stress conc factors
TRANSCRIPT
-
8/13/2019 Stress Conc Factors
1/42
Stress concentration factorsStress concentration factors
L. VerganiL. Vergani
-
8/13/2019 Stress Conc Factors
2/42
2
POLITECNICO DI MILANO
Load and sign convention
Any applied load can be classified with respect to time in the following
ways:
Static load- load is gradually applied and equilibrium is reachedin a relatively short time. The structure experiences no dynamic
effect.
Sustained load- Load (such as the weight of a structure) is
constant over a long time
Impact load- Load is rapidly applied An impact load is usuallyattributed to an energy imparted to a system
Cyclic load- Load can vary and even reverse itself in sign and
has a characteristic period with respect to time
-
8/13/2019 Stress Conc Factors
3/42
3
POLITECNICO DI MILANO
Load and sign convention
And with respect to the area over which it is applied in the following ways:
Concentrated load- Load is applied to a small area
Distributed load- Load is spread along the entire area
-
8/13/2019 Stress Conc Factors
4/42
4
POLITECNICO DI MILANO
Load and sign convention
Classification with respect the direction
Normal load: the load passes through the centroid of the
resisting areaPositive: tensile
Negative:compression
-
8/13/2019 Stress Conc Factors
5/42
5
POLITECNICO DI MILANO
Load and sign convention
Shear force: the force P is in the same direction of V.
The shear force is positive if the force direction and the
normal direction are both positive or negative
positive
-
8/13/2019 Stress Conc Factors
6/42
6
POLITECNICO DI MILANO
Load and sign convention
Bending moment: load is applied
transversely to the longitudinal axis
of the member.
Topper half of the
member positive
If the tensile stresses are in the
positive direction the bendingmoment is positive
-
8/13/2019 Stress Conc Factors
7/42
7
Load and sign convention
Torsion moment: load is applied in order to apply a
twisting motion
Right hand rule
-
8/13/2019 Stress Conc Factors
8/42
8
POLITECNICO DI MILANO
Load and sign convention
Support reaction:
If the support preventstranslation in a given
direction, a force is
developed on the memberin that direction.
If the support prevents
rotation a coupled moment
is averted on the member.
-
8/13/2019 Stress Conc Factors
9/42
9
POLITECNICO DI MILANO
Load and sign convention
Static equilibrium:
Equilibrium of a body requires:A balance of forces to prevent the body from translating along to a
straight or curved path;A balance of of moments to prevent to prevent the body from rotating
Equation:
Py= 0Px= 0 P
z= 0M x= 0 M y= 0 Mz= 0
-
8/13/2019 Stress Conc Factors
10/42
-
8/13/2019 Stress Conc Factors
11/42
11
POLITECNICO DI MILANO
Stress concentration factors
If the cross section of a load-carrying member
varies gradually the stresses are accuratelycalculated by the previous equations.
If there are abrupt changes ( as: holes,
notches, fillets) in the cross section themechanics of materials can not predict the high
values of the stresses.
-
8/13/2019 Stress Conc Factors
12/42
-
8/13/2019 Stress Conc Factors
13/42
13
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
14/42
14
POLITECNICO DI MILANO
Stress concentration factors
Kt=maximum stressnominal stress
= max
nom
nom= F
(h d)b
-
8/13/2019 Stress Conc Factors
15/42
15
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
16/42
16
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
17/42
17
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
18/42
18
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
19/42
19
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
20/42
20
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
21/42
21
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
22/42
22
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
23/42
23
POLITECNICO DI MILANO
Stress concentration factors
-
8/13/2019 Stress Conc Factors
24/42
Stress concentration factors
-
8/13/2019 Stress Conc Factors
25/42
-
8/13/2019 Stress Conc Factors
26/42
Stress concentration factors
-
8/13/2019 Stress Conc Factors
27/42
27
POLITECNICO DI MILANO
Stress concentration factors
Kt is the elastic stress concentration factor
The values are not valid if there is yielding
Kt is used to obtain the peak stress near a
notches Kt is not related to the failure of a crcaked
member
-
8/13/2019 Stress Conc Factors
28/42
Stress concentration factors
d=20 mm
D=26r=2 mm
T=100 Nm
n=?
max=?
-
8/13/2019 Stress Conc Factors
29/42
-
8/13/2019 Stress Conc Factors
30/42
30
POLITECNICO DI MILANO
Stress concentration factors
max=Ktn
max=Ktn
-n-n
-
8/13/2019 Stress Conc Factors
31/42
31
POLITECNICO DI MILANO
Stress concentration factors
schema
-
8/13/2019 Stress Conc Factors
32/42
Stress concentration factors
If we neglect the curvature
If we consider a small
thickness vessel The most stressed zones
are near the nozzles
a
c
-
8/13/2019 Stress Conc Factors
33/42
33
POLITECNICO DI MILANO
Stress concentration factors
max sn
max=Ktn=Kt
n
E
max=Ktn
max=Ktn
-
8/13/2019 Stress Conc Factors
34/42
-
8/13/2019 Stress Conc Factors
35/42
35
POLITECNICO DI MILANO
Fully plastic yielding
Ductile material
model
-
8/13/2019 Stress Conc Factors
36/42
If we apply P>Pi ?
h=110 mm
d=10 mm
b=10 mm
Fully plastic yielding
h
d
P
P
-
8/13/2019 Stress Conc Factors
37/42
37
POLITECNICO DI MILANO
Fully plastic yielding
The fully plastic yielding can be estimated
considering a perfect plastic material
Ppt=sn(h d)b
sn
Ppt
Ppt
-
8/13/2019 Stress Conc Factors
38/42
38
POLITECNICO DI MILANO
Fully plastic yielding
If the material is ductile, beyond the point of
yielding the local strains are larger thanwould be estimated from elastic analysis.
When the applied load increases a larger
volume of material yields.
When yielding spreads to the entire cross
sectional area is the fully plastic yielding.
Notch stress and strain
-
8/13/2019 Stress Conc Factors
39/42
39
Notch stress and strain
Neubers rule 2t
KKK =
nom
eff
nom
eff
K
K
max
max
=
=
E
KK nomtnomnomteffeff
222
maxmax
==
In order to evaluate the strain value:
Notch stress and strain
-
8/13/2019 Stress Conc Factors
40/42
40
Notch stress and strain
Perfect plastic material
r
r
maxeffmaxeff
max
=snloading
r=
sn- K
t
nom
r=
max- K
t
nom/E
unloading
E
K
sn
nomt
2
max
)(=
EK
sn
nomt
2
max)(=
Notch stress and strain
-
8/13/2019 Stress Conc Factors
41/42
41
Notch stress and strain
loading
unloading
E
K nt2
om )( =
E
K
E
K
sn
ntnt
2
)()( 2om2
om =
=
r
r
maxeff
maxeff
E
K
E
K
E
K
sn
nomt
sn
nomt
sn
nomt
r
2
)(
2
)()( 222max
=
==
snr =
-
8/13/2019 Stress Conc Factors
42/42
Example