von Mises Stress Failure Theory in a Spinning Solid Disk

Topic: von Mises Yield Criterion and Rotating Disk

Subject: Machine Design

Tool: Scilab

By: Gani Comia, September 2025

• Spinning Solid Disks

Rotating circular machine elements can be modeled as rotating disks to evaluate stresses. Similar to the theory of thick-walled cylinders, both tangential and radial stresses are present, but in this case, they result from inertial forces within the ring or disk. The tangential and radial stresses are governed by the following conditions.

1. The outside radius of the disk is significantly larger than its thickness, \(r_o \geq 10 \, t\).
2. The ring or disk has a uniform thickness.
3. The stresses are evenly distributed throughout the thickness.

Reference article and derivation of the stresses in the rotating rings and disks can be found in the website https://roymech.org/Useful_Tables/Mechanics/Rotating_cylinders.html.

• Tangential Stress Equation for Solid Disk

$$\sigma_t = \left(\frac{\rho \, \omega^2}{8}\right) \left[ (3 + \nu) \,  r_o^2 - (1 - 3 \, \nu) \, r^2 \right] \tag{1}$$

Where:

\(\sigma_t\) – tangential stress, \(Pa\)

\(\rho\) - density, \(kg/m^2\)

\(\omega\) – angular velocity, \(rad/s\)

\(\nu\) – Poisson’s Ratio

\(r_o\) – outside radius of cylinder, \(m\)

\(r\) – radius at the point of calculation, \(m\)

• Radial Stress Equation for Solid Disk

$$\sigma_r = \left( \frac{\rho \, \omega^2}{8} \right) (3 + \nu) (r_o^2 - r^2) \tag{2}$$

Where:

\(\sigma_r\) – radial stress, \(Pa\)

• von Mises Yield Criterion (Maximum Distortion Energy Criterion)

In engineering, the von Mises yield criterion states that yielding begins when the von Mises stress \(\sigma_v\)​, derived from the Cauchy stress tensor, equals the yield strength \(\sigma_y\)​.

The von Mises Equation for a general state of stress is

$$\sigma_{\nu} = \sqrt{\frac{1}{2} \left[ (\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{33} - \sigma_{11})^2 \right] + 3 \, (\sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2)} \tag{3}$$

For the boundary conditions,

$$\sigma_{12} = \sigma_{23} = \sigma_{31} = 0 \tag{4}$$

the von Mises defines the equations under principal stress

$$\sigma_{\nu} = \sqrt{\frac{1}{2} \left[ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right]} \tag{5}$$

For the 2D multi-axial stress, where \(\sigma_3 = 0\) as boundary condition under principal plane stress, the equivalent von Mises stress, \(\sigma_{\nu}\), becomes

$$\sigma_{\nu} = \sqrt{\sigma_1^2 + \sigma_2^2 - \sigma_1 \, \sigma_2} \tag{6}$$

• von Mises Yield Criterion and Factor of safety

The use of the von Mises criterion as a yield criterion is only exactly applicable when the material properties are isotropic, there is no considerable dynamic or unpredictable loads, and the ratio of the shear yield strength to the tensile yield strength has the value as in Equation (7).

$$\frac{\sigma_s}{\sigma_y} = \frac{1}{\sqrt{3}} \approx 0.577 \tag{7}$$

In practice it is necessary to use engineering judgement and use the above ratio.

The factor of safety based on yield strength is given by Equation (8).

$$N_y = \frac{\sigma_y}{\sigma_{\nu}} - 1 \tag{8}$$

Critical components suggest a factor of safety of \(\text{2.5}  \leq  N_y  \leq  \text{3.0} \).

• Application

A solid disk (Figure 1) is analyzed at different rotational speeds to determine safe operating limits. The design factor indicates the maximum speed at which the disk can rotate without failure. Using the von Mises Yield Criterion, the equivalent stress is evaluated from the combined tangential and radial stresses. The design factor, as given by Equation 8, defines the likelihood of structural failure for the specified parameters.

Assumed Parameters:

Material:    Alloy Steel

Density:    7,850 \(kg/m^3\)

Yield Strength:    1,250 \(MPa\)

Poisson’s Ratio:    0.29

Radius:    0.5 \(m\)

Rotational Speed:    2500, 5000, 7500, and 10000 \(rpm\)

Figure 1. Rotating Solid Disk (ChatGPT Image Sep 10, 2025).

Figure 2 presents the equivalent combined stresses according to the von Mises yield criterion. The stress distribution varies radially across the disk, reaching its maximum at the center where \(r = 0\).

Figure 2. von Mises Stress from the Point of Analysis along the Disk Radius.

A contour plot, shown in Figure 3, provides an alternative visualization of the stress distribution for a given rotational speed.

Figure 3. von Mises Stress Distribution

Figure 4 shows the design factor at various rotational speeds. The minimum factor of safety is indicated for each case. Rotational speed of \(7500  \, \text{rpm} \) and above do not satisfy the criteria for critical components.

Figure 4. Design Factor using von Mises Stress for Spinning Disk.

Refer to the following Scilab scripts for figure reproduction.

Scilab Script for Figure 2 and 4

// Copyright (C) 2025 - Gani Comia
// Date of creation: 12 Sep 2025
// Script: von Mises Yield Criterion in a Spinning Disk
clear;clc;

// Primary parameters
rho = 7850;                             // kg/m^3, density of steel
sigmaY = 1250;                          // MPa, yield strength, alloy steel
rpm = [2500 5000 7500 10000];           // rpm, rotational speed      
radps = rpm.*(2*%pi/60);                // rad/s, maximum rotational speed
nu = 0.29;                              // Poisson's ration of carbon steel
r_o = 0.5;                              // m, outer radius of cylinder

// Secondary parameters
R = linspace(0,r_o,100)
n = length(radps)

// von Mises stress distribution for a rotational speed
for i = 1:n
    // tangential stress, Pa
    sigmaT(i,:) = ((rho.*radps(i).^2)/8).*((3+nu).*(r_o).^2 - (1-3*nu).*(R.^2));
    // radial stress, Pa
    sigmaR(i,:) = ((rho.*radps(i).^2)/8).*((3+nu).*((r_o).^2 - (R.^2)));
    // von Mises stress, Pa
    sigmaV(i,:) = sqrt(sigmaT(i,:).^2 + sigmaR(i,:).^2 - (sigmaT(i,:).*sigmaR(i,:)))
    // von Mises stress, MPa
    sigmaV(i,:) = sigmaV(i,:)/1e6
end

// Plot of von Mises stress distribution for a given rotation speed
clf;
scf(0)
f = gcf()
f.figure_size = [800,800]
subplot(2,2,1)
    plot(R,sigmaV(1,:),"g-","linewidth",4.5);
    title("von Mises Stress at 2,500 rpm")
    xlabel("Disk Radius, m","fontsize",3)
    ylabel("Stress, MPa","fontsize",3)
    xgrid(color("grey"),1,7)
    mprintf("Max von Mises #1 : %0.0f\n", max(sigmaV(1,:)))
    legend("$\LARGE \sigma_{Vmax} = \text{55 MPa}$",with_box=%f)
subplot(2,2,2)
    plot(R,sigmaV(2,:),"m-","linewidth",4.5);
    title("von Mises Stress at 5,000 rpm")
    xlabel("Disk Radius, m","fontsize",3)
    ylabel("Stress, MPa","fontsize",3)
    xgrid(color("grey"),1,7)
    mprintf("Max von Mises #3 : %0.0f\n", max(sigmaV(2,:)))
    legend("$\LARGE \sigma_{Vmax} = \text{221 MPa}$",with_box=%f)
subplot(2,2,3)
    plot(R,sigmaV(3,:),"b-","linewidth",4.5);
    title("von Mises Stress at 7,500 rpm")
    xlabel("Disk Radius, m","fontsize",3)
    ylabel("Stress, MPa","fontsize",3)
    xgrid(color("grey"),1,7)
    mprintf("Max von Mises #3 : %0.0f\n", max(sigmaV(3,:)))
    legend("$\LARGE \sigma_{Vmax} = \text{498 MPa}$",with_box=%f)
subplot(2,2,4)
    plot(R,sigmaV(4,:),"r-","linewidth",4.5);
    title("von Mises Stress at 10,000 rpm")
    xlabel("Disk Radius, m","fontsize",3)
    ylabel("Stress, MPa","fontsize",3)
    xgrid(color("grey"),1,7)
    mprintf("Max von Mises #4 : %0.0f\n", max(sigmaV(4,:)))
    legend("$\LARGE \sigma_{Vmax} = \text{885 MPa}$",with_box=%f)
    a1 = xstring(0.25,860,"https://gani-mech-toolbox.blogspot.com")
    a1.font_size = 2
    
// factor-of-safety calculation and plot
for j = 1:n
    Ny(j,:) = (sigmaY./sigmaV(j,:)) - 1
end

scf(1)
g = gcf()
g.figure_size = [700,700]
plot("ln",R,Ny(1,:),"g-","linewidth",4.5)
plot("ln",R,Ny(2,:),"m-","linewidth",4.5)
plot("ln",R,Ny(3,:),"b-","linewidth",4.5)
plot("ln",R,Ny(4,:),"r-","linewidth",4.5)
title("Design Factor using von Mises Stress Criterion for Spinning Disk","fontsize",3.5)
xlabel("Disk Radius, m","fontsize",3.5)
ylabel("Design Factor, Ny","fontsize",3.5)
xgrid(color("grey"),1,7)

// minimum design factor for a given rotation annotation
format(5)
a2 = xstring(0.005,Ny(1,1),["N (min) = ",string(Ny(1,1))," at 2,500 rpm"])
a2.font_size = 3
a3 = xstring(0.005,Ny(2,1),["N (min) = ",string(Ny(2,1))," at 5,000 rpm"])
a3.font_size = 3
a4 = xstring(0.005,Ny(3,1),["N (min) = ",string(Ny(3,1))," at 7,500 rpm"])
a4.font_size = 3
a5 = xstring(0.005,Ny(4,1)-0.2,["N (min) = ",string(Ny(4,1))," at 10,000 rpm"])
a5.font_size = 3

// design parameter annotation
a6 = "$N_y = \frac{\sigma_y}{\sigma_v} - 1$"
a7 = "$\sigma_v = \sqrt{\sigma_1^2 + \sigma_2^2 - \sigma_1 \sigma_2}$"
a8 = "$\frac{\sigma_s}{\sigma_y} = \frac{1}{\sqrt{3}} \; , \; \text{assumption}$"
a9 = xstring(0.0625,10,[a6;a7;a8])
a9.font_size = 4
a11 = xstring(0.05,8,"https://gani-mech-toolbox.blogspot.com")
a11.font_size = 2.5

ax = gca()
ax.data_bounds = [0.005 0; 0.5 25];

Scilab Script for Figure 3

// Copyright (C) 2025 - Gani Comia
// Date of creation: 5 Sep 2025
// Script: von Mises Stress Distribution
clear;clc;

// Primary parameters
rho = 7850;                             // kg/m^3, density of steel
rpm = [2500 5000 7500 10000];           // rpm, rotation speed
radps = max(rpm.*(2*%pi/60));           // rad/s, maximum rotational speed
nu = 0.29;                              // Poisson's ration of carbon steel
r_o = 0.5;                              // m, outer radius of cylinder

// Create meshgrid
n = 250;
x = linspace(-r_o, r_o, n);
y = linspace(-r_o, r_o, n);
[X, Y] = ndgrid(x, y);

// Compute radial distance from center
R = sqrt(X.^2 + Y.^2);

// Define stress distribution at maximum rotational speed
sigmaT = zeros(n, n);
sigmaR = zeros(n, n);
sigmaV = zeros(n, n);

for i = 1:n
    for j = 1:n
        if R(i,j) <= r_o then
            // tangential stress
            sigmaT(i,j) = ((rho.*radps.^2)/8).*((3+nu).*(r_o).^2 - (1-3*nu).*(R(i,j)).^2);
            // radial stress
            sigmaR(i,j) = ((rho.*radps.^2)/8).*((3+nu).*((r_o).^2 - (R(i,j)).^2));
            // von Mises stress
            sigmaV(i,j) = sqrt(sigmaT(i,j).^2 + sigmaR(i,j).^2 - (sigmaT(i,j).*sigmaR(i,j))); 
        else
            sigmaT(i,j) = %nan;     // Outside the circle
            sigmaR(i,j) = %nan;
            sigmaV(i,j) = %nan;
        end
    end
end

// Stress distribution using contour plot
clf;
f = gcf()
f.figure_size = [700,700]

sigmaV = sigmaV./(1e6)
disp(max(sigmaV)); disp(min(sigmaV));

contourf(x, y, sigmaV, 60);
title("von Mises Stress Distribution in Circular Domain at 10K rpm","fontsize",3.5);
xlabel("x-coordinates","fontsize",3)
ylabel("y-coordinates","fontsize",3)
xgrid(color("grey"),1,7)
isoview on;

f.color_map = jet(64)
colorbar(750,900)

ax = gca()
ax.data_bounds = [-0.6 -0.6; 0.6 0.6]
ax.auto_ticks = ["on" "on"]

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Disclaimer: The formulas and calculations presented are for technical reference only. Users must verify the accuracy and ensure compliance with applicable engineering standards, codes, and safety requirements before practical application.

References

1. “Rotating Disks and Cylinders”. RoyMech.org. 2020. https://roymech.org/Useful_Tables/Mechanics/Rotating_cylinders.html .
2. “von Mises yield criterion”. Wikipedia The Free Encyclopedia. 18 September 2024. https://en.wikipedia.org/wiki/Von_Mises_yield_criterion .
3. J.E. Shigley and C.R. Mischke. Mechanical Engineering Design. 5th Ed. New York, New York 10020. McGraw-Hill Publishing Company. 1989.