Goodman Fatigue Safety Factor Calculator (Goodman, Soderberg, Gerber and ASME)

Shaft Fatigue Safety Factor Calculator

How to Use the Calculator

1. Choose check or design mode. To check a shaft you already have, enter its diameter at the critical section. To size a new shaft, enter a target factor of safety instead and the calculator finds the smallest diameter.
2. Enter the loads at the critical section. Take the bending moment and the torque at the point where the shaft is most likely to fail: usually a shoulder, keyway or groove near a bearing or gear. For a rotating shaft with a steady load, the bending moment is alternating and the transmitted torque is a mean load.
3. Pick the material or enter S_ut and S_y from your material certificate.
4. Set the endurance limit. Let the calculator estimate it from the Marin factors, or enter a tested value for the actual part.
5. Enter the fatigue stress concentration factors K_f and K_fs for the critical feature. Use 1 for a smooth section.
6. Read the four factors of safety and the diagram. A factor above 1 means the operating point is inside the line; it is the factor by which all loads could grow before the line is reached.

Worked Example: A Steel Shaft with a Shoulder

Problem. A 30 mm diameter shaft of cold-drawn AISI 1045 steel (S_ut = 630 MPa, S_y = 530 MPa) has a machined shoulder. At the shoulder it carries a rotating bending moment of 150 N·m and a steady torque of 120 N·m. Use K_f = 1.8 for bending and K_fs = 1.4 for torsion, and a reliability of 99%. Find the fatigue factor of safety and the diameter needed for n = 2.

1. Endurance limit of the test bar.
   S_e′ = 0.5 S_ut = 0.5 × 630 = 315 MPa
2. Marin factors.
   k_a = 4.51 × 630^−0.265 = 0.8172 (machined surface)
   k_b = (30 / 7.62)^−0.107 = 0.8636 (size)
   k_c = 1, k_d = 1, k_e = 0.814 (combined loading, room temperature, 99%)
3. Corrected endurance limit.
   S_e = 0.8172 × 0.8636 × 0.814 × 315 = 181.0 MPa
4. Von Mises stresses (16 / πd³ = 188,630 m⁻³). The bending moment is alternating and the torque is a mean load.
   σ_a′ = (16/πd³) √[4 (K_f M_a)²] = 188,630 × 540 = 101.9 MPa
   σ_m′ = (16/πd³) √[3 (K_fs T_m)²] = 188,630 × 291.0 = 54.9 MPa
5. Goodman factor of safety.
   1/n = σ_a′/S_e + σ_m′/S_ut = 101.9/181.0 + 54.9/630 = 0.5629 + 0.0871 = 0.6500
   n = 1.54
6. The other criteria for the same stresses: Soderberg 1.50, Gerber 1.74 and ASME elliptic 1.75. The first-cycle yield check gives n_y = 530 / 115.7 = 4.58, so yielding is not a concern.
7. Diameter for n = 2. Because k_b depends on d, the calculator solves for d by iteration: 32.8 mm for Goodman, and 33.1, 31.5 and 31.4 mm for Soderberg, Gerber and ASME. A standard 35 mm shaft would satisfy all four.

Load this example into the calculator

What Is Fatigue and Why Do Shafts Suffer from It?

A part that is loaded and unloaded many times can crack and break at a stress far below its yield strength. This is fatigue. A crack starts at a tiny flaw or stress concentration, grows a little with every cycle and finally leaves too little material to carry the load. Steels show an endurance limit: a stress amplitude below which the part can run for an effectively unlimited number of cycles. That is the target of an infinite-life design, and it is what this calculator checks.

A rotating shaft is the classic case. Suppose it carries a gear with a steady force. The bending moment at a section does not change, but the shaft rotates inside it, so any point on the surface passes through the tension side and then the compression side once every revolution. The bending stress at that point therefore reverses completely between a positive and a negative value: it is an alternating stress with zero mean. A steady transmitted torque, on the other hand, gives a constant shear stress, a mean stress. Real shafts have both, and the Goodman diagram is how the two are combined.

The Endurance Limit and the Marin Factors

The endurance limit measured on a small, polished test bar is, for steels, about half the tensile strength: S_e′ = 0.5 S_ut up to a tensile strength of 1400 MPa, above which it is taken as 700 MPa. A real shaft is rougher, larger and less certain, so Marin’s equation reduces the test value:

The surface factor shows why a rough finish hurts: for the same steel, a hot-rolled surface can cost much more than a ground one, and the loss grows with the strength of the steel, since stronger steels are more notch sensitive. This is one reason high-strength steel is not a cure for fatigue problems.

Stress Concentration: K_t and K_f

Shoulders, keyways, grooves and holes concentrate stress. The theoretical factor K_t comes from the geometry (for example, a sharp shoulder fillet has a large value and a generous radius a smaller one). In fatigue the real effect is usually a little smaller, so the fatigue concentration factor is

The factors are applied to the stress amplitudes of the loads. This calculator applies K_f to both the alternating and the mean bending stress and K_fs to both torsional components, which is the conservative practice used in the textbook method.

Combining Bending and Torsion: Von Mises Stresses

For a solid round shaft of diameter d, the bending stress at the surface is σ = 32 K_f M / (πd³) and the torsional shear stress is τ = 16 K_fs T / (πd³). The distortion-energy (von Mises) equivalent stress for this plane stress state is √(σ² + 3τ²). Applying it separately to the alternating and the mean components gives

The Four Failure Criteria

The criteria agree when the mean stress is small, because they all pass through the endurance limit on the vertical axis. They differ when the mean stress is large. For a shaft with fully reversed bending and a modest steady torque, the choice changes the answer by 10 to 20%, which is well inside the uncertainty of the endurance limit itself.

Sizing a shaft directly

When the endurance limit does not depend on the diameter, each criterion can be solved for d. With A = √[4(K_fM_a)² + 3(K_fsT_a)²] and B = √[4(K_fM_m)² + 3(K_fsT_m)²], the DE-Goodman result is

Do Not Forget Yielding

The Goodman and Gerber lines protect against fatigue but do not strictly prevent yielding in the first cycle. The calculator therefore also checks the peak stress, with the alternating and mean parts of every load added: n_y = S_y / σ_max′, where σ_max′ = √[ (32 K_f(M_a+M_m)/πd³)² + 3 (16 K_fs(T_a+T_m)/πd³)² ]. This is the conservative form of the “Langer” line. For shafts with a high mean load, this can be the limiting check.

Common Mistakes

Assumptions and Limitations

• Steel, infinite life. The endurance limit rules (S_e′ = 0.5 S_ut) are for steels, and the result is a factor of safety against fatigue at very high cycle counts.
• Constant-amplitude loading with in-phase alternating and mean components. Variable loads need cycle counting and damage accumulation.
• Solid round section at the critical point, with bending and torsion only. Axial loads, hollow shafts and transverse shear are not included.
• Marin factors are estimates. Real endurance limits scatter and depend on the actual surface, residual stress and environment. Use the direct endurance limit option when you have test data.
• Stress concentration factors are your input. The calculator does not look up K_t for you, and the value of q depends on the material and notch radius.
• No dynamics. Deflection, critical speed, key and press-fit fretting and bearing life are separate checks.

How This Calculator Was Verified

The stresses were checked by building the full three-dimensional stress tensor at the shaft surface and taking its von Mises value from the principal stresses. The four factors of safety were checked by finding numerically, with a root finder, where the load line from the origin meets each failure curve, with no closed-form safety factors involved. The Marin factors were recomputed from their defining coefficients, and the reliability factors from the normal distribution (the 90% value uses the textbook deviate, 1.288, which differs from the exact quantile by about 0.05% in k_e). The design mode was compared with the closed-form shaft equations for all four criteria. Across 1,500 random loading cases and 300 random design cases everything agreed to better than one part in 10⁷ (the design mode is limited by its bisection tolerance). The page’s own code is also checked live:

The “Calculator” column is computed live, by the same code that runs the calculator, in your browser.

References and Further Reading

1. Budynas, R. G., & Nisbett, J. K. Shigley’s Mechanical Engineering Design. McGraw-Hill. (Chapter 6: fatigue failure resulting from variable loading, Marin factors, failure criteria; Chapter 7: shafts and shaft components, DE-Goodman, DE-Gerber, DE-Soderberg and DE-ASME elliptic.)
2. Norton, R. L. Machine Design: An Integrated Approach. Pearson. (Fatigue design, shaft design.)
3. Juvinall, R. C., & Marshek, K. M. Fundamentals of Machine Component Design. Wiley. (Fatigue and endurance limit.)
4. Collins, J. A., Busby, H. R., & Staab, G. H. Mechanical Design of Machine Elements and Machines. Wiley.
5. Pilkey, W. D., & Pilkey, D. F. Peterson’s Stress Concentration Factors. Wiley. (Values of K_t and notch sensitivity.)
6. Marin, J. Mechanical Behavior of Engineering Materials. Prentice-Hall, 1962. (Origin of the modifying factors.)
7. Gerber, W. (1874), Goodman, J. Mechanics Applied to Engineering (1899) and Soderberg, C. R. (working-stress papers of the 1930s): the original sources of the failure lines that carry their names.