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  • The ease of erection of suspension bridges is a major factor in their use for long spans. Once¬†the main cables are in position, they furnish a stable working base or platform from which¬†the deck and stiffening truss sections can be raised from floating barges or other equipment¬†below, without the need for auxiliary falsework. For the Severn Bridge, for example, 60-ft¬†box-girder deck sections were floated to the site and lifted by equipment supported on the¬†cables.
    Until the 1960s, the field process of laying the main cables had been by spinning (Art. 15.12.3). (this term is actually a misnomer, for the wires are neither twisted nor braided, but are laid parallel to and against each other.) The procedure (Fig. 15.68) starts with the hanging  of a catwalk at each cable location for use in construction of the bridge. An overhead cableway is then installed above each catwalk. Loops of wire (two or four at a time) are carried over the span on a set of grooved spinning wheels. These are hung from an endless hauling rope of the cableway until arrival at the far anchorage. There, the loops are pulled off the spinning wheels manually and placed around a semicircular strand shoe, which connects them by an eyebar or bolt linkage to the anchorage (Fig. 15.33). The wheels then start back to the originating anchorage. At the same time, another set of wheels carrying wires starts out from that anchorage. The loops of wire on the latter set of wheels are also placed manually around a strand shoe at their anchorage destination. Spinning proceeds as the wheels shuttle back and forth across the span. A system of counterweights keeps the wires under continuous tension as they are spun.
    The wires that come off the bottom of the wheels (called dead wires) and that are held back by the originating anchorage are laid on the catwalk in the spinning process. The wires passing over the wheels from the unreelers and moving at twice the speed of the wheels, are called live wires.
    As the wheels pass each group of wire handlers on the catwalks, the dead wires are  temporarily clipped down. The live wires pass through small sheaves to keep them in correct order. Each wire is adjusted for level in the main and side spans with come-along winches, to ensure that all wires will have the same sag.
    The cable is made of many strands, usually with hundreds of wires per strand (Art. 15.12).
    All wires from one strand are connected to the same shoe at each anchorage. Thus, there are as many anchorage shoes as strands. At saddles and anchorages, the strands maintain their identity, but throughout the rest of their length, the wires are compacted together by special machines. The cable usually is forced into a circular cross section of tightly bunched parallel wires.
    The usual order of erection of suspension bridges is substructure, pylons and anchorages, catwalks, cables, suspenders, stiffening trusses, floor system, cable wrapping, and paving.
    Cables are usually coated with a protective compound. The main cables are wrapped with wire by special machines, which apply tension, pack the turns tightly against one another, and at the same time advance along the cable. Several coats of protective material, such as paint, are then applied For alternative wrapping, see Art. 15.14.

    Typical cable bands are illustrated in Figs. 15.39 and 15.40. These are usually made of paired, semicylindrical steel castings with clamping bolts, over which the wire-rope or strand suspenders are looped or attached by socket fittings.
    Cable-stayed structures are ideally suited for erection by cantilevering into the main span from the piers. Theoretically, erection could be simplified by having temporary erection hinges at the points of cable attachment to the girder, rendering the system statically determinate, then making these hinges continuous after dead load has been applied. The practical implementation of this is difficult, however, because the axial forces in the girder are larger and would have to be concentrated in the hinges. Therefore, construction usually follows conventional tactics of cantilevering the girder continuously and adjusting the cables as necessary to meet the required geometrical and statical constraints. A typical erection sequence is illustrated in Fig. 15.69.
    Erection should meet the requirements that, on completion, the girder should follow a  prescribed gradient; the cables and pylons should have their true system lengths; the pylons should be vertical, and all movable bearings should be in a neutral position. To accomplish this, all members, before erection, must have a deformed shape the same as, but opposite in direction to, that which they would have under dead load. The girder is accordingly cambered, and also lengthened by the amount of its axial shortening under dead load. The pylons and cable are treated in similar manner.
    Erection operations are aided by raising or lowering supports or saddles, to introduce prestress as required. All erection operations should be so planned that the stresses during the erection operations do not exceed those due to dead and live load when the structure is completed; otherwise loss of economy will result.

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  • For short-span structures (under about 500 ft) it is commonly assumed in seismic analysis¬†that the same ground motion acts simultaneously throughout the length of the structure. In¬†other words, the wavelength of the ground waves are long in comparison to the length of¬†the structure. In long-span structures, such as suspension or cable-stayed bridges, however,¬†the structure could be subjected to different motions at each of its foundations. Hence, in¬†assessment of the dynamic response of long structures, the effects of traveling seismic waves¬†should be considered. Seismic disturbances of piers and anchorages may be different at one¬†end of a long bridge than at the other. The character or quality of two or more inputs into¬†the total structure, their similarities, differences, and phasings, should be evaluated in dynamic¬†studies of the bridge response.
    Vibrations of cable-stayed bridges, unlike those of suspension bridges, are susceptible to a unique class of vibration problems. Cable-stayed bridge vibrations cannot be categorized as vertical (bending), lateral (sway), and torsional; almost every mode of vibration is instead a three-dimensional motion. Vertical vibrations, for example, are introduced by both longitudinal and lateral shaking in addition to vertical excitation. In addition, an understanding is needed of the multimodal contribution to the final response of the structure and in providing representative values of the response quantities. Also, because of the long spans of such structures, it is necessary to formulate a dynamic response analysis resulting from the multisupport excitation. A three-dimensional analysis of the whole structure and substructure to obtain the natural frequencies and seismic response is advisable. A qualified specialist should be consulted to evaluate the earthquake response of the structure.

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  • The wind-induced failure on November 7, 1940, of the Tacoma Narrows Bridge in the state¬†of Washington shocked the engineering profession. Many were surprised to learn that failure¬†of bridges as a result of wind action was not unprecedented. During the slightly more than ¬†12 decades prior to the Tacoma Narrows failure, 10 other bridges were severely damaged or¬†destroyed by wind action (Table 15.12). As can be seen from Table 12a, wind-induced¬†failures have occurred in bridges with spans as short as 245 ft up to 2800 ft. Other ‚Äė‚Äėmodern‚Äô‚Äô¬†cable-suspended bridges have been observed to have undesirable oscillations due to wind¬†(Table 15.12b).

    Required Information on Wind at Bridge Site

    Prior to undertaking any studies of wind instability for a bridge, engineers should investigate the wind environment at the site of the structure. Required information includes the character  of strong wind activity at the site over a period of years. Data are generally obtainable from local weather records and from meteorological records of the U.S. Weather Bureau. However, caution should be used, because these records may have been attained at a point some distance from the site, such as the local airport or federal building. Engineers should also be aware of differences in terrain features between the wind instrumentation site and the structure site that may have an important bearing on data interpretation. Data required are wind velocity, direction, and frequency. From these data, it is possible to predict high wind speeds, expected wind direction and probability of occurrence.
    The aerodynamic forces that wind applies to a bridge depend on the velocity and direction of the wind and on the size, shape, and motion of the bridge. Whether resonance will occur under wind forces depends on the same factors. The amplitude of oscillation that may build up depends on the strength of the wind forces (including their variation with amplitude of bridge oscillation), the energy-storage capacity of the structure, the structural damping, and the duration of a wind capable of exciting motion.


    The wind velocity and direction, including vertical angle, can be determined by extended observations at the site. They can be approximated with reasonable conservatism on the basis of a few local observations and extended study of more general data. The choice of the wind conditions for which a given bridge should be designed may always be largely a matter of judgment.
    At the start of aerodynamic analysis, the size and shape of the bridge are known. Its energy-storage capacity and its motion, consisting essentially of natural modes of vibration, are determined completely by its mass, mass distribution, and elastic properties and can be computed by reliable methods.
    The only unknown element is that factor relating the wind to the bridge section and its motion. This factor cannot, at present, be generalized but is subject to reliable determination in each case. Properties of the bridge, including its elastic forces and its mass and motions (determining its inertial forces), can be computed and reduced to model scale. Then, wind conditions bracketing all probable conditions at the site can be imposed on a section model.
    The motions of such a dynamic section model in the properly scaled wind should duplicate reliably the motions of a convenient unit length of the bridge. The wind forces and the rate at which they can build up energy of oscillation respond to the changing amplitude of the motion. The rate of energy change can be measured and plotted against amplitude. Thus, the section-model test measures the one unknown factor, which can then be applied by calculation to the variable amplitude of motion along the bridge to predict the full behavior of the structure under the specific wind conditions of the test. These predictions are not precise but are about as accurate as some other features of the structural analysis.

    Criteria for Aerodynamic Design

    Because the factor relating bridge movement to wind conditions depends on specific site and bridge conditions, detailed criteria for the design of favorable bridge sections cannot be written until a large mass of data applicable to the structure being designed has been accumulated.
    But, in general, the following criteria for suspension bridges may be used:
    ‚ÄĘ A truss-stiffened section is more favorable than a girder-stiffened section.
    ‚ÄĘ Deck slots and other devices that tend to break up the uniformity of wind action are likely¬†to be favorable.
    ‚ÄĘ The use of two planes of lateral system to form a four-sided stiffening truss is desirable¬†because it can favorably affect torsional motion. Such a design strongly inhibits flutter and¬†also raises the critical velocity of a pure torsional motion.
    ‚ÄĘ For a given bridge section, a high natural frequency of vibration is usually favorable:
    For short to moderate spans, a useful increase in frequency, if needed, can be attained by increased truss stiffness. (Although not closely defined, moderate spans may be regarded as including lengths from about 1,000 to about 1,800 ft.)

    For long spans, it is not economically feasible to obtain any material increase in natural frequency of vertical modes above that inherent in the span and sag of the cable.
    The possibility should be considered that for longer spans in the future, with their unavoidably low natural frequencies, oscillations due to unfavorable aerodynamic characteristics of the cross section may be more prevalent than for bridges of moderate span.
    ‚ÄĘ At most bridge sites, the wind may be broken up; that is, it may be nonuniform across¬†the site, unsteady, and turbulent. So a condition that could cause serious oscillation does¬†not continue long enough to build up an objectionable amplitude. However, bear in mind:
    There are undoubtedly sites where the winds from some directions are unusually steady and uniform.
    There are bridge sections on which any wind, over a wide range of velocity, will continue to build up some mode of oscillation.
    ‚ÄĘ An increase in stiffness arising from increased weight increases the energy-storage capacity¬†of the structure without increasing the rate at which the wind can contribute energy. The¬†effect is an increase in the time required to build up an objectionable amplitude. This may¬†have a beneficial effect much greater than is suggested by the percentage increase in¬†weight, because of the sharply reduced probability that the wind will continue unchanged¬†for the greater length of time. Increased stiffness may give added structural damping and¬†other favorable results.
    Although more specific design criteria than the above cannot be given, it is possible to design a suspension bridge with a high degree of security against aerodynamic forces. This involves calculation of natural modes of motion of the proposed structure, performance of dynamic-section-model tests to determine the factors affecting behavior, and application of these factors to the prototype by suitable analysis.
    Most long-span bridges built since the Tacoma bridge failure have followed the above procedures and incorporated special provisions in the design for aerodynamic effects. Designers of these bridges usually have favored stiffening trusses over girders. The second Tacoma Narrows, Forth Road, and Mackinac Straits Bridges, for example, incorporate deep stiffening trusses with both top and bottom bracing, constituting a torsion space truss. The Forth Road and Mackinac Straights Bridges have slotted decks. The Severn Bridge, however, has a streamlined, closed-box stiffening girder and inclined suspenders. Some designs incorporate longitudinal cable stays, tower stays, or even transverse diagonal stays (Deer Isle Bridge). Some have unloaded backstays. Others endeavor to increase structural damping by frictional or viscous means. All have included dynamic-model studies as part of the design.

    Wind-Induced Oscillation Theories

    Several theories have been advanced as models for mathematical analysis to develop an understanding of the process of wind excitation. Among these are the following.
    Negative-Slope Theory. When a bridge is moving downward while a horizontal wind is blowing (Fig. 15.66a), the resultant wind is angled upward (positive angle of attack) relative to the bridge. If the lift coefficient CL , as measured in static tests, shows a variation with wind angle  such as that illustrated by curve A in Fig. 15.66b, then, for moderate amplitudes, there is a wind force acting downward on the bridge while the bridge is moving downward. The bridge will therefore move to a greater amplitude than it would without this wind force. The motion will, however, be halted and reversed by the action of the elastic forces. Then, the vertical component of the wind also reverses. The angle of attack becomes
    negative, and the lift becomes positive, tending to increase the amplitude of the rebound.
    With increasing velocity, the amplitude will increase indefinitely or until the bridge is de stroyed. A similar, though more complicated situation, would apply for torsional or twisting motion of the bridge.

    Vortex Theory. This attributes aerodynamic excitation to the action of periodic forces having a certain degree of resonance with a natural mode of vibration of the bridge. Vortices, which form around the trailing edge of the airfoil (bridge deck), are shed on alternating sides, giving rise to periodic forces and oscillations transverse to the deck.
    Flutter Theory. The phenomenon of flutter, as developed for airfoils of aircraft and applied to suspension-bridge decks, relates to the fact that the airfoil (bridge deck) is supported so that it can move elastically in a vertical direction and in torsion, about a longitudinal axis.
    Wind causes a lift that acts eccentrically. This causes a twisting moment, which, in turn, alters the angle of attack and increases the lift. The chain reaction becomes catastrophic if the vertical and torsional motions can take place at the same coupled frequency and in appropriate phase relation.
    F. Bleich presented tables for calculation of flutter speed vF for a given bridge, based on flat-plate airfoil flutter theory. These tables are applicable principally to trusses. But the tables  are difficult to apply, and there is some uncertainty as to their range of validity.
    A. Selberg has presented the following formula for flutter speed:

    Selberg has also published charts, based on tests, from which it is possible to approximate the critical wind speed for any type of cross section in terms of the flutter speed.
    Applicability of Theories. The vortex and flutter theories apply to the behavior of suspension bridges under wind action. Flutter appears dominant for truss-stiffened bridges, whereas vortex action seems to prevail for girder-stiffened bridges. There are mounting indications, however, these are, at best, estimates of aerodynamic behavior. Much work has been done and is being done, particularly in the spectrum approach and the effects of nonuniform, turbulent winds.

    Design Indices

    Bridge engineers have suggested several criteria for practical design purposes. O. H. Ammann, for example, developed two analytical-empirical indices that were applied in the design of the Verrazano Narrows Bridge, a vertical-stiffness index and a torsional-stiffness index.
    Vertical-Stiffness Index Sv . This is based on the magnitude of the vertical deflection of the suspension system under a static downward load covering one-half the center span. The index includes a correction to allow for the effect of structural damping of the suspended structure and for the effect of different ratios of side span to center span.


    Typical values of these indices are listed in Table 15.13 for several bridges.
    Other indices and criteria have been published by D. B. Steinman. In connection with
    these, Steinman also proposed that, unless aerodynamic stability is otherwise assured, the

    Solution of these equations for the natural frequencies and modes of motion is dependent on the various possible static forms of suspension bridges involved (see Fig. 15.9). Numerous lengthy tabulations of solutions have been published.

    Damping

    Damping is of great importance in lessening of wind effects. It is responsible for dissipation of energy imparted to a vibrating structure by exciting forces. When damping occurs, one part of the external energy is transformed into molecular energy, and another part is transmitted to surrounding objects or the atmosphere. Damping may be internal, due to elastic hysteresis of the material or plastic yielding and friction in joints, or Coulomb (dry friction), or atmospheric, due to air resistance.

    Aerodynamics of Cable-Stayed Bridges

    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. However, there is a trend towards the use of the composite steel-concrete superstructure girders (Fig. 15.16) for increasingly longer spans and to reduce girder dead weight. This configuration, because of the long spans and decreased mass, can be relatively more sensitive to aerodynamic effects as compared to a torsionally stiff box.

    Stability Investigations

    It is most important to note that the validation of stability of the completed structure for expected wind speeds at the site is mandatory. However, this does not necessarily imply that the most critical stability condition of the structure occurs when the structure is fully completed.
    A more dangerous condition may occur during erection, when the joints have not been fully connected and, therefore, full stiffness of the structure has not yet been realized.
    In the erection stage, the frequencies are lower than in the final condition and the ratio of torsional frequency to flexural frequency may approach unity. Various stages of the partly erected structure may be more critical than the completed bridge. The use of welded components in pylons has contributed to their susceptibility to vibration during erection.
    Because no exact analytical procedures are yet available, wind-tunnel tests should be used to evaluate the aerodynamic characteristics of the cross section of a proposed deck girder, pylon, or total bridge. More importantly, the wind-tunnel tests should be used during the design process to evaluate the performance of a number of proposed cross sections for a particular project. In this manner, the wind-tunnel investigations become a part of the design decision process and not a postconstruction corrective action. If the wind-tunnel evaluations are used as an after-the-fact verification and they indicate an instability, there is the distinct risk that a redesign of a retrofit design will be required that will have undesirable ramifications on schedules and availability of funding.

    Rain-Wind Induced Vibration

    Well known mechanisms of cable vibration are vortex and wake galloping. Starting in approximately
    the mid-1980‚Äôs, a new phenomenon of cable vibration has been observed that¬†occurs during the simultaneous presence of rain and wind, thus, it is given the name ‚Äė‚Äėrainwind¬†vibration,‚Äô‚Äô or rain vibration.
    The excitation mechanism is the formation of water rivulets, at the top and bottom, that run down the cable oscillating tangentially as the cables vibrate, thus changing the aerodynamic profile of the cable (or the enclosing HDPE pipe). The formation of the upper rivulet appears to be the more dominant factor in the origin of the rain-wind vibration.
    In the current state-of-the-art, three basic methods of rain-wind vibration suppression are being considered or used:
    ‚ÄĘ Rope ties interconnecting the cable stays in the plane of the stays, Fig. 15.67a
    ‚ÄĘ Modification of the external surface of the enclosing HDPE pipe, Fig. 15.67b
    ‚ÄĘ Providing external damping

    The interconnection of stays by rope ties produces node points at the point of connection of the secondary tie to the cable stays. The purpose is to shorten the free length of the stay and modify the natural frequency of vibration of the stay. The modification of the surface may be such as protuberances that are axial, helical, elliptical or circular or grooves or dimples. The intent is to discourage the formation of the rivulets and/or its oscillations.
    Various types of dampers such as viscous, hydraulic, tuned mass and rubber have also been used to suppress the vibration.
    The rain-wind vibration phenomenon has been observed during construction prior to grout injection which then stabilizes after grout injection. This may be as a result of the difference in mass prior to and after grout injection (or not). It also has been noticed that the rain-wind vibration may not manifest itself until some time after completion of the bridge. This may be the results of a transition from initial smoothness of the external pipe to a roughness, sufficient to hold the rivulet, resulting from an environmental or atmospheric degradation of the surface of the pipe.
    The interaction of the various parameters in the rain-wind phenomenon is not yet well understood and an optimum solution is not yet available. It should also be noted that under similar conditions of rain and wind, the hangars of arch bridges and suspenders of suspension bridges can also vibrate.

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  • In general, the height of a pylon in a cable-stayed bridge is about 1‚ĀĄ6 to 1‚ĀĄ8 the main span.
    Depth of stayed girder ranges from 1‚ĀĄ60 to 1‚ĀĄ80 the main span and is usually 8 to 14 ft,¬†averaging 11 ft. Live-load deflections usually range from 1‚ĀĄ400 to 1‚ĀĄ500 the span.



    To achieve symmetry of cables at pylons, the ratio of side to main spans should be about 37 where three cables are used on each side of the pylons, and about 25 where two cables are used. A proper balance of side-span length to main-span length must be established if uplift at the abutments is to be avoided. Otherwise, movable (pendulum-type) tiedowns must be provided at the abutments.
    Wide box girders are mandatory as stayed girders for single-plane systems, to resist the torsion of eccentric loads. Box girders, even narrow ones, are also desirable for double-plane systems to enable cable connections to be made without eccentricity. Single-web girders, however, if properly braced, may be used.
    Since elastic-theory calculations are relatively simple to program for a computer, a formal set may be made for preliminary design after the general structure and components have been sized.

    Manual Preliminary Calculations for Cable Stays. Following is a description of a method of manual calculation of reasonable initial values for use as input data for design of a cablestayed bridge by computer. The manual procedure is not precise but does provide first-trial cable-stay areas. With the analogy of a continuous, elastically supported beam, influence lines for stay forces and bending moments in the stayed girder can be readily determined.

    From the results, stress variations in the stays and the girder resulting from concentrated loads can be estimated.
    If the dead-load cable forces reduce deformations in the girder and pylon at supports to zero, the girder acts as a beam continuous over rigid supports, and the reactions can be computed for the continuous beam. Inasmuch as the reactions at those supports equal the vertical components of the stays, the dead-load forces in the stays can be readily calculated.
    If, in a first-trial approximation, live load is applied to the same system, the forces in the stays (Fig. 15.61) under the total load can be computed from

    The reactions may be taken as Ri = ws, where w is the uniform load, kips per ft, and s, the distance between stays. At the ends of the girder, however, Ri may have to be determined by other means.
    Determination of the force Po acting on the back-stay cable connected to the abutment (Fig. 15.62) requires that the horizontal force Fh at the top of the pylon be computed first.
    Maximum force on that cable occurs with dead plus live loads on the center span and dead load only on the side span. If the pylon top is assumed immovable, Fh can be determined from the sum of the forces from all the stays, except the back stay:

    For the structure illustrated in Fig. 15.64, values were computed for a few stays from Eqs. (15.47), (15.48), (15.49), and (15.51) and tabulated in Table 15.11a. Values for the final design, obtained by computer, are tabulated in Table 15.11b.
    Inasmuch as cable stays 1, 2, and 3 in Fig. 15.64 are anchored at either side of the anchor pier, they are combined into a single back-stay for purposes of manual calculations. The edge girders of the deck at the anchor pier were deepened in the actual design, but this increase in dead weight was ignored in the manual solution. Further, the simplified manual solution does not take into account other load cases, such as temperature, shrinkage, and creep.
    Influence lines for stay forces and girder moments are determined by treating the girder as a continuous, elastically supported beam. From Fig. 15.65, the following relationships are obtained for a unit force at the connection of girder and stay:



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  • The static behavior of a cable-stayed girder can best be gaged from the simple, two-span¬†example of Fig. 15.54. 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 stays constituted relatively soft spring supports. Heavy and long, the stays could not be highly stressed. Usually, the cables were installed with significant slack, or sag. Consequently, large deflections occurred under live load as the sag decreased. In contrast, modern cables are made of high-strength steel, are relatively short and taut, and have low weight.
    Their elastic action may therefore be considered linear, and an equivalent modulus of elasticity may be used [Eq. (15.1)]. The action of such cables then produces something more nearly like a four-span beam for a structure such as the one in Fig. 15.54.

    If the pylon were hinged at its base connection with a stayed girder at B, rather than fixed, 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. The resulting girder deflection might exceed that due to the elastic stretch of the cables. In contrast, the elastic shortening of the pylon has no appreciable effect.
    Relative girder stiffness plays a dominant role in the structural action. The stayed 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 support of the cables diminishes, 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. 15.54, 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, though not as esthetically pleasing as other arrangements.



    The stiffness of the system also depends on whether the cables are fixed at the towers (at D, for example, in Fig. 15.54) or whether they run continuously over (or through) the pylons. Some early designs with more than one cable to a pylon from the main span required one of the cables to be fixed to the pylon and the others to be on movable saddle supports.
    Most contemporary designs fix all the stays to the pylon.
    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 variations in cross sections. By prestressing 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 of the cables 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.
    (‚Äė‚ÄėGuidelines for the Design of Cable-Stayed Bridges,‚Äô‚Äô Committee on Cable-Stayed¬†Bridges, American Society of Civil Engineers.)

    Static Analysis‚ÄĒElastic 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 system depends on the number of cables, types of connections (fixed or movable) of cables with the pylons, and the nature of the pylon connection at its base with the stayed girder or pier. The girder is usually made continuous over three spans. Figure 15.55 shows the order of redundancy for various single-plane systems of cables.
    If the bridge has two planes of cables, two stayed 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 of these cross members adds two redundants, 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 large structures.
    For a cable-stayed structure such as that illustrated in Fig. 15.56a, it is convenient to select as redundants the bending moments in the stayed girder at those points where the  cables and pylons support the girder. When these redundants are set equal to zero, an articulated, statically determinate base system is obtained, Fig. 15.56b. 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 indeterminate base system, with the cable forces as redundants. But computation is generally increased.

    A third method involves imposition of hinges, for example at a and b (Fig. 15.57), so placed as to 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 = Xa + Xb and Z = Xa + 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.
    A similar concept is illustrated in Fig. 15.58, which shows the application of independent symmetric and antisymmetric group stress relationships to simplify calculations for an 8- degree indeterminate system. Thus, the first redundant group X1 is the self-stressing of the lowest cables in tension to produce M1= +1 at supports.
    The above procedures also apply to influence-line determinations. Typical influence lines for two bridge types are shown in Fig. 15.59. These demonstrate that the fixed cables have a favorable effect on the girders but induce sizable bending moments in the pylons, as well as differential forces on the saddle bearings.
    Note also that the radiating system in Fig. 15.55c and d generally has more favorable bending moments for long spans than does the harp system of Fig. 15.59. Cable stresses also are somewhat lower for the radiating 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 radiating system presents esthetic problems, because of the different intersection angles when the cables are in two planes.
    Furthermore, fixity of the cables at pylons with the radiating system in Fig. 15.55c and d produces a wider range of stress than does a movable arrangement. This can adversely influence design for fatigue.
    A typical maximum-minimum moment and axial-force diagram for a harp bridge is shown in Fig. 15.60.
    The secondary effect of creep of cables (Art. 15.12) can be incorporated into the analysis.
    The analogy of a beam on elastic supports is changed thereby to that of a beam on linear viscoelastic supports. Better stiffness against creep for cable-stayed bridges than for comparable suspension bridges has been reported.

    Static Analysis‚ÄĒDeflection 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  of stayed girders is relatively small. 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 Severn Bridge is 6% for the stayed girder and less than 1% for the cables. Similarly, for the Du¨sseldorf 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 (Art. 15.19.1). 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 constant, the vertical and horizontal cable components V and H change by magnitudes V and H by virtue of the new deformed geometry. The first approximate correction determines the effects of these V and H forces on the deformed system, as well as the effects of V and H due to the changed geometry. This process is repeated until convergence, which is fairly rapid.

     

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