Maximum shear stress theory failure

Maximum shear stress theory which postulates that failure will occur in a complex stress system when the maximum shear stress reaches the value of the shear stress at failure in simple tension. 

The maximum shear-stress theory has been found to be suitable for predicting the failure of ductile materials under complex loading and is the criterion normally used in the pressure-vessel design. 

The maximum intensity of stress allowed will depend on the particular theory of failure adopted in the design method (see Section 13.3.2). The maximum shear-stress theory is normally used for pressure vessel design. 

If the maximum shear stress theory is taken as the criterion of failure (Section 13.3.2), then the maximum pressure that a monobloc vessel can be designed to withstand without failure is given by ... 

Manning (1947) has shown that the maximum shear strain energy theory of failure (due to Mises (1913)) gives a closer fit to experimentally determined failure pressures for monobloc cylinders than the maximum shear stress theory. This criterion of failure gives ... 

Lance, R.H. and Robinson, D.N., (1972), A maximum shear stress theory of plastic failure of fiber-reinforced materials. J. Mech. Phys. Solids, 19,49. 

Since its inception, the design requirements of the code have been based on the maximum-stress theory of failure. Over the past 50 years, it has been established that yielding under pressure correlates better with the maximum-shear-stress theory. Therefore, both Division 2 and Section III, Nuclear Vessels, are based on this latter theory, resulting in a more precise evaluation of the stresses in the various p s of a vessel. 

The underlying basis of Division 2 is similar to that of Section III, but simplified rules are provided for calculating the thickness of commonly used shapes. Designers may be surprised to find that under certain conditions the thickness of ellipsoidal heads will need to be greater under Division 2 than under Division 1. Simplified formulas for torispherical head design are not included because difficulties have been encountered in developing a formula based on the maximum-shear-stress theory of failure and more time is needed. 

In many situations, the yield strength is used to identify the allowable stress to which a material can be subjected. For components that have to withstand high pressures, such as those used in pressurized water reactors (PWRs), this criterion is not adequate. To cover these situations, the maximum shear stress theory of failure has been incorporated into the ASME (The American Society of Mechanical Engineers) Boiler and Pressure Vessel Code, Section m. Rules for Construction of Nuclear Pressure Vessels. The maximum shear stress theory of failure was originally proposed for use in the U S. Naval Reactor Program for PWRs. It will not be discussed in this text. 

Two basic theories of failure are used in the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section I, Section IV, Section 111 Division 1 (Subsections NC, ND, and NE), and Section VIII Division 1 use the maximum principal stress theory. Section ni Division 1 (Subsection NB and the optional part of NC) and Section VIII Division 2 use the maximum shear stress theory or the Tresca criterion. The maximum principal stress theory (sometimes called Rankine theory) is appropriate for materials such as cast iron at room temperature, and for mild steels at temperatures below the nil ductility transition (NDT) temperature (discussed in Section 3.7). Although this theory is used in some design codes (as mentioned previously) the reason is that of simplicity, in that it reduces the amount of analysis, although often necessitating large factors of safety. 

Within the context of pressure vessel design codes, the comparison of the allowable strength of the material is always done with respect to the stress intensities. This puts the comparison in terms of the appropriate failure theory either the maximum shear stress theory (Tresca criterion) or the maximum distortion energy theory (von Mises criterion). These failure theories have been discussed in some detail in Chapter 3. 

Maximum shear stress theory (Tresca) Failure occurs when the maximum shear stress at-an arbitrary point in a stressed body is equal to the maximum shear stress at failure (rupture or yield) in a uniaxial tensile test. 

Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

Engineers have known for some time that the maximum shear stress theory and the distortion energy theory predict yielding and fatigue failure in ductile materials better than does the maximum stress theory. However, the maximum stress theory is easier to apply, and with an adequate safety factor it gives satisfactory designs. But where a more exact analysis is desired, the maximum shear stress theory is used. 

This theory was first proposed by Tresca in 1865 and experimentally verified by Guest in 1900. It states that in a multiaxial stress state failure occurs when the maximum shear stress exceeds the maximum shear stress at failure in a monotonic tensile traction test. In a tensile test it is... 

The von Mises criterion defines an ellipse in the 2D principal stress plane as shown in Figure 2.4. The maximum shear stress theory is also shown in Figure 2.4. However, for metals, while the maximum shear stress criterion is conservative, not only can the von Mises criterion be derived, but it also fits the experimental data better than the maximum shear criterion, and thus is the best estimation of the failure envelope. 

By definition, a brittle material does not fail in shear failure oeeurs when the largest prineipal stress reaehes the ultimate tensile strength, Su. Where the ultimate eompressive strength, Su, and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and MeKee, 1991 Norton, 1996). The probabilistie failure eriterion is essentially the same as equation 4.55. 

For materials such as mild steel, w hich fail in shear rather than direct tension, the maximum shear theory of failure should be used. For internal pressure only, the maximum shear stress occurs on the inner surface of the cylinder. At this surface both tensile and compressive stresses are maximum. In a cylinder, the maximum tensile stress is the circumferential stress, (70. The maximum c ompressive stress is the radial stress, These stresses would be computed as... 

The fatigue evaluation methodology is based on the minimum shear stress theory of failure. This consists of finding the amplitude (one half of the range) through which the maximum shear stress fluctuates. This is obtained in the ASME Code procedure by using stress difference and stress intensities (twice the maximum shear stress). 

Maximum distortion energy (or maximum octahedral shear stress) theory (von Mises) Failure occurs when the maximum distortion energy (or maximum octahedral shear stress) at an arbitrary point in a stressed medium reaches the value equivalent to the maximum distortion energy (or maximum octahedral shear stress) at failure (yield) in simple tension... 

The maximum principle stress theory (Rankine s theory) states that the largest principle stress component, 03, in the material determines failure regardless of the value of normal or shearing stresses. The stability criterion is formulated as... 

Consider for example, the maximum shear stress (or Tresca) theory of failure with respect to yielding, which can be stated as follows A material subjected to any combination of loads will yield whenever the maximum shear stress at any point in the material exceeds the value of the maximum shear stress in a simple tensile test at yield. ... 

Part AD This part contains requirements for the design of vessels. The rules of Division 2 are based on the maximum-shear theory of failure for stress failure and yielding. Higher stresses are permitted when wind or earthquake loads are considered. Any rules for determining the need for fatigue analysis are given here. 

---