Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

132
AxIAL dEFORMATION
Uniform tensile stress σ
y
σ θ
τ rθ
σ r
r
θ
x
2a
Uniform tensile stress σ
FIGURE 5.16 Circular hole in a wide plate subjected to uniform unidirectional tension.
elasticity solution is expressed in terms of a radial stress s r , a tangential stress s θ , and a
shearing stress τ r θ , as shown in Figure 5.16. The equations are as follows:
2
2 4
s ⎛ a
a a
2 1 ⎞ s
r 2 1 4
s r = −
⎝
⎜ 2
⎠
⎟ − ⎛
⎝
⎜ − 2 4
r
+ 3r
s
τ
θ
rθ
2
s ⎛ a
a
2 1 ⎞ s
= +
⎝
⎜
r 2 1 2
⎠
⎟ + ⎛
⎝
⎜ + 4
3r
2
s ⎛ a a
2 1 2
= + −
⎝
⎜ 2 4
r
3r
4
4
⎞
sin 2θ
⎠
⎟
⎞
cos 2θ
⎠
⎟
⎞
cos 2θ
⎠
⎟
On the boundary of the hole (at r = a) these equations reduce to the following:
s = 0
r
s = s(1 + 2cos2 θ)
τ
θ
rθ
= 0
At θ = 0°, the tangential stress s θ = 3s, where s is the uniform tensile stress in the plate in
regions far removed from the hole. Thus, the stress-concentration factor associated with
this type of discontinuity is 3.
The localized nature of a stress concentration can be evaluated by considering the
distribution of the tangential stress s θ along the x axis (θ = 0°). Here,
s
θ
2
s ⎛ a a
=
2 2 + +
⎝
⎜ 2 4
r
3r
At a distance r = 3a (i.e., one hole diameter from the hole boundary), this equation yields
s θ = 1.074s. Thus, the stress that began as three times the nominal stress at the boundary
⎞
⎠
⎟
4