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).

Rectangular Bar With Central Hole
  • w = bar width
  • d = hole diameter
  • t = bar thickness
  • F = applied force (tensile or compressive)
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { F \over (w - d)t } $$
Cannot display plot -- browser is out of date.

Input Geometry:


Results:

d/w:

Kt:


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, Out-of-Plane 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 out-of-plane bending moment.

Rectangular Bar With Central Hole
  • w = bar width
  • d = hole diameter
  • t = bar thickness
  • M = applied bending moment
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { M \over S } = { 6M \over (w - d) t^2 } $$
Cannot display plot -- browser is out of date.

Input Geometry:


Results:

d/t:

d/w:

Kt:


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 U-Notches, Axial Force

This tab defines the stress concentration factor for a bar with a rectangular cross section and two U-notches. The bar has an applied axial force (tensile or compressive).

Rectangular Bar With U-Notch
  • w = bar width
  • h = notch height
  • r = notch radius
  • t = bar thickness
  • F = applied force (tensile or compressive)
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { F \over (w - 2h)t } = { F \over dt } $$
Cannot display plot -- browser is out of date.

Input Geometry:

Results:

d:

w/d:

r/d:

h/r:

Kt:

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 U-Notches, Bending

This tab defines the stress concentration factor for a bar with a rectangular cross section and two U-notches. The bar has an applied in-plane bending moment.

Rectangular Bar With U-Notch
  • w = bar width
  • h = notch height
  • r = notch radius
  • t = bar thickness
  • M = applied bending moment
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { 6M \over t (w - 2h)^2 } = { 6M \over t d^2 } $$
Cannot display plot -- browser is out of date.

Input Geometry:

Results:

d:

w/d:

r/d:

h/r:

Kt:

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).

Rectangular Bar With Fillet
  • D = width of larger section
  • d = width of smaller section
  • r = radius of fillet
  • t = bar thickness
  • F = applied force (tensile or compressive)
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { F \over dt } $$
Cannot display plot -- browser is out of date.

Input Geometry:

Results:

h:

r/d:

D/d:

h/r:

Kt:


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 in-plane bending moment.

Rectangular Bar With Fillet
  • D = width of larger section
  • d = width of smaller section
  • r = radius of fillet
  • t = bar thickness
  • M = applied bending moment
Cannot display plot -- browser is out of date.

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:

$$ \sigma_{nom} = { 6M \over t d^2 } $$
Cannot display plot -- browser is out of date.

Input Geometry:

Results:

h:

r/d:

D/d:

h/r:

Kt:


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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