Maximum-shear-stress theory

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 theory leads to the hexagonal safety domain shown in Fig. 9.4b. It is worth nothing that for ductile materials the theory may be conservative, as shown in Fig. 9.4b for steel, copper and aluminum. Combining this theory with the modified Goodman relations (5.62) fatigue failure will occur if 

An important drawback of the Tresca criterion is that it does not consider the effect of the normal stress on the damage progression (see Fig. 9.11). In fact, a component under uniaxial traction a and another under pure torsion t — all have the same equivalent stress Og but while the former loading condition results in the existence of a stress, cr , normal to the plane of Zmax opening the micro-crack as it forms, the latter condition has no normal stress and the micro-crack remains closed, as it can be easily seen from the respective Mohr circles shown in Fig. 9.5. This closure introduces friction and mechanical interlocking effects that retard the micro-crack growth. 

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For brittle materials such as glass or cast iron, the maximum shear-stress theory is usually applied. 

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 is often called Tresca s, or Guest s, theory. 

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 ... 

Departing from the maximum shear stress theory of plastic flow R. H. Lance and D. N. Robinson [6] developed yield conditions for fiber reinforced eomposite materials. The authors of [6] assumed that the material could flow plastically if (i) the shear stress on planes parallel to fibers, and in a direction perpendicular to them, reaches a critical value or... 

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. 

They are the maximum stress theory and the maximum shear stress theory. ... 

For simple analysis upon which the thickness formulas for ASME Code, Section I or Section VIII, Division I, are based, it makes little difference whether the maximum stress theory or maximum shear stress theory is used. For example, according to the maximum stress theory, the controlling stress governing the thickness of a cylinder is circumferential stress, since it is the largest of the three principal stresses. According to the maximum shear stress theory, the controlling stress would be one-half the algebraic difference between the maximum and minimum stress ... 

Division 2 stress analysis considers all stresses in a triaxial state combined in acc-ordance with the maximum shear stress theory. Division 1 and the procedures outlined in this bcx)k c-onsider a biaxial state of stress combined in accordance with the maximum stress theory. Just as you would not design a nuclear reactor to the rules of Division 1, you would not design an air receiver by the techniques of Division 2. Each has its place and applications. The following discussion on categories of stress and allowables will utilize information from Dicision 2, which can be applied in general to all vessels. 

This theory is illustrated graphically for the four states of biaxial stress shown in Figure 1-3. This theory correlates even better with ductile test specimens than the maximum shear stress theory. 

The new ASME Section VIII, Division 2, Part 5 utilizes the distortion energy theory to establish the equivalent stress in an elastic analysis where in the pre-2007 edition this was done with the maximum shear stress theory. 

According to the maximum shear stress theory, the maximum shear equals the shear stress at the elastic limit as determined from the uniaxial tension test. Here the maximum shear stress is one half the difference between the largest (say principal stresses. This is also known as the Tresca criterion, which states that pelding takes place when... 

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. 

Two basic theories of strength are used in the ASME Boiler and Pressure Vessel Code. Section I, Section IV, the ASME Code, VIII-1, and Section m. Division 1, Subsections NC, ND, and ME use the maximum stress theory. Section DB, Division 1, Subsection NB and the optional part of NC, and the ASME Code, VHI-2, use the maximum shear stress theory. 

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