INTRODUCTION

The Ruck-A-Chucky Bridge will carry a local intercounty road across the middle fork of the American River in California at a point some 10 miles air distance above the Auburn Dam. The American River, about 30 ft. deep and 100 ft. wide before damming, will have a depth of 450 ft. and a width of 1,100 ft. as a result of the construction of the dam.

In order to provide a vertical clearance of 50 ft. above high reservoir water level, the bridge proper would have a length of 1,300 ft. between the hillsides, with a slope of 40° to the horizontal. A straight bridge, with limited curvature at the ends, would require heavy approach cuts and tunnels. A curved bridge would be a much better solution. Conventionally, a curved bridge of this length has to be supported on several intermediate piers. In this case, however, such a solution would be extremely costly because of the water depth of 450 ft., which would aggravate seismic forces on the piers.

The preliminary study concluded that conventional bridges were completely unsatisfactory. The search continued until finally the hanging arc evolved (Fig. 1). The hanging arc, so-called because it is curved on plan and suspended by cables, is made up of two components: the cable stays acting in tension and the curved girder carrying the traffic and absorbing the axial compression produced by the cables. The cables are post-tensioned to control the stresses and strains. They are anchored on the slope so as to control the line of pressure in the girder. Thus, an ideal stress condition is achieved with small bending and torsional moments.

A 1,500 ft. radius was chosen for the curvature of the hanging arc layout after trade-off studies between the location of the anchoring pedestals on the slopes and the hillside excavation at both approaches. For the bridge girder, two designs were made, one using steel and one concrete. Light-weight concrete was adopted for the concrete girder to reduce cable weight and cost. A-588 steel was used for all the exterior plates of the steel girder to minimize maintenance.

CABLE LAYOUT AND DESIGN

The cable layout for this bridge seemed to lend itself well for "optimization" studies using computer programs. This proved to be not so, due to the irregular topography of the pedestal locations. Furthermore, since slight variations from the theoretical optimum generally do not result in appreciable difference in material requirements, an exact theoretical mathematical solution was not considered a necessity.

The criteria for achieving an optimum cable layout are:

1. Each cable must balance the weight of the girder in the cable's tributary loading area to result in minimum vertical moment. This must be optimized with a horizontal angle with the bridge that will result in minimum horizontal bending moment along the girder, and in a minimum amount of cable steel.
2. The cable formation must be aesthetically appealing.
3. Fire protection for the cables should be minimized.
4. Minimum constraint by local topography and geology at the cable pedestal locations.
5. A uniformity of cable inclination at the bridge to simplify anchorage details.

In a trial and error approach to meet the above goals, over 100 different cable formations were analyzed with the help of a computer program,

In consideration of size of girder elements and total weights of cables during erection, the selected spacing of cables for the steel box girder bridge was 50 ft. (total cables of 48) and 30 ft. (total cables of 80) for the concrete girder bridge (Figs. 2 & 3). These cable spacings also resulted in bending moments in the girder that required reasonable, as well as optimum, girder depths and material thicknesses.

After the cable formation was decided on, forces in the cables under different loading conditions were computed. Based on a factor of safety of 2.7 (on ultimate strength) for dead load only, and of 2.25 for dead plus live load, the cable areas were obtained. They were then checked for other loading conditions in accordance with AASHTO Specifications for stress combinations.

One consideration for the choice of these stresses is fatigue. Since the live load stresses are only a small portion of the dead load (in the order of 15%), and since the full design live load practically never occurs on a bridge of this span, there is little cause for the usual concern about live load fatigue stresses to this bridge. The heaviest live load would be an occasional logging truck equivalent to the action of an AASHTO HS-20 truck. Three types of cable steel were considered:

1. Seven-wire strands.
2. Parallel wires.
3. Bridge strands with diameter up to 4 inches.

The bridge design was based on 1/2-in. diameter, 7-wire, 270 ksi strands, since many competitive anchorage types are available for such cables. But 1/4-in. multiple straight wire cables and bridge strands will be allowed as alternates in construction bidding.

Tubing will be used to protect cables against corrosion. Several types were compared. For construction ease and cost reasons, polyethylene tubes of 1/2-in. thick were specified.

Injection of cement grout within the tubes is needed to insure protection against corrosion and fire. Grease grouting was rejected because of the difficulty of preventing leakage, and the lack of fire resistance.

Several methods of installing the cables were investigated. The recommended method consists of first installing small pilot cables between the two anchor points on which the sheathed cables are attached and pulled upward by a towline.

CONCRETE GIRDER

The concrete box girder will be fixed at the abutments to provide stability and resistance during construction. A hinge arrangement at mid-span would require complex and costly articulation details. It would also increase flexibility under seismic disturbances. It was found that: (1) temperature stresses were not significant, and could be reduced if the concrete at mid-span was keyed at a proper temperature, and (2) the shrinkage and creep stresses are quite low for this bridge. Thus, it was decided to make the concrete girder continuous at mid-span.

The depth of the concrete box girder section was made 9 ft. to provide vertical stiffness and to distribute live load and construction load on the deck to a sufficient number of adjoining cables. This height is necessary to provide easy access to the inside of the girder for inspection, maintenance, and construction purposes. The concrete girder is supported by cables at 30 ft. intervals, based on construction and aesthetic considerations.

The closed box shape (Fig. 4) is chosen for its torsional. rigidity and its minimum requirement of concrete. The wall thickness is partly governed by the placement of prestressing steel, reinforcing bar, and concrete. With transverse post-tensioning along the top and bottom surfaces of the box, the entire deck concrete is placed under biaxial prestressing and its resistance to local loading is enhanced.

A loop cable arrangement was devised for the cable-to-box attachment. The loop anchor will be buried in the concrete box and post-tensioned prior to its attachment to the cable. This arrangement will make it simple to provide for the different angle inclinations between the cable and the box.

An initial horizontal moment is provided at the abutment by the cables so as to counteract the moment produced by axial shortening. Additionally. the cables near the abutment will be over-post-tensioned to balance negative vertical moments produced by live load. Across the mid-span, continuity cables will be placed inside the box to maintain the concrete in compression and reduce cracks and also to reduce the axial stresses at the abutments.

STEEL BOX GIRDER

Similar to the concrete girder, the steel girder must be fixed at the abutment in order to resist horizontal constructional moments. Under full live load and other loading combinations, maximum bottom fiber stress at the abutment becomes critical. These stresses are reduced by adjusting the cable forces near the abutments.

A hinge at mid-span, permitting expansion and contraction in the longitudinal direction, would reduce the temperature effects. However, it would increase the mid-span displacements under vertical and lateral loads, increase the horizontal bending moment and thrust at the abutment, and give rise to hammer actions under seismic disturbance. It was decided to make the steel girder continuous across mid-span, as for the concrete girder. From the point of view of allowable stresses, A-36 steel would be sufficient at most sections. However, to reduce maintenance cost, A-588 is specified for the entire envelope,

The box section (Fig. 5) is designed to resist wheel loads and other loadings with enough torsional and lateral stiffness to meet constructional requirements. A more or less conventional orthotropic section is used to resist axial compression in addition to horizontal and vertical moments.

The depth of the section was set at 8 ft. to provide easy access within the section for construction and maintenance, sufficient bending resistance at the abutment, local rigidity, and to distribute concentrated loading to several adjoining cables.

The side and bottom plates of the box are 3/4 inch thick (Pig. 5) and the deck plate is 1/2 inch with 5/16 inch trough stiffeners. The four vertical web plates are 3/8 inch thick of A-36 steel. Because of the thin material in all plates, stiffeners are required. The plates are stiffened by Tee sections and the web plate by plate stiffeners.

To facilitate transportation, each panel 50 ft. long by 50 ft. wide, will be fabricated in five longitudinal sections, each 10 ft. wide, and detailing is made on that basis.

For all shop work, welding is preferred for the orthotropic deck. In the field, welding is also desirable from the standpoint of appearance. However, field welding of side and bottom plates would add considerably to the time and cost of erection. The design therefore calls for bolted splices at these plates and welding for the deck plate. All connections of stiffeners will be bolted.

The attachment of the cable to the girder to transfer large forces from a round cable and anchor to the thin flat plates of a steel box girder presented one of the most demanding problems of the design. The selected method consists of nose fairing on each edge of the box girder through which the cable enters to its anchorage on the horizontal bottom plate of the fairing. At the location of the cable anchorage, the horizontal bottom plate of the nose fairing passes through the sloping side plates and becomes the bottom flange of the floorbeam. Also, at the location of the cable, a lateral truss will be part of the box girder. This truss will be composed of the floorbeam, the transverse stiffeners for the side and bottom plates and added diagonal members consisting of double angles. This truss, in addition to reducing torsional warping, will transfer the vehicle wheel loads to the cables. The fairing also serves the purpose of improving the aerodynamic characteristics of the girder. It is continuous throughout the girder length and forms an integral part of the entire structure. To provide for the different cable angles at the girder, each anchor plate will be oriented differently according to its location.

CONSTRUCTION OF SUPERSTRUCTURE

The steepness of the hillside and the depth of the ravine dictate that the girder be constructed from the two abutments outward with a minimum amount of falsework near the abutments (Fig. 6). The erection platform is first made ready adjacent to the abutments to receive the first deck panel. As one panel is assembled, it is hung by a pair of cables from their hillside pedestals. These cables will be post-tensioned to predetermined stresses,

The vertical control of the erection of both steel girder and concrete girder is accomplished by post-tensioning the cables at the pedestals. The .length of the cables will be controlled so that after post-tensioning, a predicted vertical force will be applied at the panel point to balance the dead load and to place the panel point at predetermined coordinates as required under the given loading conditions.

While the cable formation is designed to achieve minimum horizontal bending moment in the completed stage, large horizontal bending moments will be produced during construction. As the erection progresses outward from the abutment, the horizontal bending moment which is greatest at the abutment would increase and reach a maximum when about one-quarter of the span is erected.

To reduce these large bending moments to within reasonable limits, two horizontal erection cables will be installed for each half of the steel girder. Each of these cables will be pulled in two stages and later relaxed also in two stages for staged moment reduction. For the concrete girder, three cables are needed for each half span to accomplish the same purpose. (Fig. 7).

Bending moments in the steel girder are more easily controlled because steel can resist tension as well as compression. The concrete girder, however, has to depend upon its ability to resist tensile fiber stresses by the amount of axial pro-compression it has received at a particular time.

The live load is only a small portion of the total load on this bridge. The maximum proportion of live-load stress to total stress (dead and live) is about 17%; the average for all cables is only 11%. The effect of live load is therefore not significant except for the negative vertical bending moment at both abutments.

Due to the large curvature in the horizontal plane and the slenderness of the girders in the vertical plane, temperature stresses are relatively low and amount to a maximum of 3 ksi near the abutment for the steel girder, and 300 psi at the abutments on the top of the deck for the concrete bridge.

Shrinkage and creep stresses in the concrete girder are not as significant as first suspected. The maximum value was found to be 400 psi occurring at the abutment on the upper deck. A reduced modulus of elasticity of 2,000.000 psi was used in the computation. Computer analysis using SAP-4 program for seismic load equal to 15%

gravity applied in the horizontal direction was made. The stresses were found to be insignificant, amounting to a maximum of only 2 ksi for the steel girder and 300 psi for the concrete girder.

As is expected, construction stresses are quite significant for this bridge, and have to be controlled by the use of horizontal erection cables. A sufficient margin of safety is provided to allow for limited possible deviation during construction. It is exceedingly important therefore that the stresses and deflections of the bridge during construction be carefully monitored to detect any unforeseen conditions that might occur,

SEISMIC RESPONSES AND MODEL STUDIES

A seismic risk analysis was conducted, taking into account all the recorded earthquakes that had taken place within a radius of 200 miles and their effect on the bridge site. A design spectrum for this bridge was then computed as a composite of all the maximum credible earthquakes. Based on this spectrum, a time history input was formulated for both the dynamic analysis and experimental studies.

A model of 1:200 scale was made of the bridge for testing, including shaking table tests at the University of California, Berkeley. Made of aluminum, the model has a solid rectangular section of 0.157" thick by 0.686" wide, and was designed to reproduce all significant elastic deformations, forces and stresses, the cable effects including their sag, and dynamic effects due to vertical, transverse and torsional vibrations. Figure 8 shows the results of the frequency of the first ten modes of vibration. Good correlation was shown between the computed values and the results from the model. Confidence in the mathematical dynamic model was established. The results of the static loading of the model also confirmed the analytical calculations.

As a result of the dynamic analyses and the model tests, the following conclusions were drawn concerning the response of this bridge to seismic disturbances at the site:

1. The bridge is exceedingly effective in resisting all horizontal components of ground motion.
2. The response of the bridge to the vertical component of ground motion is also small.
3. The linear theory is accurate enough to predict the bridge response to seismic disturbances as far as the design is concerned.

WIND STUDIES AND TESTS

Wind tunnel tests and wind stability studies were carried out at Colorado State University. The experiments included section model tests and terrain model tests.

The results show that the bridge is extremely stiff torsionally and in the horizontal direction, but it is relatively flexible in the vertical direction. The bridge girder as designed, both steel and concrete, proved to be very favorable aerodynamically, so that there is effectively no aerodynamic coupling between the bending and torsion motions of the deck cross-section. This assures that flutter will not occur with wind velocity up to 120 mph and perhaps beyond, noting that the 100-year wind velocity at the site is only 90 mph. There was also no significant indication of vortex-induced oscillation of the deck during the model tests.

The long unsupported cables of various lengths tend to be excited by winds of relatively low velocity through the phenomenon of vortex shedding, but none of their frequencies is likely to excite important girder modes. With the special anchorage design of the cables, fatigue problems are not expected to arise. Since the cables can freely vibrate without causing any harm, the use of cable dampers may be postponed as now intended until a realistic assessment is made after installation from an observation of response of the cables to the action of wind.

ABUTMENT AND PEDESTALS

An unusual feature of the abutments for this bridge is the essentially horizontal direction of the reaction it has to resist, as contrasted to vertical reaction for abutments supporting the conventional bridges and inclined reaction for arches. The vertical load on these abutments is exceedingly small.

To minimize hillside cutting, the abutments are curved to follow the road leading to the bridge. Thus, the abutments will serve as part of the approach roads and fit well into the contours of the slope. Ribs and diaphragms are built into the abutment to distribute the load from the girder into the foundation. For the concrete girder, the problem is relatively simple; but for the steel girder, a transition from steel box to concrete abutment was developed. Shear connectors are to be welded onto the steel girder at the abutments and embedded into concrete to transmit the load,

Pedestals are small concrete piers against which post-tensioning will be applied first by the rock anchors and later on by the stayed cables. When both have been post-tensioned, the net resultant force on the pedestal is almost vertical but with an inclination toward the hillside to enhance sliding resistance.

CONCLUSION

The conception and design of this Hanging Arc represents an achievement in modern bridge engineering whereby technology in its many respects is rationally and interdisciplinarily applied to transform an environmental obstacle into an asset, thus arriving at an economical as well as an aesthetic solution.

In spite of its unique formation and appearance, this bridge has been designed and will be constructed using only techniques which have been proven elsewhere. These include rock anchoring, post-tensioning, cable-stayed construction, free cantilever construction; the use of erection cables, orthotropic plates, the application of soil and rock mechanics, seismic risk and dynamic analysis, shaking table tests, dynamic wind studies, wind tunnel tests, load-balancing, stress control and computer analysis.

The present intent is to bid both the concrete and steel girder designs in competition and after bid opening to select which design will be constructed.

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