Chapter 7

Show that for a beam of arbitrary cross section where $$ I_{x}=\int_{A} r^{2} d A $$ the following relation holds: $$ I_{y y}+I_{z z}=I_{x} $$ Use this result to show that for a set of axes located in the centroid of the cross section of a solid circular shaft of radius \(r\) $$ I_{y y}=I_{z z}=\frac{I_{x}}{2}=\frac{\pi r^{4}}{4} $$

Letting \(I_{y y}\) and \(I_{z z}\) be the moments of inertia of the area \(A\) about the \(y\) and \(z\) axes through the centroid, and letting \(I_{y z}\) be the product of inertia for the same axes, derive the following relations: $$ \begin{gathered} I_{z^{\prime} z^{\prime}}=I_{z z}+\bar{y}^{2} A \\ I_{y y^{\prime}}=I_{y y}+\bar{z}^{2} A \\ I_{y z^{\prime}}=I_{y z}+\bar{y} \bar{z} A \end{gathered} $$ These relations illustrate the parallel-axis theorem.

A straight, thin steel strip of thickness \(t\) and width \(w\) is clamped to a rigid block of radius \(R\) with a length \(4 c\) extending from the clamp. The end of the strip is loaded with a force \(P\) sufficient to bring the strip into contact with the block over a distance \(c\). Assuming that \(c \ll R\), find the distribution of force between the strip and the block in the region \(B C\). Find also the magnitude of the force \(P\) in terms of the dimensions of the strip and block (and any other quantities deemed necessary).

A thin-walled cylindrical tank of radius \(r\), thickness \(t\), and length \(L\) is supported at its ends. It is filled with a heavy liquid which is vented to the atmosphere. If the weight of the tank is negligible compared with the weight of the liquid, show that the maximum bending stress in the tank is independent of the radius of the tank.

A cross section of a cilium (see Prob. 3.21) is shown in the figure. The dark areas are fibrils which are thought to be responsible for the cilium motion. The bending moment at the base of the cilium is estimated to be \(5 \times 10^{-7} \mathrm{~N} \cdot \mathrm{m}\), and an experimental value of the radius of curvature at the base is \(6 \mu \mathrm{m}\). Assuming that the bending forces are carried by the fibrils alone, estimate the elastic modulus of the fibrils. The total second moment \(I_{z z}\) of all the fibril cross-sectional areas is approximately \(4 \times\) \(10^{-8} \mathrm{~mm}^{4}\)

Under average conditions, what is the maximum bending stress in the lead of your pencil? Make your own estimate of the geometry and the loading conditions.

A beam is made of two identical metal bars soldered together. What is the ratio of the stiffness $$ k_{b}=\frac{M_{b}}{d \phi / d s} $$ of this beam to the stiffness of a beam in which the two bars are not soldered and act independently? What is the ratio of the maximum bending stresses for the two cases?

If a rectangular beam is made of a material whose stress-strain curve in both tension and compression is well represented by \(\sigma=c|\epsilon|^{n}\), derive an expression for the maximum bending stress in terms of the applied moment.

Consider a symmetrical beam that is initially curved in its plane of symmetry. Repeat the arguments of Sec. \(7.2\) to show that when a bending moment acts in the plane of initial curvature, plane cross sections remain plane; i.e., plane radial cross sections in the undeformed beam become plane radial cross sections in the deformed beam. Demonstrate that the increase in curvature of the neutral axis is $$ \frac{\Delta \phi}{R_{o} \phi}=\frac{1}{R_{1}}-\frac{1}{R_{o}} $$ Show by taking an appropriate free body that equilibrium requires the existence of radial normal stresses in the interior of the beam. Finally, decide whether or not the tangential strain distribution is linear across the radial depth of the beam. \(^{11}\)

A bookshelf is made out of \(6 \mathrm{~mm}\) plate glass. For long-time service, ordinary plate glass cannot safely be stressed to more than about \(7 \mathrm{MN} / \mathrm{m}^{2}\) in tension. If the supports are located in the optimum position, estimate the average weight of books per unit length which can be placed along the shelf.

Consider the problem of pure bending of the symmetrical composite beam which has been made by bonding together two materials of different elastic properties. Carry out a development parallel to that given in Secs \(7.2\) to \(7.5\) to obtain the deformation and the stresses in the composite beam. Show that the neutral surface is located by the distance \(y_{N}\) pictured, where (a) \(y_{N}=\frac{E_{1} \bar{y}_{1} A_{1}+E_{2} \bar{y}_{2} A_{2}}{E_{1} A_{1}+E_{2} A_{2}}\) and that the moment-curvature relation is (b) \(\frac{d \phi}{d s}=\frac{1}{\rho}=\frac{M_{b}}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\) where \(\left(I_{z z}\right)_{1}\) and \(\left(I_{z z}\right)_{2}\) are, respectively, the moments of inertia of the areas \(A_{1}\) and \(A_{2}\) about the neutral surface. Finally, show that the bending stress in the beam is given by (c) \(\quad\left(\sigma_{x}\right)_{i}=-E_{i} \frac{M_{b} y}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\) where \(i\) takes on the value of 1 or 2 , depending on which material we are interested in.

Concrete is a brittle material which has good strength in compression but very little strength in tension. Despite its low tensile strength, economic use can be made of concrete in reinforced-concrete construction in which steel bars are imbedded in the concrete to provide tensile action. For a reinforcedconcrete beam, carry out a development parallel to that given in Secs \(7.2\) to \(7.5\) under the assumptions that \(n o\) tensile stresses are carried by the concrete and that the tensile stress in the steel is uniform over the bars. Show that the neutral surface is located at a distance \(k d\) below the top of the beam, where the factor \(k\) is determined by the following quadratic equation. $$ E_{s}(d-k d) A_{s}-E_{c} \frac{b(k d)^{2}}{2}=0 $$ Show also that the tensile stress in the steel and the maximum compressive stress in the concrete are given by $$ \begin{aligned} \sigma_{s} &=\frac{M_{b}}{A_{s} d(1-k / 3)} \\ \left(\sigma_{c}\right)_{\max } &=\frac{2 M_{b}}{b d^{2} k(1-k / 3)} \end{aligned} $$

Find the location of the shear center for the slit, thin-walled tube of radius \(r\) and thickness \(t\).

The cross section of a solid circular shaft of radius \(r\) is acted on by a bending moment \(M_{b}\) and a twisting moment \(M_{t}\). Show that the maximum shear stress in the shaft is given by $$ \tau_{\max }=\frac{r}{2 I_{y v}} \sqrt{M_{b}^{2}+M_{t}^{2}} $$ Show also that this shear stress acts on planes whose normals make an angle $$ \psi=\frac{1}{2} \tan ^{-1} \frac{M_{b}}{M_{t}} $$ with the axial and tangential directions. Finally, verify that these results are valid for a hollow circular shaft (not necessarily thin-walled).