Question 7.1: An element in plane stress is subjected to stresses σx = 16,...
An element in plane stress is subjected to stresses \sigma_{x} = 16,000 psi, \sigma_{y} = 6,000 psi, and \tau_{xy} = \tau_{yx} = 4,000 psi, as shown in Fig. 7-7a.
Determine the stresses acting on an element inclined at an angle θ = 45°.
Transformation equations. To determine the stresses acting on an inclined element, we will use the transformation equations (Eqs. 7-4a and 7-4b). From the given numerical data, we obtain the following values for substitution into those equations:
\frac{\sigma_{x}+\sigma_{y}}{2} = 11,000 psi \frac{\sigma_{x}-\sigma_{y}}{2} = 5,000 psi \tau_{xy} = 4,000 psi
sin 2 θ = sin 90° = 1 cos 2 θ = cos 90° = 0
Substituting these values into Eqs. (7-4a) and (7-4b), we get
\sigma_{x_{1}}=\frac{\sigma_{x}+\sigma_{y}}{2}+\frac{\sigma_{x}-\sigma_{y}}{2}\cos 2_\ \theta+\tau_{xy}\sin 2_\ \theta (7-4a)
= 11,000 psi + (5,000 psi)(0) + (4,000 psi)(1) = 15,000 psi
\tau_{x_{1}y_{1}} =-\frac{\sigma_{x}-\sigma_{y}}{2}\sin 2_\ \theta +\tau_{xy}\cos 2_\ \theta (7-4b)
= -(5,000 psi)(1) + (4,000 psi)(0) = -5,000 psi
In addition, the stress \sigma_{y_{1}} may be obtained from Eq. (7-5):
\sigma_{y_{1}} =\frac{\sigma_{x}+\sigma_{y}}{2}-\frac{\sigma_{x}-\sigma_{y}}{2}\cos 2_\ \theta-\tau_{xy}\sin 2_\ \theta= 11,000 psi – (5,000 psi)(0) – (4,000 psi)(1) = 7,000 psi
Stress elements. From these results we can readily obtain the stresses acting on all sides of an element oriented at θ = 45° , as shown in Fig. 7-7b. The arrows show the true directions in which the stresses act. Note especially the directions of the shear stresses, all of which have the same magnitude. Also, observe that the sum of the normal stresses remains constant and equal to 22,000 psi (see Eq. 7-6).
\sigma_{x_{1}}+\sigma_{y_{1}} = \sigma_{x}+\sigma_{y} (7-6)
Note: The stresses shown in Fig. 7-7b represent the same intrinsic state of stress as do the stresses shown in Fig. 7-7a. However, the stresses have different values because the elements on which they act have different orientations.
