Page 367 - Mechanical Behavior of Materials

368 Chapter 8 Fracture of Cracked Members

S S

P P
= +
(a) K 1 K 2 K 2 — K 3
a
P
b
S S

S S
K 1 K 2
2b 2b K 2 K 3
P P P P
= +
(b) –
P
2a 2a 2a
S S

Superposition for either (a) or (b): (t = thickness)
1 P
K 1 = K 2 + K 2 − K 3 , so that K 2 = (K 1 + K 3 ), where S =
2 2bt
(a) Single crack in finite-width plate for any α = a/b:
√
P √ P F 3 πα

K 1 = F P1 √ , K 3 = F 3 S πa, K 2 = √ F P1 +
t b 2 t b 2
F P1 is F P from Fig. 8.15, and F 3 is F from Fig. 8.12(a).
(b) Infinite array of collinear cracks, exact solutions for any α = a/b:

$ %
!
!
P 1 πα P 1 1 πα
K 1 = √ √ , K 3 = S 2b tan , K 2 = √ √ + tan
t b sin πα 2 2 t b sin πα 2 2
Approximations within 10%: (Note limits for (a) or (b), respectively, below each equation.)
√
P 1 πα 0.89P

K 2 = √ √ + , K 2 = √
2 t b πα 2 t b
(α ≤ 0.32 or 0.38) (0.12 ≤ α ≤ 0.57 or 0.65)
Figure 8.23 Superposition to obtain solutions for cases of a single crack (a), or a row of
cracks (b), loaded on one side. (Arguments of trigonometric functions are in radians.)