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Stress in Two Dimensions

Figure 2.14 Transformation of stresses from the x, y coordinate system to the x ′, y ′ coordinate system. (a) Illustration of the coordinate systems. (b) Triangular element (with sides in the x, y, and y ′ directions) on which a static force balance is carried out.

Thus the shear stresses are symmetric in that their value is independent of the order of the subscripts. Three independent components of stress σxx, σyy, and σxy must be specified in order to prescribe the two-dimensional state of stress. The state of stress is dependent on the orientation of the coordinate system. We will now determine the three components of stress in a coordinatesystem x′, y′ inclined at an angle θ with respect to the x, y coordinate system as illustrated in Figure 2–14a. To determine the normal stress, we carry out a static force balance on the triangular element OAB illustrated. The sides of the triangle lie in the x, y, and y′ directions. We first write a force balance in the y direction. The force in the y direction on face AO is σyyAO, and the force in the y direction on face OB is σxyOB.

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