Distortion Energy Theory

Of all the theories dealing with the prediction of yielding in complex stress systems, the Distortion Energy Theory (also called the von Mises Failure Theory) agrees best with experimental results for ductile materials, for example mild steel and aluminium (Collins, 1993 Edwards and McKee, 1991 Norton, 1996 Shigley and Mischke, 1996). Its formulation is given in equation 4.57. The right-hand side of the equation is the effective stress, L, for the stress system. 

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) ... 

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... 

This section incorporates the unpublished work of Palmer and Weaver subsequently the fatigue analysis was included as an integral part of the FMP Shaft Design Guide which Palmer and Weaver compiled. Results are quoted, for brevity the reader is referred to references dealing with the Distortion Energy Theory of Failure (also called deviatoric stress, octahedral, von Mises, or shear strain) for a complete analysis. 

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. 

The results of the equivalent stress calculation from the distortion energy theory indicate that reduced shell thickness values may be obtained if a more rigorous analysis is performed. This is the basis for part of the design-by-analysis section in Section Vm, Division 2. 

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. 

The distortion energy theory considers failure to have occurred when the distortion energy accumulated in the component under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension test. This is also known as the von Mises criterion, which states that 5delding will take place when... 

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. 

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. 

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