F= function for determining properties and functions of uncracked portion of section.

Ro

=

=

=

radius to end of crack.

radius to center line of uncracked portion of section.

distance from upstream edge of uncracked portion of section to center of gravity of that portion of section.

T, radial thickness of uncracked portion of section.

=

=

=

=

=

=

=

area of uncracked portion of section.

moment of inertia of uncracked portion of section about a circumferential line through its center of gravity.

resultant moment about center of gravity of uncracked portion of section.

uplift force on total horizontal section.

moment of U about center of gravity of total horizontal section. vertical stress at uncracked face of cantilever.

76. Cracking at Upstream Face. If a cantilever is cracked at the upstream face, it is necessary to determine how far the cracks extend radially along each horizontal section. Then the area, moment of inertia, resultant moment, and stress can be determined for the uncracked portion of each section.

From figure 20, which shows a horizontal section of a cantilever cracked at the upstream face, the volume of the indicated stress solid is

The moment of the stress solid about the center of gravity of the original section is

By substituting equation 64 for W + U and simplifying,

tion, it is evident that the value of F determines the ratio RD/Ro for the uncracked portion of the horizontal section. A curve in figure 21, developed from equation 65, is used to evaluate the ratio RD/R。 for given values of F. If RD/Ro, as determined by the curve, is less than RD/RU, the section does not crack. Conversely, if RD/Ro is greater than R1/Rʊ, the section is cracked. If RD/R。 is greater than unity, the section is completely cracked.

1

Ꭰ

With the value of RD/Ro determined, T, can be evaluated by the equation

Equations for area, A1, and moment of inertia, I, of the uncracked

portion of the horizontal section are obtained by substituting A, for A;

1

T1 for T; Ro/Raxis for Ru/Raxis; and RD/R。 for RD/Rʊ, in equations 51 and 53. The revised formulas are

1

Values of A, and I, are obtained from curves in figure 16, using proper multipliers and following the same procedure as indicated for the uncracked section.

The moment of the redistributed forces about the center of gravity of the uncracked portion of the horizontal section is determined from the stress equation at the end of the crack.

ΣW+U
+ M,
A,

Ig,
I,

= 0

(70)

If values of A, and I1, obtained from equations 68 and 69, and the value of lg, obtained by proper symbolical substitution in equation 52, are substituted in equation 70, it reduces to

M, is evaluated by multiplying values from the proper curve in figure 21, based on equation 71, by the quantity (W + U)T 1.

The stress at the downstream face in pounds per square inch is determined by the equation,

77. Cracking at Downstream Face.-Equations and curves for a cantilever cracked at the downstream face are derived in the same manner as those for cracking at the upstream face. Figure 22 shows a horizontal

section of a cantilever cracked at the downstream face. By considering forces, moments, and stresses in figure 22, the following equations may be derived: