The steel frame (E 5 200 GPa) shown has a diagonal brace BD with an area of 1920 mm2

A 4.8-ft-long steel wire of \(\frac{1}{4}\)-in. diameter is subjected to a 750-lb tensile load. Knowing that \(E=29 \times 10^{6}\) psi, determine (a) the elongation of the wire, (b) the corresponding normal stress.

Two gage marks are placed exactly 250 mm apart on a 12-mm-diameter aluminum rod with E = 73 GPa and an ultimate strength of 140 MPa. Knowing that the distance between the gage marks is 250.28 mm after a load is applied, determine (a) the stress in the rod, (b) the factor of safety.

A nylon thread is subjected to a 2-lb tensile load. Knowing that \(E=0.7 \times 10^{6}\) psi and that the length of the thread increases by 1.1%, determine (a) the diameter of the thread, (b) the stress in the thread.

A 9-m length of 6-mm-diameter steel wire is to be used in a hanger. It is noted that the wire stretches 18 mm when a tensile force P is applied. Knowing that E = 200 GPa, determine (a) the magnitude of the force P, (b) the corresponding normal stress in the wire.

A steel rod is 2.2 m long and must not stretch more than 1.2 mm when an 8.5-kN load is applied to it. Knowing that E = 200 GPa, determine (a) the smallest diameter rod that should be used, (b) the corresponding normal stress caused by the load.

A control rod made of yellow brass must not stretch more than \(\frac{1}{8}\) in. when the tension in the wire is 800 lb. Knowing that \(E=15 \times 10^{6}\) psi and that the maximum allowable normal stress is 32 ksi, determine (a) the smallest diameter that can be selected for the rod, (b) the corresponding maximum length of the rod.

An aluminum pipe must not stretch more than 0.05 in. when it is subjected to a tensile load. Knowing that \(E=10.1 \times 10^{6}\) psi and that the allowable tensile strength is 14 ksi, determine (a) the maximum allowable length of the pipe, (b) the required area of the pipe if the tensile load is 127.5 kips.

A cast-iron tube is used to support a compressive load. Knowing that E = 69 GPa and that the maximum allowable change in length is 0.025%, determine (a) the maximum normal stress in the tube, (b) the minimum wall thickness for a load of 7.2 kN if the outside diameter of the tube is 50 mm.

A block of 10-in. length and \(1.8 \times 1.6\)-in. cross section is to support a centric compressive load P. The material to be used is a bronze for which \(E=14 \times 10^{6}\) psi. Determine the largest load that can be applied, knowing that the normal stress must not exceed 18 ksi and that the decrease in length of the block should be at most 0.12% of its original length.

A 9-kN tensile load will be applied to a 50-m length of steel wire with E = 200 GPa. Determine the smallest-diameter wire that can be used knowing that the normal stress must not exceed 150 MPa and that the increase in the length of the wire should be at most 25 mm.

The 4-mm-diameter cable BC is made of a steel with E = 200 GPa. Knowing that the maximum stress in the cable must not exceed 190 MPa and that the elongation of the cable must not exceed 6 mm, find the maximum load P that can be applied as shown.

Rod BD is made of steel \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) and is used to brace the axially compressed member ABC. The maximum force that can be developed in member BD is 0.02P. If the stress must not exceed 18 ksi and the maximum change in length of BD must not exceed 0.001 times the length of ABC, determine the smallest-diameter rod that can be used for member BD.

The specimen shown is made from a 1-in.-diameter cylindrical steel rod with two 1.5-in.-outer-diameter sleeves bonded to the rod as shown. Knowing that \(E=29 \times 10^{6}\) psi, determine (a) the load P so that the total deformation is 0.002 in., (b) the corresponding deformation of the central portion BC.

Both portions of the rod ABC are made of an aluminum for which E 5 70 GPa. Knowing that the magnitude of P is 4 kN, determine (a) the value of Q so that the deflection at A is zero, (b) the corresponding deflection of B.

The rod ABC is made of an aluminum for which E = 70 GPa. Knowing that P = 6 kN and Q = 42 kN, determine the deflection of (a) point A, (b) point B.

Two solid cylindrical rods are joined at B and loaded as shown. Rod AB is made of steel \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\), and rod BC of brass \(\left(E=15 \times 10^{6} \mathrm{psi}\right)\). Determine (a) the total deformation of the composite rod ABC, (b) the deflection of point B.

A \(\frac{1}{8}\)-in.-thick hollow polystyrene cylinder \(\left(E=0.45 \times 10^{6} \mathrm{psi}\right)\) and a rigid circular plate (only part of which is shown) are used to support a 10-in.-long steel rod AB \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) of \(\frac{1}{4}\)-in. diameter. If an 800-lb load P is applied at B, determine (a) the elongation of rod AB, (b) the deflection of point B, (c) the average normal stress in rod AB.

The 36-mm-diameter steel rod ABC and a brass rod CD of the same diameter are joined at point C to form the 7.5-m rod ABCD. For the loading shown and neglecting the weight of the rod, determine the deflection of (a) point C, (b) point D.

The steel frame (E = 200 GPa) shown has a diagonal brace BD with an area of \(1920 \mathrm{\ mm}^{2}\) . Determine the largest allowable load P if the change in length of member BD is not to exceed 1.6 mm.

For the steel truss \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) and loading shown, determine the deformations of members AB and AD, knowing that their cross-sectional areas are \(4.0 \mathrm{in}^{2}\) and \(2.8 \mathrm{in}^{2}\), respectively.

Members AB and BC are made of steel \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) with cross-sectional areas of \(0.80 \mathrm{in}^{2}\) and \(0.64 \mathrm{in}^{2}\), respectively. For the loading shown, determine the elongation of (a) member AB, (b) member BC.

Members ABC and DEF are joined with steel links (E = 200 GPa). Each of the links is made of a pair of \(25 \times 35-\mathrm{mm}\) plates. Determine the change in length of (a) member BE, (b) member CF.

Each of the links AB and CD is made of aluminum (E = 75 GPa) and has a cross-sectional area of \(125 \mathrm{\ mm}^{2}\). Knowing that they support the rigid member BC, determine the deflection of point E.

Link BD is made of brass \(\left(E=15 \times 10^{6} \mathrm{psi}\right)\) and has a cross-sectional area of \(0.40 \ \mathrm{in}^{2}\). Link CE is made of aluminum \(\left(E=10.4 \times 10^{6} \mathrm{psi}\right)\) and has a cross-sectional area of \(0.50 \ \mathrm{in}^{2}\). Determine the maximum force P that can be applied vertically at point A if the deflection of A is not to exceed 0.014 in.

An axial force of 60 kN is applied to the assembly shown by means of rigid end plates. Determine (a) the normal stress in the brass shell, (b) the corresponding deformation of the assembly.

The length of the assembly decreases by 0.15 mm when an axial force is applied by means of rigid end plates. Determine (a) the magnitude of the applied force, (b) the corresponding stress in the steel core.

The 4.5-ft concrete post is reinforced with six steel bars, each with a \(1 \frac{1}{8}\)-in. diameter. Knowing that \(E_{s}=29 \times 10^{6}\) psi and \(E_{c}=4.2 \times 10^{6}\) psi, determine the normal stresses in the steel and in the concrete when a 350-kip axial centric force P is applied to the post.

For the concrete post of Prob. 9.27, determine the maximum centric force that can be applied if the allowable normal stress is 20 ksi in the steel and 2.4 ksi in the concrete.

Three steel rods \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) support an 8.5-kip load P. Each of the rods AB and CD has a \(0.32-\mathrm{in}^{2}\) cross-sectional area and rod EF has a \(1-\mathrm{in}^{2}\) cross-sectional area. Neglecting the deformation of rod BED, determine (a) the change in length of rod EF, (b) the stress in each rod.

Two cylindrical rods, one of steel and the other of brass, are joined at C and restrained by rigid supports at A and E. For the loading shown and knowing that \(E_{s}=200 \ \mathrm{GPa}\) and \(E_{b}=105 \ \mathrm{GPa}\), determine (a) the reactions at A and E, (b) the deflection of point C.

Solve Prob. 9.30 assuming that rod AC is made of brass and rod CE is made of steel.

Two cylindrical rods, CD made of steel \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) and AC made of aluminum \(\left(E=10.4 \times 10^{6} \mathrm{psi}\right)\), are joined at C and restrained by rigid supports at A and D. Determine (a) the reactions at A and D, (b) the deflection of point C.

Three wires are used to suspend the plate shown. Aluminum wires of \(\frac{1}{8}\)-in. diameter are used at A and B while a steel wire of \(\frac{1}{12}\)-in. diameter is used at C. Knowing that the allowable stress for aluminum \(\left(E=10.4 \times 10^{6} \mathrm{psi}\right)\) is 14 ksi and that the allowable stress for steel \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) is 18 ksi, determine the maximum load P that can be applied.

The rigid bar AD is supported by two steel wires of \(\frac{1}{16}\)-in. diameter \(\left(E=29 \times 10^{6} \mathrm{psi}\right)\) and a pin and bracket at D. Knowing that the wires were initially taut, determine (a) the additional tension in each wire when a 220-lb load P is applied at D, (b) the corresponding deflection of point D.

The rigid rod ABC is suspended from three wires of the same material. The cross-sectional area of the wire at B is equal to half of the cross-sectional area of the wires at A and C. Determine the tension in each wire caused by the load P.

The rigid bar ABCD is suspended from four identical wires. Determine the tension in each wire caused by the load P.

The brass shell \(\left(\alpha_{b}=20.9 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\) is fully bonded to the steel core \(\left(\alpha_{s}=11.7 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\). Determine the largest allowable increase in temperature if the stress in the steel core is not to exceed 55 MPa.

The assembly shown consists of an aluminum shell \(\left(E_{a}=70 \mathrm{\ GPa}\right.,\left.\alpha_{a}=23.6 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\) fully bonded to a steel core \(\left(E_{s}=200 \mathrm{\ GPa}\right.,\left.\alpha_{s}=11.7\times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\) and is unstressed at a temperature of \(20^{\circ} \mathrm{C}\). Considering only axial deformations, determine the stress in the aluminum shell when the temperature reaches \(180^{\circ} \mathrm{C}\).

A 4-ft concrete post is reinforced by four steel bars, each of \(\frac{3}{4}\)-in. diameter. Knowing that \(E_{s}=29 \times 10^{6} \mathrm{psi}, \alpha_{s}=6.5 \times 10^{-6} / \mathrm{F}\) and \(E_{c}=3.6 \times 10^{6} \mathrm{psi}\)i and \(\alpha_{c}=5.5 \times 10^{-6} /{ }^{\circ} \mathrm{F}\), determine the normal stresses induced in the steel and in the concrete by a temperature rise of \(80^{\circ} \mathrm{F}\).

The steel rails for a railroad track \(\left(E_{s}=29 \times 10^{6} \mathrm{psi}, \alpha_{s}=6.5 \times\right.\left.10^{-6} / \mathrm{F}\right)\) were laid out at a temperature of \(30^{\circ} \mathrm{F}\). Determine the normal stress in the rails when the temperature reaches \(125^{\circ} \mathrm{F}\) assuming that the rails (a) are welded to form a continuous track, (b) are 39 ft long with \(\frac{1}{4}\)-in. gaps between them.

A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of brass \(\left(E_{b}=105 \mathrm{\ GPa}, \ \alpha_{b}=\right.\left.20.9 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\) and portion BC is made of aluminum \(\left(E_{a}=72 \mathrm{\ GPa}, \ \alpha_{a}=\right.\left.23.9 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\). Knowing that the rod is initially unstressed, determine (a) the normal stresses induced in portions AB and BC by a temperature rise of \(42^{\circ} \mathrm{C}\), (b) the corresponding deflection of point B.

A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of steel \(\left(E_{s}=29 \times 10^{6} \mathrm{\ psi}, \ \alpha_{s}=\right.\left.6.5 \times 10^{-6} /{ }^{\circ} \mathrm{F}\right)\) and portion BC is made of brass \(\left(E_{b}=15 \times 10^{6} \mathrm{\ psi}, \ \alpha_{b}=\right.\left.10.4 \times 10^{-6} /{ }^{\circ} \mathrm{F}\right)\). Knowing that the rod is initially unstressed, determine (a) the normal stresses induced in portions AB and BC by a temperature rise of \(65^{\circ} \mathrm{F}\), (b) the corresponding deflection of point B.

For the rod of Prob. 9.42, determine the maximum allowable temperature change if the stress in the steel portion AB is not to exceed 18 ksi and if the stress in the brass portion BC is not to exceed 7 ksi.

Determine (a) the compressive force in the bars shown after a temperature rise of \(96^{\circ} \mathrm{C}\), (b) the corresponding change in length of the bronze bar.

Knowing that a 0.5-mm gap exists when the temperature is \(20^{\circ} \mathrm{C}\), determine (a) the temperature at which the normal stress in the aluminum bar will be equal to –90 MPa, (b) the corresponding exact length of the aluminum bar.

At room temperature \(\left(70^{\circ} \mathrm{F}\right)\) a 0.02-in. gap exists between the ends of the rods shown. At a later time when the temperature has reached \(320^{\circ} \mathrm{F}\), determine (a) the normal stress in the aluminum rod, (b) the change in length of the aluminum rod.

A brass link \(\left(E_{b}=15 \times 10^{6} \mathrm{psi}, \alpha_{b}=10.4 \times 10^{-6}/{ }^{\circ} \mathrm{F}\right)\) and a steel rod \(\left(E_{s}=29 \times 10^{6} \mathrm{psi}, \alpha_{a}=6.5 \times 10^{-6}/{ }^{\circ} \mathrm{F}\right)\) have the dimensions shown at a temperature of 658F. The steel rod is cooled until it fits freely into the link. The temperature of the whole assembly is then raised to \(100^{\circ} \mathrm{F}\). Determine (a) the final normal stress in the steel rod, (b) the final length of the steel rod.

Two steel bars \(\left(E_{s}=200 \mathrm{\ Gpa} \ \text{and}\ \alpha_{s}=11.7 \times 10^{-6}/{ }^{\circ} \mathrm{C}\right)\) are used to reinforce a brass bar \(\left(E_{b}=105 \mathrm{\ Gpa} \ \text{and}\ \alpha_{b}=20.9 \times 10^{-6}/{ }^{\circ} \mathrm{C}\right)\) that is subjected to a load P = 25 kN. When the steel bars were fabricated, the distance between the centers of the holes that were to fit on the pins was made 0.5 mm smaller than the 2 m needed. The steel bars were then placed in an oven to increase their length so that they would just fit on the pins. Following fabrication, the temperature in the steel bars dropped back to room temperature. Determine (a) the increase in temperature that was required to fit the steel bars on the pins, (b) the stress in the brass bar after the load is applied to it.

In a standard tensile test a steel rod of \(\frac{7}{8}\)-in. diameter is subjected to a tension force of 17 kips. Knowing that v = 0.3 and \(E=29 \times 10^{6} \mathrm{\ psi}\), determine (a) the elongation of the rod in an 8-in. gage length, (b) the change in diameter of the rod.

A standard tension test is used to determine the properties of an experimental plastic. The test specimen is a 15-mm-diameter rod, and it is subjected to a 3.5-kN tensile force. Knowing that an elongation of 11 mm and a decrease in diameter of 0.62 mm are observed in a 120-mm gage length, determine the modulus of elasticity, the modulus of rigidity, and Poisson’s ratio of the material.

A 2-m length of an aluminum pipe of 240-mm outer diameter and 10-mm wall thickness is used as a short column and carries a centric axial load of 640 kN. Knowing that E = 73 GPa and v = 0.33, determine (a) the change in length of the pipe, (b) the change in its outer diameter, (c) the change in its wall thickness.

The change in diameter of a large steel bolt is carefully measured as the nut is tightened. Knowing that E = 200 GPa and v = 0.29, determine the internal force in the bolt if the diameter is observed to decrease by \(13 \ \mu \mathrm{m}\).

An aluminum plate (E = 74 GPa, v = 0.33) is subjected to a centric axial load that causes a normal stress \(\sigma\). Knowing that, before loading, a line of slope 2:1 is scribed on the plate, determine the slope of the line when \(\sigma=125 \mathrm{\ MPa}\).

A 600-lb tensile load is applied to a test coupon made from \(\frac{1}{16}\)-in. flat steel plate \(\left(E=29 \times 10^{6} \mathrm{psi}, \ \nu=0.30\right)\). Determine the resulting change (a) in the 2-in. gage length, (b) in the width of portion AB of the test coupon, (c) in the thickness of portion AB, (d) in the cross-sectional area of portion AB.

The aluminum rod AD is fitted with a jacket that is used to apply a hydrostatic pressure of 6000 psi to the 12-in. portion BC of the rod. Knowing that \(E=10.1 \times 10^{6} \mathrm{\ psi}\) and v = 0.36, determine (a) the change in the total length AD, (b) the change in diameter at the middle of the rod.

For the rod of Prob. 9.55, determine the forces that should be applied to the ends A and D of the rod (a) if the axial strain in portion BC of the rod is to remain zero as the hydrostatic pressure is applied, (b) if the total length AD of the rod is to remain unchanged.

A 20-mm square has been scribed on the side of a large steel pressure vessel. After pressurization, the biaxial stress condition of the square is as shown. Using the data available in App. A, for structural steel, determine the percent change in the slope of diagonal DB due to the pressurization of the vessel.

A fabric used in air-inflated structures is subjected to a biaxial loading that results in normal stresses \(\sigma_{x}=120 \mathrm{\ MPa}\) and \(\sigma_{z}=160 \mathrm{\ MPa}\). Knowing that the properties of the fabric can be approximated as E = 87 GPa and v = 0.34, determine the change in length of (a) side AB, (b) side BC, (c) diagonal AC.

In many situations it is known that the normal stress in a given direction is zero. For example, \(\sigma_{z}=0\) in the case of the thin plate shown. For this case, which is known as plane stress, show that if the strains \(\boldsymbol{\epsilon}_{x}\) and \(\boldsymbol{\epsilon}_{y}\) have been determined experimentally, we can express \(\boldsymbol{\sigma}_{x}\), \(\boldsymbol{\sigma}_{z}\) and \(\boldsymbol{\epsilon}_{z}\) as follows: \(\sigma_{x}=E \frac{\epsilon_{x}+\nu \epsilon_{y}}{1-\nu^{2}}\) \(\sigma_{y}=E \frac{\epsilon_{y}+\nu \epsilon_{x}}{1-\nu^{2}}\) \(\epsilon_{z}=-\frac{\nu}{1-\nu}\left(\epsilon_{x}+\epsilon_{y}\right)\)

In many situations physical constraints prevent strain from occurring in a given direction. For example, \(\boldsymbol{\epsilon}_{z}=0\) in the case shown, where longitudinal movement of the long prism is prevented at every point. Plane sections perpendicular to the longitudinal axis remain plane and the same distance apart. Show that for this situation, which is known as plane strain, we can express \(\sigma_{z}\), \(\boldsymbol{\epsilon}_{x}\), and \(\boldsymbol{\epsilon}_{y}\) as follows: \(\sigma_{z}=\nu\left(\sigma_{x}+\sigma_{y}\right)\) \(\boldsymbol{\epsilon}_{x}=\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{x}-\nu(1+\nu) \sigma_{y}\right]\) \(\epsilon_{y}=\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{y}-\nu(1+\nu) \sigma_{x}\right]\)

The plastic block shown is bonded to a rigid support and to a vertical plate to which a 240-kN load P is applied. Knowing that for the plastic used G = 1050 MPa, determine the deflection of the plate.

A vibration isolation unit consists of two blocks of hard rubber bonded to a plate AB and to rigid supports as shown. Knowing that a force of magnitude P = 6 kips causes a deflection \(\delta=\frac{1}{16}\) in. of plate AB, determine the modulus of rigidity of the rubber used.

A vibration isolation unit consists of two blocks of hard rubber with a modulus of rigidity G = 2.75 ksi bonded to a plate AB and to rigid supports as shown. Denoting by P the magnitude of the force applied to the plate and by \(\delta\) the corresponding deflection, determine the effective spring constant, \(k=P / \delta\), of the system.

An elastomeric bearing (G = 0.9 MPa) is used to support a bridge girder as shown to provide flexibility during earthquakes. The beam must not displace more than 10 mm when a 22-kN lateral load is applied as shown. Knowing that the maximum allowable shearing stress is 420 kPa, determine (a) the smallest allowable dimension b, (b) the smallest required thickness a.

Two holes have been drilled through a long steel bar that is subjected to a centric axial load as shown. For P = 6.5 kips, determine the maximum value of the stress (a) at A, (b) at B.

Knowing that \(\sigma_{\text {all }}=16\) ksi, determine the maximum allowable value of the centric axial load P.

Knowing that, for the plate shown, the allowable stress is 125 MPa, determine the maximum allowable value of P when (a) r = 12 mm, (b) r = 18 mm.

Knowing that P = 38 kN, determine the maximum stress when (a) r = 10 mm, (b) r = 16 mm, (c) r = 18 mm.

(a) Knowing that the allowable stress is 140 MPa, determine the maximum allowable magnitude of the centric load P. (b) Determine the percent change in the maximum allowable magnitude of P if the raised portions are removed at the ends of the specimen.

A centric axial force is applied to the steel bar shown. Knowing that \(\sigma_{\text {all }}\) is 135 MPa, determine the maximum allowable load P.

Knowing that the hole has a diameter of \(\frac{3}{8}\) in., determine (a) the radius \(r_{f}\) of the fillets for which the same maximum stress occurs at the hole A and at the fillets, (b) the corresponding maximum allowable load P if the allowable stress is 15 ksi.

For P = 8.5 kips, determine the minimum plate thickness t required if the allowable stress is 18 ksi.

The aluminum rod ABC \(\left(F=10.1 \times 10^{6} \mathrm{\ psi}\right)\), which consists of two cylindrical portions AB and BC, is to be replaced with a cylindrical steel rod DE \(\left(E=29 \times 10^{6} \mathrm{\ psi}\right)\) of the same overall length. Determine the minimum required diameter d of the steel rod if its vertical deformation is not to exceed the deformation of the aluminum rod under the same load and if the allowable stress in the steel rod is not to exceed 24 ksi.

The brass tube AB \(\left(E=15 \times 10^{6} \mathrm{\ psi}\right)\) has a cross-sectional area of \(0.22 \mathrm{\ in}^{2}\) and is fitted with a plug at A. The tube is attached at B to a rigid plate that is itself attached at C to the bottom of an aluminum cylinder \(\left(E=10.4 \times 10^{6} \mathrm{\ psi}\right)\) with a cross-sectional area of \(0.40 \mathrm{\ in}^{2}\). The cylinder is then hung from a support at D. In order to close the cylinder, the plug must move down through \(\frac{3}{64}\) in. Determine the force P that must be applied to the cylinder.

The length of the 2-mm-diameter steel wire CD has been adjusted so that with no load applied, a gap of 1.5 mm exists between the end B of the rigid beam ACB and a contact point E. Knowing that E = 200 GPa, determine where a 20-kg block should be placed on the beam in order to cause contact between B and E.

The uniform rods AB and BC are made of steel and are loaded as shown. Knowing that \(E=29 \times 10^{6}\) psi, determine the magnitude and direction of the deflection of point B when \(\theta=22^{\circ}\).

The steel bars BE and AD each have a \(6 \times 18-\mathrm{mm}\) cross section. Knowing that E = 200 GPa, determine the deflections of points A, B, and C of the rigid bar ABC.

In Prob. 9.77, the 3.2-kN force caused point C to deflect to the right. Using a \(\alpha=11.7 \times 10^{-6} /{ }^{\circ} \mathrm{C}\), determine (a) the overall change in temperature that causes point C to return to its original position, (b) the corresponding total deflection of points A and B.

An axial centric force P is applied to the composite block shown by means of a rigid end plate. Determine (a) the value of h if the portion of the load carried by the aluminum plates is half the portion of the load carried by the brass core, (b) the total load if the stress in the brass is 80 MPa.

A steel tube \(\left(E=29 \times 10^{6} \mathrm{\ psi}\right)\) with a \(1 \frac{1}{4}\)-in. outer diameter and a \(\frac{1}{8}\)-in. thickness is placed in a vise that is adjusted so that its jaws just touch the ends of the tube without exerting any pressure on them. The two forces shown are then applied to the tube. After these forces are applied, the vise is adjusted to decrease the distance between its jaws by 0.008 in. Determine (a) the forces exerted by the vise on the tube at A and D, (b) the change in length of the portion BC of the tube.

The block shown is made of a magnesium alloy for which \(E=6.5 \times 10^{6}\) psi and v = 0.35. Knowing that \(\sigma_{x}=-20\) ksi, determine (a) the magnitude of \(\boldsymbol{\sigma}_{y}\) for which the change in the height of the block will be zero, (b) the corresponding change in the area of the face ABCD, (c) the corresponding change in the volume of the block.

A vibration isolation unit consists of two blocks of hard rubber bonded to plate AB and to rigid supports as shown. For the type and grade of rubber used, \(\tau_{\text {all }}=220\) psi and G = 1800 psi. Knowing that a centric vertical force of magnitude P = 3.2 kips must cause a 0.1-in. vertical deflection of the plate AB, determine the smallest allowable dimensions a and b of the block.

A hole is to be drilled in the plate at A. The diameters of the bits available to drill the hole range from 9 to 27 mm in 6-mm increments. If the allowable stress in the plate is 145 MPa, determine (a) the diameter d of the largest bit that can be used if the allowable load P at the hole is to exceed that at the fillets, (b) the corresponding allowable load P.

(a) For P = 58 kN and d = 12 mm, determine the maximum stress in the plate shown. (b) Solve part a assuming that the hole at A is not drilled.