Analysis of Reinforced Arcs

ANALYSIS OF REINFORCED ARCS. The analysis of the reinforced concrete arch having fixed ends is substantially the same as that for a plain concrete arch, except that the moment of inertia of the homogeneous cross section should be replaced by that of the composite cross section.

The differences between the solution for the plain concrete arch and that for the reinforced arch will be considered in order.

Equations of Condition. It

is customary to place the re inforcing steel symmetrically about the neutral surface of the arch ring. In § 1302 the fiber stress of the plain concrete section is stated to be f = E dand for the symmetrically reinforced concrete section, the fiber stress in the concrete is f, = d and that in the steel is f, = E,- d Similarly, the differential moment of the stress in the plain concrete section is stated to be f z da; but for the reinforced section the elementary moment is (f, z + f, z da,). Substituting in equation a, page 672, the above values for the rein forced concrete section, instead of the corresponding values for the plain concrete section, remembering that E, = n (see § 447), and carrying the work through, equation c, page 672, becomes By analogy, the equations of conditions, equations 1, 2, and 3, then become To Make 48 _ (I, + nI,) Constant. The neutral line of

the arch ring must be divided so as to make 4s _ (I + n1,) constant. The method of making this division is the same for the reinforced section as that explained in § 1311 for the plain section. Proceeding as in the second paragraph of § 1311, construct a line similar to DF, Fig. 231, page 677, to represent I, + and I, = a b d,' + A, in which is the depth of the con crete from the neutral axis, is the thickness of the steel in the direc tion of the radius of the arch, d, is the distance from the neutral line to the center of the steel, and A, is the sum of the area of the cross section of the steel on the two sides of the neutral axis. Since it is customary to consider only a section of the arch a unit long, we may put and ordinarily we may take Next proceeding as in § 1312, divide the neutral line into a pre determined number of parts.

To Locate the Line kk.

The method of locating the line kk for the reinforced arch is exactly the same as that for the plain concrete arch—see § 1313.

To Find the Tine Equilibrium Polygon.

The method of find ing the true equilibrium polygon for the reinforced arch is exactly the same as that explained in § 1314-26.