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Design of Voussoir Arches 160

arch, thickness, crown, ring, line and span

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DESIGN OF VOUSSOIR ARCHES 160. Methods of Design.—In designing masonry arches, the form and dimensions of the arch ring are first assumed and the stability of the arch, as assumed, is then investigated. The graphical method of investigation is commonly employed, a line of pressure (see Sec tion 156) being drawn and the maximum unit compression computed. Stability requires that the line of pressure remain within the middle third of the arch ring and that the unit compression does not exceed a safe value. If the first assumpt ions are not satisfactory the shape or dimensions of the arch ring may be modified and the new assump tions tested as before.

Arches subjected to the action of moving loads should be tested for conditions of partial loading, which may cause unsymmetrical distortion of arch ring, as well as for full load over the whole arch. For ordinary loadings and spans of moderate length, it is usually suffi cient to draw the line of pressure for arch fully loaded and with live load extending over half the arch, but in large and important struc tures, or those with unusual loadings, it may be desirable to test the arch ring with live loads in other positions which seem likely to produce maximlun distortions of the line of pressure.

161. Thickness of Arch Masonry.—The choice of dimensions for the trial arch ring is necessarily baz ed upon judgment founded upon knowledge of the dimensions of existing arches, which are found to differ widely, and rules have been formulated by several authorities for the purpose of aiding in selecting the dimensions.

Crown Thickness.—Several different formulas have been proposed for determining the thickness at the crown. Traulwinc's formula for the depth of keystone of first-class cut-stone arches, whether circular or elliptical, is Depth of key in feet For second-class work this depth may be increased about one eighth part: or for brick or rubble about one-third.

Ii'.unlrine's formula for the depth of keystone for a single arch is This gives results which agree fairly well with Trautwine's formula. For an arch of a series, Rankine also recommends

These formulas make the thickness depend upon the span and rise of the arch without regard to the loading. They agree fairly well with many examples of existing arches, but make the thickness rather large for arches of moderate span.

Douglas Formulas.—In NIerriman's American Civil Engineer's Pocket Book, _\lr. Walter J. Douglas gives the following rules for thickness at crown: These formulas give smaller thickness for highway arches of short span than Trautwine's and do not vary the thickness with the rise of the arch.

Thickness at Skewba.ck.—If the arch ring be made of uniform thick ness, the unit pressure at the ends will be greater than at the crown. The pressure may often be made fairly uniform by making the thick ness at any radial joint equal to the crown thickness times the secant of the angle made by the joint with the vertical.

In the American Civil Engineer's Pocket Book, Mr. Douglas recommends that the thickness at the springing line of a masonry arch be obtained by adding the following percentages to the crown thickness: (1) Add 50 per cent for circular, parabolic, and catenarian arches having a ratio of rise to span less than one-quarter.

(2) Add 100 per cent for circular, parabolic, cat enarian, and three centered arches having a ratio of rise to span greater than one-quarter.

(3) Add 150 per cent for elliptical, five-centered and seven centered arches.

_Alr. Douglas recommends that the top thickness of abutments be assumed at five times the crown thickness. For a pier between arches in a series he suggests a thickness at top of three and one-half times the crown thickness, but places an abutment at every third or fifth span.

Trautwine gives a method for design of abutment., approximately as follows (see Fig. S6) : Thickness at springing line in feet Continue bd downward to bottom of abutment, and upward a distance 2. From e draw a tangent c—f to the extrados.

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