Mechanics of Materials

Russell C. Hibbeler

Chapter 4

Axial Load - all with Video Answers

Educators

Chapter Questions

Problem 1

The copper shaft is subjected to the axial loads shown. Determine the displacement of end $A$ with respect to end $D$. The diameters of each segment are $d_{A B}=3 \mathrm{in}$., $d_{B C}=2 \mathrm{in}$., and $d_{C D}=1 \mathrm{in}$. Take $E_{\mathrm{cu}}=18\left(10^3\right) \mathrm{ksi}$.

Chai Santi

Chai Santi

Numerade Educator

Problem 2

The assembly consists of an A-36 steel rod CB and an a 2014-T6 aluminum rod $B A$, each having a diameter of 20 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 $A$. Neglect the size of the connections at $B$ and $C$.

Paul A.

California State Polytechnic University, Pomona

Problem 3

The A-36 steel rod is subjected to the loading shown. If the cross-sectional area of the rod is $50 \mathrm{~mm}^2$, determine the displacement of its end $D$. Neglect the size of the couplings at $B, C$, and $D$.

Rashmi Sinha

Numerade Educator

Problem 4

The A-36 steel rod is subjected to the loading shown. If the cross-sectional area of the rod is $50 \mathrm{~mm}^2$, determine the displacement of $C$. Neglect the size of the couplings at $B, C$, and $D$.

Rashmi Sinha

Numerade Educator

Problem 5

The A992 steel rod is subjected to the loading shown. If the diameter of the rod is 50 mm , determine the displacement of $B$ and $A$. Neglect the size of the couplings at $B, C$, and $D$.

Rashmi Sinha

Numerade Educator

Problem 6

The A-36 steel drill shaft of an oil well extends $12,000 \mathrm{ft}$ 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 assembly consists of two 12 -mm-diameter A992 steel rods $A B$ and $C D$, a 20 -mm-diameter $6061-\mathrm{T} 6$ aluminum $\operatorname{rod} E F$, and a rigid bar $A E C$. If $P=20 \mathrm{kN}$, determine the displacement of $F$.

Chai Santi

Numerade Educator

Problem 8

The assembly consists of two 12 -mm-diameter A992 steel rods $A B$ and $C D$, a $20-\mathrm{mm}$-diameter $6061-\mathrm{T} 6$ aluminum $\operatorname{rod} E F$, and a rigid bar $A E C$. If the horizontal displacement of $F$ is 0.024 mm , determine the magnitude of $P$.

Chai Santi

Numerade Educator

Problem 9

The linkage is made of two pin-connected A-36 steel members, each having a cross-sectional area of $1.5 \mathrm{in}^2$. If a vertical force of $P=50 \mathrm{kip}$ is applied to joint $A$, determine the displacement at $A$.

Naman Kumar

Numerade Educator

Problem 10

The linkage is made of two pin-connected A-36 steel members, each having a cross-sectional area of $1.5 \mathrm{in}^2$. Determine the magnitude of the force $\mathbf{P}$ needed to displace point A 0.025 in . downward.

Naman Kumar

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 $500-\mathrm{lb}$ load if the members are originally horizontal when the load is applied. Each wire has a cross-sectional area of $0.025 \mathrm{in}^2$.

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 $500-\mathrm{lb}$ load is applied. The members are originally horizontal, and each wire has a cross-sectional area of $0.025 \mathrm{in}^2$.

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 $500 \mathrm{~mm}^2$ and is made of A-36 steel. Determine the vertical displacement of the bar at $B$ when the load is applied.

Chai Santi

Numerade Educator

Problem 14

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 $w=12 \mathrm{kN} / \mathrm{m}$ at $y=2 \mathrm{~m}$, 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.

Naman Kumar

Numerade Educator

Problem 15

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 $w=4 \mathrm{kN} / \mathrm{m}$ at $y=0$ to $w=12 \mathrm{kN} / \mathrm{m}$ at $y=2 \mathrm{~m}$, 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.

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 $d=10 \mathrm{in}$. 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 $k=12 \mathrm{kip} / \mathrm{in}$.

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 $E$.
$\psi^P$

Rashmi Sinha

Numerade Educator

Problem 18

The load of 800 lb 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 load if the members were horizontal before the load was applied. Each wire has a cross-sectional area of $0.05 \mathrm{in}^2$.

Naman Kumar

Numerade Educator

Problem 19

The load of 800 lb 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 load is applied. The members were originally horizontal, and each wire has a cross-sectional area of $0.05 \mathrm{in}^2$.

Chai Santi

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 shown. 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 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 6 kip is applied to the ring $F$, determine the horizontal displacement of point $F$.

Naman Kumar

Numerade Educator

Problem 22

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 6 kip is applied to the ring $F$, determine the angle of tilt of bar $A C$.

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 kN , determine the change in its length and its new cross-sectional dimensions at section $a-a . E_{\mathrm{st}}=200 \mathrm{GPa}, v_{\mathrm{st}}=0.29$.

Naman Kumar

Numerade Educator

Problem 24

The bar has a length $L$ and cross-sectional area $A$. Determine its elongation due to the force $\mathbf{P}$ and its own weight. The material has a specific weight $\gamma$ ( weight/volume) and a modulus of elasticity $E$.

Narayan Hari

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 $0.75 \mathrm{in}^2$. Determine the greatest load $P$ that can be applied so that the roller support at $B$ is not displaced more than 0.03 in .

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 rod has a slight taper and length $L$. It is suspended from the ceiling and supports a load $\mathbf{P}$ at its end. Show that the displacement of its end due to this load is $\delta=P L /\left(\pi E r_2 r_1\right)$. Neglect the weight of the material. The modulus of elasticity is $E$.

Chai Santi

Numerade Educator

Problem 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 $P$.

Chai Santi

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 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 .

Naman Kumar

Numerade Educator

Problem 32

. 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.

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 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 .

Naman Kumar

Numerade Educator

Problem 34

The 304 stainless steel post $A$ has a diameter of $d=2 \mathrm{in}$. and is surrounded by a red brass C83400 tube B. If a force of 5 kip is applied to the rigid cap, determine the average normal stress developed in the post and the tube.

Chai Santi

Numerade Educator

Problem 35

The 304 stainless steel post $A$ is surrounded by a red brass C83400 tube B. If a force of 5 kip is applied to the rigid cap, determine the required diameter $d$ of the steel post so that the load is shared equally between the post and tube.

Chai Santi

Numerade Educator

Problem 36

. The composite bar consists of a 20 -mm-diameter A-36 steel segment $A B$ and 50 -mm-diameter red brass C83400 end segments DA and CB. Determine the average normal stress in each segment due to the applied load.

Chai Santi

Numerade Educator

Problem 37

. The composite bar consists of a 20 -mm-diameter A-36 steel segment $A B$ and 50 -mm-diameter red brass C83400 end segments DA and CB. Determine the displacement of $A$ with respect to $B$ due to the applied load.

Chai Santi

Numerade Educator

Problem 38

The A-36 steel column, having a cross-sectional area of $18 \mathrm{in}^2$, is encased in high-strength concrete as shown. If an axial force of 60 kip is applied to the column, determine the average compressive stress in the concrete and in the steel. How far does the column shorten? It has an original length of 8 ft .

Naman Kumar

Numerade Educator

Problem 39

. The A-36 steel column is encased in high-strength concrete as shown. If an axial force of 60 kip is applied to the column, determine the required area of the steel so that the force is shared equally between the steel and concrete. How far does the column shorten? It has an original length of 8 ft .

Satpal Satpal

Numerade Educator

Problem 40

. The two pipes are made of the same material and are connected as shown. If the cross-sectional area of $B C$ is $A$ and that of $C D$ is $2 A$, determine the reactions at $B$ and $D$ when a force $\mathbf{P}$ is applied at the junction $C$.

Narayan Hari

Numerade Educator

Problem 41

. 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. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$.

Naman Kumar

Numerade Educator

Problem 42

Two A-36 steel wires are used to support the $650-\mathrm{lb}$ engine. Originally, $A B$ is 32 in . long and $A^{\prime} B^{\prime}$ is 32.008 in . long. Determine the force supported by each wire when the engine is suspended from them. Each wire has a cross-sectional area of $0.01 \mathrm{in}^2$.

Chai Santi

Numerade Educator

Problem 43

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 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 $\left(\sigma_{\text {allow }}\right)_{s t}=200 \mathrm{MPa}$ for the steel and $\left(\sigma_{\text {allow }}\right)_{\mathrm{al}}=150 \mathrm{MPa}$ for the aluminum. $E_{\mathrm{st}}=200 \mathrm{GPa}$, $E_{\text {al }}=70 \mathrm{GPa}$.

Naman Kumar

Numerade Educator

Problem 45

The bolt $A B$ 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 plates 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 plates.

Naman Kumar

Numerade Educator

Problem 46

. 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 $P=200 \mathrm{kN}$ 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 C83400 post surrounded by a 304 stainless steel tube. Before the load is applied the gap between these two parts is 1 mm . 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 rigid member is held in the position shown by three A-36 steel tie rods. Each rod has a diameter of 20 mm . Determine the forces in the rods if a turnbuckle on rods $A B$ and $C D$ undergoes one and a half revolution. The lead of the screw is 0.9 mm . Neglect the size of the turnbuckle and assume that it is rigid. Note: The lead would cause the rod, when unloaded, to shorten 0.9 mm when the turnbuckle is rotated one revolution.

Chai Santi

Numerade Educator

Problem 49

Two identical rods $A B$ and $C D$ each have a length $L$ and diameter $d$, and are used to support the rigid beam, which is pinned at $E$. If a vertical force $\mathbf{P}$ is applied to the beam, determine the normal stress developed in each rod. The rods are made of material that has a modulus of elasticity of $E$.

Chai Santi

Numerade Educator

Problem 50

Two identical rods $A B$ and $C D$ each have a length $L$ and diameter $d$, and are used to support the rigid beam, which is pinned at $E$. If a vertical force $\mathbf{P}$ is applied to the beam, determine the angle of rotation of the beam. The rods are made of material that has a modulus of elasticity of $E$.

Chai Santi

Numerade Educator

Problem 51

The press consists of two rigid heads that are held together by the two A-36 steel $\frac{1}{2}$-in.-diameter rods. A $6061-\mathrm{T} 6$ solid aluminum cylinder is placed in the press and the screw is adjusted so that it just presses up against the cylinder. If it is then tightened one-half turn, determine the average normal stress in the rods and in the cylinder. The single-threaded screw on the bolt has a lead of 0.01 in . Note: The lead represents the distance the screw advances along its axis for one complete turn of the screw.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 52

The press consists of two rigid heads that are held together by the two A-36 steel $\frac{1}{2}$-in.-diameter rods. A $6061-\mathrm{T} 6$ solid aluminum cylinder is placed in the press and the screw is adjusted so that it just presses up against the cylinder. Determine the angle through which the screw can be turned before the rods or the specimen begin to yield. The single-threaded screw on the bolt has a lead of 0.01 in . Note: The lead represents the distance the screw advances along its axis for one complete turn of the screw.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 53

The assembly consists of two 2014-T6 aluminum rods $C D$ and $E F$ having a diameter of 20 mm , an A992 steel $\operatorname{rod} A B$ having a diameter of 30 mm , and a rigid member $G$. If the supports at $A, D$, and $F$ are rigid, determine the average normal stress developed in rods $A B, C D$, and $E F$.

Chai Santi

Numerade Educator

Problem 54

. The assembly consists of two 2014-T6 aluminum rods $C D$ and $E F$ having a diameter of 20 mm , an A992 steel $\operatorname{rod} A B$ having a diameter of 30 mm , and a rigid member $G$. If the supports at $A, D$, and $F$ each have a stiffness of $k=300 \mathrm{MN} / \mathrm{m}$, determine the average normal stress developed in the rods when the load is applied.

Chai Santi

Numerade Educator

Problem 55

Rod $A B$ has a diameter $d$ and fits snugly between the rigid supports at $A$ and $B$ when it is unloaded. The modulus of elasticity is $E$. Determine the support reactions at $A$ and $B$ if the rod is subjected to the linearly distributed axial load.

Chai Santi

Numerade Educator

Problem 56

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 $150-\mathrm{kg}$ 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 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 $150-\mathrm{kg}$ mass is suspended from the ring at $A$.

Naman Kumar

Numerade Educator

Problem 58

If the $1.5-\mathrm{in}$.-diameter supporting rods are made from 2014-T6 aluminum, determine the average normal stress developed in each rod when $P=80$ kip.

Chai Santi

Numerade Educator

Problem 59

If the supporting rods of equal diameter are made from 2014-T6 aluminum, determine the required diameter of each rod to the nearest $\frac{1}{16} \mathrm{in}$. when $P=80 \mathrm{kip}$. The allowable normal stress of the steel is $\sigma_{\text {allow }}=40 \mathrm{ksi}$.

Chai Santi

Numerade Educator

Problem 60

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 $400 \mathrm{~mm}^2 \cdot E_{\mathrm{al}}=70 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 61

The distributed loading is supported by the three suspender bars. $A B$ and $E F$ are made of aluminum and $C D$ is made of steel. If each bar has a cross-sectional area of $450 \mathrm{~mm}^2$, determine the maximum intensity $w$ of the distributed loading so that an allowable stress of $\left(\sigma_{\text {allow }}\right)_{\mathrm{s}}=180 \mathrm{MPa}$ in the steel and $\left(\sigma_{\text {allow }}\right)_{\mathrm{al}}=94 \mathrm{MPa}$ in the aluminum is not exceeded. $E_{\mathrm{st}}=200 \mathrm{GPa}$, $E_{\mathrm{al}}=70 \mathrm{GPa}$. Assume $A C E$ is rigid.

Naman Kumar

Numerade Educator

Problem 62

The horizontal beam is assumed to be rigid and supports the distributed load shown. Determine the vertical reactions at the supports. Each support consists of a wooden post having a diameter of 120 mm and an unloaded (original) length of 1.40 m . Take $E_{\mathrm{w}}=12 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 63

The horizontal beam is assumed to be rigid and supports the distributed load shown. Determine the angle of tilt of the beam after the load is applied. Each support consists of a wooden post having a diameter of 120 mm and an unloaded (original) length of 1.40 m . Take $E_{\mathrm{w}}=12 \mathrm{GPa}$.

Rashmi Sinha

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 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 C83400 red-brass rod $A B$ and 2014-T6 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 $1.75 \mathrm{in}^2$.

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

A steel surveyor's tape is to be used to measure the length of a line. The tape has a rectangular cross section of 0.05 in . by 0.2 in . and a length of 100 ft when $T_1=60^{\circ} \mathrm{F}$ and the tension or pull on the tape is 20 lb . Determine the true length of the line if the tape shows the reading to be 463.25 ft when used with a pull of 35 lb at $T_2=90^{\circ} \mathrm{F}$. The ground on which it is placed is flat. $\alpha_{\mathrm{st}}=9.60\left(10^{-6}\right) /{ }^{\circ} \mathrm{F}$, $E_{\mathrm{st}}=29\left(10^3\right) \mathrm{ksi}$.

Ma Ednelyn Lim

Numerade Educator

Problem 71

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

Chai Santi

Numerade Educator

Problem 72

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

Chai Santi

Numerade Educator

Problem 73

A 6-ft-long steam pipe is made of A-36 steel with $\sigma_Y=40 \mathrm{ksi}$. It is connected directly to two turbines $A$ and $B$ as shown. The pipe has an outer diameter of 4 in . and a wall thickness of 0.25 in . The connection was made when $T_1=70^{\circ} \mathrm{F}$. If the turbines' points of attachment are assumed rigid, determine the force the pipe exerts on the turbines when the steam and thus the pipe reach a temperature of $T_2=275^{\circ} \mathrm{F}$.

Chai Santi

Numerade Educator

Problem 74

A 6 -ft-long steam pipe is made of A-36 steel with $\sigma_Y=40 \mathrm{ksi}$. It is connected directly to two turbines $A$ and $B$ as shown. The pipe has an outer diameter of 4 in . and a wall thickness of 0.25 in . The connection was made when $T_1=70^{\circ} \mathrm{F}$. If the turbines' points of attachment are assumed to have a stiffness of $k=80\left(10^3\right) \mathrm{kip} / \mathrm{in}$., determine the force the pipe exerts on the turbines when the steam and thus the pipe reach a temperature of $T_2=275^{\circ} \mathrm{F}$.

Chai Santi

Numerade Educator

Problem 75

The three bars are made of A-36 steel and form a pin-connected truss. If the truss is constructed when $T_1=50^{\circ} \mathrm{F}$, determine the force in each bar when $T_2=110^{\circ} \mathrm{F}$. Each bar has a cross-sectional area of $2 \mathrm{in}^2$.

Victor Salazar

Numerade Educator

Problem 76

The three bars are made of A-36 steel and form a pin-connected truss. If the truss is constructed when $T_1=50^{\circ} \mathrm{F}$, determine the vertical displacement of joint $A$ when $T_2=150^{\circ} \mathrm{F}$. Each bar has a cross-sectional area of $2 \mathrm{in}^2$.

Victor Salazar

Numerade Educator

Problem 77

Three bars, each made of different materials, are connected together and placed between two walls when the temperature is $T_1=12^{\circ} \mathrm{C}$. Determine the force exerted on the (rigid) supports when the temperature becomes $T_2=18^{\circ} \mathrm{C}$. The material properties and cross-sectional area of each bar are given in the figure.

Chai Santi

Numerade Educator

Problem 78

When the temperature is $30^{\circ} \mathrm{C}$, 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 each tank provides a rigid support at $A$ and $B$.

Chai Santi

Numerade Educator

Problem 79

The A-36 steel rod has a diameter of 50 mm and is lightly attached to the rigid supports at $A$ and $B$ when $T_1=80^{\circ} \mathrm{C}$. If the temperature becomes $T_2=20^{\circ} \mathrm{C}$ and an axial force of $P=200 \mathrm{kN}$ is applied to its center, determine the reactions at $A$ and $B$.

Chai Santi

Numerade Educator

Problem 80

The A-36 steel rod has a diameter of 50 mm and is lightly attached to the rigid supports at $A$ and $B$ when $T_1=50^{\circ} \mathrm{C}$. Determine the force $P$ that must be applied to the collar at its midpoint so that, when $T_2=30^{\circ} \mathrm{C}$, the reaction at $B$ is zero.

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 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 $50-\mathrm{mm}$-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 .

Naman Kumar

Numerade Educator

Problem 83

The rigid block has a weight of 80 kip and is to be supported by posts $A$ and $B$, which are made of A-36 steel, and the post $C$, which is made of C83400 red brass. If all the posts have the same original length before they are loaded, determine the average normal stress developed in each post when post $C$ is heated so that its temperature is increased by $20^{\circ} \mathrm{F}$. Each post has a cross-sectional area of $8 \mathrm{in}^2$.

Chai Santi

Numerade Educator

Problem 84

The pipe is made of A-36 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 . and the wall thickness is 0.15 in .

Chai Santi

Numerade Educator

Problem 85

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 the pipe exerts on the walls. The pipe was loosely fitted between the walls when $T=60^{\circ} \mathrm{F}$.

Chai Santi

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

Determine the maximum axial force $P$ that can be applied to the bar so as not to exceed an allowable tensile stress of $\sigma_{\text {allow }}=200 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 90

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.

Naman Kumar

Numerade Educator

Problem 91

Determine the maximum axial force $P$ that can be applied to the bar. The allowable stress is $\sigma_{\text {allow }}=21 \mathrm{ksi}$.

Chai Santi

Numerade Educator

Problem 92

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

Chai Santi

Numerade Educator

Problem 93

The member is to be made from a steel plate that is 0.25 in . thick. If a $1-\mathrm{in}$. 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

Determine the maximum axial force $P$ that can be applied to the plate. The allowable stress is $\sigma_{\text {allow }}=36 \mathrm{ksi}$.

Chai Santi

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 for this geometry?

Chai Santi

Numerade Educator

Problem 96

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

Naman Kumar

Numerade Educator

Problem 97

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 $50 \mathbf{N}$. If a horizontal force $\mathbf{P}$ is applied to the ring, determine the force in each wire if $P=20 \mathrm{~N}$. What is the smallest force $P$ that must be applied to the ring to reduce the force in wire $C B$ to zero? Take $\sigma_Y=300 \mathrm{MPa} . E_{\mathrm{st}}=200 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 98

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. 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 crosssectional 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 $w=25 \mathrm{kip} / \mathrm{ft}$ 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 $0.75 \mathrm{in}^2$ 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 .

Naman Kumar

Numerade Educator

Problem 101

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

Chai Santi

Numerade Educator

Problem 103

The wire $B C$ has a diameter of 0.125 in . and the material has the stress-strain diagram shown. Determine the vertical displacement of the handle at $D$ if the pull at the grip is slowly increased and reaches a magnitude of (a) $P=450 \mathrm{lb}$, (b) $P=600 \mathrm{lb}$. Assume the bar is rigid.

Chai Santi

Numerade Educator

Problem 104

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 $\left(\sigma_Y\right)_{\mathrm{st}}=640 \mathrm{MPa}$, and for the bronze $\left(\sigma_Y\right)_{\mathrm{br}}=520 \mathrm{MPa}$, determine the largest possible value of $P$ that can be applied to the bolt.Assume the materials to be elastic perfectly plastic. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 105

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 $\left(\sigma_Y\right)_{\mathrm{st}}=640 \mathrm{MPa}$, 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. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 106

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 $\sigma_Y=530 \mathrm{MPa}$, and $E_{\mathrm{st}}=200 \mathrm{GPa}$, 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? Assume that the steel is elastic perfectly plastic.

Chai Santi

Numerade Educator

Problem 107

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 $\sigma_Y=530 \mathrm{MPa}$, and $E_{\mathrm{st}}=200 \mathrm{GPa}$, 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. Assume that the steel is elastic perfectly plastic.

Chai Santi

Numerade Educator

Problem 108

The 2-in-diameter bar is fixed connected at its ends and supports the axial load $\mathbf{P}$. If the material is elastic perfectly plastic as shown by the stress-strain 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 $C$.

Chai Santi

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