Stress concentration
! "# $
% & ' " "( *) + , - .
/ -
' " 0 ' "
!1
-
, '
2 , 2 -
' "34 5 -) !6 3 3 - / ) 2 , 2 - "
6 5 -) ' ' - / ) 2 , 2 - " "
5 -)
' ' - / )
x y
z x
y
z x
y
z
7 , 2 -
Validity of Paterson’s formula
Stress concentration factor for an ellipse (Paterson) :
Ktg = 1 + 2 b a
with b horizontal axis, a vertical axis for a vertical tension on the plate.
type of geometry 1 + 2 ab Finite Element
circle, a=b=1mm 3 3.018
ellipse, a=0.4mm ; b=1.mm 6 5.995 ellipse, a=2.5mm ; b=1.mm 1.8 1.821
Comparison between circle and "flat" hole
Comparison between circle and "flat" hole
sig22 contour plot circle sig_inf = 100MPa r=1mm
x y
z
0 11
22 33
44 56
67 78
89 1e+02
1.1e+02 1.2e+02
1.3e+02 1.4e+02
1.6e+02 1.7e+02
1.8e+02 1.9e+02
2e+02
min:-5.537470 max:301.772552 y x z
Sat Mar 3 15:16:51 2001 /home/gc/COURS/2.002/lab4/Square/circle.ps
sig22 contour plot ellipse
sig_inf = 100MPa b=1mm, a=0.4mm
x y
z
0 11
22 33
44 56
67 78
89 1e+02
1.1e+02 1.2e+02
1.3e+02 1.4e+02
1.6e+02 1.7e+02
1.8e+02 1.9e+02
2e+02
min:-8.112257 max:599.511902 y x z
Sat Mar 3 15:16:34 2001 /home/gc/COURS/2.002/lab4/Square/ellipX.ps
circle (max=301 MPa) flat (max= 599 MPa)
Comparison between circle and "elongated" hole
Comparison between circle and "elongated" hole
sig22 contour plot circle sig_inf = 100MPa r=1mm
x y
z
0 11
22 33
44 56
67 78
89 1e+02
1.1e+02 1.2e+02
1.3e+02 1.4e+02
1.6e+02 1.7e+02
1.8e+02 1.9e+02
2e+02
min:-5.537470 max:301.772552 y x z
Sat Mar 3 15:16:51 2001 /home/gc/COURS/2.002/lab4/Square/circle.ps
sig22 contour plot ellipse
sig_inf = 100MPa b=1mm, a=2.5mm
x y
z
0 11
22 33
44 56
67 78
89 1e+02
1.1e+02 1.2e+02
1.3e+02 1.4e+02
1.6e+02 1.7e+02
1.8e+02 1.9e+02
2e+02
min:-7.905968 max:182.130478 y x z
Sat Mar 3 15:17:28 2001 /home/gc/COURS/2.002/lab4/Square/ellipY.ps
circle (max=301 MPa) elongated (max=182 MPa)
Stress concentration for a circular hole
50 100 150 200 250 300 350
1 2 3 4 5 6 7 8 9 10
(50.*(2.+(1./x)**2 + 3.*(1./x)**4)) fine mesh coarse_quad mesh coarse-lin mesh
σ22 profile on the ligament for a 1mm circular hole in a 10mm×10mm plate with various meshes
Stress concentration for a circular hole (zoom)
120 140 160 180 200 220 240 260 280 300 320
1 1.2 1.4 1.6 1.8 2
(50.*(2.+(1./x)**2 + 3.*(1./x)**4)) fine mesh coarse-quad mesh coarse-lin mesh
σ22 profile on the ligament for a 1mm circular hole in a 10mm×10mm plate with various meshes (zoom)
Stress concentration for various hole shapes
100 150 200 250 300 350 400 450 500 550 600
1 1.5 2 2.5 3 3.5 4 4.5 5
stress in the ligament (MPa)
distance to the center (mm)
circle a=1mm flat a=0.4mm high a=2.5mm
σ22 profile on the ligament for a 3 types of hole in a 10mm×10mm plate (σ∞=100MPa), b=1mm
