Theories of failure

The failure of a simple structural element under unidirectional stress (tensile or compressive) is easy to relate to the tensile strength of the material, as determined in a standard tensile test, but for components subjected to combined stresses (normal and shear stress) the position is not so simple, and several theories of failure have been proposed. The three theories most commonly used are described below  

Maximum principal stress theory which postulates that a member will fail when one of the principal stresses reaches the failure value in simple tension, or. The failure point in a simple tension is taken as the yield-point stress, or the tensile strength of the material, divided by a suitable factor of safety. 

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. 

For a system of combined stresses there are three shear stresses maxima  

The maximum shear stress will depend on the sign of the principal stresses as well as their magnitude, and in a two-dimensional stress system, such as that in the wall of a thin-walled pressure vessel, the maximum value of the shear stress may be that given by putting (73 = 0 in equations 13.3 and c. 

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Subsection A This subsection contains the general requirements applicable to all materials and methods of construction. Design temperature and pressure are defined here, and the loadings to be considered in design are specified. For stress failure and yielding, this section of the code uses the maximum-stress theory of failure as its criterion. 

The simplified failure envelopes are not derived from physical theories of failure in which the actual physical processes that cause failure on a microscopic level are integrated to obtain a failure theory. We, instead, deal with phenomenological theories in which we ignore the actual failure mechanisms and concentrate on the gross macroscopic events of failure. Phenomenological theories are based on curve-fitting, so they are failure criteria and not theories of any kind (the term theory implies a formal derivation process). 

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. 

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 ... 

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. 

Classical theories of failure are based on concepts of maximum stress, strain, or strain energy and assume that the material is homogeneous and free from defects. Stresses, strains, and strain energies are typically obtained through elastic analyses. 

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. 

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