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IMTS (ISO 9001-2008 Internationally Certified) BRIDGE ENGINEERING

BRIDGE ENGINEERING

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CONTENTS BRIDGE ENGINEERING

01-81

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1 BRIDGE ENGINEERING

Bridge engineering covers the planning, design, construction, operation, and maintenance of structures that carry facilities for movement of humans, animals, or materials over natural or created obstacles. Most of the diagrams used in this section were taken from the “Manual of Bridge Design Practice,” State of California Department of Transportation and “Standard Specifications for Highway Bridges,” American Association of State Highway and Transportation Officials. The authors express their appreciation for permission to use these illustrations from this comprehensive and author- itative publication. General Design Considerations Bridge Types Bridges are of two general types: fixed and mov- able. They also can be grouped according to the following characteristics: Supported facilities: Highway or railway bridges and viaducts, canal bridges and aqueducts, pedes- trian or cattle crossings, material-handling bridges, pipeline bridges. Bridge-over facilities or natural features: Bridges over highways and over railways; river bridges; bay, lake, slough and valley crossings. Basic geometry: In plan—straight or curved, square or skewed bridges; in elevation—low-level bridges, including causeways and trestles, or high- level bridges. Structural systems: Single-span or continuous- beam bridges, single- or multiple-arch bridges, suspension bridges, frame-type bridges. Construction materials: Timber, masonry, con- crete, and steel bridges. Design Specifications Designs of highway and railway bridges of con- crete or steel often are based on the latest editions of the “Standard Specifications for Highway Bridges” or the “Load and Resistance Factor Design Specifications” (LRFD) of the American Association of State Highway and Transportation Officials (AASHTO) and the “Manual for Rail- way Engineering” of the American Railway En- gineering and Maintenance-of-Way Association (AREMA). Also useful are standard plans issued by various highway administrations and railway companies. Length, width, elevation, alignment, and angle of intersection of a bridge must satisfy the functional requirements of the supported facilities and the geometric or hydraulic requirements of the bridged-over facilities or natural features. Figure 17.1 shows typical highway clearance diagrams. Selection of the structural system and of the construction material and detail dimensions is governed by requirements of structural safety; economy of fabrication, erection, operation, and maintenance; and aesthetic considerations. FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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Highway bridge decks should offer comfort- able, well-drained riding surfaces. Longitudinal grades and cross sections are subject to standards similar to those for open highways (Sec. 16). Provisions for roadway lighting and emergency services should be made on long bridges. Barrier railings should keep vehicles within the roadways and, if necessary, separate vehicular

Fig. 17.1 Minimum clearances for highway structures. (a) Elevation of a highway bridge showing minimum vertical clearances below it. (b) Typical bridge cross sections indicating minimum horizontal clearances. Long-span bridges may have different details and requirements. lanes from pedestrians and bicyclists. Utilities carried on or under bridges should be adequately protected and equipped to accommodate expan- sion or contraction of the structures. Most railroads require that the ballast bed be continuous across bridges to facilitate vertical track adjustments. Long bridges should be equipped with service walkways. Design Loads for Bridges Bridges must support the following loads without exceeding permissible stresses and deections: Dead load D, including permanent utilities Live load L and impact I Longitudinal forces due to acceleration or decel- eration LF and friction F Centrifugal forces CF Wind pressure acting on the structure W and the moving load WL Earthquake forces EQ Earth E, water and ice pressure ICE, stream ow SF, and uplift B acting on the substructure Forces resulting from elastic deformations, includ- ing rib shortening R Forces resulting from thermal deformations T, including shrinkage S, and secondary prestressing effects

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Highway Bridge Loads Vehicular live load of highway bridges is expressed in terms of design lanes and lane loadings. The number of design lanes depends on the width of the roadway. In the standard specifications, each lane load is represented by a standard truck with trailer (Fig. 17.3) or, alternatively, as a 10-ft-wide uniform load in combination with a concentrated load (Fig. 17.2). As indicated in Fig. 17.3, there are two classes of loading: HS20 and HS15, which represent a truck and trailer with three loaded axles. These loading designations are followed by a 44, which indicates that the loading standard was adopted in 1944. The LRFD HL-93 vehicular live load consists of a combination of the HS20-44 design truck depicted in Fig. 17.3, or the LRFD design tandem, and the LRFD live load. The LRFD design tandem is defined as a pair of 25 kip axles spaced 4.0 ft apart. The LRFD live load consists of 0.64 k/lf applied uniformly in the longitudinal and trans- verse direction. When proportioning any member, all lane loads should be assumed to occupy, within their respective lanes, the positions that produce maximum stress in that member. Table 17.1 gives maximum moments, shears, and reactions for one loaded lane. Effects resulting from the simultaneous loading of more than two lanes may be reduced by a loading factor, which is 0.90 for three lanes and 0.75 for four lanes. In design of steel grid and timber floors for HS20 loading, one axle load of 24 kips or two axle loads of 16 kips each, spaced 4 ft apart, may be used, whichever produces the greater stress, instead of the 32-kip axle shown in Fig. 17.3. For slab design, the centerline of the wheel should be assumed to be 1 ft from the face of the curb. Wind forces generally are considered as mov- ing loads that may act horizontally in any direc- tion. They apply pressure to the exposed area of the superstructure, as seen in side elevation; to traffic on the bridge, with the center of gravity 6 ft above the deck; and to the exposed areas of the sub- structure, as seen in lateral or front elevation. Wind loads in Tables 17.2 and 17.3 were taken from “Standard Specifications fo r Highway Bridges,” American Association of State Highway and Transportation Officials. They are based on 100 - mi/h wind velocity. They should be multiplied by (V/100)2 for other design velocities except for Group III loading (Art. 17.4). In investigation of overturning, add to horizon- tal wind forces acting normal to the longitudinal bridge axis an upward force of 20 lb/ft2 for the structure without live load or 6 lb/ft2 when the structure carries live load. This force should be applied to the deck and sidewalk area in plan at the windward quarter point of the transverse superstructure width. Impact is expressed as a fraction of live-load stress and determined by the formula:

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Fig. 17.2 HS loadings for simply supported spans. For maximum negative moment in continu- ous spans, an additional concentrated load of equal weight should be placed in one other span for maximum effect. For maximum positive moment, only one concentrated load should be used per lane, but combined with as many spans loaded uniformly as required for maximum effect. where l ¼ span, ft; or for truck loads on cantilevers, length from moment center to farthermost axle; or for shear due to truck load, length of loaded por- tion of span. For negative moments in continuous spans, use the average of two adjacent loaded spans. For cantilever shear, use I ¼ 30%. Impact is not figured for abutments, retaining walls, piers, piles (except for steel and concrete piles above ground rigidly framed into the superstructure), foundation pressures and footings, and sidewalk loads. Longitudinal forces on highway bridges should be assumed at 5% of the lane load plus concentrated load for moment headed in one

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Fig. 17.3 Standard truck loading. HS trucks: W ¼ combined weight on the first two axles, which is the same weight as for H trucks. V indicates a variable spacing from 14 to 30 ft that should be selected to produce maximum stress. direction, plus forces resulting from friction in bridge expansion bearings. Centrifugal forces should be computed as a percentage of design live load C

¼

where S ¼ design speed, mi/h

6: 68S2 R

(17:2)

R ¼ radius of curvature, ft These forces are assumed to act horizontally 6 ft above deck level and perpendicular to the bridge centerline. Restraint forces, generated by preventing rotations of deformations, must be considered in design.

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Thermal forces, in particular, from restraint, may cause overstress, buckling, or cracking. Provision should be made for expansion and contraction due to temperature variations, and on concrete structures, also for shrinkage. For the continental United States, Table 17.4 covers tem-perature ranges of most locations and includes the effect of shrinkage on ordinary beam-type concrete structures. The coefficient of thermal expansion for where P ¼ pressure, lb/ft2 V ¼ velocity of water, ft/s K ¼ 4⁄3 for square ends, 1⁄2 for angle ends when angle is 308 or less, and 2⁄ for both concrete and steel per 8Fahrenheit is 0.0000065 circular piers (approximately 1⁄ ). The shrinkage coefficient Ice pressure should be assumed as 400 psi. The for concrete arches and rigid frames should be assumed as 0.002, equivalent to a temperature drop of 31 8F. Stream-flow pressure on a pier should be computed from P ¼ KV2

(17:3)

design thickness should be determined locally. Earth pressure on piers and abutments should be computed by recognized soil-mechanics for- mulas, but the equivalent fluid pressure should be at least 36 lb/ft3 when it increases stresses and not more than 27 lb/ft3 when it decreases stresses. Sidewalks and their direct supports should be designed for a uniform live load of 85 lb/ft2.

The effect of sidewalk live loading on main bridge members should be computed from Impact loads, as a percentage of railroad live loads, may be computed from Table 17.5. Longitudinal forces should be computed for P ¼ 30 þ 3000 /l 55 w 50

2

60 lb=ft (17:4)

braking and traction and centrifugal forces should be computed corresponding to each axle. See the AREMA ‘‘Manual for Railway Engineering’’ for where P ¼ sidewalk live load, lb/ft2 l ¼ loaded length of sidewalk, ft w ¼ sidewalk width, ft Curbs should resist a force of 500 lb/lin ft acting 10 in above the floor. For design loads for railings, see Fig. 17.4.

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Railway Bridge Loads Live load is specified by axle -load diagrams or by the E number of a “Cooper ’s train,” consisting of two locomotives and an indefinite number of freight cars. Figure 17.5 shows the typical axle spacing and axle loads for E80 loading. Members receiving load from more than one track should be assumed to be carrying the fol- lowing proportions of live load: For two tracks, full live load; for three tracks, full live load from two more information (www.arema.org). Proportioning of Bridge Members and Sections The following groups represent various combi- nations of loads and forces to which a structure may be subjected. Each component of the structure, or the foundation on which it rests, should be proportioned to withstand safely all group combi- nations of these forces that are applicable to the particular site or type. Group loading combinations for service load design and load factor design are given by Group (N) ¼ g[bD D þ bL (L þ I) þ bC CF þ bE E þ bB B þ bS SF þ bW W tracks and half from the third track; for four tracks, full live load from two, half from one, and onefourth from the remaining one. þ bWL WL þ bL LF þ bR (R þ S þ T) þ bEQ EQ þ bICE ICE] (17:5)

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Fig. 17.4 Service loads for railings: P ¼ 10 kips, L ¼ post spacing, w ¼ 50 lb/ft. Rail loads are shown on the left, post loads on the right. (Rail shapes are for illustrative purposes only.)

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Fig. 17.5 Axle Spacing and Axle Loads for E80 loading

where N ¼ group number, or number assigned to a specific combination of loads g ¼ capacity reduction factor to provide for small adverse variations in materials, workmanship, and dimensions within acceptable tolerances b ¼ load factor (subscript indicates appli- cable type of load) See Table 17.6 for appropriate coefficients. See also Art. 17.3.1 and Secs. 8 and 9. AASHTO LRFD associates load combinations with various limit states according to design ob- jectives. The sum of the factored loads must be less than the sum of the factored resistance: X hi gi Qi

wRn

(17:6)

where hi ¼ load modifier relating to ductility, redun- dancy, and operational importance gi ¼ load factor, a statistically based multi- plier reflecting certainty in the value for force effect Qi ¼ force effect i w ¼ resistance factor, a statistically based multiplier reflecting certainty in value for particular material property

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See Table 17.7 and 17.8 for design objectives, limit state load combinations and load factors. Resistance factors vary according to material and char- acteristic such as bending, shear, bearing, torsion, etc., and are not shown. In LRFD, both the g’s and w’s have been calibrated to achieve a uniform level of safety throughout the structure. Seismic Design Seismic forces are an important loading consider- ation that often controls the design of bridges in seismically active regions. All bridges should be designed to insure life safety under the demands imparted by the Maximum Considered Earthquake (MCE). Higher levels of performance may be required by the bridge owner to provide post earthquake access to emergency facilities or when the time required to restore service after an earthquake would create a major economic impact. * For ballasted decks use 90% of calculated impact (steel bridges only) L ¼ span, ft; S ¼ longitudinal beam spacing, ft; DL ¼ applicable dead load; LL ¼ applicable live load. RE ¼ the rocking effect consisting of the percentage of downward on one rail and upward on the other rail, increasing and decreasing, respectively, the loads otherwise specified. RE shall be expressed as a percentage; either 10% of the axle load or 20% of the wheel load. ** Impact is reduced for L . 175 ft or when load is received from more than two tracks. All bridges should have a clearly identifiable system to resist forces and deformations imposed by seismic events. Experimental research and past performance has demonstrated that simple bridge features lead to more predictable seismic response. Irregular features lead to complex and less predictable seismic response and should be avoided in high seismic region whenever possible (See Table 17.9). Every effort should be made to balance the effective lateral stiffness between adjacent bents within a frame, adjacent columns within a bent, and adjacent frames. If irregular features or significant variations in lateral stiffness are unavoidable, they should be ass essed with more rigorous analysis and designed for a higher level of seismic performance. Seismic effects for box culverts and buried structures need not be considered, except when they cross active faults. Seismic Design Approach Ordinarily bridges are not designed to remain elastic during the MCE because of economic constraints and the uncertainties in predicting seismic demands. Design codes permit the designer to take advantage of ductility and post elastic strength as long as the expected deformations do not exceed the bridge’s lateral displacement capacity. Ductile

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Table 17.6

11

Capacity-Reduction and Load Factors Load factor b for*

Group

g

D

(L þ I)n

(L þ I)P

CF

E

B SF

W

WL

LF R þ S þ T

EQ

ICE

% of basic unit stresses

Service-Load Design† I IA IB II III IV V VI VII VIII IX

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1 1 1 1 1 1 1 1 1 1 1

1 2 0 0 1 1 0 1 0 1 0

0 0 1 0 0 0 0 0 0 0 0

1 0 1 0 1 1 0 1 0 1 0

bE 0 bE 1 bE bE 1 bE 1 1 1

1 0 1 1 1 1 1 1 1 1 1

1 0 1 1 1 1 1 1 1 1 1

0 0 0 1 0.3 0 1 0.3 0 0 1

0 0 0 0 1 0 0 1 0 0 0

0 0 0 0 1 0 0 1 0 0 0

0 0 0 0 0 1 1 1 0 0 0

0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 1

1 0 1 1 1 1 1 1 1 1 1

0 0 0 1 0.3 0 1 0.3 0 0 1

0 0 0 0 1 0 0 1 0 0 0

0 0 0 0 1 0 0 1 0 0 0

0 0 0 0 0 1 1 1 0 0 0

0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 1

100 150 ‡

125 125 125 140 140 133 140 150

LOAD FACTOR DESIGN I IA IB II III IV V VI VII VIII IX

1.3 1.3 1.3 1.3 1.3 1.3 1.25 1.25 1.3 1.3 1.20

bD bD bD bD bD bD bD bD bD bD bD

* D ¼ dead load L ¼ live load I ¼ live-load impact E ¼ earth pressure

1.67} 2.20 0 0 1 1 0 1 0 1 0

0 0 1 0 0 0 0 0 0 0 0

1.0 0 1.0 0 1 1 0 1 0 1 0

bE 0 bE bE bE bE bE bE bE bE bE

1 0 1 1 1 1 1 1 1 1 1

LF ¼ longitudinal force from live load (L þ I)n ¼ live load plus impact for AASHTO highway loading CF ¼ centrifugal force

B ¼ buoyancy F ¼ longitudinal force due to friction impact consistent W ¼ wind load on structure with the overload criteria of the WL ¼ wind load on live load operating agency

T ¼ temperature EQ ¼ earthquake SF ¼ stream flow pressur e ICE ¼ ice pressure (L þ I)P ¼ live load plus R ¼ rib shortening S ¼ shrinkage

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† For service-load design: No increase in allowable unit stresses is permitted for members or connections carrying wind loads only. bE ¼ 1.0 for lateral loads on rigid frames subjected to full earth pressure ¼ 0.5 when positive moment in beams and slabs is reduced by half the earth-pressure moment Check both loadings to see which one governs. Maximum unit stress ( operating rating ) Allowable basic unit stress

100

§ For load factor design: bE ¼ 1.3 for lateral earth pressure for rigid frames excluding rigid culverts ¼ 0.5 for lateral earth pressure when checking positive moments in rigid frames ¼ 1.0 for vertical earth pressure bD ¼ 0.75 when checking member for minimum axial load and maximum moment or maximum eccentricity and column design ¼ 1.0 when checking member for maximum axial load and minimum moment and column design ¼ 1.0 for flexural and tension members } bD ¼ 1.25 for design of outer roadway beam for combination of sidewalk and roadway live load plus impact, if it governs the design, but section capacity should be at least that required for bD ¼ 1.67 for roadway live load alone ¼ 1.00 for deck-slab design for D þ L þ I Table 17.7 AASHTO LRFD Load Combinations and Load Factor

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response in bridge systems is typically achieved through sustained hysteric force-deformation cycles that dissipate energy. This dissipation occurs internally, within the structural members, by the formation of flexural plastic hinges, or externally with isolation bearings or external dampers. Inelastic behavior should be limited to pre- determined locations within the bridge that can be easily inspected and repaired following an earth- quake. Preferable locations for inelastic behavior on most bridges include columns, pier walls, and abutment backwalls and wingwalls. Inelastic response in the superstructure is not desirable because it is difficult to inspect and repair and may prevent the bridge from being restored to a serviceable condition. Members not participating in the primary A ¼ Acceleration coefficient from national ground motion maps. S ¼ Site coefficient specified in Table 17.10 Seismic Demands The uniform load method can be used to determine the seismic loading for bridges that will respond principally in their fundamental mode of vibration. Equivalent static earthquake loads are calculated by multiplying the tributary permanent load by a response spectra coef ficient: C sm W where g ¼ Acceleration of gravity K ¼ Bridge lateral stiffness

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Single span bridges do not require seismic analysis. The minimum design force at the con- nections between the superstructure and substruc- ture shall not be less than the product of the site coefficient S, the acceleration coefficient A, and the tributary permanent load. The multimode spectral mode analysis method should be used if coupling between the longitudi- nal, transverse and/or vertical response is expec- ted. A three dimensional linear dynamic model should be used to represent the bridge. The elastic seismic forces and displacement generated from where pe ¼ Equivalent uniform static seismic load per unit length of bridge Csm ¼ Elastic

response

coefficient

see equation 17.8

W ¼ Dead load of the bridge superstructure and tributary substructure L ¼ Total length of bridge in ft multiple mode shapes are combined using accep- table methods such as the root-mean-square method or the complete quadratic combination method. The number of modes in the model should be at least three times the number of spans being modeled. Site-specific response spectra are often developed for multi-modal analysis that incorpor- ates the seismic source, ground attenuation, and near fault phenomena.

Csm ¼ 2=3 m 2:5A

(17:8)

When response spectra analysis is used, a maximum single seismic force is calculated by where Tm ¼ Period of vibration of the mth mode (seconds) combining two horizontal orthogonal ground motion components. These components are applied along a I 1.0 Rock of any description (shale-like or crystalline) or stiff soils (sands, gravels, stiff clays) less than 200 ft in depth overlying rock II

1.2

Stiff cohesive or deep cohesionless soils more than 200 ft in depth overlying rock

III 1.5 Soft to medium-stiff clays and sands characterized by 30 ft or more of clay with or without intervening layers of sand. IV

2.0

Soft clays or silts greater than 40 ft of depth.

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Fig. 17.6 Equivalent static earthquake loads. longitudinal axis defined by a chord intersecting the centerline of the bridge at the abutments and a normal transverse axis (See Fig. 17.7). It is uneconomical to design bridges to resist large earthquakes elastically. Columns are assumed to deform inelastically where seismic forces exceed design levels established by dividing the elastically computed moments by the appropriate response modification factors, R. (See Tables 17.11 and 17.12) The AASHTO Bridge Design Specification defines three levels of response modification factors for critical bridges, essential bridges and other bridges. The bridge owner must determine the performance level required consi- dering social/survival and security/defense requirements. More rigorous analysis such as inelastic time history analysis should be used on geometrically complex bridges, critical bridges and bridges within close proximity of earthquake faults. The nonlinear analysis provides forces and deformations as a function of time for a specified earthquake motion.

Fig. 17.7 Orthogonal bridge axis definition. A minimum of three ground motions representing the design event should be used. Nonlinear static analysis, commonly known as pushover analysis has recently been adopted by Caltrans. The inelastic displacement capacity of the piers is compared to the displacements from an elastic demand analysis

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that considers the bridge’s cracked flexural stiffness. The inelastic deformation capacity of the earthquake resisting members is calculated using moment curvature analysis utiliz- ing expected material properties and dependable materials strain limits. Displacement capacities are Table 17.11

AASHTO Substructure Response Modification Factors

Importance Category Substructure

Critical

Essential

other

Wall-type piers—larger dimension

1.5

1.5

2.0

Reinforced concrete pile bents † Vertical piles only † With batter piles

1.5 1.5

2.0 1.5

3.0 2.0

Single Columns

1.5

2.0

3.0

Steel or composite steel and concrete pile bents † Vertical pile only † With batter piles

1.5 1.5

3.5 2.0

5.0 3.0

Multiple column bents

1.5

3.5

5.0

also limited by degradation of strength and P-D effects that occur under large inelastic deformations. If the P-D moments are less than 20% of the plastic moment capacity of the member, they are typically ignored. Seismic Design of Concrete Bridge Columns Cross sectional column dimensions should be limited to the depth of the superstructures or bent cap to reduce the potential for inelastic damage migrating into the superstructure. The longitudinal reinforcement for compression members should not exceed 4% of the columns gross cross sectional area to insure adequate ductility, avoid congestion and to permit adequate anchorage of the longiConnection For rectangular sections the total gross sectional area of rectangular hoop reinforcement shall not be less than either: developed at the most probable location within the column with a rational combination of the most adverse end moments. The shear resisting mechAsh ¼ total cross sectional area of tie reinforcement, including supplementary cross ties having a vertical spacing “s” and cross section having core diameter of hc (in2) hc ¼ core dimension of tied column in the direction under consideration (in2) The potential plastic hinge region is defined as the larger of 1.5 times the cross sectional dimension in the direction of bending or the region of column where the moment exceeds 75% of the maximum plastic moment. The column design shear force should be calculated considering the flexural overstrength

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The concrete contribution is significantly dimin- ished under high ductilities and cyclic loading and is often ignored in the plastic moment regions. The flexural reinforcement in continuous or cantilever members needs to detailed to provide continuity of reinforcement at intersections with other members to develop nominal moment resistance of the joint can be developed to resist the shear depicted in Fig. Several shear design models defining the shear resisting mechanisms for columns and joints can be found in the AASHTO Design Specifications or the Caltrans Seismic Design Criteria.The unseating of girders and abutments must be avoided in all circumstances. The seat width needs to accommodate thermal movement, prestress shortening, creep, shrinkage and anticipated earth- quake displacements. The seat width should not be less then 1.5 times the elastic seismic displacement

Fig. 17.8 Joint shear stresses in T-joints.

of the superstructure at the seat or: N ¼ Support width normal to the centerline of bearing L ¼ Length of the bridge deck to the adjacent expansion joint, or the end of the bridge (ft) H ¼ Average height of the columns support- ing the bridge deck to the next expansion joint (ft) (H ¼ 0 for single span bridges) S ¼ Skew of the support measured from line normal to span (deg) (MCEER, “Recommended LRFD Guidelines for the Seismic Design of Highway Bridges,” Caltrans Seismic Design Criteria, vol. 1.1 (www.dot.ca.gov); AASHTO LRFD Bridge Design Specification (www.aashto.org).) FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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Steel Bridges Steel is competitive as a construction material for medium and long-span bridges for the following reasons: It has high strength in tension and com- pression. It behaves as a nearly perfect elastic material within the usual working ranges. It has strength reserves beyond the yield point. The high standards of the fabricating industry guarantee users uniformity of the controlling properties within narrow tolerances. Connection methods are reliable, and workers skilled in their application are available. The principal disadvantage of steel in bridge construction, its susceptibility to corrosion, is being increasingly overcome by chemical additives or improved protective coatings. Systems Used for Steel Bridges The following are typical components of steel bridges. Each functional types and structural systems listed in Art. 17.1.

may

be

applied to

any

of the

Main support: Rolled beams, plate girders, box girders, or trusses. Connections: (See also Art. 17.7.) High-strength- bolted, welded, or combinations. Timber decks are restricted to bridges on roads of minor importance. Plates of corrosion resistant steel should be used as ballast supports on through plate-girder bridges for railways. For roadway decks of stiffened steel plates, see Art. 17.13. Deck framing: Deck resting directly on main members or supported by grids of stringers and floor beams. Location of deck: On top of main members: deck spans (Fig. 17.9a); between main members, the underside of the deck framing being flush with that of the main members: through spans (Fig. 17.9b). Grades and Design Criteria for Steel for Bridges Preferred steel grades, permissible stresses, and standards of details, materials, and quality of work for steel bridges are covered in the AREMA and AASHTO specifications. Properties of the various grades of steel and the testing methods to be used to control them are regulated by specifications of ASTM. Properties of the structural steels presently preferred in bridge construction are Tabulated in Table 17.13. Dimensions and geometric properties of com- mercially available rolled plates and shapes are tabulated in the “Steel Construction Manual,” for allowable stress design and for load-and-resistancefactor design of the American Institute of Steel Construction (AISC), and in manuals issued by the major steel producers. All members, connections, and parts of steel bridges should be designed by the load-factor design method, and then checked for fatigue at service-level loads. The fatigue check should assure that all connections are within allowable stress ranges (FSR). The design strength of a beam or girder is based on the dimensional properties of the section and the spacing of compression flange bracing. The three types of member sections are (1) compact, (2)

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braced noncompact, and (3) partially braced. The AASHTO Flexural design formulas for the three types of I-Girder sections are shown in Table 17.14.

Fig. 17.9 Two-lane deck-girder highway bridge

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Design Limitations on Depth Ratios, Slenderness Ratios, Deflections n AASHTO specifications restrict the depth-to-span ratios D/L of bridge structures and the slenderness ratios l/r of individual truss or bracing members to the values in where D ¼ depth of construction, ft L ¼ span, ft, c to c bearings for simple spans or distance between points of contra- flexure for continuous spans l ¼ unsupported length of member, in r ¼ radius of gyration, in These are minimum values; preferred values are higher.

Connections with High-Strength Bolts The parts may be clamped together by bolts of quenched and tempered steel, ASTM A325. The nuts are tightened to 70% of their specified tensile strength. Details and quality of work are covered by the “Specifications for Structural Joints Using ASTM A325 and A490 Bolts,” approved by the Research Council on Structural Connections of the Engin- eering Foundation. Maximum stresses for bearing type connections are given Tensioned ASTM A325 bolts are the preferred bolt for all steel bridge connections. The nuts on Table 17.15 Dimensional Limitations for Bridge AASHTO Min depth-span ratios: For noncomposite beams or girders For simple span composite girders* For continuous composite girders* For trusses

1/25 1/22 1/25 1/10

Max slenderness ratios: For main members in compression For bracing members in compression For main members in tension For bracing members in tension

120 140 200 240

* For composite girders the depth shall include the concrete slab. tensioned A325 and A490 bolts will not loosen under vibrations associated with bridge loadings. If ASTM A307 bolts or non tensioned high strength bolts are used, provisions should be made to prevent nut loosing by the use of thread locking adhesive, self-locking nuts, or double nuts. Bolted connections subject to tension, or combined tension and shear, or stress reversal, or severe vibration, or heavy impact loads, or any other condition where joint slippage would be detrimental, shall use ten- sioned High-Strength bolts and be designed as a slip-critical connection. Slip-critical connections are designed to prevent slip at a specified overload condition in addition to meeting the strength requirements in bearing. The overload condition at which the connection is FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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required work as a friction connection (no slip) is equal to Dead load þ 1.67(Live load þ Impact). The slip strength of the connection is based on the number of slip plains, the friction coefficient of the contact surfaces, the type of hole, and the bolt tension stress. The AASHTO specifications provide equa- tions for determining the slip strength of connec- tions. Welded Connections In welding, the parts to be connected are fused at high temperatures, usually with addition of suitable metallic material. The “Structural Welding Code,” AWS D1.5, American Welding Society, regulates application of the various types and sizes of welds, permissible stresses in the weld and parent metal, permissible edge configurations, kinds and sizes of electrodes, details of quality of work, and qualification of welding procedures and welders. (For Maximum welding stresses, see Many designers favor the combination of shop welding with high-strength-bolted field connections. Pin Connections Hinges between members subject to relative rotation are usually formed with pins, machined steel cylinders. They are held in either semicircular machined recesses or smoothly fitting holes in the connected members.For fixation of the direction of the pin axis, pins up to 10-in diameter have threaded ends for recessed nuts, which bear against the connected members. Pins over 10-in diameter are held by recessed caps. These in turn are held by either tap bolts or a rod that runs axially through a hole in the pin itself and is threaded and secured by nuts at its ends. Pins are designed for bending and shear and for bearing against the connected members. (For stresses, see Art. 9.6.) Rolled-Beam Bridges The simplest steel bridges consist of rolled wide- flange beams and a traffic-carrying deck. Rolled beams serve also as floor beams and stringers for decks of plate-girder and truss bridges. Reductions in steel weight may be obtained, but with greater labor costs, by adding cover plates in the area of maximum moments, by providing continuity over several spans, by utilizing the deck in composite action, or by a combination of these measures. The principles of design and details are essentially identical with those of plate girders (Art. 17.9). Plate-Girder Bridges The term plate girder applies to structural elements of I-shaped cross section that are welded from plates. Plate girders are used as primary support- ing elements in many structural systems: as simple beams on abutments or, with overhanging ends, on piers; as continuous or hinged multispan beams; as stiffening girders of arches and suspension bridges, and in frame-type bridges. They also serve as floor beams and stringers on these other bridge systems. Their prevalent application on highway and railway bridges is in the form of deck-plate girders in combination with concrete decks (Fig. 17.9). (For design of concrete deck slabs, see Art. 17.20. For girders with steel decks (orthotropic decks), see Art 17.13. Girders with track ties mounted directly on the top flanges, open-deck girders, are used on branch railways and industrial spurs. Through plate girders (Fig. 17.9b) are now practically restricted to railway bridges where allowable structure depth is limited. FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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The two or more girders supporting each span must be braced against each other to provide stability against overturning and flange buckling, to resist transverse forces (wind, earthquake, centrifugal), and to distribute concentrated heavy loads. On deck girders, this is done by transverse bracing in vertical planes. Transverse bracing should be installed over each bearing and at intermediate locations not over 25 ft apart. This bracing may consist either of full-depth cross frames or of solid diaphragms with depth at least half the web depth for rolled beams and preferably three-quarters the web depth for plate girders. End cross frames or diaphragms should be proportioned to transfer fully all vertical and lateral loads to the bearings. On through-girder spans, since top lateral and transverse bracing systems cannot be installed, the top flanges of the girders must be braced against the floor system. For the purpose, heavy gusset plates or knee braces may be used (Fig. 17.9b). The most commonly used type of steel bridge girder is the welded plate girder. It is typically laterally braced, noncompact, and unsymmetrical, with top and bottom flanges of different sizes. Figure 17.10b shows a typical welded plate girder. Variations in moment resistance are obtained by using flange plates of different thicknesses, widths, or steel grades, butt-welded to each other in suc- cession. Web thickness too may be varied. Girder webs should be protected against buckling by transverse and, in the case of deep webs, longitu- dinal stiffeners. Transverse bearing stiffeners are required to transfer end reactions from the web into the bearings and to introduce concentrated loads into the web. Intermediate and longitudinal stiff- eners are required if the girder depth-to-thickness ratios exceed critical values (see Art. 9.13.4). Stiffeners may be plain plates, angles, or T sections. Transverse stiffeners can be in pairs or single elements. The AASHTO Specifications con- tain restrictions on width-to-thickness ratios and minimum widths of plate stiffeners (Art. 9.13.4). Web-to-flange connections should be capable of carrying the stress flow from web to flange at every section of the girder. At an unloaded point, the stress flow equals the horizontal shear per linear inch. Where a wheel load may act, for example, at upper flange -to-web connections of deck girders, the stress flow is the vectorial sum of the horizontal shear per inch and the wheel load (assumed distributed over a web length equal to twice the deck thickness). Welds connecting bearing stiffeners to the web must be designed for the full bearing reaction. Space restrictions in the shop, clearance restric- tions in transportation, and erection considerations may require dividing long girders into shorter sections, which are then joined (spliced) in the field. Individual segments, plates or angles, must be spliced either in the shop or in the field if they exceed in length the sizes produced by the rolling mills or if shapes are changed in thickness to meet stress requirements. Specifications require splices to be designed for the average between the stress due to design loads and the capacity of the unspliced segment, but for not less than 75% of the latter. In bolted design, material may have to be added at each splice to satisfy this requirement. Each splice element must be connected by a sufficient number of bolts to develop its full strength. Whenever it is possible to do so, splices of individual segments should be staggered. No splices should be located in the vicinity of the highest-stressed parts of the girder, for example, at midspan of simple-beam spans, or over the bearings on continuous beams. Figure 17.10a presents a design flow chart for welded plate girders. (F. S. Merritt and R. L. Brockenbrough, “Struc- tural Steel Designers’ Handbook,” 2nd ed., McGraw- Hill Inc., New York (books-mcgraw-hill.com).)

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Composite-Girder Bridges Installation of appropriately designed shear con- nectors between the top flange of girders or beams and the concrete deck allows use of the deck as part of the top flange (equivalent cover plate). The resulting increase in effective depth of the total section and possible reductions of the topflange steel usually allow some savings in steel compared with the noncomposite steel section. The overall economy depends on the cost of the shear connec- tors and any other additions to the girder or the deck that may be required and on possible limitations in effectiveness of the composite section as such.

Fig. 17.10 Welded plate girder. (a) Flow chart gives steps in load-factor design. (b) Typical plate girder—stiffened, braced, and noncompact.

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In areas of negative moment, composite effect may be assumed only if the calculated tensile stresses in the deck are either taken up fully by reinforcing steel or compensated by prestressing. The latter method requires special precautions to assure slipping of the deck on the girder during the prestressing operation but rigidity of connection after completion. If the steel girder is not shored up while the deck concrete is placed, computation of dead- load stresses must be based on the steel section alone. The effective flange width of the concrete slab that is used as a T-beam flange of a composite girder is the lesser of the following: 1. One-fourth of the span length of the girder 2. The center to center distance between adjacent girders 3. Twelve times the least thickness of the slab Shear connectors should be capable of resisting all forces tending to separate the abutting concrete and steel surfaces, both horizontally and vertically. Connectors should not obstruct placement and thorough compaction of the concrete. Their instal- lation should not harm the structural steel. The types of shear connectors presently pre- ferred are channels, or welded studs. Channels should be placed on beam flanges normal to the web and with the channel flanges pointing toward the girder bearings. The modular ratio recommended for stress analysis of composite girders under live loads is given in Table 17.17. For composite action under dead loads, the concrete section may be assumed to be subjected to constant compressive stress. This will cause the concrete to undergo plastic flow and thus will reduce its capacity to resist stress. This is taken into account in design of a composite girder for dead loads by multiplying by 3 the modular ratio n given in Table 17.17. Most composite girders, however, are designed for composite action only for live loads and dead loads (usually, curbs, railings, and utilities) that are added after the concrete deck has attained sufficient strength to support them. Table 17.17 Modular Ratio for Composite Girders with Live Loads

2000 – 2400 2500 – 2900 3000 – 3900 4000 – 4900 5000 or more

15 12 10 8 6

*Es ¼ elastic modulus of the steel Ec ¼ elastic modulus of the concrete

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Example—Stress Calculations for a Composite Girder: The following illustrates the procedure for determining flexural stresses in a composite welded girder for factored loads. The girder is assumed to be fabricated of M270, Grade 50, steel, with yield point Fy ¼ 50 ksi. It will not be shored during placement of the concrete deck. For the concrete, the 28-day compressive strength is assumed to be f 0 ¼ 3:25 ksi, n ¼ 10 for live loads, n ¼ 30 for dead loads. Dimensions, section properties, and bending moments are given in the following: The section properties of the steel girder alone are determined first. For the purpose, the moment of inertia I1-1 of the steel section (Fig. 17.11a) is calculated with respect to the bottom of the girder. Then, the moment of inertia INA with respect to the neutral axis is computed. Next, the section prop- erties of the composite section (Fig. 17.11b) are calculated. Stresses in the concrete are small, since the steel girder carries the weight of the deck. Specified minimum compressive strength of concrete deck f 0 , psi Modular ratio n ¼ Es =E

Fig. 17.11 Sections of composite plate girder: (a) steel section alone; (b) composite section.

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Fatigue Design of Bridge Members All members and connections should be designed so that maximum stresses induced by loads are less than allowable stresses and also so that the range of stresses induced by variations in service loads is less than the allowable fatigue stress range. If a member is entirely in compression and never is subjected to tensile stresses, a fatigue check is not required. Fatigue is an important consideration in design of all bridge components but may be especially critical for welded girders. Welding leaves residual stresses in welded regions due to heat input during the welding process and subsequent differential cooling. The types of connections that are most commonly used with welded plate girders and that type is given in Table 17.18a. Table 17.19 gives the allowable stress ranges for various stress categories. Table 17.20 lists allowable stress cycles for various types of roads and bridge members. (“Economical and

Fatigue-Resistant

Steel

Orthotropic-Deck Bridges An orthotropic deck is, essentially, a continuous, flat steel plate, with stiffeners (ribs) welded to its underside in a parallel or rectangular pattern. The term orthotropic is shortened from orthogonal anisotropic, referring to the mathematical theory used for the flexural analysis of such decks. When used on steel bridges, orthotropic decks are usually joined quasi-monolithically, by welding or highstrength bolting, to the main girders and floor beams. They then have a dual function as Live loads roadway and as structural top flange. The combination of plate or box girders with orthotropic decks allows the design of bridges of considerable slenderness and of nearly twice the span reached by girders with concrete decks. The

Fig. 17.12 Fatigue stress categories for some commonly used connections (see Table 17.13). In (c), category C applies also to transverse loading. FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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most widespread application of orthotropic decks is on continuous, two- to five-span girders on lowlevel river crossings in metropolitan areas, where approaches must be kept short and grades low. This construction has been used for main spans up to 1100 ft in cable-stayed bridges and up to 856 ft without cable stays. There also are some spectacu- lar high-level orthotropic girder bridges and some arch and suspension bridges with orthotropic stiff- ening girders. On some of the latter, girders and deck have been combined in a single lens-shaped box section that has great stiffness and low aero- dynamic resistance. Box Girders Single-web or box girders may be used for orthotropic bridges. Box girders are preferred if structure depth is restricted. Their inherent stiff- ness makes it possible to reduce, or to omit, unsightly transverse bracing systems. In cross section, they usually are rectangular, occasionally trapezoidal. Minimum dimensions of box girders are controlled by considerations of accessibility and ease of fabrication. Wide decks are supported by either single box girders or twin boxes. Wide single boxes have been built with multiple webs or secondary interior trusses. Overhanging floor beams sometimes are supported by diagonal struts. Depth-Span Ratios Girder soffits are parallel to the deck, tapered, or curved. Parallel flanges, sometimes with tapered side spans, generally are used on unbraced girders with depth-to-main-span ratios as low as 1 : 70. Parallel-flange unbraced girders are practically restricted to high-level structures with unrestricted clearance. Unbraced low-level girders usually are designed with curved soffits with minimum depth- tomain-span ratios of about 1 : 25 over the main piers and 1 : 50 at the shallowest section. Cable-Suspended Systems with Orthotropic Decks Main spans of bridges may have girders suspended from or directly supported by cables that are hung from towers, or pylons. The cables are curved if the

Table 17.18 Fatigue Stress Categories for Bridge Members (a) Stress Categories for Typical Connections Type of connection

Figure No.

Toe of transverse stiffeners Butt weld at flanges Gusset for lateral bracing

Stress

Category

17.13a Tension or reversal

C

17.13b Tension or reversal

B

17.13c

Tension or reversal

B

(assumed groove weld, R

24 in)

Flange to web

17.13d

Shear

F

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(b) Stress Categories for Weld Conditions in Fig. 17.13c Weld condition*

Category

Unequal thickness; reinforcement in place

E

Unequal thickness; reinforcement removed

D

Equal thickness; reinforcement in place

C

Equal thickness; reinforcement removed

B

(c) Stress Categories for Radii R in Fig. 17.13c Category for welds R, in†

Fillet

Groove

24 or more

d

b

From 6 to 24

d

c

From 2 to 6

d

d

2 or less

E

e

* For transverse loading, check transition radius for possible assignment of lower category. † Also applies to transverse loading.

girders are suspended at each floor beam (suspenM270, Grade 50. Minimum thickness is seldom less sion bridges); otherwise they are straight (cable than 7 stayed bridges). In cable-stayed bridges, the cables may extend from the pylons to the connections with the girders in tiers, parallel to each other (harped), or in a bundle pattern (radiating from the pylons). See Fig. 17.24. Each cable stay adds one degree of statical in- determinancy to a system. To make the actual conditions conform to design assumptions, the cable length must be adjustable either at the anchorages to the girders or at the saddles on the towers. (See also Art. 17.16.) Steel Grades The steel commonly used for orthotropic plates is weldable high-strength, low-alloy structural steel

under heavy wheel loads. The maximum thickness seldom exceeds 3⁄ in because of the decrease in permissible working stresses of high-strength low- alloy steel and the increase of fillet- and butt-weld sizes for plates of greater thickness.

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Floor Beams If, as in most practical cases, the deck spans transversely between main girders, transverse ribs are replaced by the floor beams, which are then built up of inverted T sections, with the deck plate acting as top flange. Floor-beam spacings are preferably kept constant on any given structure. They range from less than 5 ft to over 15 ft. Longer spacings have been suggested for greater economy. Table 17.19 Allowable Fatigue Stress Range FSR*, ksi, for Bridge Members

For Structures with Redundant Load Paths† Category

For 100,000 Cycles

For 500,000 Cycles

For 2,000,000 Cycles

For over 2,000,000 Cycles

A B C

63S 49 35.5

37S 29 21

24S 18 13

D E F G

28 22 16 15

16 13 9.2 12

10 8 5.8 9

24S 16 10 12‡ 7 4.5 2.6 8

Nonredundant-Load-Path Structures Category

For 100,000 Cycles

For 500,000 Cycles

For 2,000,000 Cycles

For over 2,000,000 Cycles

A B C

50S 39 28

29S 23 16

D Ex F

22 17 12

13 10 9

24S 16 10 12‡ 8 6 7

24S 16 9 11‡ 5 2.3 6

* The range of stress is defined as the algebraic difference between the maximum stress and the minimum stress. Tension stress is considered to have the opposite algebraic sign from compression stress. † Structure types with multiload paths where a single fracture in a member cannot lead to the collapse. For example, a simple supported single-span multibeam bridge or a multielement eyebar truss member has redundant load path. r welds on girder webs or flanges. x Partial-length welded cover plates should not be used on flanges more than 0.8 in thick for nonredundant-load path structures.

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S For unpainted weathering steel, A709, the category ‘A’ allowable FSR values values shown, see AASHTO Specifications.

are less then

Ribs These are either open (Fig. 17.13a) or closed (Fig. 17.13b). The spacing of open ribs is seldom less than 12 in or more than 15 in. The lower limit is determined by accessibility for fabrication and maintenance, the upper by considerations of deck-plate stiffness. To reduce deformations of the surfacing material under concentrated traffic loads, some specifications require the plate Usually, the longitudinal ribs are made continu- ous through slots or cutouts in the floor-beam webs to avoid a multitude of butt welds. Rib splices can then be coordinated with the transverse deck splices. Closed ribs, because of their greater torsional rigidity, give better load distribution and, other things being equal, require less steel and less weld- ing than open ribs. Disadvantages of closed ribs thickness to be not less than 1 of the spacing are their inaccessibility for inspection and maintenance and more complicated splicing details. between open ribs or between the weld lines of closed ribs. There have also been some difficulties in defining the weld between closed ribs and deck plate. Table 17.20 Allowable Stress Cycles for Bridge Members Main (longitudinal) load-carrying members Type of road Freeways, expressways ,

Case I

ADTT* 2500 or more

Truck loading 2,000,000‡

Lane loading† 500,000

major highways, and streets

* Average daily truck traffic (one direction). † Longitudinal members should also be checked for truck loading. ‡ Members should also be investigated for fatigue when over 2 million stress cycles are produced by a single truck on the bridge with load distributed to the girders as designated for traffic lane loading. Fabrication Orthotropic decks are fabricated in the shop in panels as large as transportation and erection facilities permit. Deck-plate panels are fabricated by butt-welding available rolled plates. Ribs and floor beams are fillet-welded to the deck plate in upside- down position. Then, the deck is welded to the girder webs. It is important to schedule all welding sequences to minimize distortion and locked-up stresses. The most effective method has been to fit up all components of a panel —deck plate, ribs, and floor beams—before starting any welding, then to place the fillet welds from rib to rib and from floor beam

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to floor beam, starting from the panel center and uniformly proceeding toward the edges. Since this sequence practically requires manual welding throughout, American fabricators prefer to join the ribs to the deck by automatic fillet welding before assembly with the floor beams. After slipping the floor -beam webs over the ribs, the fabricators weld manually only the beam webs to the deck. This method requires careful preevaluation of rib distortions, wider floor-beam slots, and conse- quently more substantial or only one-sided rib-tofloor-beam welds. Analysis Stresses in orthotropic decks are considered as resulting from systems:

a

superposition of

four

static

System I consists of the deck plate considered as an isotropic plate elastically supported by the ribs

Fig. 17.13 Rib shapes used in orthotropic-plate decks. (Fig. 17.14a). The deck is subject to bending from wheel loads between the ribs. System II combines the deck plate, as transverse element, and the ribs, as longitudinal elements. The ribs are continuous over, and elastically supported by, the floor beams (Fig. 17.14b). The orthotropic analysis furnishes the distribution of concentrated (wheel) loads to the ribs, their flexural and torsional stresses, and thereby the axial and torsional stresses of the deck plate as their top flange. System III combines the ribs with the floor beams and is treated either as an orthotropic system or as a grid (Fig. 17.14c). Analysis of this system furnishes the flexural stresses of the floor beams,

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Fig. 17.14 Orthotropic-plate deck may be considered to consist of four systems: (a) Deck plate supported on ribs; (b) rib-deck T beams spanning between floor beams; (c) floor beam with deck plate as top flange, supported on girders; (d) girder with deck plate as t op flange. including the stresses the deck plate receives as their top flange. System IV comprises the main girders with the orthotropic deck as top flange (Fig. 17.14d). Axial stresses in the deck plate and ribs and shear stresses in the deck plate are obtained from the flexural and torsional analysis of the main girders by conventional methods. Theoretically, the deck plate should be designed for the maximum principal stresses that may result from the simultaneous effect of all four systems. Practically, because of the rare coincidence of the maxima from all systems and in view of the great inherent strength reserve of the deck as a membrane (second-order stresses), a design is generally satisfactory if the stresses from any one system do not exceed 100% of the ordinarily permissible working stresses and 125% from a combination of any two systems. In the design of long-span girder bridges, special attention must be given to buckling stability of deep webs and of the deck. Also, consideration should be given to conditions that may arise at intermediate stages of construction.

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Steel-Deck Surfacing

All traffic -carrying steel decks require a covering of some nonmetallic material to protect them from accidental damage, distribute wheel loads, com- pensate for surface irregularities, and provide a nonskid, plane riding surface. To be effective, the surfacing must adhere firmly to the base and resist wear and distortion from traffic under all con - ditions. Problems arise because of the elastic and thermal properties of the steel plate, its sensitivity to corrosion, the presence of bolted deck splices, and the difficulties of replacement or repair under traffic. The surfacing material usually is asphaltic. Strength is provided by the asphalt itself (mastic- type pavements) or by mineral aggregate (asphalt- concrete pavement). The usefulness of mastic-type pavements is restricted to a limited temperature range, below which they become brittle and above which they may flow. The effectiveness of the mineral aggregate of asphalt concrete depends on careful grading and adequate compaction, which on steel decks sometimes is difficult to obtain. Asphalt properties may be improved by admixtures of highly adhesive or ductile chemicals of various plastics families. (“Design Manual for Orthotropic Steel Plate Deck Bridges,” American Institute of Steel Construction, Chicago, Ill.; F. S. Merritt and R. L. Brockenbrough, “Structural Steel Designers’ Hand- book,” 2nd ed., McGraw-Hill, Inc., New York.) Truss Bridges Trusses are lattices formed of straight members in triangular patterns. Although truss-type construction is applicable to practically every static system, the term is restricted here to beam-type structures: simple spans and continuous and hinged (cantilever) structures. For typical single- span bridge truss configurations, see Fig. 6.50. For the stress analysis of bridge trusses, see Arts. 6.46 through 6.50. Truss bridges require more field labor than comparable plate girders. Also, trusses are more costly to maintain because of the more complicated makeup of members and poor accessibility of the exposed steel surfaces. For these reasons, and as a result of changing aesthetic preferences, use of trusses is increasingly restricted to long-span bridges for which the relatively low weight and consequent easier handling of the individual members are decisive advantages. The superstructure of a typical truss bridge is composed of two main trusses, the floor system, a top lateral system, a bottom lateral system, cross frames, and bearing assemblies. Decks for highway truss bridges are usually concrete slabs on steel framing. On long-span railway bridges, the tracks are sometimes mounted directly on steel stringers, although continuity of the track ballast across the deck is usually preferred. Orthotropic decks are rarely used on truss bridges. Most truss bridges have the deck located between the main trusses, with the floor beams framed into the truss posts. As an alternative, the deck framing may be stacked on top of the top chord. Deck trusses have the deck at or above top- chord level (Fig. 17.15); through trusses, near the FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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bottom chord (Fig. 17.16). Through trusses whose depth is insufďŹ cient for the installation of a top lateral system are referred to as half through trusses or pony trusses. Figure 17.16 illustrates a typical cantilever truss bridge. The main span comprises a suspended

Fig. 17.15 Deck truss bridge. span and two cantilever arms. The side, or anchor, arms counterbalance the cantilever arms. Sections of truss members are selected to ensure effective use of material, simple details for connections, and accessibility in fabrication, erec- tion, and maintenance. Preferably, they should be symmetrical. In bolted design, the members are formed of channels or angles and plates, which are combined into open or half-open sections. Open sides are braced by lacing bars, stay plates, or perforated cover plates. Welded truss members are formed of plates. Figure 17.17 shows typical truss-member sections. For slenderness restrictions of truss members, see Art. 17.7. The design strength of tensile members is controlled by their net section, that is, by the section area that remains after deduction of rivet

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or bolt holes. In shop-welded ďŹ eld-bolted con- struction, it is sometimes economical to build up tensile members by butt-welding three sections of different thickness or steel grades. Thicker plates or higher-strength steel is used for the end sections to compensate for the section loss at the holes. The permissible stress of compression members depends on the slenderness ratio (see Art. 9.11). Design speciďŹ cations also impose restrictions on the width-to-thickness ratios of webs and cover plates to prevent local buckling. The magnitude of stress variation is restricted for members subject to stress reversal during passage of a moving load (Art. 9.20). All built-up members must be stiffened by diaphragms in strategic locations to secure their squareness. Accessibility of all members and con-

Fig. 17.16 Typical cantilever truss bridge.

nections for fabrication and maintenance should be a primary design consideration. Whenever possible, each web member should be fabricated in one piece reaching from the top to the bottom chord. The shop length of chord mem- bers may extend over several panels. Chord splices should be located near joints and may be in- corporated into the gusset plates of a joint. In most trusses, members are joined by bolting or welding with gusset plates. Pin connections, which were used frequently in earlier truss bridges, are now the exception. As a rule, the centerlines or center-of-gravity lines of all members converging at a joint intersect in a single point (Fig. 17.19).

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Stresses in truss members and connections are divided into primary and secondary stresses. Pri- mary stresses are the axial stresses in the members of an idealized truss, all of whose joints are made with frictionless pins and all of whose loads are applied at pin centers. Secondary stresses are the stresses resulting from the incorrectness of these assumptions. Somewhat higher stresses are al- lowed when secondary stresses are considered. (Some specifications require computation of the flexural stresses in compression members caused by their own weight as primary stresses.) Under ordinary conditions, secondary stresses must be computed only for members whose depth is more than one-tenth of their length. (F. S. Merritt and R. L. Brockenbrough, “Structural Steel Designers’ Handbook,” hill.com).)

2nd

ed., McGraw-Hill, Inc., New

York (books.mcgraw-

Suspension Bridges These are generally preferred for spans over 1800 ft, and they compete with other systems on shorter spans. The basic structural system consists of flexible main cables and, suspended from them, stiffening girders or trusses (collectively referred to as “stiffening beams”), which carry the deck framing. The vehicular traffic lanes are as a rule accom- modated between the main supporting systems.

Fig. 17.17 Typical sections used in steel bridge trusses.

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Sidewalks may lie between the main systems or cantilever out on both sides. Stiffening Beams Stiffening beams distribute concentrated loads, reduce local deections, act as chords for the lateral system, and secure the aerodynamic stability of the structure. Spacing of the stiffening beams is controlled by the roadway width but is seldom less longitudinal compression equal to the horizontal component of the cable tension. Continuity Single-span suspension bridges are rare in engin- eering projects. They may occur in crossings of narrow gorges where the rock on both sides provides a reliable foundation for high-level cable anchorages. than the span. The overwhelming majority of suspension bridges have main cables draped over two towers. Stiffening beams may be either plate girders, box girders, or trusses. On major bridges, their Such bridges consist, thus, of a main span and two depth is at least of the main span. side spans. Preferred ratios of side span to main span are 1 : 4 to 1 : 2. Ratios of cable sag to main span are preferably in the range of 1 : 9 to 1 : 11, seldom Anchorages The main cables are anchored in massive concrete blocks or, where rock subgrade is capable of resisting cable tension, in concrete-ďŹ lled tunnels. Or the main cables are connected to the ends of the stiffening girders, which then are subjected to less than 1 : 12. If the side spans are short enough, the main cables may drop directly from the tower tops to the anchorages, in which case the deck is carried to the abutments on independent, single-span plate girders or trusses. Otherwise, the suspension system is extended over both side spans to the

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38

Fig. 17.18

Box girder stiffening beam—Carquinez Bridge.

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Fig. 17.19 Pin joint in the lower chord of a bridge truss at a support.

next piers. There, the cables are deflected to the anchorages. The first system allows the designer some latitude in alignment, for example, curved roadways. The second requires straight side spans, in line with the main span. It is the common system for suspension bridges that are links in a chain of FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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multiple-span crossings. When side spans are not suspended, the stif- fening beam is of course restricted to the main span. When side spans are suspended, the stiffen- ing beams of the three spans may be continuous or discontinuous at the towers. The spans are typic- ally restrained to the tower at the ends. Continuity of stiffening beams is required in self-anchored suspension bridges, where the cable ends are anchored to the stiffening beams. Cable Systems The suspenders between main cables and stiffening beams are usually equally spaced and vertical. Main cables, suspenders, and stiffening beams (girders or trusses) are usually arranged in vertical planes, symmetrical with the longitudinal bridge axis. Bridges with inward- or outward-sloping cables and suspenders and with offset stiffening beams are less common. Three-dimensional stability is provided by top and bottom lateral systems and transverse frames, similar to those in ordinary girder and truss bridges. Rigid roadway decks may take the place of either or both lateral systems, especially in double decked trusses. In the United States, the main cables are usually made up of 6-gage galvanized bridge wire of 220 to 225 ksi ultimate and 82 to maximum 90 ksi working stress. The wires are usually placed parallel but sometimes in strands and compacted and wrapped with No. 9 wire. In Europe, strands containing elaborately shaped heat-treated cast-steel wires are sometimes used. Strands must be prestretched. They have a lower and less reliable modulus of elasticity than parallel wires. The heaviest cables, those of the Golden Gate Bridge, are about 36 in in diameter. Twin cables are used if larger sections are required. Suspenders may be eyebars, rods, single steel ropes, or pairs of ropes slung over the main cable. Connections to the main cable are made with cable bands. These are cast steel whose inner faces are molded to fit the main cable. The bands are clamped together with high-strength bolts. Floor System In the design of the floor system, reduction of dead load and resistance to vertical air currents should be the governing considerations. The deck is usu- ally lightweight concrete or steel grating partly filled with concrete with the exception of box sections which usually have a wearing surface. Expansion joints should be provided every 100 to 120 ft to prevent mutual interference of deck and main structure. Stringers should be made compo- site with the deck for greater strength and stiffness. Floor beams may be plate girders or trusses, depending on available clearance. With trusses, wind resistance is less. Towers The towers may be portal type, multistory, or diagonally braced frames (Fig. 17.20). They may be of cellular construction, made of steel plates and

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Fig. 17.20 Types of towers used for long suspension bridges.

shapes, or steel lattices, or of reinforced concrete. The substructure below the “spray” line is concrete. The base of steel towers is usually fixed, but it may be hinged. (Hinged towers, however, offer some erection difficulties.) The cable saddles at the top of fixed towers are sometimes placed on rollers to reduce the effect on the towers of unbalanced cable deflections. Cable bents can be considered as short towers, either fixed or hinged, whose axis coincides with the bisector of the angle formed by the cable. Analysis For gravity loads, the three elements of a suspen- sion bridge in a vertical plane—the main cable or chain, the suspenders, and the stiffening beam— are considered as a single system. The system of discrete suspenders often is idealized as one of continuous suspension. The stiffening beam is assumed stressless under dead load, a condition approximated by appropriate methods of erection. Moments and shears are produced by that part of the live load not taken up by the main cable through the suspenders. Also, moments and shears result from changes in cable length and sag due to tempera- ture variations or unbalanced loadings of adjacent spans. Deflections of the stiffening beam are strictly elastic; that is, neglecting the effect of shear, the curvature at any section of the elastic line of the loaded beam is proportional to the FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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bending moment divided by the moment of inertia of that section. The suspenders are subject to tension only. Their elongation under live load is usually neglected in the analysis. The main cable typically is assumed to have no flexural stiffness and to be subject to axial tension only. Its shape is that of a funicular polygon of the applied forces (which include the dead weight of the cable). The pole distance H, lb, which is the horizontal component of the cable tension, is constant for a given loading and a given sag. The shape of the cable under given loads, that is, its ordinate y, ft, and slope tan a at any point with abscissa x, ft, can be expressed in terms of moment Mo, ft-kips, and shear V, kips, that a simple beam of the same span L, ft, as the cable would have under the same load (Fig. 17.21). The shape of the cable under its own weight without suspended load would be a catenary; under full dead load, the cable shape is usually closer to a parabola. The difference is small. Concentrated or sectionally uniform live load superimposed on the dead load subjects the cable to additional strain and causes it to adjust its shape to the changed load configuration. The resulting deformations are not exactly proportional to the additional loading; their magnitude is influenced by the already existing dead-load stresses. If Mo is the bending moment of the stiffening beam under the applied load but without cooperation of the cable, the beam moment M with co- operation of the cable will be M ¼ Mo Hy

(17:17)

More specifically, using subscripts D and L, res- pectively, for dead and live load and considering that yL ¼ yD þ Dy

(17:18)

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Fig. 17.21 Stresses in cable and stiffening beam of a suspension bridge. one gets the following expression for the dead- plus live-load bending moment of the beam (see Fig. 17.21b):

M ¼ MD þ ML ¼ MD0 þ ML, 0 (HD þ HL )(yD þ Dy)

(17:19)

But, since MD ¼ MD0 2 HDyD ¼ 0, because dead load (ideally), M ¼ ML0 (HD þ HL )Dy HL yD

the stiffening beam has no bending moment under

(17:20)

This is the basic equation of the cable-beam system. In this equation, ML0, HD, and yD are given. HL and Dy must be so determined that the conditions of static equilibrium of all forces and geometric compatibility of all deformations are satisfied throughout the system.

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The mathematically exact solution of the prob- lem is known as the deflection theory. A less exact, older theory is known as the elastic theory. Besides these, there are several approximate methods based on observed regularities in the behavior of suspension bridges, which are sufficiently accurate to serve for preliminary design. Wind Resistance Wind acting on the main cables and on part of the suspenders is carried to the towers by the cables. Wind acting on the deck, stiffening beams, and live load is resisted mainly by the lateral bracing system and slightly by the cables because of the gravity component resulting from any elastic lateral deflection of the main supporting system. Oscillations of the structure may be generated by live load, earthquake, or wind. Live-load vibrations are insignificant in major bridges. (N. C. Raab and H. C. Wood, “Earthquake Stresses in the San Francisco – Oakland Bay Bridge,” Transactions of the American Society of Civil Engineers, vol. 106, 1941). Oscillations due to wind, however, can become dangerous if excessive amplitudes build up; that is, if the exciting impulses approach the natural frequency of the structure. Oscillating wind forces are caused by eddies, which may be generated outside the structure or by the structure itself, especially on the lee side of large plates. Oscillations of the structure may be purely flexural, purely torsional, or coupled (flutter), the last two being the more dangerous. Methods used to predict the aerodynamic behavior of suspension bridges include: Mathematical analysis of the natural frequency of the structure in flexure and torsion [F. Bleich, C. B. McCullogh, R. Rosecrans, and G. S. Vincent, “Mathematical Theory of Vibration in Suspension Bridges,” Government Printing Office, Washing - ton, D.C.: A. G. Pugsley, “Theory of Suspension Bridges,” Edward Arnold (Publishers) Ltd., London]. Wind-tunnel tests on scale models of the entire Application of Steinman’s criteria (these are controversial) (D. B. Steinman, “Rigidity and Aerodynamic Stability of Suspension Bridges,” with discussion, Transactions of the American Society of Civil Engineers, vol. 110, 1945). Tuned mass dampers and tuned liquid dampers have been used to decrease the amplitude of vortex oscillations. Tower Stresses The towers must resist the forces imposed on them by the main cables in addition to the gravity and wind loads acting directly. The following forces must be considered: The vertical components of the main cables in main and side spans under dead load, live load, temperature change, seismic, and wind acting on the main cables, both parallel and transverse to the bridge axis; reactions to longitudinal cable movements due to unbalanced loading. These reactions will develop unless the movements are taken up by hinges or friction-free roller nests. Theoretically, the magnitude of these movements will be affected by the flexural resistance Q of the towers, but this effect, being comparatively small, is usually neg- lected. Movement of the tower top generates bending moments. These increase from the top to the bottom at the rate of Mx ¼ Vy þ Qx

(17:21)

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where V ¼ vertical cable reaction x ¼ distance below top y ¼ horizontal deflection at x Q ¼ horizontal resistance at top The magnitude of Q is such that the total deflection equals the longitudinal cable movement. It is found by solving the differential equation for the elastic curve of the tower axis. Thus, Q y ¼ A sin cx þ B cos cx V x

(17:22)

structure or of typical sections (“Aerodynamic Stability of Suspension Bridges with Special Reference to the Tacoma Narrows Bridge,” University of Washington Engineering Experiment Station Bulle- tin 116). Q sin cx ¼ V c cos cL x in which c ¼ pffiVffiffiffi=ffiffiEffiffiffiIffiffi, I ¼ moment of inertia, and E ¼ modulus of elasticity of tower, if the towers have constant cross sections. The bending moment at x is Q sin cx Mx ¼ c cos cL

(17:23)

where L ¼ height of tower. If, as is usual, the tower cross section varies in several steps, the coefficients A and B in Eq. (17.22) differ from section to section. They are found from the continuity conditions at each step. Anchorages and footings should be designed for adequate safety against uplift, tipping, and sliding under any possible combination of acting forces. (S. Hardesty and H. E. Wessman, “Preliminary Design of Suspension Bridges,” Transactions of the American Society of Civil Engineers, vol. 104, 1939; R. J. Atkinson and R. V. Southard, “On the Problem of Stiffened Suspension Bridges and Its Treatment by Relaxation Methods,” Proceedings of the Institute of Civil Engineers, 1939; C. D. Crosthwaite, “The Corrected Theory of the Stiffened Suspension Bridge,” Proceedings of the Institute of Civil Engineers, 1946; Ling-Hi Tsien, “A Simplified Method of Analyzing Suspension Bridges,” Transactions of the American Society of Civil Engineers, vol. 114, 1947; F. S. Merritt and R. L. Brockenbrough, “Structural Steel Designers’ Handbook,” 2nd ed., McGraw- Hill, Inc., New York (books.mcgraw-hill.com).) Cable-Stayed Bridges* The cable-stayed bridge, also called the stayed- girder (or truss), has come into wide use since about 1950 for medium- and long-span bridges because of its economy, stiffness, aesthetic qualities, and ease of erection without falsework. Design of cable-stayed bridges utilizes taut cables

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connecting pylons to a span to provide intermedi- ate support for the span. This principle has been understood by bridge engineers for at least the last two centuries, as indicated by the bridge in

Fig. 17.22 Cable-stayed chain bridge (Hatley system, 1840). effects have to be contended with, as in the Salazar Bridge. Characteristics of Cable-Stayed Bridges The cable-stayed bridge offers a proper and eco- nomical solution for bridge spans intermediate between those suited for deck girders (usually up to 600 to 800 ft but requiring extreme depths, up to 33 ft) and the longer-span suspension bridges (over 1000 ft). The cable-stayed bridge thus finds application in the general range of 600- to 1600-ft spans but may be competitive in cost for spans as long as 2900 ft. A cable-stayed bridge has the advantage of greater stiffness over a suspension bridge. Use of single or multiple box girders gains large torsional and lateral rigidity. These factors make the structure stable against wind and aerodynamic effects. The true action of a cable-stayed bridge (Fig. 17.23) is considerably different from that of a suspension bridge. As contrasted with the rela- tively flexible cable of the latter, the inclined, taut cables of the cable-stayed structure furnish rela- tively stable point supports in the main span. Deflections are thus reduced. The structure, in effect, becomes a continuous girder over the piers, with additional intermediate, elastic (yet relatively stiff) supports in the span. As a result, the girder 1 Fig. 17.22. The Roeblings used cable stays as may be shallow. Depths usually range supplementary stiffening elements in the famous Brooklyn Bridge (1883). Many recently built and proposed suspension bridges also incorporate taut cable stays when dynamic (railroad) and long-span * Extracted with permission from F. S. Merritt and R. L. Brockenbrough, “Structural Steel Designers Handbook,” 2nd ed., McGraw-Hill, Inc., New York. 1

the main span, sometimes even as small as

1⁄ 100 the span Cable forces are usually balanced between the main and flanking spans, and the structure is internally anchored; that is, it requires no massive masonry anchorages. Analogous to the selfanchored suspension bridge, second-order effects of the type requiring analysis by a deflection theory are of relatively minor importance. Thus, static

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Fig. 17.23 Axial forces in a cable-stayed girder of (a) self-anchored, (b) fully anchored, (c) partly anchored cable-stayed bridges. analysis is simpler, and the structural behavior may be more clearly understood. The above remarks apply to the common, self- anchored type of cable-stayed bridges, character- ized by compression in the main bridge girders (Fig. 17.23a). It is possible to conceive of the opposite extreme of a fully anchored (earth- anchored) cable bridge in which the main girders are in tension. This could be achieved by pinning the girders to the abutments and providing sliding joints in the side-span girders adjacent to the pylons (Fig. 17.23b). The fully anchored system is stiffer than the self-anchored system and may be advantageously analyzed by second-order deflec- tion theory because (analogous to suspension bridges) bending moments are reduced by the deformations. A further increase in stiffness of the fully anchored system is possible by providing piers in the side spans at the cable attachments (Fig. 17.24). This is advantageous if the side spans are not used for boat traffic below, and if, as is often the case, the side spans cross over low water or land (Knieb - ru¨ cke at Du¨ sseldorf, Fig. 17.27i). A partly anchored cable-stayed system (Fig. 17.23c) has been proposed wherein some of the cables are self-anchored and some fully anchored. The axial forces in the girders are then partly compression and partly tension, but their magni- tudes are considerably reduced.

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Fig. 17.24 Anchorage of side-span cables at piers and abutments increases stiffness of the center span.

Fig. 17.25 Classification of cable -stayed bridges by arrangement of cables. (From A. Feige, “Evolution of German Cable-Stayed Bridges: An Overall Survey,” Acier-Stahl-Steel, vol. 12, 1966.) Classification of Cable-Stayed Bridges The relatively small diameter of the cables and the absolute minimum amount of overhead structure required are the principal features contributing to the excellent architectural appearance of cablestayed bridges. The functional character of the structural design produces, as a by-product, a graceful and elegant solution for a bridge crossing. This is encouraged by the wide variety of possible types, using single or multiple cables, including the bundle, harp, fan, and star configurations, as seen in elevation (Fig. 17.25). These may be symmetrical or asymmetrical. A wide latitude of choice of cross section of the bridge at the pylons is also possible (Fig. 17.26). The most significant distinction occurs between those with twin pylons (individual, portal, or A frame) and those with single pylons in the center of the roadway. The single pylons usually require a large box girder to resist the torsion of eccentric loadings, and the box is most frequently of steel with an anisotropic steel deck. The singlepylon type is advantageous in allowing a clear unobstructed view from cars passing over the bridge. The pylons may (as with suspension-bridge towers) be either fixed or pinned at their bases. In the case of fixity, this may be either with the girders or directly with the pier. Some details of cablestayed bridges are shown in the elevations and cross sections in Fig. 17.27.

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Fig. 17.26 Shapes of pylons used for cable-stayed bridges. (a) Portal-frame type with top cross member; (b) pylon fixed to pier and without top cross member; (c) pylon fixed to superstructure and without top cross member; (d) axially located pylon fixed to the superstructure; (e) A-shaped pylon; ( f ) laterally offset pylon fixed to a pier; ( g) diamond -shaped pylon. (From A. Feige, “Evolution of German Cable- Stayed Bridges: An Overall Survey,” Acier-Stahl-Steel, vol. 12, 1966.)

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Fig. 17.27 Some cable-stayed bridges, with cross sections taken at pylons. (a) Bu¨ chenauer crossing at Bruchsal, 1956; (b) Ju¨ licherstrasse crossing at Du¨ sseldorf, 1964; (c) bridge over the Stro¨ msund, Sweden, 1955; (d) bridge over the Rhine near Ludswigshafen,

Maxau,

1966; (e) bridge on the elevated highway at

1969; ( f ) Severin Bridge, Cologne, 1959; (g) bridge over the Rhine near Levenkusen, 1965; (h) North Bridge at Du¨ sseldorf, 1958; (i) Kniebru¨ cke at Du¨ sseldorf, 1969; ( j) bridge over the Rhine at Rees, 1967.

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Cable-Stayed Bridge Analysis The static behavior of a cable-stayed girder can best be gaged from the simple, two-span example of Fig. 17.28. The girder is supported by one stay cable in each span, at E and F, and the pylon is fixed to the girder at the center support B. The static system has two internal cable redundants and one external support redundant. If the cable and pylon were infinitely rigid, the structure would behave as a continuous four-span beam AC on five rigid supports A, E, B, F, and C. The cables are elastic, however, and correspond to springs. The pylon also is elastic but much stiffer because of its large cross section. If cable stiffness is reduced to zero, the girder assumes the shape of a deflected two -span beam ABC. Cable-stayed bridges of the nineteenth century differed from those of the 1960s in that their tendons constituted relatively soft spring supports. Heavy and long, the tendons could not be stressed highly. Usually, the cables were installed with significant slack or sag. Consequently, large deflections occurred under live load as the sag was diminished. Modern cables have high-strength steel, are relatively short and taut, and their weight is low. Their elastic action may therefore be considered linear, and an equivalent modulus of elasticity may be used. The action of such cables then produces something more nearly like the four-span beam for a structure like the one in Fig. 17.28. If the pylon were hinged at its base connection with the girder at B, the pylon would act as a pendulum column. This would have an important effect on the stiffness of the system, for the spring support at E would become more flexible. In magnitude, the effect might exceed that due to the elastic stretch of the cables. In contrast, the elastic shortening of the tower has no appreciable effect.

Fig. 17.28 Deflected positions (dash lines) of a cable-stayed bridge. Relative girder stiffness plays a dominant role in the structural action. The girder tends to approach a beam on rigid supports A, E, B, F, C as girder stiffness decreases toward zero. With increasing girder stiffness, however, the action of the cables becomes minor and the bridge approaches a girder supported on its piers and abutments A, B, C. In a three-span bridge, a side-span cable connected to the abutment furnishes more rigid support to the main span than does a cable attached to some point in the side span. In Fig. 17.28, for example, the support of the load P in the position shown would be improved if the cable attachment at F were shifted to C. This explains why cables from the pylon top to the abutment are structurally more efficient, although not as aesthetically pleasing as other arrangements. The stiffness of the system is also affected by whether the cables are fixed at the towers (at D, for example, in Fig. 17.28) or whether they run con- tinuously over (or through) the towers. Most designs with more than one cable to a pylon from the main span require one of the cables to be fixed to the pylon and the others to be on movable saddle supports.

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The curves of maximum-minimum girder moments for all load variations usually show a large range of stress. Designs providing for the corresponding normal forces in the girder may require large variation in cross sections. By pre- stressing the cables or by raising or lowering the support points, it is possible to achieve a more uniform and economical moment capacity. The amount of prestressing to use for this purpose may be calculated by successively applying a unit force in each cable and drawing the respective moment diagrams. Then, by trial, the proper multiples of each force are determined so that when their moments are superimposed on the maximum- minimum moment diagrams, an optimum balance results. Static Analysis— Elastic Theory Cable-stayed bridges may be analyzed by the general method of indeterminate analysis with the equations of virtual work. The degree of internal redundancy of the sys- tem depends on the number of cables, types of

Fig. 17.29 Redundants in three-span cable-stayed bridges.

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connections (fixed or movable) of cables with the pylons, and the nature of the pylon connection at its base with the girder or pier. The girder is usually made continuous over three spans. Figure 17.29 shows the order of redundancy for various single- plane systems of cables. If the bridge has two planes of cables, two girders, and double pylons, it usually also must be provided with a number of intermediate cross diaphragms in the floor system, each of which is capable of transmitting moment and shear. The bridge may also have cross girders across the top of the pylons. Each cross member adds two redun- dants, to which must be added twice the internal redundancy of the single plane structure, and any additional reactions in excess of those needed for external equilibrium as a space structure. The redundancy of the space structure is very high, usually of the order of 40 to 60. Therefore, the methods of plane statics are normally used, except for larger structures. It is convenient to select as redundants the bending moments in the girder at those points where the cables and pylons join the girder. When these redundants are set equal to zero, an articulated, statically determinate truss base sys- tem is obtained (Fig. 17.30). When the loads are applied to this choice of base system, the stresses in the cables do not differ greatly from their final values, so the cables may be dimensioned in a preliminary way. Other approaches are also possible. One is to use the continuous girder itself as a statically in- determinate base system, with the cable forces as redundants. But computation is generally in- creased. A third method involves imposition of hinges, for example at a and b (Fig. 17.31), so placed as to

Fig. 17.30 Three-span cable-stayed bridge. (a) Girder is continuous over the three spans. (b) Insertion of hinges in the girder at cable-attachment points makes the structure statically determinate. form two coupled symmetrical base systems, each statically indeterminate to the fourth degree. The influence lines for the four indeterminate cable forces of each partial base system are at the same time also the influence lines of the cable forces in the real system. The two redundant moments Xa and Xb are treated as symmetrical and antisymmetrical group loads, Y ¼ Xa þ Xb, and Z ¼ Xa 2 Xb, to calculate influence lines for the 10-degree-indeterminate structure shown. Kern moments are plotted to determine maximum effects of combined bending and axial forces. Note that the bundle system in Fig. 17.29c and d generally has more favorable bending moments for long spans than does the harp system of Fig. 17.29e and f. Cable stresses also are somewhat lower for the bundle system because the steeper cables are more effective. But the concentration of cable forces at the top of the pylon introduces detailing and construction difficulties. When viewed at an angle, the bundle system presents aesthetic problems because of the different intersection angles when the cables are in two planes. Furthermore, fixity of the cables at pylons with the bundle system in Fig. 17.29c and d produces a wider range of stress than does a movable arrangement. FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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This can influence design for fatigue. Fig. 17.31 Insertion of hinges at a and b in the center span of a three-span continuous girder reduces the degree of indeterminancy. The secondary effect of creep of cables can be incorporated into the analysis. The analogy of a beam on elastic supports is changed thereby to a beam on linear viscoelastic supports. Static Analysis— Deflection Theory Distortion of the structural geometry of a cable- stayed bridge under action of loads is considerably less than in comparable suspension bridges. The influence on stresses of distortion is relatively small for cable-stayed bridges. In any case, the effect of distortion is to increase stresses, as in arches, rather than the reverse, as in suspension bridges. This effect for the Severin Bridge is 6% for the girder and less than 1% for the cables. Similarly for the Dusseldorf North Bridge, stress increase due to distortion amounts to 12% for the girders. The calculations, therefore, most expeditiously take the form of a series of successive corrections to results from first-order theory. The magnitude of vertical and horizontal displacements of the girder and pylons can be calculated from the first -order theory results. If the cable stress is assumed con- stant, the vertical and horizontal cable components V and H change by magnitudes DV and DH by virtue of the new deformed geometry. The first approximate correction determines the effects of these DV and DH forces on the deformed system, as well as the effect of V and H due to the changed geometry. This process is repeated until conver- gence, which is fairly rapid. Dynamic Analysis— Aerodynamic Stability The aerodynamic action of cable-stayed bridges is less severe than that of suspension bridges because of increased stiffness due to the taut cables and the widespread use of torsion box decks. Preliminary Design of Cable-Stayed Bridges In general, height of a pylon in a cable-stayed bridge is about 1⁄6 to 1⁄3 the span. Depth of girder Bridge Deck n Although cable-stiffened bridges usually incorporate an orthotropic steel deck with steel box girders, to reduce the dead load, other types of construction also are in use. For the Lower Yarra River Bridge in Australia, a concrete deck was specified to avoid site welding and to reduce the amount of shop fabrication. The Maracaibo Bridge likewise incorporates a concrete deck, and the Bridge of the Isles (Canada) has a concrete-slab deck supported on longitudinal and transverse steel box girders and steel floor beams. The Bu¨ chenauer Bridge also has a concrete deck. Use of a concrete deck in place of orthotropic-plate construction is largely a matter of local economics. ranges from 1⁄ 400 to 1⁄500 the span and is usually 8 to The cost of structure to carry the added dead load 14 ft averaging 11 ft. Wide box girders are mandatory for single- plane systems to resist the torsion of eccentric loads. Box girders, even narrow ones, are also de- sirable for double-plane systems to enable cable connections to be made without eccentricity. Single-web girders, however, are occasionally used. To achieve symmetry of cables at pylons the ratio of side to main spans should be about 3 : 7 where three cables are used on each side of the pylons, and about 2 : 5 where two cables are used. A proper balance of side-span length to main-span length must be established if uplift at the abut- ments is to be avoided. Otherwise, movable (pen- dulum-type) tiedowns must be provided at the abutments. The usual range of live-load deflections is from should be compared with the lower cost per square foot of the concrete deck and other possible advantages, such as better durability and increased stability against wind.

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(W. Podolny, Jr., and J. B. Scalzi, “Construction and Design of Cable-Stayed Bridges,” 2nd ed., John Wiley & Sons, Inc., New York (www.wiley.com); “Guidelines for Design of Cable-Stayed Bridges,” ASCE Committee on Cable-Stayed Bridges (www. asce.org).) Steel Arch Bridges A typical arch bridge consists of two or (rarely) more parallel arches or series of arches, plus necessary lateral bracing and end bearings, and the span. columns or hangers for supporting the deck Since elastic-theory calculations are relatively simple to program for a computer, a formal set is usually made for preliminary design after the general structure and components have been proportioned. Design Details for Cable-Stayed Bridges These structures differ from usual long-span girder bridges in only a few details. Towers and Floor System n The towers are composed basically of two parts: the pier (below the deck) and the pylon (above the deck). The pylons are frequently of steel box cross section, although concrete may also be used. framing. Types of arches correspond roughly to positions of the deck relative to the arch ribs. Bridges with decks above the arches and clear space underneath (Fig. 17.32a) are designed as open spandrel arches on thrust-resisting abut- ments. Given enough underclearance and ade- quate foundations, this type is usually the most economical. Often, it is competitive in cost with other bridge systems. Bridges with decks near the level of the arch bearings (Fig. 17.32b) are usually designed as tied arches; that is, tie bars take the arch thrust. End bearings and abutments are similar to those for girder or truss bridges. Tied arches compete in cost with through trusses in locations where underclearances are restricted. Arches sometimes are preferred for aesthetic reasons. Unsightly overhead laterals can be avoided by using arches with

Fig. 17.32 Basic types of steel arch bridges: (a) Open spandrel arch; (b) tied arch; (c) arch with deck at an intermediate level; (d) multiple-arch bridge. FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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sufficiently high moment of inertia to resist buckling. Bridges with decks at an intermediate level (Fig. 17.32c) may be tied, may rest on thrustresisting abutments, or may be combined structu- rally with side spans that alleviate the thrust of the main span on the main piers (Fig. 17.32d). Intermediate deck positions are used for long, highrising spans on low piers. Spans of multiple-arch bridges are usually structurally separated at the piers. But such bridges may also be designed as continuous structures. Hinges Whether or not hinges are required for arch bridges depends on foundation conditions. Abutment movements may sharply increase rib stresses. Fully restrained arches are more sensitive to small abutment movements (and temperature variations) than hinged arches. Flat arches are more sensitive than high arches. If foundations are not fully reli- able, hinged bearings should be used. Complete independence from small abutment movements is achieved by installing a third hinge, usually at the crown. This hinge may be either permanent or temporary during erection, to be locked after all dead-load deformations have been accounted for. Arch Analysis The elementary analysis of steel arches is based on the elastic, or first-order, theory, which assumes that the geometric shape of the center line remains constant, irrespective of the imposed load. This assumption is never mathematically correct. The effects of deviations caused by overall flattening of the arch due to the elastic rib shortening, elastic or inelastic displacements of the abutment, and local deformation due to live-load concentrations in- crease with initial flatness of t he arch. An effort is usually made to eliminate the dead-load part of the effect of rib shortening and abutment yielding during erection by jacking the legs of an arch toward each other or the crown section apart before final closure. Arches subject to substantial defor- mation must be checked by the second-order, or deflection, theory. For heavy moving loads, it is sometimes advantageous to assign the flexural resistance of the system to special stiffening girders or trusses, analogous to those of suspension bridges (Art. 17.15). The arches themselves are then subject, essentially, to axial stresses only and can be designed as slender as buckling considerations permit. Arch Design In general, steel arches must be designed for combined stresses due to axial loads and bending. The height-to-span ratio used for steel arches varies within wide limits. Minimum values are around 1 : 10 for tied arches, 1 : 16 for open spandrel arches. In cross section, steel arches may be Ishaped, box-shaped, or tubular. Or they may be designed as space trusses.

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Deck Construction The roadway deck of steel arch bridges is usually of reinforced concrete, often of lightweight concrete, on a framing of steel floor beams and stringers. To avoid undesirable cooperation with the primary steel structure, concrete decks either are provided with appropriately spaced expansion joints or prestressed. Orthotropic decks that combine the functions of traffic deck, tie bar, stiffening girder, and lateral diaphragm have been used on some major arch bridges. (F. S. Merritt and R. L. Brockenbrough, “Structural Steel Designers’ Handbook,” 2nd ed., McGraw-Hill, Inc., New York (books.mcgraw-hill. com).) Horizontally Curved Steel Girders For bridges with curved steel girders, the effects of torsion must be taken into consideration by the designer. Also, careful attention should be given to cross frames—spacing, design, and connection details. The effects of torsion decrease the stresses in the inside girders (those nearer the center of curvature). But there is a corresponding increase in stresses in the outside girders. Although the differences are not large for multiple-girder systems, the differences in stress for two-girder systems with short-radius curves and long spans can be as high as 50%. The torsional forces translate into vertical and horizontal forces, which must be transferred from the outside to inside girders through the cross frames. An approximate method for analysis of curved girder stresses is given in the U.S. Steel “Highway Structures Design Handbook.” This approximate method has proven satisfactory for many structures, but for complex structures (those with long spans, short-radius curves, or with only two girders), it is recommended that a rigorous analysis using a computer program be used. For the structure in Fig. 17.33, the stress differentials in the two girders are 50% and the cross frames transfer up to 70 kips of vertical and horizontal forces between girders. The center of the main span rotated 4 in when the deck was placed. Such rotations should be anticipated and the girders erected “out of plumb” so that the final web position will be vertical. Design of curved-girder bridges should con- sider the following: 1. Full-depth cross frames should be used to transfer the lateral forces from the flanges. (See Fig. 17.34.)

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Fig. 17.33 Curved girders of Tuolomne River Bridge, California, were erected in pairs with their cross frames connected between them. (California Department of Transportation.) 2. The cross frames should be designed as primary stress-carrying members to transfer the loads. 3. Flange-plate width should be increased above the normal design minimums to provide sta- bility during handling and erection. 4. Cross-frame connections at the web plates are critical. The web plate should be thickened to provide bending resistance

Fig. 17.34 Cross curvature.

section

of curved-girder bridge at cross frame, showing forces resulting from

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Bridge Bearings Bearings are structural assemblies installed to secure the safe transfer of all reactions from the superstructure to the substructure. They must ful- fill two basic requirements: They must spread the reactions over adequate areas of the substructure, and they must be capable of adapting to elastic, thermal, and other deformations of the super- structure without generating harmful restraining forces. Generally, bearings are classified as fixed, expansion, or elastomeric. Fixed bearings adapt only to angular deflec- tions. They must be designed to resist both vertical and horizontal components of reactions. Expansion bearings adapt to both angular deflections and longitudinal movements of the superstructure. Except for friction, they resist only those components of the superstructure reactions perpendicular to these movements. In both types of bearings, provision must be made for the safe transfer of all forces transverse to the direction of the span. Elastomeric bearings are a very efficient bearing for short to medium span bridges. They are relatively maintenance free and are one of the safest bearings, under seismic loading. Elastomeric bearings generally consist of laminated layers of elastomer restrained at their surfaces by bonded laminas. The elastomer is a Neoprene rubber; the laminas consist of either glass-fiber fabrics or steel sheets. Steel-reinforced elastomeric bearings are usually used when anchor bolts are required through the bearing (Figs. 17.34 and 17.35). The bearing pressure of elastomeric bearings should not exceed 800 psi under a service-load

Fig. 17.35 Steel-laminated elastomeric bear- ing pad. combination of dead load and live load, not including impact. For steel-reinforced bearing pads, the pressure should not exceed 1000 psi. The minimum pressure allowed on any pad due to dead load only is 200 psi. The capacity of an elastomeric bearing to absorb angular deflections and longitudinal movements of the superstructure is a function of its thickness (or of the sum of the thicknesses of its rubber elements between steel laminas), its shape factor (area of the loaded face divided by the sum of the side areas free to bulge), and the properties of the elastomer. AASHTO Specifications limit the overall thick - ness of a laminated bearing to one-third its length or width, whichever is smaller (or one-fourth of its diameter). The thickness should be at least twice the horizontal movement. An alternative is a pot bearing, which supports the structure on a hydraulic cylinder with an elastomer as the liquid medium.

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Concrete Bridges Reinforced concrete is used extensively in highway bridges because of its economy in short and medium spans, durability, low maintenance costs, and easy adaptability to horizontal and vertical curvature. The principal types of cast-in-place sup- porting elements are the longitudinally reinforced slab, T beam or girder, and cellular or box girder. Precast construction, usually prestressed, often employs an I-beam or box-girder cross section. In long-span construction, posttensioned box girders often are used. Slab Bridges Concrete slab bridges, longitudinally reinforced, may be simply supported on piers and/or abutments, monolithic with wall supports, or continu- ous over intermediate supports. Design Span For simple spans, the design span is the distance center to center of supports but need not exceed the clear span plus slab thickness. For slabs monolithic with walls (without haunches), use the clear span. For slabs on steel or timber stringers, use the clear span plus half the stringer width. Load Distribution In design, usually a 1-ft-wide longitudinal, typical strip is selected and its thickness and reinforcing determined for the appropriate HS loading. Wheel loads may be assumed distributed over a width, ft, E ¼ 4 þ 0:06S

7

(17:24)

where S ¼ span, ft. Lane loads should be dis- tributed over a width of 2E. For simple spans, the maximum live-load moment, ft-kips, per foot width of slab, without impact, mated by M ¼ 0:9S S

50 ft

M ¼ 1:30S 20 50 . S , 100

for HS20 loading is closely approxi-

(17:25a) (17:25b)

For HS15 loading, use three-quarters of the value given by Eqs. (17.25). For longitudinally reinforced cantilever slabs, wheel loads should be distributed over a width, ft, E ¼ 0:35X þ 3:25

7 ft

(17:26)

where X ¼ distance from load to point of support, ft. The moment, ft-kips per foot width of slab, is where P ¼ 16 kips for H20 loading and 12 kips for H15.

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Reinforcement Slabs should also be reinforced transversely to distribute the live loads laterally. The amount

Fig. 17.36 Example of an elastomeric bearing.

should be at least the following percentage of the main reinforcing steel required for positive moment: 100=pffiSffiffi, but it need not exceed 50%. The slab should be strengthened at all unsupported edges. In the longitudinal direction, strengthening may consist of a slab section additionally reinforced, a beam integral with and deeper than the slab, or an integral reinforced section of slab and curb. These should be designed to resist a live-load moment, ft-kips, of 1.6S for HS20 loading and 1.2S for HS15 loading on simply supported spans. Values for continuous spans may be reduced 20%. Greater reductions are permissible if justified by more exact analysis. At bridge ends and intermediate points where continuity of the slabs is broken, the edges should be supported by diaphragms or other suitable means. The diaphragms should be designed to resist the full moment and shear produced by wheel loads that can pass over them.

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Design Procedure The following procedure may be used for design of a typical longitudinally reinforced concrete slab bridge (Fig. 17.37). Step 1. Determine the live-load distribution (effective width). For the three-span, 90-ft-long

Fig. 17.37 Three-span concrete-slab bridge. bridge in Fig. 17.37, S ¼ 30 ft and E ¼ 4 þ 0:06 30 ¼ 5:8 ft The distributed load for a 4-kip front wheel then is 4/5.8, or 0.69 kips, and for a 16-kip rear or trailer wheel load, 16/5.8, or 2.76 kips, per foot of slab width. For an alternative 12-kip wheel load, the distributed load is 12/5.8, or 2.07 kips per foot of slab width (see Fig. 17.38). Step 2. Assume a slab depth. Step 3. Determine dead-load moments for the assumed slab depth.

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Step 4. Determine live-load moment at point of maximum moment. (This is done at this stage to get a check on the assumed slab depth.) Step 5. Combine dead-load, live-load, and impact moments at point of maximum moment. Compare the required slab depth with the assumed depth. Step 6. Adjust the slab depth, if necessary. If the required depth differs from the assumed depth of Step 9. Determine distribution steel. Step 10. Determine the number of piles required at each bent. Figures 17.39 and 17.40 illustrate typical steel reinforcement patterns for a single-span and a twospan concrete-slab bridge, respectively, similar to Fig. 17.37, suitable for spans ranging from 16 to 44 ft and carrying HS20 or alternate loading. Re- inforcement parallel to traffic in the single-span bridge is mainly in the bottom of the slab (Fig. 17.39b), rather than in the top (Fig. 17.39a). The two span bridge has main steel reinforcement in the top of the slab (Fig. 17.40b) over the center bent, to resist negative moments and main steel reinforce- ment in the bottom of the slab (Fig. 17.40a) in positive-moment regions. Reinforcement in multi- span bridges is arranged similarly. Transverse distribution steel is spaced typically at 11 to 12 in. The thickness of the concrete slab and reinforcement sizes depend on the specified 28 -day concrete compressive strength f 0 and yield point of the step 2, the dead-load moments should be revised and step 5 repeated. Usually, the second assump- tion is sufficient to yield the proper slab depth. Steps 2 through 6 follow conventional structural theory. Step 7. Place live loads for maximum moments at other points on the structure to obtain inter- mediate values for drawing envelope curves of maximum moment. Step 8. Draw the envelope curves. Determine the sizes and points of cutoff for reinforcing bars.

Fig. 17.38 Wheel load per foot width of slab for bridge of Fig. 17.37.

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reinforcement steel. For skews up to 208, transverse reinforcement should be placed parallel to the bent. For larger skews, transverse reinforcement should be placed perpendicular to the center line of the bridge. Skews exceeding 508 require special design. (“Bridge Design Aids,” Division of Structures, California Department of Transportation, Sacra- mento, Calif. (www.dot.ca.gov).) Concrete T-Beam Bridges Widely used in highway construction, this type of bridge consists of a concrete slab supported on, and integral with, girders (Fig. 17.41). It is especially economical in the 50- to 80-ft range. Where falsework is prohibited, because of traffic con - ditions or clearance limitations, precast construction of reinforced or prestressed concrete may be used. Design of Transverse Slabs Since the girders are parallel to traffic, main rein- forcing in the slab is perpendicular to traffic. For simply supported slabs, the span should be the distance center to center of supports but need not exceed the clear distance plus thickness of slabs.

Fig. 17.39 Arrangement of slab reinforcement for a single-span bridge carrying HS20-44 or alternative loading.

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Fig. 17.40 loading.

64

Arrangement of slab reinforcement for a two-span bridge carrying HS20-44 or alternative

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Fig. 17.41 Four-span bridge with concrete T beams. For slabs continuous over more than two girders, the distance between girders.

span

may

be

taken

as

the

clear

The live-load moment, ft-kips, for HS20 loading on simply supported slab spans is given by M ¼ 0:5(S þ 2)

(17:28)

where S ¼ span, ft. For slabs continuous over three or more supports, multiply M in Eq. (17.28) by 0.8 for both positive and negative moment. For HS15 loading, multiply M by 3⁄4. Reinforcement also should be placed in the slab parallel to traffic to distribute concentrated live loads. The amount should be the following percentage of the main reinforcing steel required for positive moment: 220=pffiSffiffi, but need not exceed 67%.

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Where a slab cantilevers over a girder, the wheel load should be distributed over a distance, ft, parallel to the girder of E ¼ 0:8X þ 3:75

(17:29)

where X ¼ distance, ft, from load to point of support. The moment, ft-kips per foot of slab parallel to girder, is M ¼p/ E X

(17:30)

where P ¼ 16 kips for HS20 loading and 12 kips for HS15. Equations (17.28) to (17.30) apply also to concrete slabs supported on steel girders, including composite construction. In design of the slabs, a l-ft-wide strip is selected and its thickness and reinforcing determined. The dead-load moments, ft-kips, positive and nega- tive, can be assumed to be wS2/10, where w is the dead load, kips/ft2. Live-load moments are given by Eq. (17.28) with a 20% reduction for continuity. Impact is a maximum of 30%. With these values, standard charts can be developed for design of slabs on steel and concrete girders. Figure 17.42 shows a typical slab-reinforcement layout. T-Beam Design The structure shown in Fig. 17.41 is a typical four- span grade-separation structure. The structural frame assumed for analysis is shown in Fig. 17.43. Columns with a pinned base are less stiff than fixed columns which minimizes shrinkage and tempera- ture moments. In addition, foundation pressures in

Fig. 17.42 Assumed support conditions for the bridge in Fig. 17.41. pinned columns are considered fairly uniform, resulting in an economical footing size and design. For concrete girder design, curves of maximum moments for dead load plus live load plus impact may be developed to determine reinforcement. For live-load moments, truck loadings are moved across the bridge. As they move, they generate changing moments, shears, and reactions. It is necessary to accumulate maximum combinations of moments to provide an adequate design. For heavy moving loads, extensive investigation is necessary to find the maximum stresses in continuous structures. Figure 17.44 shows curves of maximum moments consisting of dead load plus live load plus impact combinations that are maximum along the span. From these curves, reinforcing steel amounts and lengths may be determined by plotting the moments developed. Figure 17.45 shows curves of maxi- mum shears. Figure 17.46 shows the girder steel

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Fig. 17.43 Typical layout of reinforcement in the deck of a concrete T-beam bridge.

Fig. 17.45 Reinforcing for T beams of Fig. 17.41 is determined from curves of maximum bending moment. Numbers at the ends of the bars are distances, ft, from the center line of the span or bent.

reinforcement layout. Maximum-shear require- ments are derived theoretically by a point-to-point study of variations. Usually, a straight line between center line and end maximums is adequate. Girder spacing ranges from about 7 to 9 ft. Usually, a deck slab overhang of about 2 ft 6 in is economical.

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When the slab is made integral with the girder, its effective width of compression ange in design may not exceed the distance center to center of girders, one-fourth the girder span, or girder web- width plus 12 times the least thickness of slab. For exterior girders, however, effective overhang width may not exceed half the clear distance to the next

Fig. 17.45 Curves of maximum shear for T beams of Fig. 17.41.

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Fig. 17.46 Reinforcement layout for T beams of Fig. 17.41. Reinforcement is symmetrical about the center lines of the bridge and bent 3. Numbers at the ends of the bars indicate distances, ft, from the center line of bent or span. girder, one-twelfth the girder span, or six times the slab thickness. Ratios of beam depths to spans used in con- tinuous T-beam bridges generally range from 0.065 to 0.075. An economical depth usually results when a small amount of compressive reinforcement is required at the interior supports. Design of intermediate supports or bents varies widely, according to the designer ’s preference. A simple twocolumn bent is shown in Fig. 17.41. But considerable shape variations in column cross sec- tion and elevation are possible. Abutments are usually seat type or a monolithic end diaphragm supported on piles. (“Bridge Design Aids,” Division of Structures, California Department of Transportation, Sacra- mento, Calif. (www.dot.ca.gov).) Concrete Box-Girder Bridges Box or hollow concrete girders (Fig. 17.47) are favored by many designers because of the smooth plane of the bottom surface, uncluttered by lines of individual girders. Provision of space in the open cells for utilities is both a structural and an aesthetic advantage. Utilities are supported by the bottom

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Fig. 17.47 Three-span, reinforced concrete box-girder bridge. For more details, see Fig. 17.51.

slab, and access can be made available for inspec- tion and repair of utilities. For sites where structure depth is not severely limited, box girders and T beams have been about equal in price in the 80-ft span range. For shorter spans, T beams usually are cheaper, and for longer spans, box girders. These cost relations hold in general, but box girders have, in some instances, been economical for spans as short as 50 ft when structure depth was restricted. Girder Design Structural analysis is usually based on two typical segments, interior and exterior girders (Fig. 17.48). An argument could be made for analyzing the

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entire cross section as a unit because of its inherent transverse stiffness. Requirements in “Standard Specifications for Highway Bridges,” American Association of State Highway and Transportation Officials, however, are based on live-load distri- butions for individual girders, and so design usu-

Fig. 17.48 Typical

design sections

(cross- hatched) for a box-girder bridge.

ally is based on the assumption that a box-girder bridge is composed of separate girders. Effective width of slab as compression flange of an interior girder may be taken as the smallest of the distance center to center of girders, one-fourth the girder span, and girder-web width plus 12 times the least thickness of slab. Effective overhang width for an exterior girder may be taken as the smallest of half the clear distance to the next girder, one-twelfth the girder span, and six times the least thickness of the slab. Usual depth-to-span ratio for continuous spans is 0.055. This may be reduced to about 0.048 with balanced spans, at some sacrifice in economy and increase in deflections. Simple spans usually require a minimum depth-to-span ratio of 0.06. A typical concrete box-girder highway bridge is illustrated in Fig. 17.47. Girder spacing is approximately 11 generally is at least 8 in. Changes should be grad- ual, spread over a distance at least 12 times the difference in web thickness. Top-slab design follows the procedure des- cribed for T-beam bridges in Art. 17.20. Bottom- slab thickness and secondary reinforcement are usually controlled by specification minimums. AASHTO Specifications require that slab thickness be at least one -sixteenth the clear distance between girders but not less than 6 in for the top slab and 51

at the intersections of all surfaces within the cells.

Minimum flange reinforcement parallel to the gird er should be 0.6% of the flange area. This steel may be distributed at top and bottom or placed in a single layer at the center of the slab. Spacing should not exceed 18 in. Minimum flange reinforcing normal to the girder should be 0.5% and simila rly distributed. Bottom-flange bars should be bent up into the exterior-girder webs and anchored using a standard 908 hook or equivalent. At least one-third of the top flange tension reinforcement should extend to the exterior face of the outside girder and should be anchored with 908 bends or, where the flange projects beyond the girder sufficiently, extended far enough to develop bar strength in bond. When the top slab is placed after the web walls have set, at least 10% of the negative-moment reinforcing should be placed in the web. The bars should extend a distance of at least one-fourth the span on each side of the intermediate supports of

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continuous spans, one-ďŹ fth the span from the res- trained ends of continuous spans, and the entire length of cantilevers. In any event, the web should have reinforcing placed horizontally in both faces, to prevent temperature and shrinkage cracks. The bars should be spaced not more than 12 inches c to c. Total area of this steel should be at least 10% of the area of exural tension reinforcement. Analysis of the structure in Fig. 17.47 for dead loads follows procedure. Assumed end conditions are shown in Fig. 17.49a.

conventional moment-distribution

Fig. 17.49 Loading patterns for maximum stresses in a box-girder bridge.

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Live loads, positioned to produce maximum negative moments in the girders over Pier 2, are shown in Fig. 17.49b to d. Similar loadings should be applied to find maximum positive and negative moments at other critical points. Moments should be distributed and points plotted on a maximum- moment diagram (for dead load plus live load plus impact), as shown in Fig. 17.50. Layout of main girder reinforcement follows directly from this diagram. Figure 17.51 shows a typical layout. (“Bridge Design Details,” Division of Structures, California Department of Transportation, Sacra- mento, Calif. (www.dot.ca.gov).) Prestressed-Concrete Bridges In prestressed-concrete construction, concrete is subjected to permanent compressive stresses of such magnitude that little or no tension develops when design loading is applied (Art. 8.42). Prestressing allows considerably better utili- zation of concrete than conventional reinforcement. It results in an overall dead-load reduction, which makes long spans possible with concrete, sometimes competitive in cost with those of steel. Prestressed concrete, however, requires greater sophistication in design, higher quality of materials (both concrete and steel), and greater refinement and controls in fabrication than does reinforced concrete. Depending on the methods and sequence of fabrication, prestressed concrete may be precast, pretensioned; precast, posttensioned; cast-in-place posttentioned; composite; or partly prestressed. In precast-beam bridges, the primary structure consists of precast-concrete units, usually I beams, channels, T beams, or box girders. They may be either pretensioned or posttensioned. Precast slabs may be solid or hollow. Precast I beams (Fig. 17.52) may be combined with fully or partly cast-in-place decks. This construction has the advantage that the deck can be shaped closely to the desired specifications. Precast slabs, incorporated into the deck, may be used in lieu of removable deck forms where accessibility is poor, for example, on over- water trestles or causeways. Precast T beams (Fig. 17.53) offer no advantage over the easier to fabricate, more compact I sections. Alignment of

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Fig. 17.50 Curves of maximum moment determine reinforcing for a box girder. Numbers at the ends of the bars indicate distances, ft, from the center line of piers or span.

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Fig. 17.51 Reinforcing layout for the box-girder bridge Fig. 17.47 of and moment curves of Fig. 17.50. Design stresses for HS20 loading: f 0 ¼ 3500 psi, fy ¼ 60 ksi. the flanges of T sections often is difficult. And as with I beams, the flanges must be connected with cast-in-place concrete. Precast box sections may be placed side by side to form a bridge span. If desired, they may be posttensioned transversely. Precast beams mainly are used for spans up to about 90 ft where erection of conventional falsework is not feasible or desirable. Such beams are particularly economical if conditions are favorable for mass fabrication, for example, in multispan viaducts or causeways or in the vicinity of centralized fabrication plants. Longer spans are possible but require increasingly heavy handling equipment. Standard designs for precast, prestressed girders have been developed by the Federal Highway Administration and state highway departments. Cast-in-place prestressed concrete often is used for low-level bridges, where ground conditions favor erection of conventional falsework. Typical cross

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sections are essentially similar to those used for conventionally reinforced sections, except that, in general, prestressing permits structures with thinner depths.

Fig. 17.52 Typical precast, prestressed I beam used in highway bridges.

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Fig. 17.53 Typical precast, prestressed T beam used in highway bridges. For fully cast-in-place single-span bridges, post- tensioning differs only quantitatively from that for precast elements. In design of multispan continu- ous bridges, the following must be considered: Frictional prestress losses depend on the draping pattern of the ducts. To reduce potential losses and increase the reliability of effective prestress, avoid continuously waving tendon patterns. Instead, use discontinuous simple patterns. Another method is to place tendons, usually bundles of cables, in the hollows of box girders and to bend the tendons at lubricated, accessible bearings. Prestressed concrete is competitive with other materials for spans of 150 to 250 ft or more. Construction techniques and improvements in pre- stressing hardware, such as smooth, lightweight conduits, which reduce friction losses, have brought prestressed concrete bridges into direct competition with structural steel, once preeminent in medium and long spans. Segmental construction, both precast and cast- in-place, has eliminated the need for expensive falsework, which previously made concrete brid- ges uneconomical in locations requiring long spans over navigation channels or deep canyons. The two types of segmental construction used most in the United States are the cast-in-place and precast balanced cantilever types. For cast-in-place construction, the movable formwork is supported by a structural framework, or traveler, which cantilevers from an adjacent completed section of the superstructure. As each section is cast, cured, and posttensioned, the framework is moved out and the process repeated. Figure 17.54 illustrates this type of construction. For precast construction the procedure is similar, except that the sections are prefabricated. Other methods, such as full-span and incre- mental launching procedures, can be used to fit site conditions. In all segmental construction, special attention should be given in the erection plan to limitation of temporary stresses and to mainte- nance of balance during erection and prior to span closing. Also important are an accurate prediction of creep and accurate calculation of deflections to ensure attainment of the desired structure profile and deck grades in the completed structure.

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Posttensioning makes possible widening or strengthening or other remodeling of existing concrete structures. For example, Fig. 17.55 shows a cross section through a double-deck viaduct. The row of columns under the upper deck had to be removed, and capacity had to be increased from H15 to HS20 loading. No interference with upper- deck traffic and a minimum of interference with lower deck traffic were permitted. This objective was accomplished by reinforcing each floor beam with precast units incorporating preformed ducts for tendons. Then the entire upper deck was prestressed transversely. This permitted the beams to span the full width of the bridge and carry the heavier loading. Similar remodeling has been done with cast-in-place concrete. Determination of stresses in prestressed bridges is similar to that for other structures. In analysis of statically indeterminate systems, however, the deformations caused by prestressing must be taken into account (see also Arts. 8.42 to 8.45). [C. A. Ballinger and W. Podolny, Jr., “Segmental Bridge Construction in Western Europe,” Transportation Research Board, Record 665, 1978; A. Grand, “Incremental Launching of Concrete Structures,” Journal of the American Concrete Institute, August 1975; W. Baur, “Bridge Erection

Fig. 17.54 Segmental cast-in-place concrete construction in progress for the Pine Valley Bridge, California. (California Department of Transportation.)

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Fig. 17.55 Double-deck viaduct strengthened by prestressing to permit removal of column and passage of heavier trucks. by Launching Is Fast, Safe, and Efficient,” Civil Engineering, March 1977; F. Leonhardt, “ New Trends in Design and Construction of Long- Span Bridges and Viaducts (Skew, Flat Slabs, Torsion Box),” Eighth Congress, International Association for Bridge and Structural Engineering, New York, Sept. 9 to 14, 1968.] Concrete Bridge Piers and Abutments Bridge piers are the intermediate supports of the superstructure of bridges with two or more openings. Abutments are the end supports and usually have the additional function of retaining earth fill for the bridge approaches. The minimum height of piers and abutments is governed by requirements of accessibility for maintenance of the superstructure, including bear- ings; of protection against spray for bridges over water; and of vertical clearance requirements for bridges over traveled ways. There is no upper limit for pier heights, except that imposed by economic considerations. One of the piers of the Europa Bridge, which carries an international freeway in Austria, for instance, soars to 492 ft above the ground surface of the valley. The top surface of piers must have adequate length and width to accommodate the bridge bearings of the superstructure. On abutments, added width is required for the back wall (curtain wall or bulkhead), which retains approach fill and protects

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the end section of the superstructure. In designing the aboveground sections of piers, res- trictions resulting from lateral-clearance require- ments of adjacent traveled ways and visibility needs may have to be taken into account. Length and width at the base level are controlled by stability, stress limitations in the pier shaft, and foundation design. For stress and stability analyses, the reactions from loadings (dead and live, but not impact) acting on the superstructure should be combined with those acting directly on the substructure. Longitudinal reactions depend on the type of bearing, whether fixed or expansion. Piers A number of basic pier shapes have been devel- oped to meet the widely varying requirements. Enumerated below are some of the more common types and their preferred uses. Trestle-type piers are preferred on low-level “causeways” carried over shallow waters or seas- onally flooded land on concrete slab or beam-and- slab superstructures. Each pier or bent consists of two or more bearing piles, usually all driven in the same plane, and a thick concrete deck or a prismatic cap into which the piles are framed (Fig. 17.37). Both cap and piles may be of timber or, for more permanent construction, of precast conventionally reinforced or prestressed concrete. Wall-type concrete piers on spread footings are generally used as supports for two-lane overcrossings over divided highways. Given adequate longitudinal support of the superstructure, these piers may be designed as pendulum walls, with joints at top and bottom; otherwise, as cantilever walls. T-shaped piers on spread footings, with or without bearing piles, may be used to advantage as supports of twin girders. The girders are seated on bearings at both tips of the cross beam atop the pier stem. T-shaped piers have been built either entirely of reinforced concrete or of reinforced concrete in various combinations with structural steel. Single-column piers of rectangular or circular cross section on spread footings may be used to support box girders, with built-in diaphragms acting as cross beams (Fig. 17.47). Portal frames may be used as piers under heavy steel girders, with bearings located directly over the portal legs (columns). When more than two girders are to be supported, the designer may choose to strengthen the portal cap beam or to add more columns. Preferably, all legs of each portal frame should rest on a common base plate. If, instead, separate footings are used, as, for instance, on separate pile clusters, adequate tie bars must be used to prevent unintended spreading. Massive masonry piers have been built since antiquity for multiple-arch river bridges, high-level aqueducts, and more recently, viaducts. In the twentieth century, their place has been taken by massive concrete construction, with or without natural stone facing. Where reduction of dead load is of the essence, hollow piers, often of heavily reinforced concrete, may be used. Steel towers on concrete pedestals may be used for high bridge piers. They may be designed either as thin-membered, special trellis or as closed box por- tals, or combinations of these (Figs. 17.20 and 17.26). Very tall piers, when used, are usually con- structed of reinforced or prestressed concrete, either solid or cellular in design (Fig. 17.33). Bridge abutments basically are piers with flanking (wing) walls. Abutments for short-span concrete bridges, such as T-beam or slab-type highway overcrossings, are frequently simple concrete trestles built monolithically with the superstructure (see Figs. 17.37 and 17.47). Abut- ments for steel bridges and for long-span concrete bridges that are subject to substantial end rotation and longitudinal movements should be designed as separate structures that provide a level area for the bridge bearings FOR MORE DETAILS VISIT US ON WWW.IMTSINSTITUTE.COM OR CALL ON +91-9999554621


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(bridge seat) and a back wall (curtain wall or bulkhead). The wall (stem) below the bridge seat of such abutments may be of solid concrete or thin-walled reinforced-concrete con- struction, with or without counterfort walls; but on rare occasions, masonry is used. Sidewalls, which retain approach fill, should have adequate length to prevent erosion and undesired spill of the backfill. They may be built either monolithically with the abutment stem and backwall in which case they are designed as cantilevers subject to two-way bending, or as self- supporting retaining walls on independent footings. Sidewalls may be arranged in a straight line with the abutment face, parallel to the bridge axis, or at any intermediate angle to the abutment face that may suit local conditions. Given adequate foundation conditions, the parallel-to-bridge-axis arrangement (U-shaped abutment) is often preferred because of its inherent stability. Abutments must be safe against overturning about the toe of the footing, against sliding on the footing, and against crushing of the underlying soil or overloading of piles. In earth-pressure compu- tations, the vehicular load on highways may be taken into account in the form of an equivalent layer of soil 2 ft thick. Live loads from railroads may be assumed to be 0.5 kip/ft2 over a 14-ft-wide strip for each track. In computations of internal stresses and stability, the weight of the fill material over an inclined or stepped rear face and over reinforced concrete spread footings should be considered as fully effective. No earth pressures however, should be assumed from the earth prism in front of the wall. Buoyancy should be taken into account if it may occur.

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