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