114. Application of Formulas.-Other revisions, in addition to those given for arch and load constants, occur in formulas for moments, thrusts, and shears due to applied loads, and in formulas for coordinates of arch points. These revisions are the same as for a variable thickness arch, given in section 109.

115.

OTHER TYPES OF ARCHES

General Statement.-Ordinarily, the design of an arch dam can be made with circular arches of uniform thickness, variable thickness, or with downstream fillets. However, in some cases it is desirable to use circular arches of the three-centered uniform-thickness type or the polycentered variable-thickness type. Noncircular types of arches are seldom used for arched concrete dams. Since such types of arches are seldom encountered, the discussions in the following sections are brief and limited to general considerations. Nevertheless, sufficient theory is given so that an analysis of each type can be made if necessary.

116. Three-centered Arch. A three-centered arch of uniform thickness, as shown in figure 38, is used to make the line of thrust correspond more nearly with the center line. This makes it possible to reduce

bending moments at the crown and abutments, and thereby eliminate the high tension stresses that would exist in a single-centered arch.

Formulas for a single-centered arch can be used for this case, but arch and load constants must be transferred from the crown of the abutment circular section, point F, to arch points in the central circular section. In the following equations for constants at a point Q in the central circular section, (') values are values at F due to load on the abutment circular sections; (") values are values at Q due to load on the central section; ML, HL, and VL refer to point F; x and y are coordinates of point F with Qas the origin; and is the angle from F to Q.

=

B (Ay+B cos ++c; sin p)x+(c; cos p-B; sin p)y+ B2 (cos2p-sin2)

B3= (Ay + 2B cosp+ 2ci sinp)y + 2 B2 sino cos + + B cos2 + C2 sin2 + B

(202)

C2 = (A x-2B, sino + 2c, cos p)x-2B, sino cos + B sin2 + C2 cos2 + + C2

(203)

D2 = (Ax-B, sino + c cos) ML-(B) x - B3 sino + B2 cos $) H2

+(cx- B2 sino + C2 cos p) V1 + (D' x + Decos p-D sin) + D2"

(205)

D3 = (Ay + Bicos + ci sin p) ML-(By + B3 cos + + B2 sin) HL

L

+(c' y + B2 cos & + c2 sin p) VL + (D, y + D2 sin + D's cos ) + D (206) The ML, HL, and V terms in these equations are determined in the same way as corresponding values for fillet arches at the fillet crown. Equations for these quantities are given in section 113, equations 195, 196, and 197.

In addition to equations for arch and load constants, formulas are needed for x and y values used in computing effect of abutment yield on points in the central circular section. These abutment coordinates, with the origin at any point Q in the central circular section, are as follows:

in which

x = ra sin (+APB) + (rc-ra) sin

y = rc- ra cos($+ APB) - (rc - ra) cos &

rc radius to center line of central circular section.

TB radius to center line of abutment circular section.

=

(207)

(208)

It is also necessary to have equations for ML, HL, and VL for points in the abutment circular section. These are developed by transferring the ML, HL, and VL at F to points in the abutment circular section, giving the equations,

in which BML, BHL, and BVL are moments, thrusts, and shears of the load on the abutment circular section only; FML, FHL, and FVL refer to values at F; and лË, уB, and B are coordinates from F to points in the abutment circular section.

117. Polycentered Arch. The polycentered arch of variable thickness, shown in figure 39, offers a more complete means of reducing the undesirable tensile stresses that would exist in single-centered or threecentered types. However, it is seldom necessary to go to this refinement in the design of arch dams.

The method of analysis consists of dividing the central and abutment sections into four voussoirs, or less, and considering each section as a circular variable-thickness arch. Formulas for a variable-thickness arch can be applied to the two variable-thickness sections. Additional formulas needed to consider the two sections acting in conjunction with each other are the same as given for the three-centered arch of uniform thickness, except the coordinate equations 207 and 208, which must be revised to read.

radius to upstream face in central section.

radius to upstream face in abutment section.

radius to arch center line at any point Q in central section

=

CRU-T/2.

radius to arch center line at abutment

=

BRU-TA/2.

Arch and load constants are given in equations 198 to 206, inclusive. Equations for moments, thrusts, and shears due to external load are the same as equations 209, 210, and 211. This method of analysis can also be used if either section is of uniform thickness.

118. Noncircular Arches.-Arches that do not have an extrados composed of circular segments can be analyzed by the voussoir summation method. Since tabulated arch and load constants cannot be used in the voussoir method, the summations for different trial loads require considerable time.

In the summation method integrals in general arch equations are replaced by mechanical summations of the various quantities calculated for the separate voussoirs, usually ten, into which each side of the arch is divided. Otherwise, equations for solution of crown forces and calculation of arch deflections are identical with those for the integration method.

119. Cracked Arches. In the design of arched dams it is difficult to eliminate tension at the extrados at the abutments and at the intrados at the crown. If the customary assumption that concrete is incapable of withstanding tension is made, sections of the arch subjected to tension are disregarded in the analysis. In arch analyses based on this assumption, it has been found that deflections are not greatly different from those for an uncracked arch. Therefore, analyses of cracked arches have only a small influence on trial load adjustments of deflections. The small effect on the analysis leads to the omission of the use of cracked arches in all but special cases. Cracking of vertical cantilevers, however, does have an important effect on deflection adjustments. Consequently, cantilever cracking is usually considered in trial load analyses of arch dams where preliminary analyses indicate the occurrence of vertical tension stresses.

If it is decided that effects of cracking should be considered in analyzing arch elements, amounts of cracking, or portions of the arch to be neglected, are estimated by trial. Several trial analyses are usually necessary, since one trial does not definitely determine the largest interior arch which will be free from tension.

For the analysis of a cracked arch, the voussoir summation method described in the preceding section is used. In analyzing an interior arch, it should be remembered that radial loads are applied at the upstream face of the original arch and not at the upstream face of the interior arch.