Chapter 5

Show that for small strains the fractional change in volume is the sum of the normal strain components associated with a set of three perpendicular axes.

Prove that in a linear isotropic material \(\tau_{x y}\) component of shear stress cannot produce a \(\gamma_{y z}\) component of shear strain.

Prove that in a linear isotropic material a \(\tau_{x y}\) component of shear stress cannot produce a uniform expansion or contraction consisting of three equal normal components of strain.

Discuss stress changes which may be found in a continuous railroad rail due to temperature changes during a 24 -hr period. What forces must be applied to hold the rail in place? Where must these forces be applied and how do they depend upon the length of the rail?

A cantilever bridge is built out from piers at either end and is to be joined in the middle. Discuss with a numerical example the problem of lining up the two halves of the span if the truss on one side of the roadway is in the sun and that on the other side is in the shade.

A long, thin-walled cylindrical tank of length \(L\) just fits between two rigid end walls when there is no pressure in the tank. Estimate the force exerted on the rigid walls by the tank when the pressure in the tank is \(p\) and the material of which the tank is made follows Hooke's law.

A long, thin-walled cylinder tank just fits in a rigid cylindrical cavity when there is no pressure in the tank. Estimate the tangential stress in the cylindrical-tank wall when the pressure in the tank is \(p\) and the material of which the tank is made follows Hooke's law.

A thin-walled cylindrical tank with its ends closed by thick plates is put under internal pressure. Derive an expression for the ratio of the change in length to the change in diameter.

Find expressions for the elastic displacements in a uniform bar under tensile loading. Show that your solution satisfies the 15 equations of the theory of elasticity.

A batch of 2024-T4 aluminum alloy yields in uniaxial tension at the stress \(\sigma_{o}=330 \mathrm{MN} / \mathrm{m}^{2}\). If this material is subjected to the following state of stress, will it yield according to (a) the Mises criterion, and (b) the maximum shearstress criterion? $$ \begin{array}{ll} \sigma_{x}=138 \mathrm{MN} / \mathrm{m}^{2} & \tau_{x y}=138 \mathrm{MN} / \mathrm{m}^{2} \\ \sigma_{y}=-69 \mathrm{MN} / \mathrm{m}^{2} & \tau_{y z}=0 \\ \sigma_{z}=0 & \tau_{z x}=0 \end{array} $$

The most commonly quoted results of a tensile test are the yield point or yield strength, the tensile strength, the percentage elongation, and the percentage of reduction of area. How can one tell from these results whether or not any necking occurred before fracture?

There are many practical situations where it is desirable to shrink-fit an external member on a shaft. The inner diameter of the external member is usually made slightly less than the outer diameter of the shaft. The external member is then expanded by heating, slipped over the shaft, and allowed to cool. A steel shaft with an outer diameter of \(450 \mathrm{~mm}\) and an inner diameter of \(30 \mathrm{~mm}\) has a steel tube 75 -mm-thick shrunk-fit onto it. The inner diameter of the tube is machined to be \(1.25 \mathrm{~mm}\) less than the outer diameter of the shaft. Determine the expressions for the stresses in the shaft.

Show by symmetry arguments that in linear orthotropic materials a shearstress component referred to the structural axes will produce only the corresponding shear-strain component referred to the structural axes.

In the structure shown, all three bars have the same cross-sectional area \(A\) and are compelled to have the same length \(L\), although this common length is free to expand or contract as the temperature changes. The bar materials have unequal thermal expansion coefficients and elastic moduli: $$ \begin{array}{ll} \alpha_{1}=\alpha & & \alpha_{2}=2 \alpha \\ E_{1}=E & & E_{2}=2 E \end{array} $$ Material 1 is elastic-plastic, being ideally plastic beyond a strain of \(\epsilon_{Y}\), while material 2 can be taken to remain elastic throughout the excursions described below. The system is assembled at temperature \(T=0\) with no stress in the bars. The problem is to analyze the behavior of the structure as the temperature \(T\) is increased. Specifically, answer the following questions: (a) If \(T\) is small enough, the entire system is elastic and the displacement \(u\) returns to zero when \(T\) returns to zero. Find the limiting temperature \(T_{Y}\) for which, as soon as \(T>T_{Y}\), some yielding occurs in material 1 . (b) When \(T=T_{Y}\), what is the corresponding displacement \(u ?\) (c) If now the temperature is raised to \(T=2 T_{Y}\), what is the displacement \(u\) at this temperature? (d) When the temperature is \(T=2 T_{Y}\), what is the plastic strain in the material 1 ?

A materials test is performed by pressurizing the chamber shown. The specimen is machined to have crosssectional area \(A\) at the ends and area \(k A\) in the test section \((0

A circular plate (or short cylinder) of outer radius \(a\) has a small central hole of radius \(c\), where \(c \ll a .\) The plate is subjected to an outer pressure \(p\) as shown; there is no axial force. (a) Calculate the maximum normal stress existing in the plate in the limit as \(c / a \rightarrow 0\), that is, as the hole becomes microscopic in size. (b) Calculate the ratio of this maximum stress to the maximum stress which would exist in a solid circular plate of radius \(a\) loaded with an outer pressure \(p\) This ratio gives the stress concentration factor for a small hole in a plate which is subjected to hydrostatic (equal in all directions) stress in the plane of the plate.

Consider the indicial form (see Sec. 4.15) of the complete equations of elasticity in Sec. 5.6. Show that (5.6)-(5.8) can be written in the form $$ \begin{aligned} &\sigma_{i j, i}+X_{j}=0 \\ &e_{i j}=\frac{1}{2}\left(u_{t, j}+u_{j, i}\right) \\ &e_{i j}=\frac{1+v}{E} \sigma_{i j}-\frac{v}{E} \delta_{i j} \theta+\alpha \delta_{i j} \Delta T \end{aligned} $$ where \(\delta_{i j}\) is called the Kronecker delta and is equal to 1 when \(i=j\) and equal to zero when \(i \neq j\), and where $$ \theta=\sigma_{11}+\sigma_{22}+\sigma_{33}=\sigma_{i i} $$

Show that the strain energy stored in a body (5.17) can be written in indicial notation in the form $$ U=\frac{1}{2} \int\left(\sigma_{i j} e_{i j}\right) d V $$

Show that the strain-energy expression (5.17) for an isotropic material can be written in terms of stresses in the form $$ U=\int\left[\begin{array}{l} \frac{1}{2 E}\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}\right)-\frac{v}{E}\left(\sigma_{x} \sigma_{y}+\right. \\ \left.\sigma_{y} \sigma_{z}+\sigma_{z} \sigma_{x}\right)+\frac{1}{2 G}\left(\tau_{x y}^{2}+\tau_{y z}^{2}+\tau_{z x}^{2}\right) \end{array}\right] d V $$ or in terms of the strain $$ \begin{array}{r} U=\int\left\\{\frac{E(1-v)}{2(1+v)(1-2 v)}\left(\epsilon_{x}+\epsilon_{y}+\epsilon_{z}\right)^{2}-\frac{E}{(1+v)}\left[\epsilon_{x} \epsilon_{y}+\epsilon_{y} \epsilon_{z}+\epsilon_{z} \epsilon_{x}\right.\right. \\\ \left.\left.\quad-\frac{1}{4}\left(\gamma_{x y}^{2}+\gamma_{y z}^{2}+\gamma_{z x}^{2}\right)\right]\right\\} d V \end{array} $$

A composite material is made by aligning continuous fibers of boron \(d=0.005 \mathrm{~cm}\) diameter and bonding them together in a linear array with an epoxy resin as shown in the sketch. The modulus of elasticity of the boron fibers is \(350 \mathrm{GPa}\) and that of the epoxy resin \(3.5 \mathrm{GPa}\). (a) What are the Young's moduli of the composite material in the 1 and 2 directions for a volume fraction of 40 percent of the boron fibers (the densities of the boron and resin may be assumed equal)? (b) If a second composite layer with fibers lined up parallel to the 2 direction is glued on top of the first layer, what would the new moduli be in the 1 and 2 directions? (c) Does the structure described in part (b) possess isotropy in the plane?

A threaded steel rod is subject to a mean tensile stress of \(210 \mathrm{MPa}\). The ultimate strength for the material is \(900 \mathrm{MPa}\), and \(K_{f}\) for the threaded section is \(2.6\). Estimate the maximum alternating stress that can be applied.

A closed-end cylindrical tank of \(25 \mathrm{~cm}\) diameter is made of \(0.1 \mathrm{~cm}\) thick steel having an endurance limit of \(300 \mathrm{MPa}\). The tank is supplied with air from a pump in such a fashion that there are alternating pressure pulses, equal in amplitude to 15 percent of the mean pressure. For a safety factor of 3 , what maximum mean pressure would you recommend? (Assume that due to fittings and so forth, \(K_{f}\) may approach 3.)