Solved: The resulting stress distribution along section AB

The A992 steel rod is subjected to the loading shown. If the cross-sectional area of the rod is \(??80 \mathrm{~mm}^{2}\), determine the displacement of B and A. Neglect the size of the couplings at B and C.

The copper shaft is subjected to the axial loads shown. Determine the displacement of end A with respect to end D if the diameters of each segment are \(d_{A B}=0.75 \text { in., } d_{B C}=1 \text { in., }\) \(\text { and } d_{C D}=0.5 \text { in. Take } E_{\text {cu }}=18\left(10^{3}\right) \mathrm{ksi} \text {. }\)

The composite shaft, consisting of aluminum, copper, and steel sections, is subjected to the loading shown. Determine the displacement of end A with respect to end D and the normal stress in each section. The cross-sectional area and modulus of elasticity for each section are shown in the figure. Neglect the size of the collars at B and C.

The composite shaft, consisting of aluminum, copper, and steel sections, is subjected to the loading shown. Determine the displacement of B with respect to C. The cross-sectional area and modulus of elasticity for each section are shown in the figure. Neglect the size of the collars at B and C.

The 2014-T6 aluminium rod has a diameter of 30 mm and supports the load shown. Determine the displacement of end A with respect to end E. Neglect the size of the couplings.

The A-36 steel drill shaft of an oil well extends 12 000 ft into the ground. Assuming that the pipe used to drill the well is suspended freely from the derrick at A, determine the maximum average normal stress in each pipe string and the elongation of its end D with respect to the fixed end at A. The shaft consists of three different sizes of pipe, AB, BC, and CD, each having the length, weight per unit length, and cross-sectional area indicated.

The truss is made of three A-36 steel members, each 4 having a cross-sectional area of \(??400 \mathrm{~mm}^{2}\). Determine the horizontal displacement of the roller at C when P = 8 kN.

The truss is made of three A-36 steel members, each having a cross-sectional area of \(??400 \mathrm{~mm}^{2}\). Determine the magnitude P required to displace the roller to the right 0.2 mm.

The assembly consists of two 10-mm diameter red brass C83400 copper rods AB and CD, a 15-mm diameter 304 stainless steel rod EF, and a rigid bar G. If P = 5 kN, determine the horizontal displacement of end F of rod EF.

The assembly consists of two 10-mm diameter red brass C83400 copper rods AB and CD, a 15-mm diameter 304 stainless steel rod EF, and a rigid bar G. If the horizontal displacement of end F of rod EF is 0.45 mm, determine the magnitude of P.

The load is supported by the four 304 stainless steel wires that are connected to the rigid members AB and DC. Determine the vertical displacement of the 500-lb load if the members were originally horizontal when the load was applied. Each wire has a cross-sectional area of \(??0.025 \mathrm{~in}^{2}\).

The load is supported by the four 304 stainless steel wires that are connected to the rigid members AB and DC. Determine the angle of tilt of each member after the 500-lb load is applied. The members were originally horizontal, and each wire has a cross-sectional area of \(??0.025 \mathrm{~in}^{2}\).

The rigid bar is supported by the pin-connected rod CB that has a cross-sectional area of \(??14 \mathrm{~mm}^{2}\). and is made from 6061-T6 aluminum. Determine the vertical deflection of the bar at D when the distributed load is applied. 14 \mathrm{~mm}^{2}

The post is made of Douglas fir and has a diameter of 100 mm. If it is subjected to the load of 20 kN and the soil provides a frictional resistance distributed around the post that is triangular along its sides; that is, it varies from w = 0 at y = 0 to w = 12 kN/m at y = 2 m, determine the force F at its bottom needed for equilibrium. Also, what is the displacement of the top of the post A with respect to its bottom B? Neglect the weight of the post.

The post is made of Douglas fir and has a diameter of 100 mm. If it is subjected to the load of 20 kN and the soil provides a frictional resistance that is distributed along its length and varies linearly from w = 4 kN/m at y = 0 to w =12kN/m at y = 2m, determine the force F at its bottom needed for equilibrium. Also,what is the displacement of the top of the post A with respect to its bottom B? Neglect the weight of the post.

The coupling rod is subjected to a force of 5 kip. Determine the distance d between C and E accounting for the compression of the spring and the deformation of the bolts. When no load is applied the spring is unstretched and d = 10 in. The material is A-36 steel and each bolt has a diameter of 0.25 in. The plates at A, B, and C are rigid and the spring has a stiffness of k = 12 kip/in.

The pipe is stuck in the ground so that when it is pulled upward the frictional force along its length varies 4 linearly from zero at B to \(f_{\max }(\text { force/length) }\) at C. Determine the initial force P required to pull the pipe out and the pipe’s elongation just before it starts to slip. The pipe has a length L, cross-sectional area A, and the material from which it is made has a modulus of elasticity E.

The linkage is made of three pin-connected A992 steel members, each having a diameter of \(1 \frac{1}{4}\)in. If a horizontal 4 force of P = 60 kip is applied to the end B of member AB, determine the displacement of point B.

The linkage is made of three pin-connected A992 steel members, each having a diameter of \(1 \frac{1}{4}\)in. Determine the magnitude of the force P needed to displace point B 0.25 in. to the right.

The assembly consists of three titanium (Ti-6A1-4V) rods and a rigid bar AC. The cross-sectional area of each rod is given in the figure. If a force of 60 kip is applied to the ring F, determine the horizontal displacement of point F.

The rigid beam is supported at its ends by two A-36 steel tie rods. If the allowable stress for the steel is \(\sigma_{\text {allow }}=16.2 \mathrm{ksi},\), the load w = 3 kip/ft, and x = 4 ft, determine the smallest diameter of each rod so that the beam remains in the horizontal position when it is loaded.

The rigid beam is supported at its ends by two A-36 steel tie rods. The rods have diameters \(d_{A B}=0.5 \text { in }\). and \(d_{C D}=0.5 \text { in }\). If the allowable stress for the steel is \(\sigma_{\text {allow }}=16.2 \mathrm{ksi},\) , determine the largest intensity of the distributed load w and its length x on the beam so that the beam remains in the horizontal position when it is loaded.

The steel bar has the original dimensions shown in the figure. If it is subjected to an axial load of 50 kN, determine the change in its length and its new cross-sectional dimensions at section a–a. \(E_{\mathrm{st}}=200 \mathrm{GPa}, \nu_{\mathrm{st}}=0.29\).

Determine the relative displacement of one end of the tapered plate with respect to the other end when it is subjected to an axial load P.

The assembly consists of two rigid bars that are originally horizontal. They are supported by pins and 0.25-in.-diameter A-36 steel rods. If the vertical load of 5 kip is applied to the bottom bar AB, determine the displacement at C, B, and E.

The truss consists of three members, each made from A-36 steel and having a cross-sectional area of \(0.75 \mathrm{in}^{2}\). Determine the greatest load P that can be applied so that the roller support at B is not displaced more than 0.03 in.

Solve Prob. 4–26 when the load P acts vertically downward at C.

The observation cage C has a weight of 250 kip and through a system of gears, travels upward at constant velocity along the A-36 steel column, which has a height of 200 ft. The column has an outer diameter of 3 ft and is made from steel plate having a thickness of 0.25 in. Neglect the weight of the column, and determine the average normal stress in the column at its base, B, as a function of the cage’s position y. Also, determine the displacement of end A as a function of y.

Determine the elongation of the aluminum strap when it is subjected to an axial force of 30 kN. \(\mathrm{E}_{\mathrm{al}}=70 \mathrm{GPa}\).

The ball is truncated at its ends and is used to support the bearing load P. If the modulus of elasticity for the material is E, determine the decrease in the ball’s height when the load is applied.

The column is constructed from high-strength concrete and eight A992 steel reinforcing rods. If the column is subjected to an axial force of 200 kip, determine the average normal stress in the concrete and in each rod. Each rod has a diameter of 1 in.

The column is constructed from high-strength concrete and eight A992 steel reinforcing rods. If the column is subjected to an axial force of 200 kip, determine the required diameter of each rod so that 60% of the axial force is carried by the concrete.

The A-36 steel pipe has a 6061-T6 aluminum core. It is subjected to a tensile force of 200 kN. Determine the average normal stress in the aluminum and the steel due to this loading. The pipe has an outer diameter of 80 mm and an inner diameter of 70 mm.

If column AB is made from high strength precast concrete and reinforced with four \(\frac{3}{4}\) in. diameter A-36 steel rods, determine the average normal stress developed in the concrete and in each rod. Set P = 75 kip.

If column AB is made from high strength precast concrete and reinforced with four \(\frac{3}{4}\) in. diameter A-36 steel rods, determine the maximum allowable floor loadings P. The allowable normal stresses for the concrete and the steel are \(\left(\sigma_{\text {allow }}\right)_{\text {con }}=2.5 \mathrm{ksi} \text { and }\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=24 \mathrm{ksi}\), respectively.

Determine the support reactions at the rigid supports A and C. The material has a modulus of elasticity of E.

If the supports at A and C are flexible and have a stiffness k, determine the support reactions at A and C. The material has a modulus of elasticity of E.

The load of 2000 lb is to be supported by the two vertical steel wires for which \(\sigma_{Y}=70 \mathrm{ksi} .\) Originally wire AB is 60 in. long and wire AC is 60.04 in. long. Determine the force developed in each wire after the load is suspended. Each wire has a cross-sectional area of \(0.02 \mathrm{in}^{2} . E_{\text {st }}=29.0\left(10^{3}\right) \mathrm{ksi} .\)

The load of 2000 lb is to be supported by the two vertical steel wires for which \(\sigma_{Y}=70 \mathrm{ksi} .\) Originally wire AB is 60 in. long and wire AC is 60.04 in. long. Determine the cross-sectional area of AB if the load is to be shared equally between both wires. Wire AC has a cross-sectional area of 0.02 in2. \(E_{\mathrm{st}}=29.0\left(10^{3}\right) \mathrm{ksi}\).

The A-36 steel pipe has an outer radius of 20 mm and an inner radius of 15 mm. If it fits snugly between the fixed walls before it is loaded, determine the reaction at the walls when it is subjected to the load shown.

The 10-mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. If the yield stress for the steel is\(\left(\sigma_{Y}\right)_{\mathrm{st}}=640 \mathrm{MPa}\), and for the bronze \(\left(\sigma_{Y}\right)_{\mathrm{br}}=520 \mathrm{MPa}\), determine the magnitude of the largest elastic load P that can be applied to the assembly. \(E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}\).

The 10-mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. If the bolt is subjected to a compressive force of P = 20 kN, determine the average normal stress in the steel and the bronze. \(E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}\).

The assembly consists of two red brass C83400 copper rods AB and CD of diameter 30 mm, a stainless 304 steel alloy rod EF of diameter 40 mm, and a rigid cap G. If the supports at A, C, and F are rigid, determine the average normal stress developed in the rods.

The rigid beam is supported by the three suspender bars. Bars AB and EF are made of aluminum and bar CD is made of steel. If each bar has a cross-sectional area of 450 mm2, determine the maximum value of P if the allowable stress is \(\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=200 \mathrm{MPa} \text { for the steel and }\left(\sigma_{\text {allow }}\right)_{\mathrm{al}}=150 \mathrm{MPa}\) for the aluminum. \(E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}\).

The bolt AB has a diameter of 20 mm and passes through a sleeve that has an inner diameter of 40 mm and an outer diameter of 50 mm. The bolt and sleeve are made of A-36 steel and are secured to the rigid brackets as shown. If the bolt length is 220 mm and the sleeve length is 200 mm, determine the tension in the bolt when a force of 50 kN is applied to the brackets.

If the gap between C and the rigid wall at D is initially 0.15 mm, determine the support reactions at A and D when the force P = 200 kN is applied. The assembly is made of solid A-36 steel cylinders.

The support consists of a solid red brass C83400 copper post surrounded by a 304 stainless steel tube. Before the load is applied the gap between these two parts is 1 mm. Given the dimensions shown, determine the greatest axial load that can be applied to the rigid cap A without causing yielding of any one of the materials.

The specimen represents a filament-reinforced matrix system made from plastic (matrix) and glass (fiber). If there are n fibers, each having a cross-sectional area of \(A_{f}\) and modulus of \(E_{f}\), embedded in a matrix having a cross-sectional area of A and modulus of E\(m\), determine the stress in the matrix and in each fiber when the force P is applied on the specimen.

The rigid bar is pinned at A and supported by two aluminum rods, each having a diameter of 1 in., a modulus of elasticity \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\), and yield stress of \(\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}\). If the bar is initially vertical, determine the displacement of the end B when the force of 20 kip is applied.

The rigid bar is pinned at A and supported by two aluminum rods, each having a diameter of 1 in. a modulus of elasticity \(E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\), and yield stress of \(\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}\). If the bar is initially vertical, determine the angle of tilt of the bar when the 20-kip load is applied.

The rigid bar is pinned at A and supported by two aluminum rods, each having a diameter of 1 in. and a 4 modulus of elasticity \(E_{\mathrm{al}}=10\left(10^{3}\right)\) ksi. If the bar is initially vertical, determine the displacement of the end B when the force of 2 kip is applied.

The rigid bar is pinned at A and supported by two aluminum rods, each having a diameter of 1 in. and a modulus of elasticity \(E_{\mathrm{al}}=10\left(10^{3}\right)\). If the bar is initially vertical, determine the force in each rod when the 2-kip load is applied.

The 2014-T6 aluminum rod AC is reinforced with the firmly bonded A992 steel tube BC. If the assembly fits snugly between the rigid supports so that there is no gap at C, determine the support reactions when the axial force of 400 kN is applied.The assembly is attached at D.

The 2014-T6 aluminum rod AC is reinforced with the firmly bonded A992 steel tube BC. When no load is applied to the assembly, the gap between end C and the rigid support is 0.5 mm. Determine the support reactions when the axial force of 400 kN is applied.

The three suspender bars are made of A992 steel and have equal cross-sectional areas of \(450 \mathrm{~mm}^{2}\). Determine the average normal stress in each bar if the rigid beam is subjected to the loading shown.

The three A-36 steel wires each have a diameter of 2 mm and unloaded lengths of \(L_{A C}=1.60 \mathrm{~m}\) and \(L_{A B}=L_{A D}=2.00 \mathrm{~m}\). Determine the force in each wire after the 150-kg mass is suspended from the ring at A.

A-36 steel wires AB and AD each have a diameter of 2 mm and the unloaded lengths of each wire are \(L_{A C}=1.60 \mathrm{~m}\) and \(L_{A B}=L_{A D}=2.00 \mathrm{~m}\). Determine the required diameter of wire AC so that each wire is subjected to the same force when the 150-kg mass is suspended from the ring at A.

The post is made from 606l-T6 aluminum and has a diameter of 50 mm. It is fixed supported at A and B, and at its center C there is a coiled spring attached to the rigid collar. If the spring is originally uncompressed, determine the reactions at A and B when the force P = 40 kN is applied to the collar.

The post is made from 606l-T6 aluminum and has a diameter of 50 mm. It is fixed supported at A and B, and at its center C there is a coiled spring attached to the rigid collar. If the spring is originally uncompressed, determine the compression in the spring when the load of P = 50 kN is applied to the collar.

The bracket is held to the wall using three A-36 steel bolts at B, C, and D. Each bolt has a diameter of 0.5 in. and an unstretched length of 2 in. If a force of 800 lb is placed on the bracket as shown, determine the force developed in each bolt. For the calculation, assume that the bolts carry no shear; rather, the vertical force of 800 lb is supported by the toe at A. Also, assume that the wall and bracket are rigid. A greatly exaggerated deformation of the bolts is shown.

The bracket is held to the wall using three A-36 steel bolts at B, C, and D. Each bolt has a diameter of 0.5 in. and an unstretched length of 2 in. If a force of 800 lb is placed on 4 the bracket as shown, determine how far, s, the top bracket at bolt D moves away from the wall. For the calculation, assume that the bolts carry no shear; rather, the vertical force of 800 lb is supported by the toe at A. Also, assume that the wall and bracket are rigid. A greatly exaggerated deformation of the bolts is shown.

The rigid bar is supported by the two short white spruce wooden posts and a spring. If each of the posts has an unloaded length of 1 m and a cross-sectional area of \(600 \mathrm{~mm}^{2}\), and the spring has a stiffness of k = 2 MN/m and an unstretched length of 1.02 m, determine the force in each post after the load is applied to the bar.

The rigid bar is supported by the two short white spruce wooden posts and a spring. If each of the posts has an unloaded length of 1 m and a cross-sectional area of \(600 \mathrm{~mm}^{2}\), and the spring has a stiffness of k = 2 MN/m and an unstretched length of 1.02 m, determine the vertical displacement of A and B after the load is applied to the bar.

The assembly consists of two posts AB and CD each made from material 1 having a modulus of elasticity of \(E_{1}\) and a cross-sectional area \(A_{1}\) , and a central post made from material 2 having a modulus of elasticity \(E_{2}\) and cross- sectional area \(A_{2}\) . If a load P is applied to the rigid cap, determine the force in each material.

The assembly consists of two posts AB and CD each made from material 1 having a modulus of elasticity of \(E_{1}\) and a cross-sectional area \(A_{1}\) , and a central post EF made from material 2 having a modulus of elasticity \(E_{2}\) and a cross-sectional area \(A_{2}\). If posts AB and CD are to be replaced by those having a material 2, determine the required cross-sectional area of these new posts so that both assemblies deform the same amount when loaded.

The assembly consists of two posts AB and CD each made from material 1 having a modulus of elasticity of \(E_{1}\) and a cross-sectional area \(A_{1}\), and a central post EF made from material 2 having a modulus of elasticity \(E_{2}\) and a cross-sectional area \(A_{2}\). If post EF is to be replaced by one having a material 1, determine the required cross-sectional area of this new post so that both assemblies deform the same amount when loaded.

The wheel is subjected to a force of 18 kN from the axle. Determine the force in each of the three spokes. Assume the rim is rigid and the spokes are made of the same material, and each has the same cross-sectional area.

The C83400-red-brass rod AB and 2014-T6- aluminum rod BC are joined at the collar B and fixed connected at their ends. If there is no load in the members when \(T_{1}=50^{\circ} \mathrm{F}\), determine the average normal stress in each member when \(T_{2}=120^{\circ} \mathrm{F}\). Also, how far will the collar be displaced? The cross-sectional area of each member is \(1.75 \mathrm{in}^{2}\).

The assembly has the diameters and material indicated. If it fits securely between its fixed supports when the temperature is \(T_{1}=70^{\circ} \mathrm{F}\), determine the average normal stress in each material when the temperature reaches \(T_{2}=110^{\circ} \mathrm{F}\)

The rod is made of A992 steel and has a diameter of 0.25 in. If the rod is 4 ft long when the springs are compressed 0.5 in. and the temperature of the rod is \(T=40^{\circ} \mathrm{F}\), determine the force in the rod when its temperature is \(T=160^{\circ} \mathrm{F}\).

The two cylindrical rod segments are fixed to the rigid walls such that there is a gap of 0.01 in. between them when \(T_{1}=60^{\circ} \mathrm{F}\). What larger temperature \(T_{2}\) is required in order to just close the gap? Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if \(T_{2}=300^{\circ} \mathrm{F}\). Take \(\alpha_{\text {al }}=13\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}\) \(\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}\)

The two cylindrical rod segments are fixed to the rigid walls such that there is a gap of 0.01 in. between them when T1 = 60°F. Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if T = 400????F, and also calculate the new length of the aluminum segment. \(\alpha_{\mathrm{al}}=13\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right)\) \(\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /{ }^{\circ} \mathrm{F},\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}\) \(E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}\)

The pipe is made of A992 steel and is connected to 4 the collars at A and B. When the temperature is 60°F, there is no axial load in the pipe. If hot gas traveling through the pipe causes its temperature to rise by \(\Delta T=(40+15 x)^{\circ} \mathrm{F}\), where x is in feet, determine the average normal stress in the pipe. The inner diameter is 2 in., the wall thickness is 0.15 in.

The bronze C86100 pipe has an inner radius of 0.5 in. and a wall thickness of 0.2 in. If the gas flowing through it changes the temperature of the pipe uniformly from \(T_{A}=200^{\circ} \mathrm{F}\) at A to \(T_{B}=60^{\circ} \mathrm{F}\) at B, determine the axial force it exerts on the walls. The pipe was fitted between the walls when \(T_{B}=60^{\circ} \mathrm{F}\).

The 40-ft-long A-36 steel rails on a train track are laid with a small gap between them to allow for thermal expansion. Determine the required gap d so that the rails just touch one another when the temperature is increased from \(T_{1}=-20^{\circ} \mathrm{F} \text { to } T_{2}=90^{\circ} \mathrm{F}\). Using this gap, what would be the axial force in the rails if the temperature rises to \(T_{3}=110^{\circ} \mathrm{F}\)? The cross-sectional area of each rail is \(5.10 \mathrm{in}^{2}\).

The device is used to measure a change in temperature. Bars AB and CD are made of A-36 steel and 2014-T6 aluminum alloy, respectively.When the temperature is at \(75^{\circ} \mathrm{F}\), ACE is in the horizontal position. Determine the vertical displacement of the pointer at E when the temperature rises to \(150^{\circ} \mathrm{F}\).

The bar has a cross-sectional area A, length L, modulus of elasticity E, and coefficient of thermal expansion a. The temperature of the bar changes uniformly along its length from \(T_{A} \text { at } A \text { to } T_{B} \text { at } B\) so that at any point x along the bar \(T=T_{A}+x\left(T_{B}-T_{A}\right) / L\) Determine the force the bar exerts on the rigid walls. Initially no axial force is in the bar and the bar has a temperature of \(T_{A}\).

When the temperature is at \(30^{\circ} \mathrm{C}\), the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, the temperatures at ends A and B rise to \(130^{\circ} \mathrm{C}\_ and \(80^{\circ} \mathrm{C}\), respectively. If the temperature drop along the pipe is linear, determine the average normal stress developed in the pipe. Assume each tank provides a rigid support at A and B.

When the temperature is at \(30^{\circ} \mathrm{C}\), the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, the temperatures at ends A and B rise to \(130^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\), respectively. If the temperature drop along the pipe is linear, determine the average normal stress developed in the pipe. Assume the walls of each tank act as a spring, each having a stiffness of k = 900 MN/m.

When the temperature is at \(30^{\circ} \mathrm{C}\), the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, it causes the temperature to vary along the pipe as \(T=\left(\frac{5}{3} x^{2}-20 x+120\right)^{\circ} \mathrm{C}\) where x is in meters. Determine the normal stress developed in the pipe. Assume each tank provides a rigid support at A and B.

The 50-mm-diameter cylinder is made from Am 1004-T61 magnesium and is placed in the clamp when the temperature is \(T_{1}=20^{\circ} \mathrm{C}\). If the 304-stainless-steel carriage bolts of the clamp each have a diameter of 10 mm, and they hold the cylinder snug with negligible force against the rigid jaws, determine the force in the cylinder when the temperature rises to \(T_{2}=130^{\circ} \mathrm{C}\).

The 50-mm-diameter cylinder is made from Am 1004-T61 magnesium and is placed in the clamp when the temperature is \(T_{1}=15^{\circ} \mathrm{C}\). If the two 304-stainless-steel carriage bolts of the clamp each have a diameter of 10 mm, and they hold the cylinder snug with negligible force against the rigid jaws, determine the temperature at which the average normal stress in either the magnesium or the steel first becomes 12 MPa.

The wires AB and AC are made of steel, and wire AD is made of copper. Before the 150-lb force is applied, AB and AC are each 60 in. long and AD is 40 in. long. If the temperature is increased by \(80^{\circ} \mathrm{F},\) determine the force in each wire needed to support the load. Each wire has a cross-sectional area of 0.0123 in2. TAKE \(E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}\),\(E_{\mathrm{cu}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}, \alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}\)

The cylinder CD of the assembly is heated from \(T_{1}=30^{\circ} \mathrm{C} \text { to } T_{2}=180^{\circ} \mathrm{C}\) using electrical resistance. At the lower temperature \(T_{1}\) the gap between C and the rigid bar is 0.7 mm. Determine the force in rods AB and EF caused by the increase in temperature. Rods AB and EF are made of steel, and each has a cross-sectional area of \(125 \mathrm{~mm}^{2}\). CD is made of aluminum and has a cross-sectional area of \(375 \mathrm{~mm}^{2}\). \(E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa} \text {, and } \alpha_{\mathrm{al}}=23\left(10^{-6}\right) /{ }^{\circ} \mathrm{C}\).

The cylinder CD of the assembly is heated from \(T_{1}=30^{\circ} \mathrm{C} \text { to } T_{2}=180^{\circ} \mathrm{C}\) using electrical resistance. Also, the two end rods AB and EF are heated from \(T_{1}=30^{\circ} \mathrm{C} \text { to } T_{2}=50^{\circ} \mathrm{C}\). At the lower temperature \(T_{1}\) the gap between C and the rigid bar is 0.7 mm. Determine the force in rods AB and EF caused by the increase in temperature. Rods AB and EF are made of steel, and each has a cross-sectional area of \(125 \mathrm{~mm}^{2}\). CD is made of aluminum and has a cross- sectional area of \(375 \mathrm{~mm}^{2}\). Est = 200 GPa, \(E_{\mathrm{al}}=70 \mathrm{GPa}\), \(\alpha_{\mathrm{st}}=12\left(10^{-6}\right) /{ }^{\circ} \mathrm{C}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{-6}\right) /{ }^{\circ} \mathrm{C}\).

The metal strap has a thickness t and width w and is subjected to a temperature gradient T1 to T2 (T1 6 T2). This causes the modulus of elasticity for the material to vary linearly from E1 at the top to a smaller amount E2 at the bottom. As a result, for any vertical position y, measured from the top surface,E = [(E - E )>w]y + E .Determine the position d where the axial force P must be applied so that the bar stretches uniformly over its cross section.

Determine the maximum normal stress developed in the bar when it is subjected to a tension of P = 8 kN.

If the allowable normal stress for the bar is \(\sigma_{\text {allow }}=120 \mathrm{MPa}\), determine the maximum axial force P that can be applied to the bar

The steel bar has the dimensions shown. Determine the maximum axial force P that can be applied so as not to exceed an allowable tensile stress of \(\sigma_{\text {allow }}=150 \mathrm{MPa}\).

The A-36 steel plate has a thickness of 12 mm. If \(\sigma_{\text {allow }}=150 \mathrm{MPa}\)., determine the maximum axial load P that it can support. Calculate its elongation, neglecting the effect of the fillets.

Determine the maximum axial force P that can be applied to the bar. The bar is made from steel and has an allowable stress of \(\sigma_{\text {allow }}=21 \mathrm{ksi}\).

Determine the maximum normal stress developed in the bar when it is subjected to a tension of P = 2 kip.

The member is to be made from a steel plate that is 0.25 in. thick. If a 1-in.hole is drilled through its center, determine the approximate width w of the plate so that it can support an axial force of 3350 lb. The allowable stress is \(\sigma_{\text {allow }}=22 \mathrm{ksi}\).

The resulting stress distribution along section AB for the bar is shown. From this distribution, determine the approximate resultant axial force P applied to the bar. Also, what is the stress concentration factor?

The resulting stress distribution along section AB for the bar is shown. From this distribution, determine the approximate resultant axial force P applied to the bar. Also, what is the stress concentration factor?

The weight is suspended from steel and aluminum wires, each having the same initial length of 3 m and cross-sectional area of \(4 \mathrm{~mm}^{2}\). If the materials can be assumed to be elastic perfectly plastic, with \(\left(\sigma_{Y}\right)_{\mathrm{st}}=120 \mathrm{MPa}\) and \(\left(\sigma_{Y}\right)_{\mathrm{al}}=70 \mathrm{MPa}\) determine the force in each wire if the weight is (a) 600N and (b) 720N. \(E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa} .\)

The weight is suspended from steel and aluminum wires, each having the same initial length of 3 m and cross-sectional area of \(4 \mathrm{~mm}^{2}\). If the materials can be assumed to be elastic perfectly plastic, with \(\left(\sigma_{Y}\right)_{\mathrm{st}}=120 \mathrm{MPa}\) and \(\left(\sigma_{Y}\right)_{\mathrm{al}}=70 \mathrm{MPa}\) determine the force in each wire if the weight is (a) 600N and (b) 720N. \(E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa} .\)

The bar has a cross-sectional area \(0.5 \mathrm{~in}^{2}\) and is made of a material that has a stress–strain diagram that can be approximated by the two line segments. Determine the elongation of the bar due to the applied loading.

The distributed loading is applied to the rigid beam which is supported by the three bars. Each bar has a cross-sectional area of \(1.25 \mathrm{~in}^{2}\) and is made from a material having a stress–strain diagram that can be approximated by the two line segments. If a load of w = 25 kip/ft is applied to the beam, determine the stress in each bar and the vertical displacement of the beam.

The distributed loading is applied to the rigid beam, which is supported by the three bars. Each bar has a cross-sectional area of \(0.75 \mathrm{~in}^{2}\) and is made from a material having a stress–strain diagram that can be approximated by the two line segments. Determine the intensity of the distributed loading w that will cause the beam to displace downward 1.5 in.

The rigid lever arm is supported by two A-36 steel wires having the same diameter of 4 mm. If a force of P = 3 kN is applied to the handle, determine the force developed in both wires and their corresponding elongations. Consider A-36 steel as an elastic perfectly plastic material.

The rigid lever arm is supported by two A-36 steel wires having the same diameter of 4 mm. Determine the smallest force P that will cause (a) only one of the wires to yield; (b) both wires to yield. Consider A-36 steel as an elastic perfectly plastic material.

The 300-kip weight is slowly set on the top of a post made of 2014-T6 aluminum with an A-36 steel core. If both materials can be considered elastic perfectly plastic, determine the average normal stress in each material.

The rigid beam is supported by three 25-mm diameter A-36 steel rods. If the beam supports the force of P = 230 kN, determine the force developed in each rod. Consider the steel to be an elastic perfectly plastic material.

The rigid beam is supported by three 25-mm 4 diameter A-36 steel rods. If the force of P = 230 kN is applied on the beam and removed, determine the residual stresses in each rod. Consider the steel to be an elastic perfectly plastic material.

The rigid beam is supported by the three posts A, B, and C of equal length. Posts A and C have a diameter of 75 mm and are made of a material for which E = 70 GPa and \(\sigma_{Y}=20 \mathrm{MPa}\). Post B has a diameter of 20 mm and is made of a material for which E’ = 100 GPa and \(\sigma_{Y}^{\prime}=590 \mathrm{MPa}\). Determine the smallest magnitude of P so that (a) only rods A and C yield and (b) all the posts yield.

The rigid beam is supported by the three posts A, B, and C. Posts A and C have a diameter of 60 mm and are made of a material for which E = 70 GPa and \(\sigma_{Y}=20 \mathrm{MPa}\). Post B is made of a material for which E’ = 100 GPa and \(\sigma_{Y}^{\prime}=590 \mathrm{MPa}\). If P = 130 kN, determine the diameter of post B so that all three posts are about to yield.

The bar having a diameter of 2 in. is fixed connected at its ends and supports the axial load P. If the material is elastic perfectly plastic as shown by the stress strain diagram, determine the smallest load P needed to cause segment CB to yield. If this load is released, determine the permanent displacement of point C.

Determine the elongation of the bar in Prob. 4–108 when both the load P and the supports are removed.

The rigid beam is supported by three A-36 steel wires, each having a length of 4 ft. The cross-sectional area of AB and EF is \(0.015 \mathrm{in}^{2}\), and the cross-sectional area of CD is \(0.006 \mathrm{in}^{2}\). Determine the largest distributed load w that can be supported by the beam before any of the wires begin to yield. If the steel is assumed to be elastic perfectly plastic, determine how far the beam is displaced downward just before all the wires begin to yield.