A plot of the cyclic stress capacity of a material is called a fatigue (or endurance) curve. These curves are generated through multiple cyclic tests at different stress levels. The number of cycles to failure usually increases as the applied cyclic stress decreases, often until a threshold stress (known as the endurance limit) is reached below which no fatigue failure occurs, regardless of the number of applied cycles. The endurance limit (for those metals that possess one) is usually quantified as the value of the cyclic stress level which may be applied for at least 10^8 cycles without failure. Typical ratios of the endurance limit to the ultimate tensile strength of various materials are 0.5 for cast and wrought steels; about 0.35 for several nonferrous metals such as nickel, copper and magnesium; and 0.2 to 0.3 for rough or corroded steel surfaces (depending on the degree of stress intensification).
An endurance curve for carbon and low alloy steels, taken from the ASME Section VIII Division 2 Pressure Vessel Code is shown in Figure 1-18.
Note that according to the fatigue curve, the material doesn't fail upon initial loading, despite enormously high stresses that appear to be well above the ultimate tensile stress of typical carbon and low alloy steels. The reasons for this are:
The highly stressed areas under fatigue loading are normally very localized. Catastrophic failure under one-time loading will normally occur only when the gross cross-section is overloaded.
Fatigue curves are usually generated through cyclic application of displacement, rather than force, loading. Displacement loads are "self-limiting". If a pipe is overloaded with an imposed displacement, plastic stresses will develop, deform-ing the pipe to its displaced position. At that point there will be no further tendency for displacements to occur, and therefore no continuation of the load, or further deformation leading to catastrophic failure. In the case of an applied force (which is not a self- limiting load), deformation of the pipe does not cause the force to subside, so deformation continues until failure.
The stress shown in a fatigue curve is a calculated stress, based upon the assumption that Hooke's law is applicable throughout the range of applied loading; i.e., S = E ε , where:
E = modulus of elasticity of material, psi
ε = strain in material, in/in
In reality, once the material begins to yield, stress is no longer proportional to the induced strain, and actually is much lower than that calculated.
Introduction to Pipe Stress Analysis