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.