Strength of material . lesson 7

Section 7
Stress Concentration
Stress concentrations produced by discontinuities in structures such as holes, notches, and
fillets will be introduced in this section. The stress concentration factor will be defined. The
concept of fracture toughness will also be introduced.
© Loughborough University 2010. This work is licensed under a Creative Commons Attribution 2.0 Licence.
Contents
•
Stress Concentration
•
Stress Concentration – Definition
•
Stress Concentration chart – Central hole
•
Stress Concentration Factor Formulae
•
Basic Design Rule – Yield Limited Design
•
Fatigue
•
Fracture Toughness
•
Example: Fracture Toughness
•
Credits & Notices
Stress Concentration
Engineering stress σ = P/A or the far-field stress is invalid.
–
In immediately vicinity where the external load is applied (St Venant)
–
Around discontinuities such as: holes, notches, or fillets

These irregularities (stress concentrations or stress raisers) cause a
disruption to stress flow and stresses concentrations in localised
regions. The change of section concentrates stress most strongly
where the curvature of the surface is greatest. The far-field stresses
are less useful as they underestimate the actual local stress.
Hole
Notch
Fillet
Stress Concentration
Abrupt change
Stress “flow lines” crowd
together causing high stress
concentration in transition
zone
Smooth change
“Flow lines” more evenly
distributed causing lower
stress concentration in
transition zone
Fillet
Stress Concentration
( )
( ) ( )
2 4
max
2 4
max
Mean stress over the irregular regions is given by:
= =
-

The local stress can be calculated roughly by
3
1
2 - 2 -
It can be seen that when 2 then =3
On the contrary,
m
m
m
P P
A b d h
d d
b d b d
b d
σ
σ σ
σ σ
 
 ÷
= + +
 ÷
 
=
max
when 0 (vanishing hole) then the stress
concentration becomes non-existent as .
m
d
σ σ
→

≈
(Thickness h)
P
P
b
d
Stress Concentration - Definition
max
Stress concentration factor defined as:
=
con
m
K
K
σ
σ
Stress concentration factors are:
•
Dependent on irregularity, dimensions of irregularity, overall dimensions, loading
•
Obtained experimentally, analytically, etc
•
Published in charts (e.g. Roark’s Formulas)
•
Very important in brittle materials
•
In ductile materials:
–
Important in fatigue calculation.
–

Important if safety is critical.
–
Localized yielding hardens material (strain hardening).
–
Redistributes stress concentration.
Stress Concentration Chart – Central hole
P
P
d
b
Stress Concentration Factor Formulae
     
= +
 ÷  ÷  ÷
     
   
= +
 ÷  ÷
   
2 3
2
Flat bar of width with central circular hole diamter
3 - 3.13 3.66 - 1.53
Flat bar of width with non-central circular hole diamter
3 - 3.13 3.66 -
2 2
con
con
b d
d d d

K
b b b
b d
d d
K
b b
 
 ÷
 
     
= +
 ÷  ÷  ÷
     
=
3
2 3
1.53
2
Flat bar of width with single semicircular notch of depth
2.99 - 7.30 9.74 - 4.43
Flat bar of width with two semicircular notches of depth / 2
3.07 - 3.
con
con
d
b
b h
h h h
K
b b b

b d
K
     
+
 ÷  ÷  ÷
     
     
= +
 ÷  ÷  ÷
     
2 3
2 3
37 0.65 - 0.66
Round bar of diameter with mid section notch of depth / 2
3.04 - 5.42 6.27 - 2.89
con
d d d
b b b
D d
d d d
K
D D D