Page 356 - Mechanical Behavior of Materials

Section 8.4 Application of K to Design and Analysis 357

For example, if F does not change very much between a and a c , so that F ≈ F c , then
2 √
Eq. 8.17 reduces to X a = X , so that X K = X a . Hence, a safety factor of X a = 4 is needed
K
to achieve X K = 2, and X a = 9 is needed to achieve X K = 3. This is very important in design, as
it means that crack lengths must be guaranteed to be quite small compared with the critical value a c
for fracture.
If the crack length expected to occur in actual service is relatively small, safety factors against
yielding may be calculated simply by comparing the service stress S g with the material’s yield
strength σ o :

X o = σ o /S g (8.18)
However, for applied stresses that are multiaxial, S g must be replaced by an effective stress ¯σ for one
of the yield criteria of Chapter 7. Since S g is the stress on the gross area, the preceding calculation
gives the safety factor against yielding as if no crack were present.
A more advanced method for calculating the safety factor against yielding is to compare the
applied load with the fully plastic limit load. The latter is an estimate of the load necessary to
cause yielding over the entire cross section that remains after subtracting the crack area, so that
the effect of the crack in reducing the cross-sectional area is included. See Section A.7.2 for
more explanation, and note that Fig. A.16 gives fully plastic forces and moments, P o and M o ,for
some simple cases of cracked members. Hence, this type of safety factor against yielding is given
by one of

X = P o /P, X = M o /M (8.19)
o o
as may apply for a given case, where P and M are values of force and moment for actual service.
Values chosen for safety factors must reflect the consequences of failure and whether or not
the values of the variables that affect the calculation are well known, as well as sound engineering
judgment. If possible, statistical information should be employed for variables such as stress, crack
size and shape, and materials properties. (See Section B.4 for a discussion of statistical variation in
materials properties.) Also, minimum safety factors may be set by design code, company policy,
or governmental regulation. Where the applied loads are well known and there are no unusual
circumstances, reasonable values for safety factors in stress are three against fracture and two against
yielding. The larger value for fracture is suggested because of the greater statistical scatter in K Ic
compared with yield strength, and also because brittle fracture is a more sudden, catastrophic mode
of failure than yielding.

Example 8.2
Consider the situation of Ex. 8.1, where a center-cracked plate of 2014-T651 aluminum, with
dimensions b = 50 and t = 5 mm, is subjected in service to a force of P = 50 kN.

(a) What is the largest crack length a that can be permitted for a safety factor against
fracture of 3.0 in stress?
(b) What safety factor on crack length results from the safety factor in stress of (a)?
(c) What is the safety factor against yielding?