A diagram expressing the effect of mean stress. It joins the fully reversed fatigue limit and the tensile strength with a straight line, taking σa = σw(1 − σm/σB) as a guide to the allowable stress amplitude.
Japanese name: 修正グッドマン線図
Modified Goodman diagram appears in 2 practice questions across 2 topics, including Mechanical properties and fracture, Fracture mechanics and fatigue.
With mean stress σm on the horizontal axis and stress amplitude σa on the vertical axis, the modified Goodman line is the straight line joining the fatigue limit at zero mean stress σw (the fully reversed fatigue limit) on the vertical axis and the tensile strength σB on the horizontal axis. A stress combination below this line is judged to be on the safe side of the fatigue limit. If the mean stress reaches the tensile strength, the part fails statically even with zero amplitude, so the end point on the horizontal axis is the tensile strength. A diagram of this form, showing constant life in terms of mean stress and amplitude, is generally also called a Haigh diagram, and a line can be drawn for each life.
With the modified Goodman line, the safety factor n is found from 1/n = σa/σw + σm/σB. For example, if a cyclic stress with σm = 200 MPa and σa = 120 MPa acts on a steel with σw = 250 MPa and σB = 700 MPa, then 1/n = 0.48 + 0.29 = 0.77, and n is about 1.3. This means the line is reached if the mean stress and the amplitude are both increased by about 1.3 times in the same ratio. However, if the maximum stress σm + σa exceeds the yield stress, plastic deformation occurs on the first loading, so the yield condition σm + σa ≤ σy is also checked.
There are other lines representing the effect of mean stress. The Gerber line is the parabola σa = σw{1 − (σm/σB)²}, which often passes through the middle of test data for ductile steels; the modified Goodman line lies inside it, on the safe side. The Soderberg line is a straight line whose end point on the horizontal axis is the yield stress σy, giving an even more conservative estimate. In design, the modified Goodman line is often used because it is simple and conservative. However, for brittle materials test data may fall inside the modified Goodman line, so it is important to check against data for the material.
For σw in the diagram, the fatigue limit of test specimens is not used as is but is lowered to account for the size of the part, the surface finish and the service environment. For notched parts, the stress amplitude is multiplied by the fatigue notch factor Kf, and the treatment of mean stress is decided according to the ductility of the material. Residual stresses from quenching or welding also act as mean stress, so they are included in the evaluation. And although compressive mean stress often benefits fatigue, its effect scatters, so in design the allowable amplitude on the compressive side is usually kept at the fully reversed value and the benefit is not counted.
Answer: A. Carbon steels often show a fatigue limit, a stress amplitude below which they rarely fail however many cycles are applied
For carbon and low-alloy steels, the S-N curve often flattens out at around 10⁶–10⁷ cycles, showing a fatigue limit below which failure becomes unlikely. FCC metals such as aluminum and copper alloys rarely show a distinct flat portion, so they are rated by the strength at a set number of cycles such as 10⁷ (the fatigue strength). A tensile mean stress acts to open cracks and speeds their growth, so the life is shorter at the same stress amplitude, and corrections are made with tools such as the Goodman diagram. Note that in corrosive environments or in the very-high-cycle regime beyond 10⁸ cycles, even steels may not show a clear fatigue limit.
Answer: B. About 225 MPa
σm/σB = 200/800 = 0.25, so σa = 300×(1 − 0.25) = 225 MPa. A tensile mean stress acts to open cracks, so the allowable stress amplitude is smaller than the fully reversed value (σm = 0). The modified Goodman diagram is a conservative guide that joins σw on the vertical axis and σB on the horizontal axis with a straight line. About 281 MPa is the value from the parabolic Gerber relation, which squares the mean stress ratio, 300×(1 − 0.25²), and is a less conservative estimate than Goodman. About 100 MPa divides by σw instead of σB. A compressive mean stress often works in favor of fatigue resistance, but designs usually do not take credit for that effect.
For ductile steels, test data often fall near the Gerber line, but in design the conservative modified Goodman line is commonly used. If test data for the material are available, choose the line that fits them; with little data or for brittle materials, the basic rule is to use the conservative line.
The relation Goodman presented in his 1899 book is said to have differed from today's diagram in both its assumptions and its form. Because it was later revised into the present form, which represents the relation between mean stress and stress amplitude by a straight line, it is called modified, although accounts of who devised which diagram differ.
Yes. Instead of σw on the vertical axis, take the fully reversed fatigue strength at the target number of cycles and join it to the tensile strength; the line then represents the combinations allowed for that life. Drawing a line for each life gives a diagram from which the life can be read for any combination of mean stress and amplitude.
A tensile mean stress acts to open cracks and speeds their growth, so the life is shorter at the same stress amplitude, and corrections are made with tools such as the Goodman diagram.
Which statement about fatigue S-N curves is correct?The modified Goodman diagram is a conservative guide that joins σw on the vertical axis and σB on the horizontal axis with a straight line.
A cyclic stress with a mean stress σm = 200 MPa is applied …Fatigue limit Energy release rate Fatigue striations Griffith theory Miner's rule Paris law Stress concentration factor Stress intensity factor