K t

Report
STRESS CONCENTRATION AT NOTCHES
One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”)
is the consideration of stress concentration
Stress concentration at geometrical notches are always present in a real structure
Notches introduce inhomogeneous stress distribution with a stress concentration at the root of
the notch
Stress concentration factor:
 peak
Kt 
 nom
Kt is referred as the theoretical stress concentration
factor: it is based in the assumption of linear elastic
material behavior
Kt describes the severity of the notch and depends on the
geometry of the notch configuration (shape factor of the
notch)
Common examples of stress
concentration
(a) Gear teeth
(b) Shaft keyway
(c) Bolt threads
(d) Shaft shoulder
(e) Riveted or bolted joint
(f) Welded joint
all these components might
be subjected to cyclic loads !
DEFINITIONS:
Kt 
 peak
 nom
K tg 
usual
 peak
S
alternative
For the previous example:
therefore:
K tg 
 nom
S
W
Kt 
Kt
W D
Ktg  Kt
In general Kt is the preferred factor to indicate stress concentration
following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York
(1974)
THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheet
S
r0
r

2
2
2

r0
r0 
S  r0 


 rr  1  2  1  3 2  4 2  cos 2 
2  r
r
r 


4
2
r0
r0 
S
 r   1  3 4  2 2  sin 2
2
r
r 
2
4

r0 
S  r0 
   1  2  1  3 4  cos 2 
2  r
r 


S
Along the edge of the circular hole:
  S 1  2 cos2   rr  0
 
max
  
   2
  0  S
 for  = 0 ?
3S
 r  0
Kt  3
compression at  = 0
 = 30°
or
j =  /2 –   60° from  max
Stress Profiles along the normal to the edge of the circular hole:
We are interested in evaluating:
a)
Situation for compressive remote stress (–S): presence of tensile stress ?  YES
b)
Gradient of Stress in the direction normal to the edge of the hole at the location of speak: strong gradient
c)
Gradient of Stress along the edge of the hole at the location of speak : slow decrease of stress along the
edge of the notch
d)
Volume of material subjected to high Stress around the root of the notch: larger for larger notches!
(significant to understand notch size effects on fatigue)
a) Presence of Local Tensile Stress upon Remote Compression Stress
* Fatigue Crack Growth under Compressive Stress?
* Effect on Brittle Materials: Failure Criteria Based on Maximum Normal Stress for
Brittle Materials
C
How would a cylinder of brittle material
with a distribution of defects fail under
compression?
Spherical voids and sharp like cracks are often
produced during processing of brittle materials:
- Spherical voids  sintering of ceramic powders
(remnants of initial porosity)
- Microcraks  thermal expansion mismatch
C
b) Gradient of Stress in the direction NORMAL to the edge of the hole at the location of peak
* Although the peak stress is of great importance, it is also interesting to know how fast the
stress decreases away from the root of the notch  Stress Gradient
for the circular hole:
 peak
 d  


 dr  
r0

 r0 ;
2

1
2



K
t 

The Stress Gradient at the root of a notch should give an indication of the volume of material
under high stresses  estimate the distance d along the normal to the root for a drop from
peak to 0.9 peak (10 % decrease) for a circular hole with r0 = 2.5 mm:
 peak
 d  


 dr  
r0

 r0 ;
2
0.9 peak   peak
 1
2



 3 
d
d = 0.1 mm= 100 mm
if grain size 50 mm  the depth d
corresponds only to few grains
The grains at the noth root surface are subjected to high loads and this is very
important for fatigue
c) y d) Gradient of Stress ALONG the edge of the hole at the location of peak
As fatigue crack nucleation is a surface
phenomenon it is of interest to know how fast
the tangential stress along the edge of the notch
is decreasing
S
* Slow decrease of the stress along the edge
compared with the decrease from the edge at
the location of peak
* Larger notches have a larger material
surface along the root of the notch  very
important to understand notch size effects on
fatigue
S
Note: even when a particular case was analyzed here (circular hole in an infinite plate)
the conclusions are of general validity  similar peak stresses and notch root radii
give comparable stress distribution around the root of an arbitrary notch
Effect of notch geometry on Kt
Geometrically similar specimens have the
same Kt (Kt is dimensionless) …
… but different stress gradients (stress
gradient is not dimensionless)
Larger specimens have larger volumes
and larger notch surface areas of highly
stressed material
Reason of the existence of Notch Size
Effects in Fatigue
Also: Importance of surface quality (method of
production / fabrication)  surface defects
due to manufacturing in a highly stressed
region along the wall of a hole
Further examples / further aspects of stress concentrators 
The elliptical hole in an infinite sheet
S
 peak   

x0
y 0

a
a


 S 1  2   S 1  2
b
r 


 S
a
a
Kt  1  2  1  2
b
r
b2
r
a
S
use large radii on surface
parallel to applied stress to
reduce stress concentration !
a/b
1/3
1
3
r/ a
9
1
1/9
Kt
1.67
3
7
Stress Concentration for an elliptical hole under biaxial loading:
S A  S 1  2 a b   S
S
S B   S 1  2 b a   S
* For the case of a thin walled pressure vessel
under pressure  = 0.5 and for the case of a
circular hole (a = b):
S A  2.5 S
: biaxiality
ratio
compare with the square hole
(dashed line) with rounded
corners with r 10% of hole width:
Kt = 4.04 for  = 0.5
S B  0.5 S
lower than 3 S for uniaxial loading
* Same case but elliptical hole with b/a = 2:
S A  1.5 S
SB  1.5 S
lower than 3 S for uniaxial loading (actually,
 = 1.5 S along the edge of the hole)
Stress Concentration for a circular hole in a plate under pure shear:
Kt  4
Fatigue cracks growing from holes in a shaft subjected to cyclic torsion
!
Pin - loaded hole:
Conection between a lug and a clevis:
Comparison of Kt values for a lug and an open hole
* Lugs are fatigue critical parts (also prone to fretting corrosion)  values of d/W
below 1/3 are usually avoided to keep Kt below 3.5
Superposition of notches:
If a relativelly small notch is added to the root
of the main notch
 Effect of superposition of notches:
 peak  Kt 2 * Kt1S 
K t  K t1 K t 2
This overestimates Kt because the
small notch is not completely
embedded in an homogeneous
stress field of magnitude Kt1
Technique for estimating conservative limiting value for Kt for
superposed notches:
“Fill” the notch (cross hatched
area) leaving a single deep
narrow notch
The theoretical stress concentration factor for the single deep
narrow notch will always be greater than the Kt for the multiple notch
(see Kt for Edge Notches two transparencies later)
Examples of Superposition of notches:
Lug with small lubrication hole to the lug hole
Cross section of a fatigue crack at a sharp corner
Edge notches and Corrosion Pits
Corrosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm.
Equivalent shape gives very high Kt values
Further information of the type that can be found in
the clasical handbook of R.E. Peterson, Stress Concentration
Factors, 1974
Stress concentration factors for
a shaft with a grove subjected to:
Axial
Load
Bending
Torsion
THE FATIGUE STRENGHT OF NOTCHED SPECIMENS
* STRESS – LIFE APPROACH: Notch Effects on the Fatigue Limit (Sm = 0)
Similarity Principle: if Sa = Se is the fatigue limit of
the smooth specimen, then Speak should give the
fatigue limit SeK of the notched specimen:
SeK
Sa = Se
S peak  Se  Kt SeK
Se
S

meaning that:
eK
Kt
… but this is not the case !
 The Fatigue Strength Reduction Factor
or Fatigue Notch Factor Kf is introduced:
unnotchedfatigue limit
Kf 
notchedfatigue limit
In general: Kf < Kt  fatigue limit of different materials are less notch sensitive to fatigue
than predicted by Kt
Examples:
Effect of a Notch on S - N behavior (Tryon and Day, 2003)
Mechanical Behaviour of Materials (Dowling, 1999)
The examples illustrates a general observation for different materials: the finite life region is
also less notch sensitive to fatigue than predicted by Kt
In general Kf
< Kt
and
Kt depends on geometry and mode of loading
Kf also depends on material and notch size
Blunting effects in soft materials:
Yielding at the notch root reduces peak
stress from the values predicted by Kt
Fatigue strength of a notched component
depends on the volume of highly stressed
material near the notch  also effect of stress
gradient on crack growth.
For engineering applications, the fatigue strength reduction fator Kf can be empirically
related to the elastic stress concentration facto Kt by a Notch Sensitivity Factor defined as:
q
K f 1
Kt 1
K f  1  Kt 1 q
0  q 1
q = 1  material fully notch sensitive: Kf = Kt
q = 0  material not notch sensitive: Kf = 1
Empirical equations for q were proposed by different authors:
* Peterson (1959)
* Neuber (1946)
* Siebel and Stiele (1955)
* Peterson assumed that fatigue damage occurs when a the stress at a point
located at a critical distance ap away from the notch root is equal to the fatigue
strength of a smooth specimen and obtained the following empirical equation:
1
q
aP
1
r
* for high strenght steels
with SU > 560 MPa:
Effect of notch
root radius on Kf
- r is the notch root radius
- aP is a material constant related with material strength and
ductility.
Also, q can be obtained in graphical form:
Peterson ´s notch sensitivity for steels
* Neuber assumed that fatigue failure occurs if the average stress over a length aN
from the notch root is equal to the fatigue limit of a smooth specimen and proposed
the following empirical equation:
q
1
aN
1
r
Neuber´s Notch
Sensitivity curves for
Al alloys
Relation with Hall-Petch?
- r is the notch root radius
- aN is the Neuber´s material constant related to the grain size
* Siebel and Stiele (1955) introduced the Relative Stress Gradient (RSG)  to
characterize the effects of fatigue strength reduction (instead of using the notch radius!)

1
 e max
 d e ( x) 


dx

 x 0
for the circular hole:

 d  



 peak 
r0
 dr  r0 ; 
2
1
where:   2 
Kt
 
1
Similar dependencies are found for other
geometries and for typical Kt values: 2 < Kt <5

No significant effect of Kt on 
Testing the Fatigue Strength of smooth and notched specimens they generated
empirical curves relating Kf / Kt vs. 
Equation for the
Curves:
Kt
 1  Css  
Kf
where Css is a material constant related with Sy
RSG () values
calculated from
Siebel and Stiele by
the theory of
elasticity for different
notched members
Examples:
* STRESS – LIFE APPROACH: Notch Effect on Finite Life (S–N curve)
In the Finite Life region
the notch may be less
sensitive to what is
predicted by Kf for the
Fatigue Limit
Use Kf for the whole S-N
curve? :
 to much conservative !
define a Fatigue Sensitivity
Factor at a particular Life
(and interpolate / extrapolate)
analogously to what was made for estimating S – N curves for smooth specimens  Kf for N1000 : K’f
An Empirical Fatigue Notch Sensitivity Factor (q’1000) can be defined at 1000 cycles:
 
q1000

K f 1
Kt 1
* Estimate of Fatigue Life for Notched Component: different approaches
Examples:
A)
“Juvinall”
approach
LCF
REGION!
B)
Fatigue Notch
Sensitivity
Factor Kf
depends on
Cycles to
Failure Nf
Example:
Juvinall
for the following loading cases:
(1) Reversed bending loading
(2) Reversed axial loading (negligible bending)
(3) Reverse torsional loading

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