Past practice in design of new highways often located bridges first, then aligned the roadway with them. Current practice, in contrast, usually fits bridges into the desired highway alignment.
Since curved crossings are sometimes unavoidable, and curved ramps at interchanges often must span other highways, railroads, or structures, bridges in those cases must be curved. Plate or box girders usually are the most suitable type of framing for such bridges.
Though the deck may be curved in accordance with the highway alignment, the girders may be straight or curved between skewed supports. Straight girders require less steel and have lower fabrication costs. But curved girders offer better appearance, and often the overall cost of a bridge with such girders may not be greater than that of a structure with straight members. Curved girders may reduce the number of foundations required because longer spans may be used; deck design and construction is simpler, because girder spacing and deck overhangs may be kept constant throughout the span; and cost savings may accrue from use of continuous girders, which may not be feasible with straight, skewed girders.
Consequently, curved girders are generally used in curved bridges.
Curved girders introduce a new dimension in bridge design. The practice used for straight stringers of distributing loads to an individual stringer, as indicated in a standard specification, and then analyzing and designing the stringer by itself, cannot be used for curved bridges. For these structures, the entire superstructure must be designed as a unit. Diaphragms or cross frames as well as the stringers serve as main load-carrying members, because of the torsion induced by the curvature.
Analyses of such grids are very complicated, because they are statically indeterminate to a high degree. Computer programs, however, have been developed for performing the analyses.
In addition, experience with rigorous analyses indicates that under certain conditions approximate methods give sufficiently accurate results.
The approximate methods described in this article are suitable for manual computations.
They appear to be applicable to concentric, circular stringers where the arc between supports subtends an angle not much larger than about 0.5 radian, or about 30 . Also, where the spans
are continuous, the methods may be used if the sum of the central angles subtended by each span does not exceed 90
. Accuracy of these methods, however, also seems to depend on the flexural rigidity of the deck in the radial direction and of the diaphragms.
The limitation of central angle indicates that the maximum span, along the arc, for a radius of curvature of 300 ft is about 300 x 0.5 = 150 ft for the approximate analysis. If the curved span is 200 ft, the approximate method should not be used unless the radius is at least 200/ 0.5 = 400 ft.
Each simply supported or continuous girder should have at least one torsionally fixed support.
For box girders, in addition, accuracy depends on the ratio of bending stiffness to torsional rigidity EI/GK, where E is the modulus of elasticity. G the shearing modulus, I the moment of inertia for longitudinal bending, and K the torsional constant for the radial cross section.
(For a hollow, rectangular tube,
For inclusion in the summation in the denominator, a concrete slab in composite construction should be transformed into an equivalent steel plate by dividing the concrete cross-sectional area by the modular ratio n.) If the central angle of a curved span is about 0.5 radian, the approximate method should give satisfactory results if the weighted average of EI/GK in the span does not exceed 2.5.
A curved-girder bridge may have open framing, closed framing, or a combination of the two types. In open framing, curved plate girders are assisted in resisting torsion only by cross frames, diaphragms, or floorbeams at intervals along the span. In closed framing, the curved members may be box girders or plate girders assisted in resisting torsion by horizontal lateral bracing as well as by cross frames, diaphragms, or floorbeams.
Approximate Analysis of Open Framing
The approximate method for open framing derives from a rigorous method based on consistent deformations. Various components of the structure when distorted by loads must retain geometric compatibility with each other and simultaneously stay in equilibrium. The equations developed for these conditions can be satisfied only by a unique set of internal forces.
In the rigorous method, a large number of such equations must be solved simultaneously. In the approximate method, considerable simplification is achieved by neglecting the stiffness of the plate girders in St. Venant (pure) torsion.
In the following, girders between the bridge centerline and the center of curvature are called inner girders. The rest are called outer girders.
The method will be described for a bridge with concentric circular stringers, equally spaced. Thus, for the four girders shown in Fig. 12.22a, if the distance from outer girder G1 to inner girder G4 is D, the girder spacing is D/3. The radius of the bridge centerline is R and of any girder Gn, Rn. Diaphragms are equally spaced at distance d apart along the centerline and placed radially between the girders.
Initially, the girders are assumed to be straight, and the span of each girder is taken as its developed length between supports. Preliminary moments Mp and shears Vp are computed as for straight girders.
Initially, the girders are assumed to be straight, and the span of each girder is taken as its developed length between supports. Preliminary moments Mp and shears Vp are computed as for straight girders.
These values must be corrected for the effects of curvature. The primary effect is a torque acting on every radial cross section of each girder. The torque per unit length at any section of a girder Gn is given approximately by
This torque must be resisted by end moments in the diaphragms (Fig. 12.22b).
For equilibrium, the end moments on a diaphragm must be balanced by end shears forming an oppositely directed couple. For example, the diaphragm between G2 and G3 in Fig. 12.22b is subjected to end shears V23. Also, the diaphragm between G1 and G2 is subjected to shears V12. Consequently, G2 is acted on by a net downward force V2, called a V load, at the diaphragm
V2 = V12 + V23
where upward forces are taken as positive and downward forces as negative.
The V loads applied by the diaphragms are treated as additional loads on the girders.
For a bridge with two girders, the V load on the inner girder equals that on the outer girder, at a specific diaphragm, but is oppositely directed. Determined by equilibrium conditions at the diaphragm, this V load may be computed from
where Mvn bending moment produced by V loads. Similarly, the shear due to the V loads must be added to the preliminary shears to yield the final shears. Stresses are computed in the same way as for straight girders.
Between diaphragms, the girder flanges resist the torsion. At any section, the stresses in the top and bottom flanges of a girder provide a couple equal to the torque but oppositely directed. The forces comprising this couple induce lateral bending in the flanges. If qn is the force per unit length of flange in girder Gn resisting torque,
under this loading as a continuous beam spanning between diaphragms. The maximum negative moment for design purposes may be taken as
Approximate Analysis of Closed Framing
Analysis of bridges with box girders or similar boxlike framing must take into account the torsional stiffness of these members. The method to be described is based on the following assumptions:
Girder cross sections are symmetrical about the vertical axis. Supports are radial. Curvature may vary so long as it does not change direction within a span. Diaphragms prevent distortion of the cross sections. Secondary stresses due to torsional warping are negligible.
Differential equations for determining the internal forces acting on a curved girder can be obtained from the equilibrium conditions for a differential segment (Fig. 12.23). Because upward and downward vertical forces must balance, the shear V is related to the loading w by
where t = applied torque. With the bending moments throughout the girder known, the torque at any section can be found from Eq. (12.41). [For a more rigorous solution, Eqs. (12.38), (12.39), and (12.41) may be solved simultaneously. This can be done by differentiating Eq. (12.39), solving for dT/ds, substituting the result in Eq. (12.41), and then solving the resulting second-order differential equation.]
Equation (12.41) indicates that the change in torque between any two sections of the girder equals the area of the M/R - t diagram between those sections. Consequently, the torque on a curved girder of the closed-framing type can be determined by a method similar to the conjugate-beam method for determining deflections. In the approximate method, however, the moments M determined from Eq. (12.40) are used instead of those from the more complex rigorous solution.
Thus, first the bending-moment diagram (Fig. 12.24b) is obtained for the vertical loading on the developed length of the girder (Fig. 12.24a). Then, all ordinates are divided by the radius R. Next, the applied-torque diagram (Fig. 12.24d) is plotted for the twisting moments applied by the loading to the girder (Fig. 12.24c). The ordinates of this diagram are subtracted from the corresponding ordinates of the M/R diagram. The resulting M/R - t diagram then is used as a loading diagram on the developed length of the girder (Fig. 12.24e). The resulting shears (Fig. 12.24Æ’) equal the torques T in the curved girder. Note that positive M/R - t is equivalent to an upward load on the conjugate beam.
The conjugate beam shown in Fig. 12.24c is simply supported. This requires that the angle of twist at the supports be zero. Hence, for this case, the curved girder is torsionally fixed at the supports. This condition is attained with a line of diaphragms at each support and a bearing under each web capable of resisting uplift, a common practice. Sometimes, interior supports of a continuous box girder are not fixed against torsion, for example, where a single bearing is placed under a diaphragm. In such cases, the span of the conjugate beam should be taken as the developed length of girder between supports that are fixed against torsion.
Loading
Dead loads may be distributed to curved girders in the same way as for straight girders. For live loads, the designer also may use any method commonly used for straight girders. If the distribution procedure of the AASHTO Standard Specifications for Highway Bridges is used, however, a correction factor should be applied. The sum of the AASHTO live-load distribution factors for all the girders in a curved grid will usually exceed the number of wheel loads required for the roadway width. Hence, if these factors were used to compute the live-load moments in the girders at a line of diaphragms, the V loads there would be too??large, because they are proportional to the sum of those moments. One way to correct the V loads determined with the AASHTO factors is to multiply the V loads by the ratio of the number of wheel loads required for the roadway width to the number of wheel loads determined by the sum of the AASHTO factors.
Impact may be taken into account, in the same way as for straight girders, as a percent increase in live load.
Centrifugal forces comprise a horizontal, radial loading on curved structures that does not apply to straight bridges. These forces are determined as a percentage of the live load, without impact (Art. 11.4). But the live load is restricted to one standard truck placed for maximum loading in each design lane.
Assumed to act 6 ft above the roadway surface, measured from the roadway centerline, centrifugal forces induce torques and horizontal shears in the superstructure. The shears may be assumed to be resisted by the concrete deck within its plane. The torques, however, must be resisted by the girders. In open-framing systems, the primary effect is on the preliminary bending moments. Resisting couples comprise upward and downward vertical forces in the girders. These forces increase bending moments in the outer girders (those farthest from the center of curvature) and decrease moments in the inner girders. The effect of centrifugal forces on V loads, however, is small, because V loads are determined by the sum of girder moments at a line of diaphragms and this sum is not significantly changed by centrifugal forces.
Sizing of Girders
Design rules for proportioning straight girders generally are applicable to curved girders, depth-span ratios, for example. But curvature does produce effects that should be considered for maximum economy. For instance, girder flanges in open-framing systems should be made as wide as practical to minimize lateral bending stresses. In some cases, where these stresses become too large, a reduction in spacing of diaphragms or cross frames may be desirable.
If curvature causes large adjustments to the preliminary moments in open framing, deepening of girders farthest from the center of curvature may be advantageous. This may be done without overall increase of the floor system, because of the superelevation of the deck.
Girder webs, in some cases, may have to be thicker than for straight girders with corresponding span, spacing, and loadings, because of the effects of curvature on shear. Reactions, too, may be significantly changed, and the effects on substructure design should be taken into account. For some sharply curved bridges, tie-downs may be required to prevent uplift at supports of girders closest to the center of curvature.
If horizontal lateral bracing is placed in an open-framing system, the effects of curvature should be examined more closely. Connections at frequent intervals can convert the system into the closed-framing type.
Fabrication
Curved plate girders usually are produced in one of two ways. One way is to mechanically bend the web to the desired curvature and then weld to it flange plates that have been flamecut to the required shape. The procedure differs from fabrication of plate girders in handling procedures, layout for fabrication, and web-to-flange welding methods.
Alternatively, girders may be curved by selectively heating the flanges of members initially fabricated straight. In this method, less steel is required. The heating and cooling induce residual stresses, but research indicates that they do not affect fatigue strength.
Mechanical bending is sometimes used for curving rolled beams.
