Section 3.4 Solid Mechanics Part I Kelly 493.4 Equilibrium of Stress Consider two perpendicular planes passing through a point p. The stress components acting on these planes are as shown in Fig. 3.4.1a. These stresses are usually shown together acting on a small material element of finite size, Fig. 3.4.1b. It has been seen that the stress may vary from point to point in a material but, if the element is very small, the stresses on one side can be taken to be equal to the stresses acting on the other side. Figure 3.4.1: stress components acting on two perpendicular planes through a point; (a) two perpendicular surfaces at a point, (b) small material element at the pointIt will be shown below that the stress components acting on anyother plane through pcan be evaluated from a knowledge of only these stress components. 3.4.1 Symmetry of the Shear Stress Consider the material element shown in Fig. 3.4.1b, reproduced in Fig. 3.4.2a below. The element has dimensions is yxΔ×Δand is subjected to arbitrary uniform stresses over its sides. The resultant forces of the stresses acting on each side of the element act through the side-centres, and are shown in Fig. 3.4.2b. The stresses shown are positive, but note how positive stresses can lead to negative forces, depending on the definition of the yx−axes used. The resultant force on the complete element is seen to be zero. yyσyyσxxσxxσyxσyxσxyσxyσ)a()b(xy•pxxσyxσxxσyxσyyσxyσyyσxyσ
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