Page 363 - Mechanical Behavior of Materials

364 Chapter 8 Fracture of Cracked Members

except that these are limited to a/t < 0.75 for a/c < 0.2. The foregoing equations are from
Newman (1986). This and other sources, notably Raju (1982, 1986) and Murakami (1987), together
give K solutions for a wide variety of cases of elliptical, half-elliptical, and quarter-elliptical cracks
in plates, shafts, and tubes, under both tension and bending loads.

Example 8.4
A pressure vessel made of ASTM A517-F steel operates near room temperature and has a wall
thickness of t = 50 mm. A surface crack was found in the vessel wall during an inspection. It has
an approximately semi-elliptical shape, as in Fig. 8.19(b), with surface length 2c = 40 mm and
depth a = 10 mm. The stresses in the region of the crack, as calculated without considering the
presence of the crack, are approximately uniform through the thickness and are S z = 300 MPa
normal to the crack plane and S x = 150 MPa parallel to the crack plane, where the coordinate
system of Fig. 8.19 is used. What is the safety factor against brittle fracture? Would you remove
the pressure vessel from service?
Solution From Table 8.1(a), we see that this material has a fracture toughness of K Ic =
√
187 MPa m and a yield strength of σ o = 760 MPa at room temperature. The K for the given
stresses and crack can be estimated from Fig. 8.19(b). Since c = 20 mm, we have a/c = 0.5.
Also, we have a/t = 0.2 and large b,for which F D = 1.12 is a reasonable approximation. The
quantity Q is needed:
a 1.65

Q = 1 + 1.464 = 1.466
c
Hence, the maximum K, which occurs at the point of maximum depth of the elliptical crack,
is approximately

!
!
πa π(0.010 m) √
K = K D = F D S z ≈ 1.12(300 MPa) = 49.2MPa m
Q 1.466
The stress-based safety factor against brittle fracture is
187
K Ic
X K = = = 3.80 Ans.
K 49.2
This is a reasonably high value, so it would be safe to continue using the pressure vessel until
repairs are convenient. However, the crack should be checked frequently to be sure that it is not
growing. In addition, the ASME or other design code for pressure vessels is likely to apply, and
it should be consulted in this situation.
Comment Stresses parallel to the plane of a crack do not affect K, so the given S x does not
enter the calculation. (See Section 8.5.4 for further discussion of this point.)

8.5.2 Cracks Growing from Notches

Another situation that is often of practical interest is a crack growing from a stress raiser, such as a
hole, notch, or fillet. The example of a pair of cracks growing from a circular hole in a wide plate