S

ФА

FIGURE 32-LEFT PART OF ARCH

length along center line of arch from crown to any arch point.

= angle from crown to any arch point.

x,y coordinates of any arch point with origin at crown.

=

Consider a differential element of length, ds, in the left part of the arch, as shown in figure 32. From mechanics, equations for arch movements at the crown, due to a moment, thrust, and shear acting on the differential elements, are:

The first term in these equations gives the movement due to bending, the second term gives the movement due to rib-shortening, the third term gives the movement due to shear detrusion, and the last term gives the movement due to temperature. If the equations are integrated from the crown to the abutment, and rock abutment movements added, the following equations result:

A)

-SX sinoas

-(a ya + a2 sin þ) MA - HA BcOS &A-(sin þa + a2 ya)VA + *ct cos & ds (86)

By substituting,

M = Mo + Hoy + VOX-ML

H = H2cos - Vo sin &+ HL

V = Ho sin & + VO COS -VL

and using K/Gc equals 3/Ec, as discussed in section 84, equations 84, 85, and 86 for the left part of the arch become

+ axAYA + a2 XA SiNPA - B SindA COSTA + Y SIN PA COSA +α ¿YA COSA

Ec A

Ec A

+ ax2

-S ̊ct sin & ds + AML (axA + α2COS$A)

sin (r

a2XA

+2α2XACOS PA + B sin2 PA+ T COS2 PA

Inspection shows that some of the multipliers for Mo, Ho, and Vo are duplicated in the equations. If symbols A1, B1, B2, B3, C1, C2, D1, D2, and D, are used for these multipliers and for the terms depending on loads, the equations may be written

3

Leo = LA, MO + LB, Ho + LC, Vo - LD,

Laro = LC, Mo + LB2 Ho + LC2 Vo - LD2

LASO = LB, Mo - LB3 Ho - LB2 Vo + LD3

Equations for the right part, developed in the same manner, are

(90)

(91)

(92)

89. Solution of Crown Forces.-Movements of each part of the arch, cut at the crown, may be determined by equations 90 to 95, inclusive. Using these equations and following the method outlined in section 85, the three general equations needed to solve the unknown crown forces, Mo, Ho, and Vo are

By combining constants in the brackets so that [LA1 + RA1] [£B1 + rB1] =B1, and so forth, the equations reduce to

1

The various quantities included in the A1, B1, C1, B2, C2, B3, D1, D2, and D3 terms in these equations are given in table 3. Quantities in the table are obtained from equations 87, 88, and 89 for the left part of the arch, and from similar equations for the right part.

By solving equations 99, 100, and 101 simultaneously, formulas for Mo, Ho, and Vo are derived.

Mo = [D, (Bs C2 - B2 ) - Ds (B, C2 - G, B2 ) - D2 ( B3 C - B, B2 )]

(102)

K = A, (B3 C2-B2)-B, (B, C2-C, B2)-C, (B, C, -B, B2)
For a symmetrical arch, these equations reduce to

(105)

90. Arch Deflections. In order to derive general equations for arch deflections at any arch point, the following meanings are assigned to terms used in equations 116 to 124, inclusive:

The origin is assumed to be at the point where the deflections are desired.