Lecture 4 Stress concentrations and stress gradients at...
Transcript of Lecture 4 Stress concentrations and stress gradients at...
1SCF_Gradient
Lecture 4Stress concentrations andstress gradients at notches
Stress distribution of plain
(smooth or unnotched) specimens
2SCF_Gradient
Stress-amplitude distribution of ‘smooth’
axially loaded specimen of circular cross section
Stress-amplitude distribution of ‘smooth’ axially
loaded specimen of rectangular cross section
3SCF_Gradient
Stress-amplitude distribution of
‘smooth’ rotating bending specimen
Stress concentration at notches under
elastic strain
4SCF_Gradient
Stress concentration in notched flat beam
M M
SS
bruttotverrsnitt nettotverrsnitt
sr
gross
section
net section
Ring-shaped GJS-600 specimen subjected
to diametral fatigue loading
5SCF_Gradient
Max principal stress in test ring of GJS-600
under compression (left) and tension (right)
Francis hydraulic turbine
6SCF_Gradient
Francis turbine runner
T-joint test specimen simulating transition between
blade and ring of a Francis turbine runner
7SCF_Gradient
TMM9 1/12-11, Problem 1: ‘Best’ transition?
‘Optimisation’
of T-joint
transition
8SCF_Gradient
FE stress analysis of T-joint (1)
Standard circular Approach A - lowest Kt
FE stress analysis of T-joint (2)
Approach D : test specimen Stress gradients
0.0 0.1 0.2
0.4
0.6
0.8
1.0
1.2
circularABCD
Distance from surface d/t1
σ y/σn
9SCF_Gradient
Test specimen simulating transition
between trailing edge of blade and ring of
Francis turbine runner
Optimisation strategy for
trailing edge transition
10SCF_Gradient
FE stress analysis of trailing edge
Standard circular Optimised shape (splines)
Stress concentration at elliptical hole in wide plate
2w
S
S
S
2a
2b
s
r
x
11SCF_Gradient
Kolosov (1909)-Inglis (1913) solution
Origin of a ξ-y co-ordinate system is located at the centre of
the elliptic hole, i.e., 2 2 2, .x a c a bξ = + = −
.
Normal stress (in the vertical y-direction) along the x-axis
(ξ ≥ a) is given by:
( ) ( )( ) ( ) ( )
( ) ( )
2 2 2 2 2
2 2 2 2 2
21
ya a b c c ab a b
S a b c c
ξ ξ ξ ξσ ξ
ξ ξ
− − − − + −= +
− − −
Relative stress gradient at the point of maximum stress
( ){ } ( ) { } ( )2
td d 2 1y yaa b a K
ξχ σ ξ ξ σ ξ ρ ρ
== = = = = +
Inglis’ solution for elliptical holes with a/b = 1/3,
1 and 3, Schijve (1980)
12SCF_Gradient
Kirsch’s solution (1898) for circular hole
in wide plate
.
Normal stress (in the vertical y-direction) along the x-axis
(ξ ≥ a) is given by:
Relative stress gradient at the point of maximum stress
( ){ } ( ) 13d d 2y ya
aξ
χ σ ξ ξ σ ξ ρ=
= = =
( ) 2 4
1 31 .
2 2
y a a
S
σ ξ
ξ ξ
= + +
in full agreement with Inglis’ solution.
Kirsch’s solution for circular hole in wide plate (2)
13SCF_Gradient
Representative notched members
(grooves, hole).
Semi-analytical equation for the SCF of symmetrically notched members,
Dubbels Taschenbuch für den Maschinenbau
Neuber’s rule for stress and strain
concentration at notches under inelastic
strain (SSY, LSY, creep)
14SCF_Gradient
Neuber’s rule for small (SSY) and large-scale
yielding (LSY), graphs
.
Neuber’s rule for small (SSY) and large-scale
yielding (LSY), models
.
( )
e e
* *
e e e e
2
te
ne ref L e
SSY
LSY
Nominal stress = Reference stress
K K K
P P R
σ ε
σ ε σ ε
σ σ
=
=
= =
15SCF_Gradient
Stress and strain concentration factors at U-groove
in tensile specimen
.
Stress and strain concentration factors at U-groove
in tensile specimen according to
- Neuber’s rule
- invariant total strain energy density (TSED)
- invariant strain energy density (SED)
.
16SCF_Gradient
Neuber’s rule for large-scale creep
.
( )e e
e e e e
2
te
d d
d d
0
K K Kt t
K K K K
σ ε
σ ε σ ε
= ⇒
+ =& &
References
� H.-J. Huth, Fatigue design of hydraulic turbine runners. PhD Thesis, NTNU, Trondheim, 2005.
� G. Härkegård, T. Mann, Neuber prediction of elastic-plastic strain concentration in notched tensile specimens under large-scale yielding, Journal of Strain Analysis for Engineering Design, Vol. 38, No. 1, 2003, pp. 79-94 (CEGB Prize 2004).
� G. Härkegård, S. SørbøApplicability of Neuber's rule to the analysis of stress and strain concentration under creep conditions, Trans. ASME,Journal of Engineering Materials and Technology, Vol. 120,
July 1998, pp. 224-229.