Chapter 4

Show that if the particles of a solid are acted on by "body forces" which are distributed over the volume with intensities \(X, Y\), and \(Z\) per unit volume, then the requirement of \(\Sigma \mathbf{F}=0\) leads to $$ \begin{aligned} &\frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{x y}}{\partial y}+\frac{\partial \tau_{z x}}{\partial z}+X=0 \\ &\frac{\partial \tau_{x y}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+\frac{\partial \tau_{y z}}{\partial z}+Y=0 \\ &\frac{\partial \tau_{z x}}{\partial x}+\frac{\partial \tau_{y z}}{\partial y}+\frac{\partial \sigma_{z}}{\partial z}+Z=0 \end{aligned} $$

Show that if a state of plane stress is to be described in terms of polar coordinates, the requirement that \(\Sigma F=0\) leads to the following two cquations: $$ \begin{aligned} &\frac{\partial \sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{r \theta}}{\partial \theta}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \\ &\frac{\partial \tau_{r \oplus}}{\partial r}+\frac{1}{r} \frac{\partial \sigma_{\theta}}{\partial \theta}+2 \frac{\tau_{r \oplus}}{r}=0 \end{aligned} $$ Note that the length of the curved boundary on the outer edge of the clement is \((r+\Delta r) \Delta \theta\)

Show that if a general state of stress is to be described in cylindrical coordinates, the requirement that \(\Sigma \mathbf{F}=0\) leads to the following three equations: Prob. \(4.4\) $$ \begin{aligned} &\frac{\partial \sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{r \theta}}{\partial \theta}+\frac{\partial \tau_{z r}}{\partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \\ &\frac{\partial \tau_{r \oplus}}{\partial r}+\frac{1}{r} \frac{\partial \sigma_{\theta}}{\partial \theta}+\frac{\partial \tau_{\theta z}}{\partial z}+2 \frac{\tau_{r \theta}}{r}=0 \\ &\frac{\partial \tau_{\pi}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{\theta z}}{\partial \theta}+\frac{\partial \sigma_{z}}{\partial z}+\frac{\tau_{x c}}{r}=0 \end{aligned} $$

Find the principal stresses and the oricntation of the principal axes of stress for the following cases of plane stress. $$ \begin{array}{llll} \text { (a) } \sigma_{x}=40 \mathrm{MN} / \mathrm{m}^{2} & \text { (b) } \sigma_{x}=140 \mathrm{MN} / \mathrm{m}^{2} & \text { (c) } \sigma_{x}=-120 \mathrm{MN} / \mathrm{m}^{2} \\ \sigma_{y}=0 & \sigma_{y}=20 \mathrm{MN} / \mathrm{m}^{2} & \sigma_{y}=50 \mathrm{MN} / \mathrm{m}^{2} \\ \tau_{x y}=80 \mathrm{MN} / \mathrm{m}^{2} & \tau_{x y}=-60 \mathrm{MN} / \mathrm{m}^{2} & \tau_{x y}=100 \mathrm{MN} / \mathrm{m}^{2} \\ \text { (d) } \sigma_{x}=70 \mathrm{MN} / \mathrm{m}^{2} & \text { (e) } \sigma_{x}=-70 \mathrm{MN} / \mathrm{m}^{2} & \\ \sigma_{y}=30 \mathrm{MN} / \mathrm{m}^{2} & \sigma_{y}=140 \mathrm{MN} / \mathrm{m}^{2} & \\ \tau_{x y}=60 \mathrm{MN} / \mathrm{m}^{2} & \tau_{x y}=-40 \mathrm{MN} / \mathrm{m}^{2} & \end{array} $$

Consider a thin-walled cylinder of internal radius \(r\) and thickness \(t\). If the cylinder is subjected to an internal pressure \(p\) and an axial force \(F\), show that the \(r, \theta, z\) directions are the principal stress directions. Show also that if the wall is so thin that t/r \(\leqslant 1\), then the stresses in the pipe wall are given approximately by $$ \begin{aligned} &\sigma_{r}=0 \\ &\sigma_{\theta}=\frac{p r}{t} \\ &\sigma_{z}=\frac{F}{2 \pi r t} \end{aligned} $$

Consider a thin-walled cylindrical shell of intemal radius \(r\) and thickness \(t\), with ends which will contain pressure. Show that the principal stresses in the cylinder wall are given approximately by the following when the cylinder contains an intemal pressure \(p\) : $$ \begin{aligned} &\sigma_{r}=0 \\ &\sigma_{\theta}=\frac{p r}{t} \\ &\sigma_{z}=\frac{p r}{2 t} \end{aligned} $$

Show that in a thin-walled sphere of intemal radius \(r\) and thickness \(t\) subjected to intemal pressure \(p\) the principal stresses are given approximately by the following: $$ \begin{aligned} \sigma_{r} &=0 \\ \sigma_{\theta} &=\frac{p r}{2 t} \\ \sigma_{\phi} &=\frac{p r}{2 t} \end{aligned} $$

Show in a general three-dimensional displacement that if the displaceme components in the \(x, y\), and \(z\) directions are \(u, v\) and \(w\), respectively, the stra componcnts referred to the \(x y z\) axes are $$ \begin{aligned} &\epsilon_{x}=\frac{\partial u}{\partial x} \quad \epsilon_{y}=\frac{\partial v}{\partial y} \quad \epsilon_{z}=\frac{\partial w}{\partial z} \\ &\gamma_{x y}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \quad \gamma_{j z}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z} \quad \gamma_{x x}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \end{aligned} $$

In a case of plane strain in which each point displaces radially in a rotationally symmetric fashion about the origin \(O\), the displacement can be expressed by a single displacement component \(u\) in the radial direction. Show that the strain components referred to the radial, tangential \((r, \theta)\) set of axes are $$ \epsilon_{r}=\frac{d u}{d r} \quad \epsilon_{\theta}=\frac{u}{r} \quad \gamma_{r_{0}}=0 $$

A gcneral defomation in plane strain can be described in polar coordinates by expressing the displacement of each point as the vector sum of a radial component \(u\) and a tangential component \(v\). Show that in such a case the strain components referred to the \(r, \theta\) set of axes are $$ \begin{aligned} &\epsilon_{r}=\frac{\partial u}{\partial r} \\ &\epsilon_{\theta}=\frac{1}{r} \frac{\partial v}{\partial \theta}+\frac{u}{r} \\\ &\gamma_{r_{0}}=\frac{\partial v}{\partial r}+\frac{1}{r} \frac{\partial u}{\partial \theta}-\frac{v}{r} \end{aligned} $$

In a state of plane strain in the \(x y\) plane the strain components associated with the \(x y\) axes are $$ \begin{aligned} \epsilon_{x} &=800 \times 10^{-6} \\ \epsilon_{y} &=100 \times 10^{-6} \\ y_{x y} &=-800 \times 10^{-6} \end{aligned} $$ Find the magnitude of the principal strains and the orientation of the principal strain directions.

At a point in a body the principal strains are. $$ \begin{aligned} \epsilon_{1} &=700 \times 10^{-6} \\ \epsilon_{11} &=300 \times 10^{-6} \\ \epsilon_{111} &=-300 \times 10^{-6} \end{aligned} $$ What is the maximum shear-strain componcnt at the point? What is the oricntation of the axes which experiences the maximum shear strain?

The readings of a \(45^{\circ}\) strain rosette (Fig. \(\left.4.40(b)\right)\) are (a) $$ \begin{aligned} &\epsilon_{a}=100 \times 10^{-6} \\ &\epsilon_{L}=200 \times 10^{-6} \\ &\epsilon_{c}=900 \times 10^{-6} \end{aligned} $$ $$ \begin{aligned} \epsilon_{b} &=400 \times 10^{-6} \\ \epsilon_{c} &=60 \times 10^{-6} \end{aligned} $$ Find the magnitude of the principal strains in the plane of the rosette.

If the boundary portion \(A B\) of a structure under plane stress is stress-free as illustrated, the stress vector acting on the portion \(A B\) must be zero. Express the condition that the components of the stress vector must vanish in terms of the stress components with respect to the coordinate axes and the angle \(\alpha\).

Recall from Chapter 3 that the cquilibrium cquations for a slender beam \((3.11)\) and \((3.12)\) are $$ \frac{d V}{d x}+q=0 \quad \frac{d M_{b}}{d x}+V=0 $$ Show that an integration of the equilibrium equations \((4.13)\) across the thickness of a beam in plane stress reduces to the above equations where $$ \begin{aligned} &V=\int_{-h / 2}^{h / 2} \tau_{x y} d y \quad M=-\int_{-h / 2}^{h / 2} y \sigma_{x} d y \\ &q=\sigma_{y}\left(y=\frac{h}{2}\right)-\sigma_{y}\left(y=-\frac{h}{2}\right) \end{aligned} $$