Question 7.2: A plane-stress condition exists at a point on the surface of...
A plane-stress condition exists at a point on the surface of a loaded structure, where the stresses have the magnitudes and directions shown on the stress element of Fig. 7-8a.
Determine the stresses acting on an element that is oriented at a clockwise angle of 15° with respect to the original element.
The stresses acting on the original element (Fig. 7-8a) have the following values:
\sigma_{x} = -46 MPa \sigma_{y} = 12 MPa \tau_{xy} = -19 MPa
An element oriented at a clockwise angle of 15° is shown in Fig. 7-8b, where the x_{1} axis is at an angle θ = -15° with respect to the x axis. (As an alternative, the x_{1} axis could be placed at a positive angle θ = 75° .)
Transformation equations. We can readily calculate the stresses on the x_{1} face of the element oriented at θ = -15° by using the transformation equations (Eqs. 7-4a and 7-4b). The calculations proceed as follows:
\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)
\tau_{x_{1}y_{1}} =-\frac{\sigma_{x}-\sigma_{y}}{2}\sin 2_\ \theta +\tau_{xy}\cos 2_\ \theta (7-4b)
\frac{\sigma_{x}+\sigma_{y}}{2} = -17 MPa \frac{\sigma_{x}-\sigma_{y}}{2} = -29 MPa
sin 2 θ = sin (-30°) = -0.5 cos 2 θ = cos (-30°) = 0.8660
Substituting into the transformation equations, 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
= -17 MPa + (-29 MPa)(0.8660) + (-19 MPa)(-0.5)
= -32.6 MPa
\tau_{x_{1}y_{1}} =-\frac{\sigma_{x}-\sigma_{y}}{2}\sin 2_\ \theta +\tau_{xy}\cos 2_\ \theta
= -(-29 MPa)(-0.5) + (-19 MPa)(0.8660)
= -31.0 MPa
Also, the normal stress acting on the y_{1} face (Eq. 7-5) is
\sigma_{y_{1}}=\frac{\sigma_{x}+\sigma_{y}}{2}-\frac{\sigma_{x}-\sigma_{y}}{2}\cos 2_\ \theta-\tau_{xy}\sin 2_\ \theta
= -17 MPa – (-29 MPa)(0.8660) – (-19 MPa)(-0.5)
= -1.4 MPa
This stress can be verified by substituting θ = 75° into Eq. (7-4a). As a further check on the results, we note that \sigma_{x_{1}} + \sigma_{y_{1}} = \sigma_{x} + \sigma_{y}.
The stresses acting on the inclined element are shown in Fig. 7-8b, where the arrows indicate the true directions of the stresses. Again we note that both stress elements shown in Fig. 7-8 represent the same state of stress.
