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Safety Factor against J oint Separation
Figure 15.15, page 690
If a forcePjpulls on the joint, then the safety factor againstseparation is
)1( kj
isj
CP
Pn
= ( 15.32)
Pi =preload (Eq 15.33)
Pj = external load
Ck=joint constant(dimensionless stiffness parameter)
jb
bk
kk
kC
+
= ( 15.17)
kb =bolt stiffness
kj =joint stiffness
(More on this later)
Connection Simulated as Bolt and J oint Springs
Text Reference: Figure 15.9, page 680
bolt
joint
Preload
Pi = Tension force in bolt =
compression force in joint.
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Force vs. Deflection with External Load
Figure 15.11, page 681
P
LoadPincreases tension in bolt fromPi to PCPP kib +=
Ck= kb/(kb + kj) = stiffness parameter.
Bolt tension Pb.
Text Reference: Figure 15.16, page 691
Contributions of preloadPi and external loadP on bolt tensionPb.
Force vs. Deflection with External Load
Figure 15.11, page 681
P
LoadPincreases tension in bolt fromPito PCPP kib +=
Ck= kb/(kb + kj) = stiffness parameter.
Only a fractionof
external load
contributes to bolt
tension.
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Safety factor against bolt fatigue
m
a
i
Se/Kf
Sut
m
a
A
B
C
a=
m-
im
ut
fe
f
e
aS
KS
K
S =
Safety factor against fatigue = (Length of AB)/(Length of BC).
m
ut
fe
f
e
imS
KS
K
S =
+
+
=
ut
fe
i
f
e
m
S
KS
K
S
1
At point A,
feut
iut
a
fe
a
aA
fatigueKSS
SKSn
+
==
aA
Fatigue Stress Concentration Factors
SAE gradeMetricgrade
Rolledthreads
Cutthreads
Fillet
0-24-8
3.6-5.86.6-10.9
2.23.0
2.83.8
2.12.3
Table 15.8 Fatigue stress concentration factors for threaded elements.
Safety factor against fatigue: Non-modified Goodmandiagram results in
a = Alternating bolt stress = Pba/At ;
m= Mean bolt stress = Pbm/Ati = Bolt prestress = Pi/At
At = effective
tensile area
Kf= Fatigue stress concentration factor from table below:
feut
iut
a
fe
fatigueKSS
SKSn
+
=
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Effect of Preload on Fatigue Safety Factor
Preload stress i may lower the safety factor. However, the safety
factor can be much worse without adequate preload.
Example showing relationship between preload and fatigue:
Minimum external load = 0. Maximum external load = 3000 lb.
Bolt and joint stiffnesses kb = 0.6Mlb/in. kj = 1.8Mlb/in.
Tensile cross section areaAt= 0.0775 in2.
Ultimate strength Su = 120 kpsi.
Modified endurance limit Se/Kf= 14.21 kpsi.Calculate factor of safety against fatigue for:
a)Preload = 4941 lb.
b)Preload = 1200 lb (Remember joint separation).
feut
iut
a
fefatigue
KSSSKSn+ =
4.94k
P, lb
3k
Defl, in
1.2k
P, lb
3k
Defl, in
Pmax = 4.94 + 0.25(3) = 5.69 klb.
Pmin = 4.94 klb.
Minimum bolt stress i = _____
Altern bolt stress a = _______
Adequate preload Inadequate preload
Pmax = 3.0 klb.
Pmin = 1.2 klb.
Minimum bolt stress i = _____
Altern bolt stress a = ______
feut
iut
a
fe
fatigueKSS
SKSn
+
=
nfatigue = ________ nfatigue = ________
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Safety factor against static failure
k
ipt
sbPC
PSAn
=
Bolt proof strength Preload
Stiffness parameterBolt loading
If a forcePpulls on the bolt in addition to the preload, thenthe safety factor is
( 15.31)
Preload reduces the safety factor against static failure.
However, static failure seldom matters, and less important thanjoint separation.
What is more important than static failure is fatigue.
Effect of joint constantCkon safety
factors
Text Reference: Figure 15.17, page 694
)1( k
i
sjCP
Pn
=
k
ipt
sbPC
PSAn
=
Against joint separation:
Against static failure:
Against fatigue:feut
iut
a
fe
a
aAfatigue
KSS
SKSn
+
==
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