Stress Concentration Factors
This page provides a set of interactive plots for common stress concentration factors. See the reference section for more details.
Rectangular Bar With Central Hole, Axial Force
This tab defines the stress concentration factor for a bar with a rectangular cross section and a central circular hole. The bar has an applied axial force (tensile or compressive).
 w = bar width
 d = hole diameter
 t = bar thickness
 F = applied force (tensile or compressive)
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { F \over (w  d)t } $$Input Geometry:
Results:
d/w:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = 3  3.13 \left({ d \over w }\right) + 3.66 \left({ d \over w }\right)^2  1.53 \left({ d \over w }\right)^3 $$Sources:
Rectangular Bar With Central Hole, OutofPlane Bending
This tab defines the stress concentration factor for a bar with a rectangular cross section and a central circular hole. The bar has an applied outofplane bending moment.
 w = bar width
 d = hole diameter
 t = bar thickness
 M = applied bending moment
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { M \over S } = { 6M \over (w  d) t^2 } $$Input Geometry:
Results:
d/t:
d/w:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = \left[ 1.79 + { 0.25 \over 0.39 + (d/t) } + { 0.81 \over 1 + (d/t)^2 }  { 0.26 \over 1 + (d/t)^3 } \right] \left[ 1  1.04 \left({d \over w }\right) + 1.22 \left({d \over w }\right)^2 \right] $$Sources:
Rectangular Bar With UNotches, Axial Force
This tab defines the stress concentration factor for a bar with a rectangular cross section and two Unotches. The bar has an applied axial force (tensile or compressive).
 w = bar width
 h = notch height
 r = notch radius
 t = bar thickness
 F = applied force (tensile or compressive)
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { F \over (w  2h)t } = { F \over dt } $$Input Geometry:
Results:
d:
w/d:
r/d:
h/r:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = C_1 + C_2 \left({ 2h \over w }\right) + C_3 \left({ 2h \over w }\right)^2 + C_4 \left({ 2h \over w }\right)^3 $$where the coefficients in the equation above are calculated from "Peterson's Stress Concentration Factors":
$$ 0.1 \le h/r \le 2.0 $$  $$ 2.0 \le h/r \le 50.0 $$  

$$ C_1 $$  $$ 0.955 + 2.169 \sqrt{h/r}  0.081 (h/r) $$  $$ 1.037 + 1.991 \sqrt{h/r} + 0.002 (h/r) $$ 
$$ C_2 $$  $$ 1.557  4.046 \sqrt{h/r} + 1.032 (h/r) $$  $$ 1.886  2.181 \sqrt{h/r}  0.048 (h/r) $$ 
$$ C_3 $$  $$ 4.013 + 0.424 \sqrt{h/r}  0.748 (h/r) $$  $$ 0.649 + 1.086 \sqrt{h/r} + 0.142 (h/r) $$ 
$$ C_4 $$  $$ 2.461 + 1.538 \sqrt{h/r}  0.236 (h/r) $$  $$ 1.218  0.922 \sqrt{h/r}  0.086 (h/r) $$ 
Sources:
Rectangular Bar With UNotches, Bending
This tab defines the stress concentration factor for a bar with a rectangular cross section and two Unotches. The bar has an applied inplane bending moment.
 w = bar width
 h = notch height
 r = notch radius
 t = bar thickness
 M = applied bending moment
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { 6M \over t (w  2h)^2 } = { 6M \over t d^2 } $$Input Geometry:
Results:
d:
w/d:
r/d:
h/r:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = C_1 + C_2 \left({ 2h \over w }\right) + C_3 \left({ 2h \over w }\right)^2 + C_4 \left({ 2h \over w }\right)^3 $$where the coefficients in the equation above are calculated from "Peterson's Stress Concentration Factors":
$$ 0.1 \le h/r \le 2.0 $$  $$ 2.0 \le h/r \le 50.0 $$  

$$ C_1 $$  $$ 1.024 + 2.092 \sqrt{h/r}  0.051 (h/r) $$  $$ 1.113 + 1.957 \sqrt{h/r} $$ 
$$ C_2 $$  $$ 0.630  7.194 \sqrt{h/r} + 1.288 (h/r) $$  $$ 2.579  4.017 \sqrt{h/r}  0.013 (h/r) $$ 
$$ C_3 $$  $$ 2.117 + 8.574 \sqrt{h/r}  2.160 (h/r) $$  $$ 4.100 + 3.922 \sqrt{h/r} + 0.083 (h/r) $$ 
$$ C_4 $$  $$ 1.420  3.494 \sqrt{h/r} + 0.932 (h/r) $$  $$ 1.528  1.893 \sqrt{h/r}  0.066 (h/r) $$ 
Sources:
Rectangular Bar With Fillet, Axial Force
This tab defines the stress concentration factor for a bar with a rectangular cross section and a fillet. The bar has an applied axial force (tensile or compressive).
 D = width of larger section
 d = width of smaller section
 r = radius of fillet
 t = bar thickness
 F = applied force (tensile or compressive)
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { F \over dt } $$Input Geometry:
Results:
h:
r/d:
D/d:
h/r:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = C_1 + C_2 \left({ 2h \over D }\right) + C_3 \left({ 2h \over D }\right)^2 + C_4 \left({ 2h \over D }\right)^3 $$where the coefficients in the equation above are calculated from "Roark's Formulas for Stress and Strain":
$$ 0.1 \le h/r \le 2.0 $$  $$ 2.0 \le h/r \le 20.0 $$  

$$ C_1 $$  $$ 1.007 + 1.000 \sqrt{h/r}  0.031 (h/r) $$  $$ 1.042 + 0.982 \sqrt{h/r}  0.036 (h/r) $$ 
$$ C_2 $$  $$ 0.114  0.585 \sqrt{h/r} + 0.314 (h/r) $$  $$ 0.074  0.156 \sqrt{h/r}  0.010 (h/r) $$ 
$$ C_3 $$  $$ 0.241  0.992 \sqrt{h/r}  0.271 (h/r) $$  $$ 3.418 + 1.220 \sqrt{h/r}  0.005 (h/r) $$ 
$$ C_4 $$  $$ 0.134 + 0.577 \sqrt{h/r}  0.012 (h/r) $$  $$ 3.450  2.046 \sqrt{h/r} + 0.051 (h/r) $$ 
Sources:
Rectangular Bar With Fillet, Bending
This tab defines the stress concentration factor for a bar with a rectangular cross section and a fillet. The bar has an applied inplane bending moment.
 D = width of larger section
 d = width of smaller section
 r = radius of fillet
 t = bar thickness
 M = applied bending moment
The maximum stress is calculated as \( \sigma_{max} = K_t \cdot \sigma_{nom} \), where \(K_t\) is the stress concentration factor as determined from the plot below, and \( \sigma_{nom} \) is calculated as:
$$ \sigma_{nom} = { 6M \over t d^2 } $$Input Geometry:
Results:
h:
r/d:
D/d:
h/r:
K_{t}:
The value of the stress concentration factor is calculated by:
$$ K_t = C_1 + C_2 \left({ 2h \over D }\right) + C_3 \left({ 2h \over D }\right)^2 + C_4 \left({ 2h \over D }\right)^3 $$where the coefficients in the equation above are calculated from "Roark's Formulas for Stress and Strain":
$$ 0.1 \le h/r \le 2.0 $$  $$ 2.0 \le h/r \le 20.0 $$  

$$ C_1 $$  $$ 1.007 + 1.000 \sqrt{h/r}  0.031 (h/r) $$  $$ 1.042 + 0.982 \sqrt{h/r}  0.036 (h/r) $$ 
$$ C_2 $$  $$ 0.270  2.404 \sqrt{h/r} + 0.749 (h/r) $$  $$ 3.599 + 1.619 \sqrt{h/r}  0.431 (h/r) $$ 
$$ C_3 $$  $$ 0.677 + 1.133 \sqrt{h/r}  0.904 (h/r) $$  $$ 6.084  5.607 \sqrt{h/r} + 1.158 (h/r) $$ 
$$ C_4 $$  $$ 0.414 + 0.271 \sqrt{h/r} + 0.186 (h/r) $$  $$ 2.527 + 3.006 \sqrt{h/r}  0.691 (h/r) $$ 
Sources:
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