Design of Arch Ribs and Ties

Computers greatly facilitate preliminary and final design of all structures. They also make possible consideration of many alternative forms and layouts, with little additional effort, in preliminary design. Even without the aid of a computer, however, experienced designers can, with reasonable ease, investigate alternative layouts and arrive at sound decisions for final arrangements of structures.
Rise-Span Ratio. The generally used ratios of rise to span cover a range of about 1:5 to 1:6. For all but two of the arch examples in Art. 14.8, the range is from a maximum of 1:4.7 to a minimum of 1:6.3. The flatter rise is more desirable for through arches, because appearance will be better. Cost will not vary materially within the rise limits of 1:5 to 1:6. These rise ratios apply both to solid ribs and to truss arches with rise measured to the bottom chord.
Panel Length. For solid-ribbed arches fabricated with segmental chords, panel length should not exceed 1⁄15 of the span. This is recommended for esthetic reasons, to prevent too large angular breaks at panel points. Also, for continuously curved axes, bending stresses in solid-ribbed arches become fairly severe if long panels are used. Other than this limitation, the best panel length for an arch bridge will be determined by the usual considerations, such as economy of deck construction.
Ratio of Depth to Span. In the examples in Art. 14.8, the true arches (without ties) with constant-depth solid ribs have depth-span ratios from 1:58 to 1:79. The larger ratio, however, is for a short span. A more normal range is 1:70 to 1:80. These ratios also are applicable to solid-ribbed tied arches with shallow ties. In such cases, since the ribs must carry substantial bending moments, depth requirements are little different from those for a true arch. For structures with variable-depth ribs, the depth-span ratio may be relatively small (Fig. 14.7).
For tied arches with solid ribs and deep ties, depth of rib may be small, because the ties carry substantial moments, thus reducing the moments in the ribs. For a number of such structures, the depth-span ratio ranges from 1:140 to 1:190, and for the Fremont Bridge, Portland, Ore., is as low as 1:314. Note that such shallow ribs can be used only with girder or trussed ties of considerable depth.
For truss arches, whether true or tied, the ratio of crown depth to span may range from 1:25 to 1:50. Depth of tie has little effect on depth of truss required. Except for some unusual arrangement, the moment of inertia of the arch truss is much larger than the moment of  inertia of its tie, which primarily serves as a tension member to carry the thrust. Hence, an arch truss carries substantial bending moments whether or not it is tied, and required depth is not greatly influenced by presence or absence of a tie.

Single-Web or Box Girders. For very short arch spans, single-web girders are more economical than box girders. For all the solid-ribbed arches in Art. 14.8, however, box girders were used for the arch ribs. These examples include a minimum span of 193 ft. Welded construction greatly facilitates use of box members in all types of structures.
For tied arches for which shallow ties are used, examples in Art. 14.8 show use of members made up of web plates with diaphragms and rolled shapes with post-tensioned strands. More normally, however, the ties, like solid ribs, would be box girders.
Truss Arches. All the usual forms of bolted or welded members may be used in truss arches but usually sealed, welded box members are preferred. These present a clean-cut appearance. There also is an advantage in the case of maintenance.

Another variation of truss arches that can be considered is use of Vierendeel trusses (web system without diagonals). In the past, complexity of stress analysis for this type discouraged their use. With computers, this disadvantage is eliminated. Various forms of Vierendeel truss might well be used for both arch ribs and ties. There has been some use of Vierendeel trusses for arch bracing, as shown in the examples in Art. 14.8 This design provides an uncluttered, good-looking bracing system.
Dead-Load Distribution. It is normal procedure for both true and tied solid-ribbed arches to use an arch axis conforming closely to the dead-load thrust line. In such cases, if the rib is cambered for dead load, there will be no bending in the rib under that load. The arch will be in pure compression. If a tied arch is used, the tie will be in pure tension. If trusses are used, the distribution of dead-load stress may be similarly controlled. Except for three-hinged arches, however, it will be necessary to use jacks at the crown or other stress-control procedures to attain the stress distribution that has been assumed.
Live-Load Distribution. One of the advantages of arch construction is that fairly uniform live loading, even with maximum-weight vehicles, creates relatively low bending stresses in either the rib or the tie. Maximum bending stresses occur only under partial loading not likely to be realized under normal heavy traffic flow. Maximum live-load deflection occurs in the vicinity of the quarter point with live load over about half the span.
Wind Stresses. These may control design of long-span arches carrying two-lane roadways or of other structures for which there is relatively small spacing of ribs compared with span length. For a spacing-span ratio larger than 1:20, the effect of wind may not be severe. As this ratio becomes substantially smaller, wind may affect sections in many parts of the structure.
Thermal Stresses. Temperature causes stress variation in arches. One effect sometimes neglected but which should be considered is that of variable temperature throughout a structure.
In a through, tied arch during certain times of the day or night, there may be a large difference in temperature between rib and tie due to different conditions of exposure. This difference in temperature easily reaches 30F and may be much larger.
Deflection. For tied arches of reasonable rigidity, deflection under live load causes relatively minor changes in stress (secondary stresses). For a 750-ft span with solid-ribbed arches 7 ft deep at the springing line and 4 ft deep at the crown and designed for a maximum liveload deflection of 1⁄800 of the span, the secondary effect of deflections was computed as less than 2% of maximum allowable unit stress. For a true arch, however, this effect may be considerably larger and must be considered, as required by design specifications.

Dead-Load to Total-Load Ratios. For some 20 arch spans checked, the ratio of dead load to total load varied within the narrow range of 0.74 to 0.88. A common ratio is about 0.85.
This does not mean that the ratio of dead-load stress to maximum total stress will be 0.85.
This stress ratio may be fairly realistic for a fully loaded structure, at least for most of the members in the arch system. For partial live loading, however, which is the loading condition causing maximum live-load stress, the ratio of dead to total stress will be much lower, particularly as span decreases.
For most of the arches checked, the ratio of weight of arch ribs or, in the case of tied arches, weight of ribs and ties to, total load ranged from about 0.20 to 0.30. This is true despite the wide range of spans included and the great variety of steels used in their construction.
Use of high-strength steels helps to maintain a low ratio for the longer spans. For example, for the Fort Duquesne Bridge, Pittsburgh, a double-deck structure of 423-ft span with a deep truss as a tie, the ratio of weight of arch ribs plus truss ties to total load is about 0.22, or a normal factor within the range previously cited. For this bridge, arch ribs and trusses were designed with 77% of A440 steel and the remainder A36. These are suitable strength steels for this length of span.
For the Fort Pitt Bridge, Pittsburgh, with a 750-ft span and the same arrangement of structure with shallow girder ribs and a deep truss for the ties, the ratio of weight of steel in ribs plus trussed ties to total load is 0.33. The same types of steel in about the same percentages were used for this structure as for the Fort Duquesne Bridge. A higher-strength steel, such as A514, would have resulted in a much lower percentage for weight of arch ribs
and trusses and undoubtedly in considerable economy. When the Fort Pitt arch was designed, however, the owner decided there had not been sufficient research and testing of the A514 steel to warrant its use in this structure.

For a corresponding span of 750 ft designed later for the Glenfield Bridge at Pittsburgh, a combination of A588 and A514 steels was used for the ribs and ties. The ratio of weight of ribs plus ties to total load is 0.19.
Incidentally, the factors for this structure, a single-deck bridge with six lanes of traffic plus full shoulders, are almost identical with the corresponding factors for the Sherman Minton Bridge at Louisville, Ky., an 800-ft double-deck structure with truss arches carrying three lanes of traffic on each deck. The factors for the Pittsburgh bridge are 0.88 for ratio of dead load to total load and 0.19 for ratio of weight of ribs plus ties to total load. The corresponding factors for the Sherman Minton arch are 0.85 and 0.19. Although these factors are almost identical, the total load for the Pittsburgh structure is considerably larger than that for the Louisville structure. The difference may be accounted for primarily by the double-deck structure for the latter, with correspondingly lighter deck construction.
For short spans, particularly those on the order of 250 ft or less, the ratio of weight of arch rib to total load may be much lower than the normal range of 0.20 to 0.30. For example, for a short span of 216 ft, this ratio is 0.07. On the other hand, for a span of only 279 ft, the ratio is 0.18, almost in the normal range.
A ratio of arch-rib weight to total load may be used by designers as one guide in selecting the most economical type of steel for a particular span. For a ratio exceeding 0.25, there is an indication that a higher-strength steel than has been considered might reduce costs and its use should be investigated, if available.
Effect of Form on Economy of Construction. For solid-ribbed arches, a smooth-curve axis is preferable to a segmental-chord axis (straight between panel points) so far as appearance is concerned. The curved axis, however, involves additional cost of fabrication. At the least, some additional material is required in fabrication of the arch because of the waste in cutting the webs to the curved shape. In addition to this waste, some material must be added to the ribs to provide for increased stresses due to bending. This occurs for the following reason:
Since most of the load on the rib is applied at panel points, the thrust line is nearly straight between panel points. Curving the axis of the rib causes eccentricity of the thrust line with  respect to the axis and thus induces increased bending moments, particularly for dead load.

All these effects may cause an increase in the cost of the curved rib on the order of 5 to 10%.
For tied solid-ribbed arches for which it is necessary to use a very shallow tie, costs are larger than for shallow ribs and deep ties. (A shallow tie may be necessary to meet underclearance restrictions and vertical grades of the deck.) A check of a 750-ft span for two alternate designs, one with a 5-ft constant-depth rib and 12.5-ft-deep tie and the other with a 10-ft-deep rib and 4-ft-deep tie, showed that the latter arrangement, with shallow tie, required about 10% more material than the former, with deep tie. The actual increased construction cost might be more on the order of 5%, because of some constant costs for fabrication and erection that would not be affected by the variation in weight of material.
Comparison of a tapered rib with a constant-depth rib indicates a small percentage saving in material in favor of the tapered rib. Thus, costs for these two alternatives would be nearly equal.

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