Administration (FHWA) precast, pretensioned box sections. (From
FederalHighwayAdministration, StandardPlans forHighwayBridges, Vol. 1,ConcreteSuperstructures,
U.S. Department of Transportation, Washington, D.C., 1990. With permission.)
4. Segmental Bridge
The segmentally constructed bridges have been successfully developed by combining the
concepts of prestressing, box girder, and the cantilever construction [2, 20]. The first
prestressed segmental box girder bridge was built in Western Europe in 1950. California’s
Pine Valley Bridge, as shown in Figure 10.16 (composed of three spans of 340 ft [103.6
m], 450 ft [137.2 m], and 380 ft [115.8 ft] with the pier height of 340 ft [103.6 m]), was
the first cast-in-place segmental bridge built in the U.S., in 1974.
The prestressedsegmentalbridges with precast orcast-in-place segmental canbeclassified
by the construction methods: (1) balanced cantilever, (2) span-by-span, (3) incremen-
tal launching, and (4) progressive placement. The selection between cast-in-place and

precast segmental, and among various construction methods, is dependent on project
features, site conditions, environmental and public constraints, construction time for the
project, and equipment available. Table 10.2 lists the range of application of segmental
bridges by span lengths [20].
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FIGURE 10.16:a Pine Valley Bridge, California. Construction state. (From California Department of
Transportation. With permission.)
FIGURE 10.16:b Pine Valley Bridge, California. Construction completed. (From California Depart-
ment of Transportation. With permission.)
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FIGURE 10.17: A flanged section at nominal moment capacity state.
TABLE 10.2 Range of Application of Segmental Bridge Type by Span Length
Span
ft (m) Bridge types
0–150 (0–45.7) I-type pretensioned girder
100–300 (30.5–91.4) Cast-in-place post-tensioned box girder
100–300 (30.5–91.4) Precast-balanced cantilever segmental, constant depth
200–600 (61.0–182.9) Precast-balanced cantilever segmental, variable depth
200–1000 (61.0–304.8) Cast-in-place cantilever segmental
800–1500 (243.8–457.2) Cable-stay with balanced cantilever segmental
5. Design Consideration
Comparedto reinforced concrete, the main design features of prestressed concrete arethat
stresses forconcrete andprestressingsteel anddeformationof structures at each stage(i.e.,
during construction, stressing, handling, transportation, and erection as well as during
the service life)and stressconcentrations needtobeinvestigated. In thefollowing, weshall
briefly discuss the AASHTO-LRFD [1] requirements for stress limits, nominal flexural

resistance, and shear resistance in designing a prestressed member.
a) Stress Limits
Calculations of stresses for concrete and prestressing steel are based mainly on the elastic theory.
Tables 10.3 to 10.5 list the AASHTO-LRFD [1] stress limits for concrete and prestressing tendons.
b) Nominal Flexural Resistance, M
n
Flexural strength isbased on theassumptions that (1)the strain is linearlydistributedacross across-
section (except for deep flexural member); (2) the maximum usable strain at extreme compressive
fiber is equal to 0.003; (3) the tensile strength of concrete is neglected; and (4) a concrete stress of
0.85 f

c
is uniformly distributed over an equivalent compression zone. For a member with a flanged
section (Figure 10.17) subjected to uniaxial bending, the equations of equilibrium are used to give a
nominal moment resistance of:
M
n
= A
ps
f
ps

d
p
−
a
2

+ A
s

f
y
(d
s
−
a
2
)
− A

s
f

y

d

s
−
a
2

+ 0.85f

c
(b − b
w
)β
1
h

f

a
2
−
h
f
2

(10.1)
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TABLE 10.3 Stress Limits for Prestressing Tendons
Prestressing tendon type
Stress-relieved
strand and plain Deformed
Stress Prestressing high-strength Low Relaxation high-strength
type method bars strand bars
At jacking Pretensioning 0.72f
pu
0.78f
pu
—
(f
pj
) Post-tensioning 0.76f
pu
0.80f
pu

0.75f
pu
After Pretensioning 0.70f
pu
0.74f
pu
—
transfer Post-tensioning
(f
pt
) At anchorages
and couplers 0.70
f
pu
0.70f
pu
0.66f
pu
immediately
after anchor set
General 0.70
f
pu
0.74f
pu
0.66f
pu
At service After all losses
limit 0.80
f

py
0.80f
py
0.80f
py
state (f
pe
)
FromAmerican Association of State Highway and Transportation Officials, AASHTO LRFD Bridge
Design Specifications, First Edition, Washington, D.C., 1994. With permission.
a = βc (10.2)
c =
A
ps
f
pu
+ A
s
f
y
− A

s
f

y
− 0.85β
1
f


c
(b − b
w
)h
f
0.85β
1
f

c
b
w
+ kA
ps
f
pu
d
p
(10.3)
f
ps
= f
pu

1 − k
c
d
p

(10.4)

k = 2

1.04 −
f
py
f
pu

(10.5)
where A represents area; f is stress; b is the width of the compression face of member; b
w
is
the web width of a section; h
f
is the compression flange depth of a cross-section; d
p
and d
s
are
distances from extreme compression fiber to the centroid of prestressing tendons and to centroid of
tension reinforcement, respectively; subscripts c and y indicate specified strength for concrete and
steel, respectively; subscripts p and s signify prestressing steel and reinforcement steel, respectively;
subscripts ps, py, and pu correspond to states of nominal moment capacity, yield, and specified
tensile strength of prestressing steel, respectively; superscript prime (

) represents compression; and
β
1
is the concrete stress block factor, equal to 0.85 f


c
≤ 4000 psi and 0.05 less for each 1000 psi
of f

c
in excess of 4000 psi, and minimum β
1
= 0.65. The above equations also can be used for a
rectangular section in which b
w
= b is taken.
Maximum reinforcement limit:
c
d
e
≤ 0.42 (10.6)
d
e
=
A
ps
f
ps
d
p
+ A
s
f
y
d

s
A
ps
f
ps
+ A
s
f
y
(10.7)
Minimum reinforcement limit:
φM
n
≥ 1.2M
cr
(10.8)
in which φ is the flexural resistance factor 1.0 for prestressed concrete and 0.9 for reinforced concrete,
and M
cr
is the cracking moment strength given by the elastic stress distribution and the modulus of
rupture of concrete.
c

1999 by CRC Press LLC
TABLE 10.4 Temporary Concrete Stress Limits at Jacking State Before Losses Due to Creep and
Shrinkage—Fully Prestressed Components
Stress Stress
type Area and condition ksi (MPa)
Compressive Pretensioned 0.60f


ci
Post-tensioned 0.55f

ci
Precompressed tensile zone without bonded reinforcement N/A
Area other than the precompressed tensile zones and without
bonded auxiliary reinforcement
0.0948

f

ci
≤ 0.2

0.25

f

ci
≤ 1.38

Tensile
Nonsegmental
bridges
Area with bonded reinforcement which is sufficient to resist 120%
of the tension force in the cracked concrete computed
0.22

f


ci
on the basis of uncracked section

0.58

f

ci

Handling stresses in prestressed piles

0.158

f

ci


0.415

f

ci

Type A joints with minimum bonded aux-
iliary reinforcement through the
0.0948

f


ci
max. tension
Longitudinal stress
through joint in
precompressed
joints which is sufficient to carry the cal-
culated tensile force at a stress of 0.5
f
y
with internal tendons
(0.25

f

ci
max. tension)
tensile zone Type A joints without the minimum
bonded auxiliary reinforcement through
the joints with internal tendons
No tension
Type B with external tendons 0.2 min. compression
(1.38 min. compression)
Segmental Transverse stress For any type of joint 0.0948

f

c
max. tension
bridges through joints (0.25


f

c
max. tension)
Without bonded non-prestressed rein-
forcement
No tension
Other area Bonded reinforcement is sufficient to
carry the calculated tensile force in the
0.19

f

ci
concrete on the assumption of an un-
cracked section at a stress of 0.5
f
sy
(0.50

f

ci
)
Note: Type A joints are cast-in-place joints of wet concrete and/or epoxy between precast units. Type B joints are dry joints
between precast units.
From American Association of State Highway and Transportation Officials, AASHTO LRFD Bridge Design Specifications, First
Edition, Washington, D.C., 1994. With permission.
c) Nominal Shear Resistance, V
n

The nominal shear resistance shall be determined by the following formulas:
V
n
= the lesser of

V
c
+ V
s
+ V
p
0.25f

c
b
ν
d
ν
+ V
p
(10.9)
where
V
c
=

0.0316β

f


c
b
ν
d
ν
(ksi)
0.083β

f

c
b
ν
d
ν
(MPa)
(10.10)
V
s
=
A
ν
f
y
d
ν
(
cos θ + cosα
)
sin α

s
(10.11)
where b
ν
is the effective web width determined by subtracting the diameters of ungrouted ducts
or one-half the diameters of grouted ducts; d
ν
is the effective depth between the resultants of the
tensile and compressive forces due to flexure, but not less than the greater of 0.9 d
e
or 0.72h; A
ν
is the area of transverse reinforcement within distance s; s is the spacing of the stirrups; α is the
angle of inclination of transverse reinforcement to the longitudinal axis; β is a factor indicating the
c

1999 by CRC Press LLC
TABLE 10.5 Concrete Stress Limits at Service Limit State After All Losses—Fully Prestressed
Components
Stress Stress
type Area and condition ksi (MPa)
Nonsegmental bridge at service state 0.45f

c
Compressive Nonsegmental bridge during shipping and handling 0.60f

c
Segmental bridge during shipping and handling 0.45f

c

With bonded prestressing tendons 0.19

f

c
other than piles (0.50

f

c
)
Precompressed Subjected to severe corrosive
Tensile tensile zone assuming
uncracked sections
conditions 0.0948

f

c
Nonsegmental

0.25

f

c

bridges With unbonded prestressing tendon No tension
Type A joints with minimum bonded aux-
iliary reinforcement through the joints

which is sufficient to carry the calculated
tensile force at a stress of 0.5
f
y
with in-
ternal tendons
0.0948

f

c
(0.25

f

c
)
Longitudinal stress in
precompressed tensile
zone
Type A joints without the minimum
bonded auxiliary reinforcement through
the joints
No tension
Type B with external tendons 0.2 min. compression
(1.38 min. compression)
Segmental
bridges
Transverse stress in
precompressed tensile

zone
For any type of joint 0.0948

f

c

0.25

f

c

Type A joint without minimum bonded
auxiliary reinforcement through joints
No tension
Other area
(without bonded
reinforcement)
Bonded reinforcement is sufficient to
carry the calculated tensile force in the
concrete on the assumption of an un-
cracked section at a stress of 0.5
f
sy
0.19

f

c


0.50

f

c

Note: Type A joints are cast-in-place joints of wet concrete and/or epoxy between precast units. Type B joints are dry joints
between precast units.
From American Association of State Highway and Transportation Officials, AASHTO LRFD Bridge Design Specifications, First
Edition, Washington, D.C., 1994. With permission.
ability of diagonally cracked concrete to transmit tension; andθ is the angle of inclination of diagonal
compressive stresses (Figure 10.18). The values of β and θ for sections with transverse reinforcement
are given in Table 10.6. In this table, the shear stress, ν, and strain, ε
x
, in the reinforcement on the
flexural tension side of the member are determined by:
ν =
V
u
− φV
p
φb
ν
d
ν
(10.12)
ε
x
=

M
u
d
ν
+ 0.5N
u
+ 0.5V
u
cot θ − A
ps
f
po
E
s
A
s
+ E
p
A
ps
≤ 0.002 (10.13)
where M
u
and N
u
are the factored moment and axial force (taken as positive if compressive),
respectively, associated with V
u
, and f
po

is the st ress in prestressing steel when the stress in the
surrounding concrete is zero and can be conservatively taken as the effective stress after losses, f
pe
.
When the valueof ε
x
calculated from theabove equation isnegative, its absolutevalueshall be reduced
c

1999 by CRC Press LLC