Maximum principal stress theory failure

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. 

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. 

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. 

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. 

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... 

It is worthwhile to consider whether the classical theories (or criteria) of failure can still be applied if the stress (or strain) concentration effects of geometric discontinuities (eg., notches and cracks) are properly taken into account. In other words, one might define a (theoretical) stress concentration factor, for example, to account for the elevation of local stress by the geometric discontinuity in a material and still make use of the maximum principal stress criterion to predict its strength, or load-carrying capability. 

When the material behavior is brittle rather than ductile, the mechanics of the failure process are much different. Instead of the slow coalescence of voids associated with ductile rupture, brittle fracture proceeds by the high-velocity propagation of a crack across the loaded member. If the material behavior is clearly brittle, fracture may be predicted with reasonable accuracy through use of the maximum normal stress theory of failure. Thus failure is predicted to occur in the multi-axial state of stress when the maximum principal normal stress becomes equal to or exceeds the maximum normal stress at the time of failure in a simple uniaxial stress test using a specimen of the sane material. 

The von Mises criterion defines an ellipse in the 2D principal stress plane as shown in Figure 2.4. The maximum shear stress theory is also shown in Figure 2.4. However, for metals, while the maximum shear stress criterion is conservative, not only can the von Mises criterion be derived, but it also fits the experimental data better than the maximum shear criterion, and thus is the best estimation of the failure envelope. 

The criterion used here is the approach used in the ANSI B31.3 Petroleum Refinery Piping, that is, to limit the maximum calculated principal stress to the allowable stress range rather than using the shear theory of failure. The value 1.25(.S a + provides a safety margin against the possibility of fatigue due to localized stresses and other stress conditions (see Fig. 2.1). Obviously, stresses must be computed both with and without thermal expansion, since allowable stresses are much smaller for conditions without thermal expansion. As an additional safety precaution, the computed stresses are usually based on the modulus of elasticity E at room temperature [ 8 ]. 

In order to apply the maximum stress theory of failure, the stress components must be determined in principal material directions. Using the transformation Equation 8.56, the following relationships are obtained ... 

One of the shortcomings of the maximum stress theory of failure is that there are no terms which account for interaction between stress components for the case of biaxial (or off-axis) loading. Another is that five independent equations must be satisfied. The Tsai-Hill theory of failure for anisotropic materials overcomes both of the above mentioned shortcomings. This theory can be expressed in terms of principal material stress components as follows ... 

Solving for the uniaxial stress 0 c in terms of principal material stress components and applying the maximum stress failure theory constraints shown in Equation 9.9 gives the following five conditions that must be met for allowable stress For tension ... 

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