Mechanics of Materials

R. C. Hibbeler

Chapter 4

Axial Load - all with Video Answers

Educators

Chapter Questions

Problem 1

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 $C$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

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

Chai Santi

Numerade Educator

Problem 3

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 $C$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

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 $C$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

The 2014 - $\mathrm{T} 6$ aluminium rod has a diameter of $30 \mathrm{mm}$ and supports the load shown. Determine the displacement of end $A$ with respect to end $E .$ Neglect the size of the couplings.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

The A-36 steel drill shaft of an oil well extends $12000 \mathrm{ft}$ 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, $A B$ $B C,$ and $C D,$ each having the length, weight per unit length, and cross-sectional area indicated.

Rashmi Sinha

Numerade Educator

Problem 7

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

Naman Kumar

Numerade Educator

Problem 8

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 $0.2 \mathrm{mm}$

Naman Kumar

Numerade Educator

Problem 9

The assembly consists of two 10 -mm diameter red brass $\mathrm{C} 83400$ copper rods $A B$ and $C D,$ a $15-\mathrm{mm}$ diameter 304 stainless steel rod $E F,$ and a rigid bar $G .$ If $P=5 \mathrm{kN}$ determine the horizontal displacement of end $F$ of rod $E F$

Naman Kumar

Numerade Educator

Problem 10

The assembly consists of two 10 -mm diameter red brass $\mathrm{C} 83400$ copper rods $A B$ and $C D,$ a $15-\mathrm{mm}$ diameter 304 stainless steel rod $E F$ and a rigid bar $G$. If the horizontal displacement of end $F$ of rod $E F$ is $0.45 \mathrm{mm}$, determine the magnitude of $P$

Chai Santi

Numerade Educator

Problem 11

The load is supported by the four 304 stainless steel wires that are connected to the rigid members $A B$ and $D C$ 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 in $^{2}$.

Naman Kumar

Numerade Educator

Problem 12

The load is supported by the four 304 stainless steel wires that are connected to the rigid members $A B$ and $D C$ Determine the angle of tilt of each member after the $500-1 b$ load is applied. The members were originally horizontal, and each wire has a cross-sectional area of 0.025 in $^{2}$

Chai Santi

Numerade Educator

Problem 13

The rigid bar is supported by the pin-connected rod $C B$ 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.

Naman Kumar

Numerade Educator

Problem 14

The post is made of Douglas fir and has a diameter of $100 \mathrm{mm} .$ If it is subjected to the load of $20 \mathrm{kN}$ 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 $w=12 \mathrm{kN} / \mathrm{m}$ at $y=2 \mathrm{m},$ determine the force $\mathbf{F}$ 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.

Naman Kumar

Numerade Educator

Problem 15

The post is made of Douglas fir and has a diameter of $100 \mathrm{mm} .$ If it is subjected to the load of $20 \mathrm{kN}$ and the soil provides a frictional resistance that is distributed along its length and varies linearly from $w=4 \mathrm{kN} / \mathrm{m}$ at $y=0$ to $w=12 \mathrm{kN} / \mathrm{m}$ at $y=2 \mathrm{m},$ determine the force $\mathbf{F}$ 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.

Naman Kumar

Numerade Educator

Problem 16

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.

Naman Kumar

Numerade Educator

Problem 17

The pipe is stuck in the ground so that when it is pulled upward the frictional force along its length varies linearly from zero at $B$ to $f_{\max }$ (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 $E$

Rashmi Sinha

Numerade Educator

Problem 18

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

Naman Kumar

Numerade Educator

Problem 19

The linkage is made of three pin-connected $\mathrm{A} 992$ steel members, each having a diameter of $1 \frac{1}{4}$ in. Determine the magnitude of the force $\mathbf{P}$ needed to displace point $B$ 0.25 in. to the right.

Naman Kumar

Numerade Educator

Problem 20

The assembly consists of three titanium (Ti-6A1-4V) rods and a rigid bar $A C .$ 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 $F$

Naman Kumar

Numerade Educator

Problem 21

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 $w=3 \mathrm{kip} / \mathrm{ft},$ and $x=4 \mathrm{ft},$ determine the smallest diameter of each rod so that the beam remains
in the horizontal position when it is loaded.

Naman Kumar

Numerade Educator

Problem 22

The rigid beam is supported at its ends by two A-36 steel tie rods. The rods have diameters $d_{A B}=0.5$ in. and $d_{C D}=0.3$ 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.

Naman Kumar

Numerade Educator

Problem 23

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

Naman Kumar

Numerade Educator

Problem 24

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 $P$

Chai Santi

Numerade Educator

Problem 25

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 $A B$, determine the displacement at $C, B,$ and $E$

Naman Kumar

Numerade Educator

Problem 26

The truss consists of three members, each made from $A-36$ steel and having a cross-sectional area of 0.75 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.

Naman Kumar

Numerade Educator

Problem 27

Solve Prob. $4-26$ when the load $\mathbf{P}$ acts vertically downward at $C$

Naman Kumar

Numerade Educator

Problem 28

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 $3 \mathrm{ft}$ 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 $y$

Naman Kumar

Numerade Educator

Problem 29

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

Naman Kumar

Numerade Educator

Problem 30

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

Rashmi Sinha

Numerade Educator

Problem 31

The column is constructed from high-strength concrete and eight $\mathrm{A} 992$ 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.

Naman Kumar

Numerade Educator

Problem 32

The column is constructed from high-strength concrete and eight $\mathrm{A} 992$ 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.

Naman Kumar

Numerade Educator

Problem 33

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

Naman Kumar

Numerade Educator

Problem 34

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

Naman Kumar

Numerade Educator

Problem 35

If column $A B$ 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 $\mathbf{P}$ The allowable normal stresses for the concrete and the steel $\operatorname{are}\left(\sigma_{\text {allow }}\right)_{\text {con }}=2.5 \mathrm{ksi}$ and $\left(\sigma_{\text {allow }}\right)_{\mathrm{st}}=24 \mathrm{ksi},$ respectively.

Naman Kumar

Numerade Educator

Problem 36

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

Chai Santi

Numerade Educator

Problem 37

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 $E$

Chai Santi

Numerade Educator

Problem 38

The load of 2000 lb is to be supported by the two vertical steel wires for which $\sigma_{Y}=70$ ksi. Originally wire $A B$ is 60 in. long and wire $A C$ 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 in $^{2} . E_{\mathrm{st}}=29.0\left(10^{3}\right) \mathrm{ksi}$

Naman Kumar

Numerade Educator

Problem 39

The load of 2000 lb is to be supported by the two vertical steel wires for which $\sigma_{Y}=70$ ksi. Originally wire $A B$ is 60 in. long and wire $A C$ is 60.04 in. long. Determine the cross-sectional area of $A B$ if the load is to be shared equally between both wires. Wire $A C$ has a cross-sectional area of 0.02 in $^{2} . E_{\mathrm{st}}=29.0\left(10^{3}\right) \mathrm{ksi}$

Naman Kumar

Numerade Educator

Problem 40

The $A-36$ steel pipe has an outer radius of $20 \mathrm{mm}$ and an inner radius of $15 \mathrm{mm}$. 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.

Naman Kumar

Numerade Educator

Problem 41

The 10 -mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is $20 \mathrm{mm},$ and its inner diameter is $10 \mathrm{mm}$. 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. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$

Naman Kumar

Numerade Educator

Problem 42

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

Naman Kumar

Numerade Educator

Problem 43

The assembly consists of two red brass $\mathrm{C} 83400$ copper rods $A B$ and $C D$ of diameter $30 \mathrm{mm},$ a stainless 304 steel alloy rod $E F$ of diameter $40 \mathrm{mm}$, and a rigid cap $G$. If the supports at $A, C,$ and $F$ are rigid, determine the average normal stress developed in the rods.

Chai Santi

Numerade Educator

Problem 44

The rigid beam is supported by the three suspender bars. Bars $A B$ and $E F$ are made of aluminum and bar $C D$ is made of steel. If each bar has a cross-sectional area of $450 \mathrm{mm}^{2}$ determine the maximum value of $P$ if the allowable stress is $\left(\sigma_{\text {allow }}\right)_{\text {st }}=200$ MPa for the steel and $\left(\sigma_{\text {allow }}\right)_{\text {al }}=150$ MPa for the aluminum. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}$

Naman Kumar

Numerade Educator

Problem 45

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

Naman Kumar

Numerade Educator

Problem 46

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

Naman Kumar

Numerade Educator

Problem 47

The support consists of a solid red brass $\mathrm{C} 83400$ copper post surrounded by a 304 stainless steel tube. Before the load is applied the gap between these two parts is $1 \mathrm{mm}$ 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.

Chai Santi

Numerade Educator

Problem 48

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 in each fiber when the force $P$ is applied on the specimen.

Naman Kumar

Numerade Educator

Problem 49

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.

Naman Kumar

Numerade Educator

Problem 50

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.

Naman Kumar

Numerade Educator

Problem 51

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 $E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi} .$ If the bar is initially vertical, determine the displacement of the end $B$ when the force of 2 kip is applied.

Naman Kumar

Numerade Educator

Problem 52

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 $E_{\mathrm{al}}=10\left(10^{3}\right)$ ksi. If the bar is initially vertical, determine the force in each rod when the 2 -kip load is applied.

Naman Kumar

Numerade Educator

Problem 53

The 2014-T6 aluminum rod $A C$ is reinforced with the firmly bonded $\mathrm{A} 992$ steel tube $B C .$ 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 $400 \mathrm{kN}$ is applied. The assembly is attached at $D$

Naman Kumar

Numerade Educator

Problem 54

The 2014 - $\mathrm{T} 6$ aluminum rod $A C$ is reinforced with the firmly bonded $\mathrm{A} 992$ steel tube $B C .$ When no load is applied to the assembly, the gap between end $C$ and the rigid support is $0.5 \mathrm{mm} .$ Determine the support reactions when the axial force of $400 \mathrm{kN}$ is applied.

Naman Kumar

Numerade Educator

Problem 55

The three suspender bars are made of $\mathrm{A} 992$ 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.

Naman Kumar

Numerade Educator

Problem 56

The three A-36 steel wires each have a diameter of $2 \mathrm{mm}$ 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 $A$

Naman Kumar

Numerade Educator

Problem 57

The A-36 steel wires $A B$ and $A D$ each have a diameter of $2 \mathrm{mm}$ 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 $A C$ so that each wire is subjected to the same force when the 150 -kg mass is suspended from the ring at $A$

Naman Kumar

Numerade Educator

Problem 58

The post is made from 6061-T6 aluminum and has a diameter of $50 \mathrm{mm} .$ 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 $P=40 \mathrm{kN}$ is applied to the collar.

Naman Kumar

Numerade Educator

Problem 59

The post is made from 6061-T6 aluminum and has a diameter of $50 \mathrm{mm} .$ 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 $P=50 \mathrm{kN}$ is applied to the collar.

Rashmi Sinha

Numerade Educator

Problem 60

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.

Naman Kumar

Numerade Educator

Problem 61

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 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.

Naman Kumar

Numerade Educator

Problem 62

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 $1 \mathrm{m}$ and a cross-sectional area of $600 \mathrm{mm}^{2}$ and the spring has a stiffness of $k=2 \mathrm{MN} / \mathrm{m}$ and an unstretched length of $1.02 \mathrm{m}$, determine the force in each post after the load is applied to the bar.

Naman Kumar

Numerade Educator

Problem 63

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 $1 \mathrm{m}$ and a cross-sectional area of $600 \mathrm{mm}^{2}$ and the spring has a stiffness of $k=2 \mathrm{MN} / \mathrm{m}$ and an unstretched length of $1.02 \mathrm{m},$ determine the vertical displacement of $A$ and $B$ after the load is applied to the bar.

Naman Kumar

Numerade Educator

Problem 64

The assembly consists of two posts $A B$ and $C D$ 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 crosssectional area $A_{2} .$ If a load $\mathbf{P}$ is applied to the rigid cap, determine the force in each material.

Naman Kumar

Numerade Educator

Problem 65

The assembly consists of two posts $A B$ and $C D$ each made from material 1 having a modulus of elasticity of $E_{1}$ and a cross-sectional area $A_{1},$ and a central post $E F$ made from material 2 having a modulus of elasticity $E_{2}$ and a cross-sectional area $A_{2} .$ If posts $A B$ and $C D$ 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.

Naman Kumar

Numerade Educator

Problem 66

The assembly consists of two posts $A B$ and $C D$ each made from material 1 having a modulus of elasticity of $E_{1}$ and a cross-sectional area $A_{1},$ and a central post $E F$ made from material 2 having a modulus of elasticity $E_{2}$ and a cross-sectional area $A_{2} .$ If post $E F$ 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.

Naman Kumar

Numerade Educator

Problem 67

The wheel is subjected to a force of $18 \mathrm{kN}$ 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.

Naman Kumar

Numerade Educator

Problem 68

The $\quad$ C83400-red-brass rod $A B$ and $2014-\mathrm{T} 6-$ aluminum rod $B C$ 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 in $^{2}$.

Chai Santi

Numerade Educator

Problem 69

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 $T_{2}=110^{\circ} \mathrm{F}$

Naman Kumar

Numerade Educator

Problem 70

The rod is made of $\mathrm{A} 992$ 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 $T=160^{\circ} \mathrm{F}$

Chai Santi

Numerade Educator

Problem 71

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
\[
\begin{array}{l}
T_{2}=300^{\circ} \mathrm{F} . \text { Take } \alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi} \\
\left(\sigma_{Y}\right)_{\mathrm{al}}=40 \mathrm{ksi}, \alpha_{\mathrm{cu}}=9.4\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}, \text { and } \\
\left(\sigma_{Y}\right)_{\mathrm{cu}}=50 \mathrm{ksi}
\end{array}
\]

Naman Kumar

Numerade Educator

Problem 72

The two cylindrical rod segments are fixed to the rigid walls such that there is a gap of 0.01 in. between them when $T_{1}=60^{\circ} \mathrm{F}$. Each rod has a diameter of 1.25 in. Determine the average normal stress in each rod if $T_{2}=400^{\circ} \mathrm{F},$ and also calculate the new length of the aluminum segment. Take $\alpha_{\mathrm{al}}=13\left(10^{-6}\right) /^{\circ} \mathrm{F}, E_{\mathrm{al}}=10\left(10^{3}\right) \mathrm{ksi}$
\begin{array}{l}
\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}, \text { and } \\
E_{\mathrm{cu}}=15\left(10^{3}\right) \mathrm{ksi}
\end{array}

James Kiss

Numerade Educator

Problem 73

The pipe is made of $A 992$ 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 $\Delta T=(40+15 x)^{\circ} \mathrm{F}$, 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.

Chai Santi

Numerade Educator

Problem 74

The bronze 686100 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 $T=60^{\circ} \mathrm{F}$

Naman Kumar

Numerade Educator

Problem 75

The 40 -ft-long $\mathrm{A}-36$ steel rails on a train track are laid with a small gap betwocn 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 rises to $T_{3}=110^{\circ} \mathrm{F} ?$ The cross-sectional area of each rail is 5.10 in $^{2}$

Naman Kumar

Numerade Educator

Problem 76

The device is used to measure a change in temperature. Bars $A B$ and $C D$ are made of $A-36$ steel and $2014-$ T 6 aluminum alloy, respectively. When the temperature is at $75^{\circ} \mathrm{F}, A C E$ is in the horizontal position. Determine the vertical displacement of the pointer at $E$ when the temperature rises to $150^{\circ} \mathrm{F}$

Naman Kumar

Numerade Educator

Problem 77

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 $T=T_{A}+x\left(T_{B}-T_{A}\right) / L .$ 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 $T_{A}$

Chai Santi

Numerade Educator

Problem 78

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 $B$

Chai Santi

Numerade Educator

Problem 79

When the temperature is at $30^{\circ} \mathrm{C}$, the $\mathrm{A}-36$ 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 $k=900 \mathrm{MN} / \mathrm{m}$

Chai Santi

Numerade Educator

Problem 80

When the temperature is at $30^{\circ} \mathrm{C}$, the $\mathrm{A}-36$ 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 $B$

Chai Santi

Numerade Educator

Problem 81

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 $10 \mathrm{mm}$ and they hold the cylinder snug with negligible force against the rigid jaws, determine the force in the cylinder when the temperature rises to $T_{2}=130^{\circ} \mathrm{C}$

Naman Kumar

Numerade Educator

Problem 82

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 $10 \mathrm{mm}$ 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 $12 \mathrm{MPa}$

Naman Kumar

Numerade Educator

Problem 83

The wires $A B$ and $A C$ are made of steel, and wire $A D$ is made of copper. Before the 150 -lb force is applied, $A B$ and $A C$ are each 60 in. long and $A D$ 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 crosssectional area of 0.0123 in $^{2} .$ Take $E_{\mathrm{st}}=29\left(10^{3}\right) \mathrm{ksi}$
\[
E_{\mathrm{cu}}=17\left(10^{3}\right) \mathrm{ksi}, \alpha_{\mathrm{st}}=8\left(10^{-6}\right) /^{\circ} \mathrm{F}, \alpha_{\mathrm{cu}}=9.60\left(10^{-6}\right) /^{\circ} \mathrm{F}
\]

Naman Kumar

Numerade Educator

Problem 84

The cylinder $C D$ of the assembly is heated from $T_{1}=30^{\circ} \mathrm{C}$ 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 $0.7 \mathrm{mm} .$ Determine the force in rods $A B$ and $E F$ caused by the increase in temperature. Rods $A B$ and $E F$ are made of steel, and each has a cross-sectional area of $125 \mathrm{mm}^{2} . \mathrm{CD}$ is made of aluminum and has a cross-sectional area of $375 \mathrm{mm}^{2}$
\[
E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{al}}=70 \mathrm{GPa}, \text { and } \alpha_{\mathrm{al}}=23\left(10^{-6}\right) /^{\circ} \mathrm{C}
\]

Naman Kumar

Numerade Educator

Problem 85

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

Naman Kumar

Numerade Educator

Problem 86

The metal strap has a thickness $t$ and width $w$ and is subjected to a temperature gradient $T_{1}$ to $T_{2}\left(T_{1}<T_{2}\right)$. 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 $y,$ measured from the top surface, $E=\left[\left(E_{2}-E_{1}\right) / w\right] y+E_{1} .$ Determine the position $d$ where the axial force $P$ must be applied so that the bar stretches uniformly over its cross section.

Naman Kumar

Numerade Educator

Problem 87

Determine the maximum normal stress developed in the bar when it is subjected to a tension of $P=8 \mathrm{kN}$

Chai Santi

Numerade Educator

Problem 88

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.

Chai Santi

Numerade Educator

Problem 89

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 $\sigma_{\text {allow }}=150 \mathrm{MPa}$

Chai Santi

Numerade Educator

Problem 90

The $A-36$ steel plate has a thickness of $12 \mathrm{mm}$. 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.

Naman Kumar

Numerade Educator

Problem 91

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 $\sigma_{\text {allow }}=21 \mathrm{ksi}$

Chai Santi

Numerade Educator

Problem 92

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

Chai Santi

Numerade Educator

Problem 93

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 $\sigma_{\text {allow }}=22 \mathrm{ksi}$

Naman Kumar

Numerade Educator

Problem 94

The resulting stress distribution along section $A B$ 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?

Naman Kumar

Numerade Educator

Problem 95

The resulting stress distribution along section $A B$ 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?

Naman Kumar

Numerade Educator

Problem 96

The weight is suspended from steel and aluminum wires, each having the same initial length of $3 \mathrm{m}$ 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 $(a) 600 N$ and
(b) $720 \mathrm{N} . E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}$

Naman Kumar

Numerade Educator

Problem 97

The weight is suspended from steel and aluminum wires, each having the same initial length of $3 \mathrm{m}$ 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) $600 \mathrm{N}$ and
(b) $720 \mathrm{N} . E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}$

Naman Kumar

Numerade Educator

Problem 98

The bar has a cross-sectional area of 0.5 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.

Chai Santi

Numerade Educator

Problem 99

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 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.

Naman Kumar

Numerade Educator

Problem 100

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 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.

Naman Kumar

Numerade Educator

Problem 101

The rigid lever arm is supported by two A-36 steel wires having the same diameter of $4 \mathrm{mm} .$ If a force of $P=3 \mathrm{kN}$ 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

Chai Santi

Numerade Educator

Problem 102

The rigid lever arm is supported by two A-36 steel wires having the same diameter of $4 \mathrm{mm}$. Determine the smallest force $\mathbf{P}$ 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.

Chai Santi

Numerade Educator

Problem 103

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

Naman Kumar

Numerade Educator

Problem 104

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

Chai Santi

Numerade Educator

Problem 105

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

Chai Santi

Numerade Educator

Problem 106

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

Chai Santi

Numerade Educator

Problem 107

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

Naman Kumar

Numerade Educator

Problem 108

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

Naman Kumar

Numerade Educator

Problem 109

Determine the elongation of the bar in Prob. $4-108$ when both the load $\mathbf{P}$ and the supports are removed.

Naman Kumar

Numerade Educator

Problem 110

The rigid beam is supported by three $A-36$ steel wires, each having a length of 4 ft. The cross-sectional area of $A B$ and $E F$ is 0.015 in $^{2}$, and the cross-sectional area of $C D$ is 0.006 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.

Naman Kumar

Numerade Educator