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FIGURE 1.7 Tapered axially loaded member.

Over the years, various yield functions f have been proposed to describe the yield condition of a variety of materials (see, e.g., Chen 1982; Chen and Baladi 1985; Chakrabarty 1987; Chen and Han 1988). For ductile materials (e.g., most metals), the Tresca and von Mises yield criteria are often used. A brief discussion of these two criteria is given below:

1. Tresca criterion. According to the Tresca yield criterion, yielding occurs when the maximum shear stress at a point calculated using Equations 1.24 reaches a critical value equal to syl2, where ay is the yield stress of the material obtained from a simple tension test. Mathematically, the Tresca yield criterion is expressed as i?!SP1 " SP2!l Sy , , max< 11 Sp2 - Sp31 V = -2 (1.43)

2. von Mises criterion. Despite its simplicity, one drawback of the Tresca yield criterion is that it does not take into consideration the effect of the intermediate principal stress. One method to include the effect of this principal stress in the yield function is to use the octahedral shearing stress (or the strain energy of distortion) as the key parameter to describe yielding in the materials. The von Mises yield criterion is one example. The von Mises yield criterion has the form

where ay is the yield stress obtained from a simple tension test.

It should be noted that both the Tresca and the von Mises yield criteria are independent of hydrostatic pressure effect. As a result, they should be used only for materials that are pressure insensitive. For pressure dependent materials (e.g., soils), other yield (or failure) criteria should be used. A few of these criteria are given below:

1. Rankine criterion. This criterion is often used to describe the tensile (fracture) failure of a brittle material. It has the form

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