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Mechanics of Materials 7th

Ferdinand P. Beer, E. Russell Johnston, Jr., John T. DeWolf

Chapter 1

Introduction—Concept of Stress

Educators


Problem 1

Two solid cylindrical rods $A B$ and $B C$ are welded together at $B$ and loaded as shown. Knowing that $d_{1}=30 \mathrm{mm}$ and $d_{2}=50 \mathrm{mm}$ find the average normal stress at the midsection of $(a) \operatorname{rod} A B$
$(b) \operatorname{rod} B C$.

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Problem 2

Two solid cylindrical rods $A B$ and $B C$ are welded together at $B$ and loaded as shown. Knowing that the average normal stress must not exceed 150 MPa in either rod, determine the smallest allowable values of the diameters $d_{1}$ and $d_{2}$.

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Problem 3

Two solid cylindrical rods $A B$ and $B C$ are welded together at $B$ and loaded as shown. Knowing that $P=10$ kips, find the average normal stress at the midsection of $(a) \operatorname{rod} A B,(b) \operatorname{rod} B C$.

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Problem 4

Two solid cylindrical rods $A B$ and $B C$ are welded together at $B$ and loaded as shown. Determine the magnitude of the force $\mathbf{P}$ for which the tensile stresses in rods $A B$ and $B C$ are equal.

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Problem 5

A strain gage located at $C$ on the surface of bone $A B$ indicates that the average normal stress in the bone is $3.80 \mathrm{MPa}$ when the bone is subjected to two 1200 -N forces as shown. Assuming the cross section of the bone at $C$ to be annular and knowing that its outer diameter is $25 \mathrm{mm},$ determine the inner diameter of the bone's cross section at $C$.

Rashmi S.
Numerade Educator

Problem 6

Two brass rods $A B$ and $B C$, each of uniform diameter, will be brazed together at $B$ to form a nonuniform rod of total length $100 \mathrm{m}$ that will be suspended from a support at $A$ as shown. Knowing that the density of brass is $8470 \mathrm{kg} / \mathrm{m}^{3},$ determine
(a) the length of rod $A B$ for which the maximum normal stress in $A B C$ is minimum,
(b) the corresponding value of the maximum normal stress.

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Problem 7

Each of the four vertical links has an $8 \times 36$ -mm uniform rectangular cross section, and each of the four pins has a 16 -mm diameter. Determine the maximum value of the average normal stress in the links connecting $(a)$ points $B$ and $D,(b)$ points $C$ and $E$.

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Problem 8

Link $A C$ has a uniform rectangular cross section $\frac{1}{8}$ in. thick and 1 in. wide. Determine the normal stress in the central portion of the link.

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Problem 9

Three forces, each of magnitude $P=4 \mathrm{kN},$ are applied to the structure shown. Determine the cross-sectional area of the uniform portion of rod $B E$ for which the normal stress in that portion is $+100 \mathrm{MPa}$.

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Problem 10

Link BD consists of a single bar 1 in. wide and $\frac{1}{2}$ in. thick. Knowing that each pin has a $\frac{3}{6}$ -in. diameter, determine the maximum value of the average normal stress in $\operatorname{link} B D$ if $(a) \theta=0,(b) \theta=90^{\circ}$.

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Problem 11

For the Pratt bridge truss and loading shown, determine the average normal stress in member $B E$, knowing that the cross-sectional area of that member is 5.87 in $^{2}$.

Rashmi S.
Numerade Educator

Problem 12

The frame shown consists of four wooden members, $A B C, D E F$ $B E,$ and $C F .$ Knowing that each member has a $2 \times 4$ -in. rectangular cross section and that each pin has a $\frac{1}{2}$ -in. diameter, determine the maximum value of the average normal stress
(a) in member $B E$
(b) in member $C F$

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Problem 13

An aircraft tow bar is positioned by means of a single hydraulic cylinder connected by a 25 -mm-diameter steel rod to two identical arm-and-wheel units $D E F$. The mass of the entire tow bar is $200 \mathrm{kg},$ and its center of gravity is located at $G$. For the position shown, determine the normal stress in the rod.

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Problem 14

Two hydraulic cylinders are used to control the position of the robotic arm $A B C$. Knowing that the control rods attached at $A$ and $D$ each have a 20 -mm diameter and happen to be parallel in the position shown, determine the average normal stress in
$(a)$ member $A E,(b)$ member $D G$.

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Problem 15

Determine the diameter of the largest circular hole that can be punched into a sheet of polystyrene $6 \mathrm{mm}$ thick, knowing that the force exerted by the punch is $45 \mathrm{kN}$ and that a $55-\mathrm{MPa}$ average shearing stress is required to cause the material to fail.

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Problem 16

Two wooden planks, each $\frac{1}{2}$ in. thick and 9 in. wide, are joined by the dry mortise joint shown. Knowing that the wood used shears off along its grain when the average shearing stress reaches 1.20 ksi, determine the magnitude $P$ of the axial load that will cause the joint to fail.

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Problem 17

When the force $\mathbf{P}$ reached $1600 \mathrm{Ib}$, the wooden specimen shown failed in shear along the surface indicated by the dashed line. Determine the average shearing stress along that surface at the time of failure.

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Problem 18

A load $\mathbf{P}$ is applied to a steel rod supported as shown by an aluminum plate into which a 12 -mm-diameter hole has been drilled. Knowing that the shearing stress must not exceed $180 \mathrm{MPa}$ in the steel rod and $70 \mathrm{MPa}$ in the aluminum plate, determine the largest load $\mathbf{P}$ that can be applied to the rod.

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Problem 19

The axial force in the column supporting the timber beam shown is $P=20$ kips. Determine the smallest allowable length $L$ of the bearing plate if the bearing stress in the timber is not to exceed 400 psi.

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Problem 20

Three wooden planks are fastened together by a series of bolts to form a column. The diameter of each bolt is $12 \mathrm{mm}$ and the inner diameter of each washer is $16 \mathrm{mm}$, which is slightly larger than the diameter of the holes in the planks. Determine the smallest allowable outer diameter $d$ of the washers, knowing that the average normal stress in the bolts is $36 \mathrm{MPa}$ and that the bearing stress between the washers and the planks must not exceed $8.5 \mathrm{MPa}$.

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Problem 21

A 40 -kN axial load is applied to a short wooden post that is $s u p$ ported by a concrete footing resting on undisturbed soil. Determine $(a)$ the maximum bearing stress on the concrete footing, (b) the size of the footing for which the average bearing stress in the soil is $145 \mathrm{kPa}$.

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Problem 22

An axial load $\mathbf{P}$ is supported by a short $\mathrm{W} 8 \times 40$ column of crosssectional area $A=11.7$ in and is distributed to a concrete foundation by a square plate as shown. Knowing that the average normal stress in the column must not exceed 30 ksi and that the bearing stress on the concrete foundation must not exceed $3.0 \mathrm{ksi}$ determine the side $a$ of the plate that will provide the most economical and safe design.

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Problem 23

Link $A B,$ of width $b=2$ in. and thickness $t=\frac{1}{4}$ in., is used to support the end of a horizontal beam. Knowing that the average normal stress in the link is -20 ksi and that the average shearing stress in each of the two pins is 12 ksi determine ( $a$ ) the diameter $d$ of the pins, (b) the average bearing stress in the link.

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Problem 24

Determine the largest load $\mathbf{P}$ that can be applied at $A$ when $\theta=60^{\circ},$ knowing that the average shearing stress in the 10 -mmdiameter pin at $B$ must not exceed 120 MPa and that the average bearing stress in member $A B$ and in the bracket at $B$ must not exceed $90 \mathrm{MPa}$.

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Problem 25

Knowing that $\theta=40^{\circ}$ and $P=9 \mathrm{kN}$, determine (a) the smallest allowable diameter of the pin at $B$ if the average shearing stress in the pin is not to exceed $120 \mathrm{MPa}$, ( $b$ ) the corresponding average bearing stress in member $A B$ at $B$, (c) the corresponding average bearing stress in each of the support brackets at $B$.

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Problem 26

The hydraulic cylinder $C F,$ which partially controls the position of rod $D E,$ has been locked in the position shown. Member $B D$ is $15 \mathrm{mm}$ thick and is connected at $C$ to the vertical rod by a 9-mm-diameter bolt. Knowing that $P=2 \mathrm{kN}$ and $\theta=75^{\circ},$ determine $(a)$ the average shearing stress in the bolt, $(b)$ the bearing stress at $C$ in member $B D$.

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Problem 27

For the assembly and loading of Prob. $1.7,$ determine ( $a$ ) the average shearing stress in the pin at $B,(b)$ the average bearing stress at $B$ in member $B D,(c)$ the average bearing stress at $B$ in member $A B C,$ knowing that this member has a $10 \times 50$ -mm uniform rectangular cross section.

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Problem 28

Two identical linkage-and-hydraulic-cylinder systems control the position of the forks of a fork-lift truck. The load supported by the one system shown is 1500 lb. Knowing that the thickness of member $B D$ is $\frac{5}{8}$ in., determine $(a)$ the average shearing stress in the $\frac{1}{2}$ -in.-diameter pin at $B$, (b) the bearing stress at $B$ in member $B D$.

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Problem 29

Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice shown. Knowing that $P=11 \mathrm{kN},$ determine the normal and shearing stresses in the glued splice.

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Problem 30

Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice shown. Knowing that the maximum allowable shearing stress in the glued splice is $620 \mathrm{kPa}$ determine $(a)$ the largest load $\mathbf{P}$ that can be safely applied,
(b) the corresponding tensile stress in the splice.

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Problem 31

The 1.4 -kip load $\mathbf{P}$ is supported by two wooden members of uniform cross section that are joined by the simple glued scarf splice shown. Determine the normal and shearing stresses in the glued splice.

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Problem 32

Two wooden members of uniform cross section are joined by the simple scarf splice shown. Knowing that the maximum allowable tensile stress in the glued splice is 75 psi, determine ( $a$ ) the largest load $\mathbf{P}$ that can be safely supported, (b) the corresponding shearing stress in the splice.

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Problem 33

A centric load $\mathbf{P}$ is applied to the granite block shown. Knowing that the resulting maximum value of the shearing stress in the block is 2.5 ksi, determine $(a)$ the magnitude of $\mathbf{P}$, ( $b$ ) the orientation of the surface on which the maximum shearing stress occurs, (c) the normal stress exerted on that surface, (d) the maximum value of the normal stress in the block.

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Problem 34

A 240 -kip load $\mathbf{P}$ is applied to the granite block shown. Determine the resulting maximum value of $(a)$ the normal stress,
(b) the shearing stress. Specify the orientation of the plane on which each of these maximum values occurs.

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Problem 35

A steel pipe of $400-\mathrm{mm}$ outer diameter is fabricated from $10-\mathrm{mm}$ thick plate by welding along a helix that forms an angle of $20^{\circ}$ with a plane perpendicular to the axis of the pipe. Knowing that a 300 -kN axial force $\mathbf{P}$ is applied to the pipe, determine the normal and shearing stresses in directions respectively normal and tangential to the weld.

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Problem 36

A steel pipe of $400-\mathrm{mm}$ outer diameter is fabricated from $10-\mathrm{mm}$ thick plate by welding along a helix that forms an angle of $20^{\circ}$ with a plane perpendicular to the axis of the pipe. Knowing that the maximum allowable normal and shearing stresses in the directions respectively normal and tangential to the weld are $\sigma=60 \mathrm{MPa}$ and $\tau=36 \mathrm{MPa}$, determine the magnitude $P$ of the largest axial force that can be applied to the pipe.

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Problem 37

A steel loop $A B C D$ of length 5 ft and of $\frac{3}{8}$ -in. diameter is placed as shown around a 1 -in.-diameter aluminum rod $A C$. Cables $B E$ and $D F$, each of $\frac{1}{2}$ -in. diameter, are used to apply the load $Q$. Knowing that the ultimate strength of the steel used for the loop and the cables is $70 \mathrm{ksi}$, and that the ultimate strength of the aluminum used for the rod is 38 ksi, determine the largest load $\mathbf{Q}$ that can be applied if an overall factor of safety of 3 is desired.

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Problem 38

Link $B C$ is $6 \mathrm{mm}$ thick, has a width $w=25 \mathrm{mm},$ and is made of a steel with a 480 -MPa ultimate strength in tension. What is the factor of safety used if the structure shown was designed to support a 16 -kN load $\mathbf{P} ?$

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Problem 39

Link $B C$ is $6 \mathrm{mm}$ thick and is made of a steel with a $450-\mathrm{MPa}$ ultimate strength in tension. What should be its width $w$ if the structure shown is being designed to support a 20 -kN load $\mathbf{P}$ with a factor of safety of $3 ?$

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Problem 40

Members $A B$ and $B C$ of the truss shown are made of the same alloy. It is known that a 20 -mm-square bar of the same alloy was tested to failure and that an ultimate load of $120 \mathrm{kN}$ was recorded. If a factor of safety of 3.2 is to be achieved for both bars, determine the required cross-sectional area of $(a)$ bar $A B$ $(b)$ bar $A C$.

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Problem 41

Members $A B$ and $B C$ of the truss shown are made of the same alloy. It is known that a 20 -mm-square bar of the same alloy was tested to failure and that an ultimate load of $120 \mathrm{kN}$ was recorded. If bar $A B$ has a cross-sectional area of $225 \mathrm{mm}^{2}$, determine (a) the factor of safety for bar $A B,(b)$ the cross-sectional area of bar $A C$ if it is to have the same factor of safety as bar $A B$.

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Problem 42

Link $A B$ is to be made of a steel for which the ultimate normal stress is 65 ksi. Determine the cross-sectional area of $A B$ for
which the factor of safety will be 3.20 . Assume that the link will be adequately reinforced around the pins at $A$ and $B$.

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Problem 43

Two wooden members are joined by plywood splice plates that are fully glued on the contact surfaces. Knowing that the clearance between the ends of the members is $6 \mathrm{mm}$ and that the ultimate shearing stress in the glued joint is $2.5 \mathrm{MPa}$, determine the length $L$ for which the factor of safety is 2.75 for the loading shown.

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Problem 44

For the joint and loading of Prob. $1.43,$ determine the factor of safety when $L=180 \mathrm{mm}$.

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Problem 45

Three $\frac{3}{4}$ -in.- diameter steel bolts are to be used to attach the steel plate shown to a wooden beam. Knowing that the plate will support a load $P=24$ kips and that the ultimate shearing stress for the steel used is 52 ksi, determine the factor of safety for this design.

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Problem 46

Three steel bolts are to be used to attach the steel plate shown to a wooden beam. Knowing that the plate will support a load $P=28$ kips, that the ultimate shearing stress for the steel used is $52 \mathrm{ksi},$ and that a factor of safety of 3.25 is desired, determine the required diameter of the bolts.

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Problem 47

A load $\mathbf{P}$ is supported as shown by a steel pin that has been inserted in a short wooden member hanging from the ceiling. The ultimate strength of the wood used is $60 \mathrm{MPa}$ in tension and $7.5 \mathrm{MPa}$ in shear, while the ultimate strength of the steel is $145 \mathrm{MPa}$ in shear. Knowing that $b=40 \mathrm{mm}, c=55 \mathrm{mm},$ and $d=12 \mathrm{mm},$ determine the load $\mathbf{P}$ if an overall factor of safety of 3.2 is desired.

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Problem 48

For the support of Prob. 1.47 , knowing that the diameter of the pin is $d=16 \mathrm{mm}$ and that the magnitude of the load is $P=20 \mathrm{kN},$ determine $(a)$ the factor of safety for the pin (b) the required values of $b$ and $c$ if the factor of safety for the wooden member is the same as that found in part $a$ for the pin.

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Problem 49

A steel plate $\frac{1}{4}$ in. thick is embedded in a concrete wall to anchor a high-strength cable as shown. The diameter of the hole in the plate is $\frac{3}{4}$ in., the ultimate strength of the steel used is 36 ksi. and the ultimate bonding stress between plate and concrete is 300 psi. Knowing that a factor of safety of 3.60 is desired when $P=2.5$ kips, determine (a) the required width $a$ of the plate,
(b) the minimum depth $b$ to which a plate of that width should be embedded in the concrete slab. (Neglect the normal stresses between the concrete and the end of the plate.)

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Problem 50

Determine the factor of safety for the cable anchor in Prob. 1.49 when $P=2.5$ kips, knowing that $a=2$ in. and $b=6$ in.

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Problem 51

Link $A C$ is made of a steel with a 65 -ksi ultimate normal stress and has a $\frac{1}{4} \times \frac{1}{2}$ -in. uniform rectangular cross section. It is connected to a support at $A$ and to member $B C D$ at $C$ by $\frac{3}{4}$ -in. -diameter pins, while member $B C D$ is connected to its support at $B$ by a $\frac{5}{16}$ -in.- diameter pin. All of the pins are made of a steel with a 25 -ksi ultimate shearing stress and are in single shear. Knowing that a factor of safety of 3.25 is desired, determine the largest load $\mathbf{P}$ that can be applied at $D$. Note that link $A C$ is not reinforced around the pin holes.

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Problem 52

Solve Prob. 1.51 , assuming that the structure has been redesigned to use $\frac{5}{16}$ -in.-diameter pins at $A$ and $C$ as well as at $B$ and that no other changes have been made.

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Problem 53

Each of the two vertical links $C F$ connecting the two horizontal members $A D$ and $E G$ has a $10 \times 40$ -mm uniform rectangular cross section and is made of a steel with an ultimate strength in tension of $400 \mathrm{MPa}$, while each of the pins at $C$ and $F$ has a 20 -mm diameter and are made of a steel with an ultimate strength in shear of $150 \mathrm{MPa}$. Determine the overall factor of safety for the links $C F$ and the pins connecting them to the horizontal members.

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Problem 54

Solve Prob. $1.53,$ assuming that the pins at $C$ and $F$ have been replaced by pins with a $30-$ mm diameter.

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Problem 55

In the structure shown, an 8 -mm-diameter pin is used at $A$, and 12-mm-diameter pins are used at $B$ and $D$. Knowing that the ultimate shearing stress is $100 \mathrm{MPa}$ at all connections and that the ultimate normal stress is $250 \mathrm{MPa}$ in each of the two links joining $B$ and $D,$ determine the allowable load $P$ if an overall factor of safety of 3.0 is desired.

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Problem 56

In an alternative design for the structure of Prob. $1.55,$ a pin of 10-mm-diameter is to be used at $A$. Assuming that all other specifications remain unchanged, determine the allowable load $\mathbf{P}$ if an overall factor of safety of 3.0 is desired.

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Problem 57

A 40 -kg platform is attached to the end $B$ of a 50 -kg wooden beam $A B,$ which is supported as shown by a pin at $A$ and by a slender steel rod $B C$ with a 12 -kN ultimate load. (a) Using the Load and Resistance Factor Design method with a resistance factor $\phi=0.90$ and load factors $\gamma_{D}=1.25$ and $\gamma_{L}=1.6,$ determine the largest load that can be safely placed on the platform.
(b) What is the corresponding conventional factor of safety for rod $B C ?$

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Problem 58

The Load and Resistance Factor Design method is to be used to select the two cables that will raise and lower a platform supporting two window washers. The platform weighs 160 lb and each of the window washers is assumed to weigh 195 lb with equipment. since these workers are free to move on the platform, $75 \%$ of their total weight and the weight of their equipment will be used as the design live load of each cable.
(a) Assuming a resistance factor $\phi=0.85$ and load factors $\gamma_{D}=1.2$ and $\gamma_{L}=1.5$ determine the required minimum ultimate load of one cable.
(b) What is the corresponding conventional factor of safety for the selected cables?

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Problem 59

In the marine crane shown, link $C D$ is known to have a uniform cross section of $50 \times 150 \mathrm{mm}$. For the loading shown, determine the normal stress in the central portion of that link.

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Problem 60

Two horizontal 5-kip forces are applied to pin $B$ of the assembly shown. Knowing that a pin of 0.8 -in. diameter is used at each connection, determine the maximum value of the average normal stress $(a)$ in $\operatorname{link} A B,(b)$ in $\operatorname{link} B C$.

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Problem 61

For the assembly and loading of Prob. $1.60,$ determine (a) the average shearing stress in the pin at $C$, (b) the average bearing stress at $C$ in member $B C$, (c) the average bearing stress at $B$ in member $B C$.

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Problem 62

Two steel plates are to be held together by means of $16-\mathrm{mm}$ diameter high-strength steel bolts fitting snugly inside cylindrical brass spacers. Knowing that the average normal stress must not exceed $200 \mathrm{MPa}$ in the bolts and $130 \mathrm{MPa}$ in the spacers, determine the outer diameter of the spacers that yields the most economical and safe design.

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Problem 63

A couple $\mathrm{M}$ of magnitude $1500 \mathrm{N} \cdot \mathrm{m}$ is applied to the crank of an engine. For the position shown, determine (a) the force $\mathbf{P}$ required to hold the engine system in equilibrium, (b) the average normal stress in the connecting rod $B C$, which has a $450-\mathrm{mm}^{2}$ uniform cross section.

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Problem 64

Knowing that link $D E$ is $\frac{1}{8}$ in. thick and 1 in. wide, determine the normal stress in the central portion of that link when
$(a) \theta=0,(b) \theta=90^{\circ}$.

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Problem 65

A $\frac{5}{8}$ -in. -diameter steel rod $A B$ is fitted to a round hole near end $C$ of the wooden member $C D$. For the loading shown, determine $(a)$ the maximum average normal stress in the wood, $(b)$ the distance $b$ for which the average shearing stress is 100 psi on the surfaces indicated by the dashed lines, (c) the average bearing stress on the wood.

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Problem 66

In the steel structure shown, a 6 -mm-diameter pin is used at $C$ and 10 -mm-diameter pins are used at $B$ and $D$. The ultimate shearing stress is $150 \mathrm{MPa}$ at all connections, and the ultimate normal stress is 400 MPa in link $B D$. Knowing that a factor of safety of 3.0 is desired, determine the largest load $\mathbf{P}$ that can be applied at $A$. Note that $\operatorname{link} B D$ is not reinforced around the pin holes.

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Problem 67

Member $A B C$, which is supported by a pin and bracket at $C$ and a cable $B D$, was designed to support the 16 -kN load $P$ as shown. Knowing that the ultimate load for cable $B D$ is $100 \mathrm{kN}$, determine the factor of safety with respect to cable failure.

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Problem 68

A force $\mathbf{P}$ is applied as shown to a steel reinforcing bar that has been embedded in a block of concrete. Determine the smallest length $L$ for which the full allowable normal stress in the bar can be developed. Express the result in terms of the diameter $d$ of the bar, the allowable normal stress $\sigma_{\mathrm{all}}$ in the steel, and the average allowable bond stress $\tau_{\text {all }}$. between the concrete and the cylindrical surface of the bar. (Neglect the normal stresses between the concrete and the end of the bar.)

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Problem 69

The two portions of member $A B$ are glued together along a plane forming an angle $\theta$ with the horizontal. Knowing that the ultimate stress for the glued joint is $2.5 \mathrm{ksi}$ in tension and $1.3 \mathrm{ksi}$ in shear, determine (a) the value of $\theta$ for which the factor of safety of the member is maximum, ( $b$ ) the corresponding value of the factor of safety. (Hint: Equate the expressions obtained for the factors of safety with respect to the normal and shearing stresses.)

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Problem 70

The two portions of member $A B$ are glued together along a plane forming an angle $\theta$ with the horizontal. Knowing that the ultimate stress for the glued joint is $2.5 \mathrm{ksi}$ in tension and $1.3 \mathrm{ksi}$ in shear, determine the range of values of $\theta$ for which the factor of safety of the members is at least 3.0.

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