Failure Prediction for Static Loading
6.2 Stress Concentration
6.2.1 Stress Concentration Factor Charts
As discussed above, the stress concentration factor is a func- tion of the type and shape of the discontinuity (hole, fillet, or groove), and the type of loading being experienced. Con- sideration here will be limited to only two geometries, a flat plate and a round bar. Figures 6.2 to 6.4 display the stress concentration factor due to bending and axial load for a flat plate with a hole, fillet, or groove. These figures also give the expressions for nominal stresses. Many of these curves are developed from photoelastic studies. Note from Fig. 6.2a that a small hole in a plate loaded in tension (d/b→0) leads toKc = 3.0, which is consistent with Eq. (6.2). Figures 6.5 and 6.6 show the stress concentration factor for a round bar with a fillet and a groove, respectively. Figure 6.7 shows the stress concentration factor for a flat groove, such as is used as a seat for retaining rings (see Section 11.7). Figure 6.8 shows the effect of a radial hole in a shaft. These examples are by no means all the possible geometries, but are those most often encountered in practice; for other geometries, refer to Pilkey and Pilkey [2008] or Young and Budynas [2001].
From these figures, the following observations can be made about stress concentration factors:
1. The stress concentration factor,Kc, isindependentof ma- terial properties.
2. Kcis significantly affected by part geometry. Note that as the radius of the discontinuity is decreased, the stress concentration is increased.
3. Kcis also affected by the type of discontinuity; the stress concentration factor is considerably lower for a fillet (Fig. 6.3) than for a hole (Fig. 6.2).
The stress concentration factors given in Figs. 6.2 to 6.8 were determined on the basis of static loading, with the ad- ditional assumption that the stress in the material does not exceed its proportional limit. In practice, this is usually ap- proximated by the yield stress. If the material is brittle, the proportional limit is the rupture stress, so failure for this part will begin at the point of stress concentration when the proportional limit is reached. It is thus important to apply stress concentration factors when using brittle materials. On the other hand, if the material is ductile and subjected to a static load, designers often neglect stress concentration fac- tors, since a stress that exceeds the proportional limit will not result in a crack. Instead, the ductile material will flow plasti- cally and can strain harden. Furthermore, as a material yields near a stress concentration, deformation results in blunting of notches, so that the stress concentration is reduced. In appli- cations where stiff designs and tight tolerances are essential,
Stress concentration factor, KcStress concentration factor, Kc
(b) 1.00
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Diameter-to-width ratio, λ = d/b
0.1 0.2 0.3 0.4 0.5 0.6
d/h = 0 0.25 0.5 1.0 2.0
∞ 1.0
2.0 3.0
0
Diameter-to-width ratio, λ = d/b
0.1 0.2 0.3
(a)
0.4 0.5 0.6 4.0
5.0 6.0
P b P/2
d h P/2 c
––––Mc
I –––––––6M (b – d)h2
M b M
σavg = = h d
P
P h d b
–––––––P (b – d)h σnom = = ––P
A
Figure 6.2: Stress concentration factors for rectangular plate with central hole. (a) Axial load and pin-loaded hole; (b) bending.
stress concentration should be considered regardless of ma- terial ductility.
Example 6.1: Theoretical Stress Concentration Factor
Given:A flat plate made of a brittle material and a width of b= 20mm, a major height ofH = 100mm, a minor height ofh= 50mm, and a fillet radius ofr= 10mm.
Find: The stress concentration factor and the maximum stress for the following conditions:
(a) Axial loading withP= 10,000N (b) Pure bending withM = 100Nm
(c) Axial loading ofP = 10,000N, with fillet radius re- duced to 5 mm.
Solution:
(a) Axial loading. Note from the geometry that H
h =100 50 = 2.0.
(a) 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
H/h = 3 21.5 1.151.05 1.01 0
Radius-to-height ratio, r/h
0.05 0.10 0.15 0.20 0.25 0.30 ––P
A ––P σavg = = bh
P P
H h
r b
Stress concentration factor, Kc
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
(b)
1.011.05 1.22 H/h = 6
0
Radius-to-height ratio, r/h
0.05 0.10 0.15 0.20 0.25 0.30 Stress concentration factor, Kc
M M
Mc –––
I –––6M bh2
σavg = =
H h
r b
Figure 6.3: Stress concentration factors for rectangular plate with fillet. (a) Axial load; (b) bending.
H h = 100
50 = 2.0.
Also,
r h =10
50= 0.2.
From Fig. 6.3a,Kc = 1.8. From Eq. (6.1) and Fig. 6.3a, the maximum stress is
σmax= 1.8σavg= 1.8(10,000)
(0.02)(0.05) = 18MPa.
(b) Pure bending. From Fig. 6.3b,Kc= 1.5. The maximum stress is
σmax= 1.56M
bh2 = 9(100)
(0.02)(0.05)2 = 18MPa.
(c) Axial loading but with fillet radius changed to 5 mm.
For this case
r h = 5
50= 0.1.
From Fig. 6.3a,Kc= 2.1. The maximum stress is σmax=2.1P
bh = (2.1)(10,000)
(0.02)(0.05) = 21MPa.
Thus, reducing the fillet radius by one-half increases the maximum stress by around 17%.
(a) 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
0
Radius-to-height ratio, r/h
0.05 0.10 0.15 0.20 0.25 0.30
Stress concentration factor, Kc
H/h = ∞ 1.51.15 1.05 1.01 P
r b h H P
––P A ––P σavg = = bh
r b
h H
M M
Mc–––
I –––6M bh2 σavg = =
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
(b) 0
Radius-to-height ratio, r/h
0.05 0.10 0.15 0.20 0.25 0.30
Stress concentration factor, Kc
H/h = ∞ 1.51.15 1.051.01
Figure 6.4: Stress concentration factors for rectangular plate with groove. (a) Axial load; (b) bending.
Example 6.2: Allowable Loads in the Presence of a Stress Concentration
Given: A 50-mm-wide, 5-mm-high rectangular plate has a 5-mm-diameter central hole. The allowable tensile stress is 700 MPa.
Find:
(a) The maximum tensile force that can be applied.
(b) The maximum bending moment that can be applied to reach the maximum stress.
(c) The maximum tensile force and the maximum bending moment if the hole is not present. Express the results as a ratio when compared to parts (a) and (b).
Solution:
(a) The diameter-to-width ratio isd/b= 5/50 = 0.1. The cross-sectional area with the hole is
A= (b−d)h= (50−5)5 = 225mm2= 0.225×10−3m2. From Fig. 6.2a ford/b= 0.1 the stress concentration fac- torKc = 2.70for axial loading. The maximum force is
Pmax=(700×106)(0.225×10−3)
2.70 = 58,330N.
Stress Concentration 137
Radius-to-diameter ratio, r/d (a)
0 0.1 0.2 0.3 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
D/d = 2 1.5 1.21.05 1.01 Stress concentration factor, Kc
0
Radius-to-diameter ratio, r/d
0.1 0.2 0.3
(b) Stress concentration factor, Kc
3.0
1.0 1.2 1.4 1.6 2.2 2.4 2.6 2.8
D/d = 6 31.5 1.11.03 1.01 2.0
1.8
Radius-to-diameter ratio, r/d (c)
0 0.1 0.2 0.3 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
D/d = 2 1.2 1.09 Stress concentration factor, Kc
––P A –––4P
πd2
σavg = = r
D d
P P
M
–––Mc I
––––32M πd3 σavg = =
r
D d
M
Tc ––
J 16T –––
πd3 τavg = =
r
D d
T T
Figure 6.5: Stress concentration factors for round bar with fillet. (a) Axial load; (b) bending; (c) torsion.
0
Radius-to-diameter ratio, r/d
0.1 0.2 0.3
(a) 3.0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
1.01 D/d > 2 1.11.03 Stress concentration factor, Kc
Radius-to-diameter ratio, r/d 3.0
(b) 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
1.01 D/d > 2 1.11.03 Stress concentration factor, Kc
(c)
Radius-to-diameter ratio, r/d 1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
D/d ≥ 2 1.11.01 Stress concentration factor, Kc
r
D d
P P
––P A –––4P
πd2
σavg = =
r
D d M
M
Mc –––
I 32M ––––
πd3 σavg = =
0.05 0.15 0.25
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.05 0.15 0.250.1 0.2 0.3 r
D d
Tc ––
J 16T –––
πd3
τavg = =
T T
Figure 6.6: Stress concentration factors for round bar with groove. (a) Axial load; (b) bending; (c) torsion.
Stress Concentration 139
(a)
(b)
r
D d
P P
––P A –––4P
πd2
σavg = =
M
M b
––––32M πd3 + t
Stress concentration factor, Kc
Width-to-depth ratio, b/t
0.5 1.0 2.0 3.0 4.0 5.0 6.0 10.0
8.0
6.0
4.0
2.0 1.0 3.0 9.0
7.0
5.0
r/t = 0.03 0.04 0.05 0.07 0.10 0.15 0.20 0.40 0.60 1.00
Tc––
J 16T–––
πd3 τavg = =
T T
r
D b d
t
Width-to-depth ratio, b/t
0.5 1.0 2.0 3.0 4.0 5.0 6.0 r/t = 0.03
0.04 0.06 0.10 0.20 Stress concentration factor, Kc
6.0
4.0
2.0
1.0 3.0 5.0
Figure 6.7: Stress concentration factors for round bar with a flat groove.
1.00 1.2 1.4 1.6 1.8
Stress concentration factor, Kc
2.0 2.2 2.4 2.6 2.8 3.0
0.1 0.2 0.3 Hole diameter-to-bar diameter ratio, d/D
T M M T
D d
P P
Mc –––
σnom= = (πDI 3/32) - (dDM 2/6) ––P
σavg = = A P (πD2/4) - Dd
Tc ––
τavg = = J T (πD3/16) - (dD2/6) Axial load:
Bending (plane shown is critical):
Torsion:
Nominal stresses:
Axial Be
nding
Torsion
Figure 6.8: Stress concentration factors for round bar with hole.
(b) From Fig. 6.2b for bending whend/b= 0.1 andd/h= 5/5 = 1, the stress concentration factor is Kc = 2.04.
The maximum bending moment is Mmax = (b−d)h2σall
6Kc
= (0.225×10−3)(5×10−3)(700×106) 6(2.04)
= 64.34Nm.
(c) The cross-sectional area without the hole is
A=bh= (50)5 = 250mm2= 0.250×10−3m2. Therefore,
Pmax=σallA= 700
×106 0.250
×10−3
= 175kN.
The force ratio is 175/58.33=3.00. For bending, Mmax = σallbh2
6 = σallAh 6
= (700×106)(0.25×10−3)(5×10−3) 6
= 145.8Nm.
The bending moment ratio is145.8/64.34 = 2.266.