Maximum distortion energy theory

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

To evaluate the load capacity of drill pipe (e.g., allowable tensile load while simultaneously a torque is applied), the maximum distortion energy theory is... 

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. 

The stress, L, determined using the Modified Mohr method effeetively aeeounts for all the applied stresses and allows a direet eomparison to a materials strength property to be made (Norton, 1996), as was established for the Distortion Energy Theory for duetile materials. The set of expressions to determine the effeetive or maximum stress are shown below and involve all three prineipal stresses (Dowling, 1993) ... 

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. 

Division 1 and the procedures outlined in this book consider a biaxial state of stress combined in accordance with die maximum principal stress theory. Division 2 considers triaxial stresses evaluated in accordance with the maximum shear stress theory and distortion energy 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 rules of Division 2. Each has its place and application. The following discussion on categories of stress and allowable stresses will utilize information from Division 2, which can be applied in general to all vessels. 

Von MiSOS Yiold Critorion. The Von Mises yield criterion (also known as the maximum distortional energy criterion or the octahedral stress theory) (25) states that yield will occur when the elastic shear-strain energy density reaches a critical value. There are a number of ways of expressing this in terms of the principal stresses, a common one being... 

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... 

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. 

The stress distribution given by Eq. 15.1 is shown in Fig. 15.1 for a vessel with r /fj = 2.2, The maximum stress is in the hoop direction and is at the inner surface where r = r. As the pressure is increased, the stresses increase until they reach a maximum limiting stress where failure is assumed to occur. For thin vessels the ASME Code assumes that failure occurs when the yield point is reached. This failure criterion is convenient and is called the maximum principal stress theory. In thick vessels the criterion usually applied for ductile materials is the energy of distention theory. This theory states that the inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to die strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under a state of uniaxial stress as occurs in a simple tension test. The equation that expresses this theory is given by... 

The octahedral shear stress criterion, also called the maximum energy of distortion criterion or the Von Mises theory, would predict the following torque at yield. For the shaft the octahedral shear stress would be (from Equation 3)... 

The authors apparently found that the energy of the maximum decreases upon an asymmetric distortion of the geometry, because they concluded, "If there is a hi transition state at this level of theory, it occurs in symmetry." The authors proposed that higher-level correlation effects and excursions into symmetry would eliminate the barrier entirely, in agreement with the previous, lower-level studies.si,62... 

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