The rigid member is held in the position shown by three

The A992 steel rod is subjected to the loading shown. If the cross-sectional area of the rod is \(60 \mathrm{~mm}^{2}\), determine the displacement of B and A, Neglect the size of the couplings at B , C , and D.

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 \(d_{A B}=0.75 \text { in., } \ d_{B C}=1 \text { in }\)., and \(d_{C D}=0.5 \text { in }\). Take \(E_{\mathrm{cu}}=18\left(10^{3}\right) \ \mathrm{ksi}\).

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.

Determine the displacement of B with respect to C of the composite shaft in Prob. 4–3.

The assembly consists of a steel rod CB and an aluminum rod BA , each having a diameter of 12 mm. If the rod is subjected to the axial loadings at A and at the coupling B, determine the displacement of the coupling B and the end A. The unstretched length of each segment is shown in the figure. Neglect the size of the connections at B and C, and assume that they are rigid. \(E_{\mathrm{st}}=200 \ \mathrm{GPa}, E_{\mathrm{al}}=70 \ \mathrm{GPa}\).

The bar has a cross-sectional area of \(3 \ \mathrm{in}^{2}\), and \(E=35\left(10^{3}\right) \ \mathrm{ksi}\). Determine the displacement of its end A when it is subjected to the distributed loading.

If \(P_{1}=50 \text { kip }\) and \(P_{2}=150 \text { kip }\), determine the vertical displacement of end A of the high strength precast concrete column.

If the vertical displacements of end A of the high strength precast concrete column relative to B and B relative to C are 0.08 in. and 0.1 in., respectively, determine the magnitudes of \(\mathbf{P}_{1}\) and \(\mathbf{P}_{2}\).

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.

The post is made of Douglas fir and has a diameter of 60 mm. If it is subjected to the load of 20 kN and the soil provides a frictional resistance that is uniformly distributed along its sides of w = 4 kN/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 60 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 = 0 at y = 0 to w = 3 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 hanger consists of three 2014-T6 aluminum alloy rods, rigid beams AC and BD, and a spring. If the hook supports a load of P = 60 kN, determine the vertical displacement of F . Rods AB and CD each have a diameter of 10 mm, and rod EF has a diameter of 15 mm. The spring has a stiffness of k = 100 MN/m and is unstretched when P = 0.

The hanger consists of three 2014-T6 aluminum alloy rods , rigid beams AC and BD , and a spring. If the vertical displacement of end F is 5 mm, determine the magnitude of the load P. Rods AB and CD each have a diameter of 10 mm, and rod EF has a diameter of 15 mm. The spring has a stiffness of k = 100 MN/m and is unstretched when P = 0.

Collar A can slide freely along the smooth vertical guide. If the supporting rod AB is made of 304 stainless steel and has a diameter of 0.75 in., determine the vertical displacement of the collar when P = 10 kip.

Collar A can slide freely along the smooth vertical guide. If the vertical displacement of the collar is 0.035 in. and the supporting 0.75 in. diameter rod AB is made of 304 stainless steel, determine the magnitude of P.

The A992 steel drill shaft of an oil well extends 12000 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 segment 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.

A spring-supported pipe hanger consists of two springs which are originally unstretched and have a stiffness of k = 60 kN/m , three 304 stainless steel rods, AB and CD, which have a diameter of 5 mm, and EF, which has a diameter of 12 mm, and a rigid beam GH. If the pipe and the fluid it carries have a total weight of 4 kN, determine the displacement of the pipe when it is attached to the support.

A spring-supported pipe hanger consists of two springs, which are originally unstretched and have a stiffness of k = 60 kN/m , three 304 stainless steel rods, AB and CD , which have a diameter of 5 mm, and EF , which has a diameter of 12 mm, and a rigid beam GH. If the pipe is displaced 82 mm when it is filled with fluid, determine the combined weight of the pipe and fluid.

The rod has a slight taper and length L. It is suspended from the ceiling and supports a load P at its end. Show that the displacement of its end due to this load is \(\delta=PL/(\pi Er_2 r_1)\). Neglect the weight of the material. The modulus of elasticity is E.

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 .

Determine the elongation of the A-36 steel member when it is subjected to an axial force of 30 kN. The member is 10 mm thick. Use the result of Prob. 4–24 .

Determine the elongation of the tapered A992 steel shaft when it is subjected to an axial force of 18 kip. Hint: Use the result of Prob. 4–23 .

The circular bar has a variable radius of \(r=r_0e^{ax}\) and is made of a material having a modulus of elasticity of E . Determine the displacement of end A when it is subjected to the axial force P.

Bone material has a stress–strain diagram that can be defined by the relation \(\sigma=E[\epsilon/(1+KE \epsilon)]\), where k and E are constants. Determine the compression within the length L of the bone, where it is assumed the cross-sectional area A of the bone is constant.

The weight of the kentledge exerts an axial force of P = 1500 kN on the 300-mm diameter high-strength concrete bore pile. If the distribution of the resisting skin friction developed from the interaction between the soil and the surface of the pile is approximated as shown, and the resisting bearing force F is required to be zero, determine the maximum intensity \(p_0\) kN/m for equilibrium. Also, find the corresponding elastic shortening of the pile. Neglect the weight of the pile.

The weight of the kentledge exerts an axial force of P = 1500 kN on the 300-mm diameter high-strength concrete bore pile. If the distribution of the resisting skin friction developed from the interaction between the soil and the surface of the pile is approximated as shown, determine the resisting bearing force F for equilibrium. Take \(p_0 = 180~ \mathrm{kN/m}\). Also, find the corresponding elastic shortening of the pile. Neglect the weight of the pile.

The concrete column is reinforced using four steel reinforcing rods, each having a diameter of 18 mm. Determine the stress in the concrete and the steel if the column is subjected to an axial load of 800 kN. \(E_\text{st} = 200 \text{ GPa}\), \(E_\text{c} = 25 \text{ GPa}\).

The column is constructed from high-strength concrete and four A-36 steel reinforcing rods. If it is subjected to an axial force of 800 kN, determine the required diameter of each rod so that one-fourth of the load is carried by the steel and three-fourths by the concrete. \(E_\text{st} = 200 \text{ GPa}\), \(E_\text{c} = 25 \text{ GPa}\).

The steel pipe is filled with concrete and subjected to a compressive force of 80 kN. Determine the average normal stress in the concrete and the steel due to this loading. The pipe has an outer diameter of 80 mm and an inner diameter of 70 mm. \(E_\text{st} = 200 \text{ GPa}\), \(E_\text{c} = 24 \text{ GPa}\).

If column AB is made from high strength pre-cast 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 pre-cast concrete and reinforced with four \(\frac{3}{4}\) in. diameter A-36 steel rods, determine the maximum allowable floor loadings P . The allowable normal stress for the high strength concrete and the steel are \(\left(\sigma_\text{allow}\right)_\text{con}=2.5 \text{ ksi}\) and \(\left(\sigma_\text{allow}\right)_\text{st}=24 \text{ ksi}\), 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 2800 lb is to be supported by the two essentially vertical A-36 steel wires. If originally wire AB is 60 in. long and wire AC is 40 in. long, determine the force developed in each wire after the load is suspended. Each wire has a cross-sectional area of \(0.02~ \mathrm{in^2}\).

The load of 2800 lb is to be supported by the two essentially vertical A-36 steel wires. If originally wire AB is 60 in. long and wire AC is 40 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 \(0.02~ \mathrm{in^2}\).

The rigid member is held in the position shown by three A-36 steel tie rods. Each rod has an unstretched length of 0.75 m and a cross-sectional area of \(125~ \mathrm{mm^2}\). Determine the forces in the rods if a turnbuckle on rod EF undergoes one full turn. The lead of the screw is 1.5 mm. Neglect the size of the turnbuckle and assume that it is rigid. Note: The lead would cause the rod, when unloaded, to shorten 1.5 mm when the turnbuckle is rotated one revolution.

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 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 rods AB, CD and EF.

The assembly consists of two red brass C83400 copper rods AB and CD having a diameter of 30 mm, a 304 stainless steel rod EF having a diameter of 40 mm, and a rigid member G. If the supports at A, C and F each have a stiffness of k = 200 MN/m, determine the average normal stress developed in the rods when the load is applied.

The bolt has a diameter of 20 mm and passes through a tube that has an inner diameter of 50 mm and an outer diameter of 60 mm. If the bolt and tube are made of A-36 steel, determine the normal stress in the tube and bolt when a force of 40 kN is applied to the bolt. Assume the end caps are rigid.

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 A-36 steel.

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 \(A_m\) and modulus of \(E_m\), determine the stress in the matrix and each fiber when the force P is imposed on the specimen.

The tapered member is fixed connected at its ends A and B and is subjected to a load P = 7 kip at x = 30 in. Determine the reactions at the supports. The material is 2 in. thick and is made from 2014-T6 aluminum.

The tapered member is fixed connected at its ends A and B and is subjected to a load P. Determine the greatest possible magnitude for P without exceeding an average normal stress of \(\sigma_\text{allow} = 4 \text{ ksi}\) anywhere in the member, and determine the location x at which P would need to be applied. The member is 2 in. thick.

The rigid bar supports the uniform distributed load of 6 kip/ft. Determine the force in each cable if each cable has a cross-sectional area of \(0.05~ \mathrm{in^2}\), and \(E=31(10^3) \text{ ksi}\).

The rigid bar is originally horizontal and is supported by two cables each having a cross-sectional area of \(0.05~ \mathrm{in^2}\), and \(E=31(10^3) \text{ ksi}\). Determine the slight rotation of the bar when the uniform load is supplied.

Each of the three A-36 steel wires has the same diameter. Determine the force in each wire needed to support the 200-kg load.

The 200-kg load is suspended from three A-36 steel wires each having a diameter of 4 mm. If wire BD has a length of 800.25 mm before the load is applied, determine the average normal stress developed in each wire.

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 rigid bar supports the 800-lb load. Determine the normal stress in each A-36 steel cable if each cable has a cross-sectional area of \(0.04~ \mathrm{in^2}\).

The rigid bar is originally horizontal and is supported by two A-36 steel cables each having a cross- sectional area of \(0.04~ \mathrm{in^2}\). Determine the rotation of the bar when the 800-lb load is applied.

Two identical rods AB and CD each have a length L and diameter d, and are used to support the rigid beam, which is pinned at F. If a vertical force P is applied at the end of the beam, determine the normal stress developed in each rod. The rods are made of material that has a modulus of elasticity of E.

Two identical rods AB and CD each have a length L and diameter d, and are used to support the rigid beam, which is pinned at F. If a vertical force P is applied at the end of the beam, determine the angle of rotation of the beam. The rods are made of material that has a modulus of elasticity of E.

The assembly consists of two posts AD and CF made of A-36 steel and having a cross-sectional area of \(1000~ \mathrm{mm^2}\), and a 2014-T6 aluminum post BE having a cross-sectional area of \(1500~ \mathrm{mm^2}\) If a central load of 400 kN is applied to the rigid cap, determine the normal stress in each post. There is a small gap of 0.1 mm between the post BE and the rigid member ABC.

The three suspender bars are made of the same material and have equal cross-sectional areas A. Determine the average normal stress in each bar if the rigid beam ACE is subjected to the force P.

If the 2-in. diameter supporting rods are made from A992 steel, determine the average normal stress developed in each rod when P = 100 kip.

If the supporting rods of equal diameter are made from A992 steel, determine the required diameter to the nearest \(\frac{1}{8}\) in. of each rod when P = 100 kip. The allowable normal stress of the steel is \(\sigma_\text{allow} = 24 \text{ ksi}\).

The center post B of the assembly has an original length of 124.7 mm, whereas posts A and C have a length of 125 mm. If the caps on the top and bottom can be considered rigid, determine the average normal stress in each post. The posts are made of aluminum and have a cross-sectional area of \(400~ \mathrm{mm^2}\). \(E_\text{al}=70 \text{ GPa}\).

Initially the A-36 bolt shank fits snugly against the rigid caps E and F on the 6061-T6 aluminum sleeve. If the thread of the bolt shank has a lead of 1 mm, and the nut is tightened \(\frac{3}{4}\) of a turn, determine the average normal stress developed in the bolt shank and the sleeve. The diameter of bolt shank is d = 60 mm.

Initially the A-36 bolt shank fits snugly against the rigid caps E and F on the 6061-T6 aluminum sleeve. If the thread of the bolt shank has a lead of 1 mm, and the nut is tightened \(\frac{3}{4}\) of a turn, determine the required diameter d of the shank and the force developed in the shank and sleeve so that the normal stress developed in the shank is four times that of the sleeve.

The assembly consists of a 6061-T6-aluminum member and a C83400-red-brass member that rest on the rigid plates. Determine the distance d where the vertical load P should be placed on the plates so that the plates remain horizontal when the materials deform. Each member has a width of 8 in. and they are not bonded together.

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 makeup 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 \(T = 160^\circ \mathrm F\).

If the assembly fits snugly between two rigid supports A and C when the temperature is at \(T_1\), determine the normal stress developed in both rod segments when the temperature rises to \(T_2\). Both segments are made of the same material, having a modulus of elasticity of E and coefficient of thermal expansion of \(\alpha\).

If the assembly fits snugly between the two supports A and C when the temperature is at \(T_1\), determine the normal stress developed in both segments when the temperature rises to \(T_2\). Both segments are made of the same material having a modulus of elasticity of E and coefficient of the thermal expansion of \(\alpha\). The flexible supports at A and C each have a stiffness k.

The pipe is made of A992 steel and is connected to the collars at A and B. When the temperature is \(60^\circ \mathrm F\), there is no axial load in the pipe. If hot gas traveling through the pipe causes its temperature to rise by \(\Delta T=(40+15x)^\circ \mathrm F\), 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 \(T = 60^\circ \mathrm F\).

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 \(\delta\) so that the rails just touch one another when the temperature is increased from \(T_1 = -20^\circ \mathrm F\) to \(T_2 = 90^\circ \mathrm F\). Using this gap, what would be the axial force in the rails if the temperature were to rise 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 \(\alpha\). The temperature of the bar changes uniformly along its length from \(T_A\) at A to \(T_B\) at B so that at any point x along the bar \(T = T_A + x(T_B - T_A )/L\). 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 \(T = (\frac{5}{3} x^2 ? 20 x + 120)^\circ \mathrm C\), 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!F, determine the force in each wire needed to support the load. Take \(E_\text{st} = 29(10^3)\text{ ksi}\), \(E_\text{cu} = 17(10^3)\text{ ksi}\), \(\alpha_\text{st} = 8(10^{-6})/^\circ \mathrm F\), \(\alpha_\text{cu} = 9.60(10^{-6})/^\circ \mathrm F\). Each wire has a cross-sectional area of \(0.0123~ \mathrm{in^2}\).

The center rod CD of the assembly is heated from \(T_1=30^\circ \mathrm C\) to \(T_2=180^\circ \mathrm C\) using electrical resistance heating. 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 \(375~ \mathrm{mm^2}\). \(E_\text{st}=200 \text{ GPa}\), \(E_\text{al}=70 \text{ GPa}\), and \(\alpha_\text{al}=23(10^{-6})/^\circ \mathrm C\).

The center rod CD of the assembly is heated from \(T_1 = 30^\circ \mathrm C\) to \(T_2 = 180^\circ \mathrm C\) using electrical resistance heating. Also, the two end rods AB and EF are heated from T1 = 30°C to T2 = 50°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 \(375~ \mathrm{mm^2}\). \(E_\text{st}=200 \text{ GPa}\), \(E_\text{al}=70 \text{ GPa}\), \(\alpha_\text{st}=12(10^{-6})/^\circ \mathrm C\) and \(\alpha_\text{al}=23(10^{-6})/^\circ \mathrm C\).

The metal strap has a thickness t and width w and is subjected to a temperature gradient \(T_1\) to \(T_2 ( T_1 < T_2 )\). This causes the modulus of elasticity for the material to vary linearly from \(E_1\) at the top to a smaller amount \(E_2\) at the bottom. As a result, for any vertical position y, \(E = [( E_2 - E_1 )/w ] y + E_1\). 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 \(\sigma_\text{allow}=120 \text{ MPa}\), 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 \(\sigma_\text{allow}=150 \text{ MPa}\).

Determine the maximum axial force P that can be applied to the steel plate. The allowable stress is \(\sigma_\text{allow}=21 \text{ ksi}\).

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 \(\sigma_\text{allow}=21 \text{ ksi}\).

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

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

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 for this geometry?

The 10-mm-diameter shank of the steel bolt has a bronze sleeve bonded to it. The outer diameter of this sleeve is 20 mm. If the yield stress for the steel is \((\sigma_Y)_\text{st}= 640 \text{ MPa}\), and for the bronze \((\sigma_Y)_\text{br}= 520 \text{ MPa}\), determine the largest possible value of P that can be applied to the bolt. Assume the materials to be elastic perfectly plastic. \(E_\text{st} = 200 \text{ GPa}\), \(E_\text{br} = 100 \text{ GPa}\).

The 10-mm-diameter shank of the steel bolt has a bronze sleeve bonded to it. The outer diameter of this sleeve is 20 mm. If the yield stress for the steel is \((\sigma_Y)_\text{st}= 640 \text{ MPa}\), and for the bronze \((\sigma_Y)_\text{br}= 520 \text{ MPa}\), determine the magnitude of the largest elastic load P that can be applied to the assembly. \(E_\text{st} = 200 \text{ GPa}\), \(E_\text{br} = 100 \text{ GPa}\).

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 \((\sigma_Y)_\text{st}= 120 \text{ MPa}\) and \((\sigma_Y)_\text{al}= 70 \text{ MPa}\), determine the force in each wire if the weight is (a) 600 N and (b) 720 N. \(E_\text{al} = 70 \text{ GPa}\), \(E_\text{st} = 200 \text{ GPa}\).

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

The rigid beam is supported by a pin at A and two steel wires, each having a diameter of 4 mm. If the yield stress for the wires is \(\sigma_Y=530 \text{ MPa}\), and \(E_\text{st}=200 \text{ GPa}\), determine the intensity of the distributed load w that can be placed on the beam and will just cause wire EB to yield. What is the displacement of point G for this case? For the calculation, assume that the steel is elastic perfectly plastic.

The rigid beam is supported by a pin at A and two steel wires, each having a diameter of 4 mm. If the yield stress for the wires is \(\sigma_Y=530 \text{ MPa}\), and \(E_\text{st}=200 \text{ GPa}\), determine (a) the intensity of the distributed load w that can be placed on the beam that will cause only one of the wires to start to yield and (b) the smallest intensity of the distributed load that will cause both wires to yield. For the calculation, assume that the steel is elastic perfectly plastic.

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.

Two steel wires, each having a cross-sectional area of \(2~ \mathrm{mm^2}\) are tied to a ring at C, and then stretched and tied between the two pins A and B. The initial tension in the wires is 50 N. If a horizontal force P is applied to the ring, determine the force in each wire if P = 20 N. What is the smallest force P that must be applied to the ring to reduce the force in wire CB to zero? Take \(\sigma_Y=300 \text{ MPa}\). \(E_\text{st}=200 \text{ GPa}\).

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 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.

A material has a stress–strain diagram that can be described by the curve \(\sigma = c \epsilon^{1/2}\). Determine the deflection of the end of a rod made from this material if it has a length L, cross-sectional area A, and a specific weight \(\gamma\).

Solve Prob. 4–106 if the stress–strain diagram is defined by \(\sigma = c \epsilon^{3/2}\).

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 \(\sigma_Y=20 \text{ MPa}\). Post B has a diameter of 20 mm and is made of a material for which \(E^\prime = 100 \text{ GPa}\) and \(\sigma_Y^\prime = 590 \text{ MPa}\). 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 \(\sigma_Y=20 \text{ MPa}\). Post B is made of a material for which \(E^\prime = 100 \text{ GPa}\) and \(\sigma_Y^\prime = 590 \text{ MPa}\). If P = 130 kN, determine the diameter of post B so that all three posts are about to yield. (Do not assume that the three posts have equal uncompressed lengths.)

The wire BC has a diameter of 0.125 in. and the material has the stress–strain characteristics shown in the figure. Determine the vertical displacement of the handle at D if the pull at the grip is slowly increased and reaches a magnitude of (a) P = 450 lb, (b) P = 600 lb. Assume the bar is rigid.

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. 4–111 when both the load P and the supports are removed.

The three bars are pinned together and subjected to the load P. If each bar has a cross-sectional area A, length L, and is made from an elastic perfectly plastic material, for which the yield stress is \(\sigma_Y\), determine the largest load (ultimate load) that can be supported by the bars, i.e., the load P that causes all the bars to yield. Also, what is the horizontal displacement of point A when the load reaches its ultimate value? The modulus of elasticity is E.

The assembly consists of two A992 steel bolts AB and EF and an 6061-T6 aluminum rod CD. When the temperature is at \(30^\circ \text{ C}\), the gap between the rod and rigid member AE is 0.1 mm. Determine the normal stress developed in the bolts and the rod if the temperature rises to \(130^\circ \text{ C}\). Assume BF is also rigid.

The assembly shown consists of two A992 steel bolts AB and EF and an 6061-T6 aluminum rod CD . When the temperature is at \(30^\circ \text{ C}\), the gap between the rod and rigid member AE is 0.1 mm. Determine the highest temperature to which the assembly can be raised without causing yielding either in the rod or the bolts. Assume BF is also rigid.

The rods each have the same 25-mm diameter and 600-mm length. If they are made of A992 steel, determine the forces developed in each rod when the temperature increases by \(50^\circ \text{ C}\).

Two A992 steel pipes, each having a cross-sectional area of \(0.32~\mathrm{in}^2\), are screwed together using a union at B as shown. Originally the assembly is adjusted so that no load is on the pipe. If the union is then tightened so that its screw, having a lead of 0.15 in., undergoes two full turns, determine the average normal stress developed in the pipe. Assume that the union at B and couplings at A and C are rigid. Neglect the size of the union. Note: The lead would cause the pipe, when unloaded, to shorten 0.15 in. when the union is rotated one revolution.

The force P is applied to the bar, which is composed of an elastic perfectly plastic material. Construct a graph to show how the force in each section AB and BC (ordinate) varies as P (abscissa) is increased. The bar has cross-sectional areas of \(1~\mathrm{in}^2\) in region AB and \(4~\mathrm{in}^2\) in region BC , and \(\sigma_Y=30 \text{ ksi}\).

The 2014-T6 aluminum rod has a diameter of 0.5 in and is lightly attached to the rigid supports at A and B when \(T_1 = 70^\circ \text{F}\). If the temperature becomes \(T_2 = -10^\circ \text{F}\), and an axial force of P = 16 lb is applied to the rigid collar as shown, determine the reactions at A and B.

The 2014-T6 aluminum rod has a diameter of 0.5 in. and is lightly attached to the rigid supports at A and B when \(T_1 = 70^\circ \text{F}\). Determine the force P that must be applied to the collar so that, when \(T= 0^\circ \text{F}\), the reaction at B is zero.

The rigid link is supported by a pin at A and two A-36 steel wires, each having an unstretched length of 12 in. and cross-sectional area of \(0.0125~\mathrm{in}^2\). Determine the force developed in the wires when the link supports the vertical load of 350 lb.

The joint is made from three A992 steel plates that are bonded together at their seams. Determine the displacement of end A with respect to end B when the joint is subjected to the axial loads shown. Each plate has a thickness of 5 mm.