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Unformatted text preview: MAT 200 SUMMARY OF LECTURE ON YIELDING UNDER MULTIAXIAL LOADING Under conditions of unia‘xial loading (for example in the tension test performed in the laboratories) the material yields when the normal stress (tensile stress in the laboratory experiment) reaches a certain value we call yield strength 0'0. At this point the maximum shear stress induced in the material is equal to 60/2 and acts on planes making i45° with the tensile (or compression) axis. In real life situations we may expect materials to be subjected to combinations of stresses, normal and shear. However, there always exist a set of orthogonal axis 1, 2 and 3 so that stresses referred to these axes consist only of normal stresses designated as principal stresses, 0'1, 02 and 63. Several theories have been proposed that predict the stress conditions under which yielding occurs. Three such theories are described below. 1. Masimmmnnalsnrssthem: This theory predicts that yielding will take place when the principal stress of the largest absolute value Om (largest of 01, 62 or 0'3) reaches the uniaxial yield strength.- So what this means is that the other two stresses (that are not the marrimum) play no role at all in yielding. 2- I I . I I . I l I C . . : According to Tresca it is the shear stress — not the normal stresses per se -—- that determine when the material yields, and this happens when the induced shear stress reaches the'shear stress at yielding under uniaxial loading which is era/2. Under principal stresses 0'1, 0'2 and 03 the shear stresses developed (be careful with signs) are i 01—02/2. i (IT-6312 and i 63—6112. So, when the largest of these numerically reaches 60/2 yielding will occur. Notice that if one of the three principal stresses is zero, this theory reduces the first one above. Also notice that it is only the two principal stresses with the largest difference between them that determine yielding regardless of the third (which obviously has an intermediate value). 3-“.1..1 EXHCH: It can be proven that the maximum energy of distortion is proportional to [(61—62)2 + (ctr—o3)2 + (03 — oﬁz] and since we know that the material yields when 0‘1 = 0‘0 and 02 = 63 = 0 then by substitution in the above expression we get that at yield (61-62)2 + (oz-092 + (03-431)2 = 2 0'02. This criterion thus predicts that yielding depend on the value of each of the three principal stresses . Experimental results are closer to'the predictions of Von Mises criterion and generally above those of Tresca. Thus the Tresca criterion is more conservative and the Von Mises criterion more realistic as is shown the ﬁgure below for yield under biaxial loading of steel. copper and nickel. allallll Maximum distortion energy ’r-ﬁ Maximum normal stress + Cast iron 0 Steel 0 Copper A Aluminum Comparison of failure criteria with test. ...
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