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Unformatted text preview: Section 3.4 Solid Mechanics Part I Kelly 49 3.4 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 point It will be shown below that the stress components acting on any other plane through p can 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 y x and 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 y x axes used. The resultant force on the complete element is seen to be zero. yy yy xx xx yx yx xy xy ) a ( ) b ( x y p xx yx xx yx yy xy yy xy Section 3.4 Solid Mechanics Part I Kelly 50 Figure 3.4.2: stress components acting on a material element; (a) stresses, (b) Figure 3....
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