105. Load Constants.-Load constants are procured in much the same manner as arch constants. The contribution of a voussoir to a load constant at a point is the product of the multiplier for that voussoir, in terms of Ru, rv, Tv, and Iv, and the increment of the uniform thickness tabular values for the angles from the point considered to the limits of the voussoir. Since uniform thickness tabular values include the effect of the load from the point to the rock abutment, fractional portions of tabular values must be used to obtain correct increments. Section 134 and table 9, chapter VI, show how voussoir D-terms are computed by using circular arch functions.

Equations for load constants in the following sections are for any point and represent the summation of the various voussoir contributions at the point. Separate formulas are given for radial, tangential, and twist loads because each type has different multipliers and different points of application.

106. Radial Load Constants.-Radial loads are applied normally at the upstream face. Equations for their load constants are,

107. Tangential Load Constants.-Tangential loads act along the arch center line, theoretically, but for the purpose of simplification in this method of analyzing a variable thickness arch, they are assumed to be applied along the arc through the crown center line. Consequently, the eccentricity of this load application must be taken into account in computing D-terms. The moment of a tangential load at an arch point, as K in figure 36, is He' greater than the moment at point J which is on the arc through the crown center line. Terms involving this additional moment, He', are included in the following equations for tangential load constants.

108. Twist Load Constants.-Twist loads act along the center line of the arch, theoretically; but for purposes of evaluating D-terms, they are assumed to be applied along the center line arcs of the voussoirs. This assumption, which applies a discontinuous load, can only introduce a small and negligible error in the final results, since the total length of the voussoir center lines is very nearly equal to the length of the true center line of the arch. Application of twist load to the voussoir center line produces an eccentricity with reference to other arch points and causes an additional tangential movement. This factor is included in the equation for D.. Equations for total D-terms for twist loads are,

109. Application of Arch Formulas. Besides the arch and load constants, revisions of several other quantities in circular arch formulas are necessary to make them applicable to variable thickness arches. The revisions occur in formulas for moments, thrusts, and shears due to unit loads, and in formulas for coordinates of arch points.

U

Formulas for ML, HL, and VL are the same as for uniform thickness circular arches with the following revisions. The radius, r, in radial load formulas is the radius to the arch point, Rʊ —T/2; and the radius, r, in twist and tangential load formulas is the radius to the crown center line arc. Since the H of the tangential load is applied along the crown center line arc, the quantity, Hep, must be combined with M, quantity from the load formula to give the correct value for M at an arch point. The complete formula for M1 at an arch point is ML, from the load formula, minus HLеP.

rp sin

L

L

and y

=

= r vers

The coordinates x and y are changed from x= r sin & and y to x Torp cos . These are used in the formulas for solving crown forces. In formulas 119 to 124 for arch deflections at any arch point, coordinates x and y for the abutment, with the arch points as orgins, are xА

=

ra sin Ф4 and YA

=

TA COS PA.

ARCH WITH INTRADOS FILLETS

110. General Statement. If arch abutment stresses are much greater than crown stresses, they can be reduced by providing fillets at the ends of the intrados curves. This reduces abutment stresses because the arch thickness is increased and the resultant thrust is closer to the center line of the arch. If it is necessary to extend the fillet beyond the 1/2 point of the arch, the arch is more conveniently designed as a variable thickness arch. Short radius fillets should be avoided because of indefinite excavation limits. If excavation is extended further into the canyon walls than anticipated, short radius fillets may not intersect the rock abutments.

An arch with a fillet section, as shown in figure 37, is a special case of a variable thickness arch. In the analysis of such an arch, the left and right parts are each divided into two sections, a fillet section and a uniform thickness section. By making the fillet section and the uniform section subtend central angles of whole degrees, the uniform thickness arch data with certain revisions can be used for the analysis.

Arch constants are obtained in the same manner as for a variable thickness arch. Load constants are determined independently for the fillet section and the uniform thickness section. The transfer of load effects at the fillet crown, point F, figure 37, to points in the uniform thickness section introduces the influence of the fillet on that section.

111. Notation. The notation for uniform and variable thickness arches applies to arches with fillets with the following additions:

F

Q

RU

=

=

=

crown of fillet section.

arch point in uniform thickness section.

radius to upstream face and is the same for both sections. r-radius to center line of uniform thickness section.

112. Arch Constants.-Arch constants are determined by the following procedure. The fillet section is divided into four voussoirs and the uniform section is included as an additional voussoir. Using these five voussoirs, equations 177 to 182 for variable thickness arch constants can be used for the fillet type of arch.

113. Load Constants.-Load constants at a point in the uniform thickness section, as Q in figure 37, are the sum of three quantities: first, the effect of load acting on the uniform section; second, the effect of forces and moments, due to load on the uniform section, acting on the fillet section; and, third, the effect of applied load between F and the abutment acting on the fillet section. The first quantity is evaluated in the same way as for uniform thickness arch D-terms. The second quantity is obtained by transferring the deflections at F, products of unit-load moments, thrusts, and shears, and the proper arch constants, to Q. The third quantity is determined by transferring to Q the D-terms at F due to the portion of the load on the fillet section.

In the following equations for D-terms at Q in the uniform thickness section, (') values are values at F due to load on the fillet section; (") values are values at Q due to load on the uniform section; ML, HL, and V refer to F; x and y are coordinates of F with Q as the origin; and is the angle from F to Q.

L

D1 = [A; M1- Bi H1 + C; V1 ] + D; + Di'

(192)

Dε = [A; x-B sin+C; cos )ML-(B; x - B sino + B2 cos$) HL

+(C¦ x - B1⁄2 sino + C'2 cos $)v. ] + [Dix + D2 cos & - D3 sin &] + D2

(193)

D3 = [(A; y + Bicos + ci sino) ML-(By + B cos + B sin) HL

+(C¦ & sino)V1

+(ci y + B2 cos + + C2 sin p)v.] + [D; y + D sino + D's cosp]+D"

(194)

Quantities ML, HL, and VL in these equations may be evaluated by equations for uniform thickness arches if Q is the crown point. If Q is between the fillet section and the crown of the arch, the effect of the load between Q and the crown must be eliminated. This has been done in the following equations which apply to radial, tangential, and twist loads. The subscripts F and Q preceding the terms refer to values at points F and Q, respectively, for the entire load between F or Q and the crown of the arch.