Maximum principal stress theory

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. 

Eigure 1-4 is an overlay of Figures 1-1,1-2, and 1-3 and will illustrate the major differences between the three theories. For the case of biaxial stress state, all three theories are in agreement where their bounded areas graphically overlap. The bounded area by each theory indicates the elastic range of which there is no yielding, however, it is important to note that in quadrants II and IV that the maximum principal stress theory provides unconservative results. For example, consider point B at the midpoint of the line in Figure 1-2. It shows shear stress is equal to ((72 - (-<7i))/2, which equals (<72 -I- (7i)/2 or one-half the stress which would... 

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 principal stress theory or maximum shear stress theory is used. For example, according to the maximum principal stress theory, for a cylinder only under internal pressure the controlling stress governing the thickness of a cylinder is (T0, the circumferential stress, since it is the largest of the three principal stresses. According to the maximum shear stress theory, the controlling stress would be... 

Therefore, the stress used in the maximum principal stress theory is PR... 

Three points are obvious from a comparison of the maximum principal stress theory and the maximum shear stress theory ... 

For thick-walled vessels (Rm/t < 10), the radial stress becomes significant in defining the ultimate failure of the vessel. The maximum principal stress theory is unconservative for designing these vessels. For this reason, this book has limited most of its application to thin-walled vessels where a biaxial state of stress is assumed to exist. 

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 results for the sealant component were evaluated using the maximum principal stress theory valid for brittle materials. The stress analyses were performed for aU three cases used in the CFD analyses. The stress results are illustrated at the fillet regions, in order to evaluate the geometric effect on the stress behavior. 

Maximum principal stress theory Maximum shear stress theory... 

According to the maximum principal stress theory, failure occurs when one of the three principal stresses reaches a stress value of elastic limit as determined from a uniaxial tension test. This theory is meaningful for brittle fracture situations. 

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. 

In the preceding se tions of this chapter essentially one ipelhod of combining the stresses has teen employed. This method is known as the maximum-principal-stress theory and is the method most widely used. Other theories may be use[Pg.177]

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. 

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... 

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