Stress

Abstract

The forces per unit area on each face of a cubical element are the stresses on that face. There are two types of stresses; normal stresses are perpendicular to the face of the element and shear stresses are parallel to the element face.

Appendix: Solutions

• 2.8.1 Confirm equation 2.2.

Solution

Equation (2.2) $\Rightarrow$

$$\sigma_{{x^{{\prime }} x^{{\prime }} }} = \sigma_{xx} \,{ \cos }^{2} \theta + \sigma_{yy} \,{ \sin }^{2} \theta + 2\tau_{xy} \,{ \sin }\,\theta \,{ \cos }\,\theta$$

Summation of forces in the $x^{{\prime }} - x^{{\prime }}$-direction gives:

$$\begin{aligned} & \sum {F_{{x^{{\prime }} x^{{\prime }} }} = 0} \\ & \sigma_{{x^{{\prime }} x^{{\prime }} }} A = \sigma_{xx} \,{ \cos }\,\theta \,A{ \cos }\,\theta + \sigma_{yy} \,{ \sin }\,\theta \,A{ \sin }\,\theta + \tau_{xy} \,{ \cos }\,\theta \,A{ \sin }\,\theta + \tau_{xy} A\,{ \cos }\,\theta \,{ \sin }\,\theta \\ & \sigma_{{x^{{\prime }} x^{{\prime }} }} = \sigma_{xx} \,{ \cos }^{2} \theta + \sigma_{yy} \,{ \sin }^{2} \theta + 2\tau_{xy} \,{ \cos }\theta \,A{ \sin }\theta \\ \end{aligned}$$

• 2.8.2 Confirm equation 2.5.

Equation (2.5) $\Rightarrow$

$$\sigma_{{x^{{\prime }} x^{{\prime }} }} = \frac{{\sigma_{xx} + \sigma_{yy} }}{2} + \frac{{\sigma_{xx} - \sigma_{yy} }}{2}{ \cos }\,2\theta + \tau_{xy} \,{ \sin }\,2\theta$$

Solution

Using the trigonometric identities in (2.2):

$$\begin{aligned} { \sin }2\theta & = 2{ \sin }\theta { \cos }\theta \\ { \sin }^{2} \theta & = \frac{{1 - { \cos }2\theta }}{2} \\ { \cos }^{2} \theta & = \frac{{1 + { \cos }2\theta }}{2} \\ \end{aligned}$$

$$\Rightarrow$$

$$\begin{aligned} \sigma_{x'x'} & = \sigma_{xx} \frac{{1 + { \cos }2\theta }}{2} + \sigma_{yy} \frac{{1 - { \cos }2\theta }}{2} + \tau_{xy} { \sin }2\theta \\ \sigma_{{x^{{\prime }} x^{{\prime }} }} & = \frac{{\sigma_{xx} + \sigma_{yy} }}{2} + \frac{{\sigma_{xx} - \sigma_{yy} }}{2}{ \cos }2\theta + \tau_{xy} { \sin }2\theta \\ \end{aligned}$$

• 2.8.3 Confirm equation 2.7.

Equation (2.7) $\Rightarrow$

$$\tau_{{x^{{\prime }} y^{{\prime }} }} = - \left( {\frac{{\sigma_{xx} - \sigma_{yy} }}{2}} \right){ \sin }\,2\theta + \tau_{xy} \,{ \cos }\,2\theta$$

Solution

Summation of forces in the $y^{'} - y^{'}$-direction and using trigonometric identities gives:

$$\begin{aligned} & \sum {F_{{y^{{\prime }} y^{{\prime }} }} = 0} \quad \Rightarrow \\ & \tau_{{x^{{\prime }} y^{{\prime }} }} A = A\,{ \cos }\,\theta \,\left( {\sigma_{xx} \,{ \sin }\,\theta - \tau_{xy} \,{ \cos }\,\theta } \right) + A\,{ \sin }\,\theta \,(\tau_{xy} \,{ \sin }\,\theta - \sigma_{yy} \,{ \cos }\,\theta ) \\ \end{aligned}$$

$$\begin{aligned} \tau_{{x^{{\prime }} y^{{\prime }} }} & = (\sigma_{xx} - \sigma_{yy} )\,{ \sin }\,\theta \,{ \cos }\,\theta + \tau_{xy} \left( {{ \sin }^{2} \theta - { \cos }^{2} \theta } \right) \\ \tau_{{x^{{\prime }} y^{{\prime }} }} & = (\sigma_{xx} - \sigma_{yy} )\frac{{{ \sin }\,2\theta }}{2} + \tau_{xy} \,{ \cos }\,2\theta \\ \end{aligned}$$

• 2.8.4 Confirm equation 2.9.

Equation (2.9) $\Rightarrow$

$${ \tan }\left( {2\theta_{P} } \right) = \frac{{2\tau_{xy} }}{{\sigma_{xx} - \sigma_{yy} }}$$

Solution

Normal stress on arbitrary plane is from (2.5) $\Rightarrow$

$$\sigma_{{x^{{\prime }} x^{{\prime }} }} = \frac{{\sigma_{xx} + \sigma_{yy} }}{2} + \frac{{\sigma_{xx} - \sigma_{yy} }}{2}{ \cos }\,2\theta + \tau_{xy} \,{ \sin }\,2\theta$$

Setting derivative with respect to $\theta = 0$ for maximum and minimum $\Rightarrow$

$$\begin{aligned} \frac{{d\sigma_{{x^{{\prime }} x^{{\prime }} }} }}{d\theta } & = 0 = - 2\sigma_{xx} \,{ \sin }\,\theta \,{ \cos }\,\theta + 2\sigma_{yy} \,{ \sin }\,\theta \,{ \cos }\,\theta + 2\tau_{xy} \left( {{ \cos }^{2} \theta - { \sin }^{2} \theta } \right) \\ 0 & = \frac{{ - \sigma_{xx} \,{ \sin }\, 2\theta }}{2} + \frac{{\sigma_{yy} \,{ \sin }\,2\theta }}{2} + \tau_{xy} \left( {\frac{{1 + { \cos }\,2\theta }}{2} - \frac{{1 - { \cos }\,2\theta }}{2}} \right) \\ 0 & = \frac{{{ \sin }\,2\theta }}{2}\left( {\sigma_{yy} - \sigma_{xx} } \right) + \tau_{xy} \,{ \cos }\,2\theta \\ { \tan }\,2\theta & = \frac{{2\tau_{xy} }}{{\sigma_{xx} - \sigma_{yy} }} \\ \end{aligned}$$

• 2.8.5 Plot the variation of normal stress on planes passing through a point if it is known that the state of stress is planar with $\sigma_{xx} = 40,\;\sigma_{yy} = - 20,\;\tau_{xy} = 10$

Solution

Plotting Eq. (2.2) $\Rightarrow$