Mechanics of Materials

R. C. Hibbeler

Chapter 11

Design of Beams and Shafts - all with Video Answers

Educators

Chapter Questions

Problem 1

The beam is made of timber that has an allowable bending stress of $\sigma_{\text {alloe }}=6.5 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=500 \mathrm{kPa}$. Determine its dimensions if it is to be rectangular and have a height-to-width ratio of 1.25 Assume the beam rests on smooth supports.

Hast Aggarwal

Hast Aggarwal

Numerade Educator

Problem 3

Solve Prob. $11-2$ if $P=10$ kip.

Chai Santi

Numerade Educator

Problem 4

The brick wall exerts a uniform distributed load of $1.20 \mathrm{kip} / \mathrm{ft}$ on the beam. If the allowable bending stress is $\sigma_{\text {allow }}=22 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=12 \mathrm{ksi}$ select the lightest wide-flange section with the shortest depth from Appendix B that will safely support the load. If there are several choices of equal weight, choose the one with the shortest height.

Hast Aggarwal

Numerade Educator

Problem 5

Select the lightest-weight wide-flange beam from Appendix B that will safely support the machine loading shown. The allowable bending stress is $\sigma_{\text {allow }}=24 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=14 \mathrm{ksi}$,/

Hast Aggarwal

Numerade Educator

Problem 6

The spreader beam $A B$ is used to slowly lift the $3000-$ lb pipe that is centrally located on the straps at $C$ and $D .$ If the beam is a $\mathrm{W} 12 \times 45,$ determine if it can safely support the load. The allowable bending stress is $\sigma_{\text {allow }}=22 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=12 \mathrm{ksi}$.

Chai Santi

Numerade Educator

Problem 7

Select the lightest-weight wide-flange beam with the shortest depth from Appendix B that will safely support the loading shown. The allowable bending stress is $\sigma_{\text {allow }}=24 \mathrm{ksi}$ and the allowable shear stress of $\tau_{\text {allow }}=14 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator

Problem 8

Select the lightest-weight wide-flange beam from Appendix B that will safely support the loading shown. The allowable bending stress $\sigma_{\text {allow }}=24$ ksi and the allowable shear stress of $\tau_{\text {allow }}=14 \mathrm{ksi}$

Hast Aggarwal

Numerade Educator

Problem 9

Select the lightest W360 wide-flange beam from Appendix B that can safely support the loading. The beam has an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=80 \mathrm{MPa}$. Assume there is a pin at $A$ and a roller support at $B$

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Numerade Educator

Problem 10

Investigate if the $\mathrm{W} 250 \times 58$ beam can safely support the loading. The beam has an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=80 \mathrm{MPa}$. Assume there is a pin at $A$ and a roller support at $B$

Hast Aggarwal

Numerade Educator

Problem 11

The beam is constructed from two boards, If each nail can support a shear force of $200 \mathrm{lb}$, determine the maximum spacing of the nails, $s, s^{\prime},$ and $s^{\prime \prime},$ to the nearest $\frac{1}{8}$ inch for regions $A B, B C,$ and $C D,$ respectively.

Hast Aggarwal

Numerade Educator

Problem 12

Determine the minimum width of the beam to the nearest $\frac{1}{4}$ in. that will safely support the loading of $P=8$ kip. The allowable bending stress is $\sigma_{\text {allow }}=24 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=15 \mathrm{ksi}$

Chai Santi

Numerade Educator

Problem 12

The joists of a floor in a warehouse are to be selected using square timber beams made of oak. If each beam is to be designed to carry $90 \mathrm{lb} / \mathrm{ft}$ over a simply supported span of $25 \mathrm{ft}$, determine the dimension $a$ of its square cross section to the nearest $\frac{1}{4}$ in. The allowable bending stress is $\sigma_{\text {allow }}=4.5 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {ples }}=125$ psi.

Hast Aggarwal

Numerade Educator

Problem 13

The timber beam has a width of 6 in. Determine its height $h$ so that it simultaneously reaches its allowable bending stress $\sigma_{\text {allow }}=1.50 \mathrm{ksi}$ and an allowable shear stress of $\tau_{\text {allow }}=50$ psi. Also, what is the maximum load $P$ that the beam can then support?

Hast Aggarwal

Numerade Educator

Problem 14

The beam is constructed from four boards. If each nail can support a shear force of 300 Ib, determine the maximum spacing of the nails, $s, s^{\prime}$ and $s^{\prime \prime},$ for regions $A B$ $B C,$ and $C D,$ respectively.

Hast Aggarwal

Numerade Educator

Problem 15

The beam is constructed from two boards. If each nail can support a shear force of 200 lb, determine the maximum spacing of the nails, $s, s^{\prime},$ and $s^{\prime \prime},$ to the nearest $\frac{1}{8}$ in. for regions $A B, B C,$ and $C D,$ respectively.

Hast Aggarwal

Numerade Educator

Problem 16

If the cable is subjected to a maximum force of $P=50 \mathrm{kN},$ select the lightest $\mathrm{W} 310$ wide-flange beam that can safely support the load. The beam has an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=85 \mathrm{MPa}$

Hast Aggarwal

Numerade Educator

Problem 17

If the $\mathrm{W} 360 \times 45$ wide-flange beam has an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=85 \mathrm{MPa}$, determine the maximum cable force P that can safely be supported by the beam.

Hast Aggarwal

Numerade Educator

Problem 18

If $P=800$ Ib, determine the minimum dimension $a$ of the beam's cross section to the nearest $\frac{1}{8}$ in. to safely support the load. The wood has an allowable normal stress of $\sigma_{\text {allow }}=1.5 \mathrm{ksi}$ and an allowable shear stress of $\tau_{\text {allow }}=$ $150 \mathrm{psi}$

Hast Aggarwal

Numerade Educator

Problem 19

If $a=3$ in and the wood has an allowable normal stress of $\sigma_{\text {-2low }}=1.5 \mathrm{ksi}$, and an allowable shear stress of $\tau_{\text {allow }}=150$ psi, determine the maximum allowable value of $P$ that can act on the beam.

Hast Aggarwal

Numerade Educator

Problem 20

The beam is constructed from three plastic strips. If the glue can support a shear stress of $\tau_{\text {allow }}=8 \mathrm{kPa}$ determine the largest magnitude of the loads $\mathbf{P}$ that the beam can support

Hast Aggarwal

Numerade Educator

Problem 21

If the allowable bending stress is $\sigma_{\text {allow }}=6 \mathrm{MPa}$, and the glue can support a shear stress of $\tau_{\text {allow }}=8 \mathrm{kPa}$ determine the largest magnitude of the loads $\mathbf{P}$ that can be applied to the beam.

Hast Aggarwal

Numerade Educator

Problem 22

allowable bending stress of $\sigma_{\text {allow }}=1.1 \mathrm{ksi}$ and an allowable shear stress of $\tau_{\text {allow }}=0.70$ ksi. Determine the width $b$ if the height $h=2 b$

Hast Aggarwal

Numerade Educator

Problem 23

Select the lightest-weight wide-flange beam from Appendix B that will safely support the loading The allowable bending stress is $\sigma_{\text {allow }}=24 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=14 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator

Problem 24

Draw the shear and moment diagrams for the the shear and moment diggans shaft, and determine its required diameter to the nearest $\frac{1}{8}$ in. if $\sigma_{\text {allow }}=30 \mathrm{ksi}$ and $\tau_{\text {allow }}=15 \mathrm{ksi}$. The journal bearings at $A$ and $C$ exert only vertical reactions on the shaft. Take $P=6 \mathrm{kip}$

Hast Aggarwal

Numerade Educator

Problem 24

Draw the shear and moment diagrams for the shaft, and determine its required diameter to the nearest $\frac{1}{8}$ in. if $\sigma_{\text {allow }}=30 \mathrm{ksi}$ and $\tau_{\text {allow }}=15 \mathrm{ksi}$. The journal bearings at $A$ and $C$ exert only vertical reactions on the shaft. Take $P=6 \mathrm{kip}$.

Hast Aggarwal

Numerade Educator

Problem 25

Draw the shear and moment diagrams for the shaft. and determine its required diameter to the nearest $\frac{1}{4}$ in. if $\sigma_{\text {allow }}=30 \mathrm{ksi}$ and $\tau_{\text {allow }}=15 \mathrm{ksi}$. The journal bearings at $A$ and $C$ exert only vertical reactions on the shaft. Take $P=12 \mathrm{kip}$.

Hast Aggarwal

Numerade Educator

Problem 26

Select the lightest-weight wide-flange beam from Appendix B that will safely support the loading. The allowable bending stress is $\sigma_{\text {allow }}=22 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=12$ ksi.

Hast Aggarwal

Numerade Educator

Problem 27

$\mathrm{W} 14 \times 22$ wide-flange beam and check if the beam will safely support the loading. The allowable bending stress is $\sigma_{\text {allow }}=30 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=15 \mathrm{ksi}$

Hast Aggarwal

Numerade Educator

Problem 28

$=11-28 .$ Select the lightest-weight W16 wide-flange beam from Appendix B that will safely support the loading. The allowable bending stress is $\sigma_{\text {allow }}=30 \mathrm{ksi}$ and the allowable shear stress is $\tau_{\text {allow }}=15 \mathrm{ksi}$

Hast Aggarwal

Numerade Educator

Problem 29

The beam is to be used to support the machine, which exerts the forces of 6 kip and 8 kip. If the maximum bending stress is not to exceed $\sigma_{\text {allow }}=22 \mathrm{ksi}$, determine the required width $b$ of the flanges.

Chai Santi

Numerade Educator

Problem 30

The steel beam has an allowable bending stress $\sigma_{\text {allow }}=140 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=90$ MPa. Determine the maximum load that can safely be supported.

Hast Aggarwal

Numerade Educator

Problem 31

Determine the variation in the width $w$ as a function of $x$ for the cantilevered beam that supports a concentrated force $\mathbf{P}$ at its end so that it has a maximum constant thickness $t$.

Hast Aggarwal

Numerade Educator

Problem 32

If it is made from a plate and has a constant width $b$ determine the absolute maximum bending stress in the beam.

Hast Aggarwal

Numerade Educator

Problem 33

The tapered beam supports the concentrated force $\mathbf{P}$ at its center. Determine the absolute maximum bending stress in the beam. The reactions at the supports are vertical.

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Numerade Educator

Problem 34

The beam is made from a plate that has a constant thickness $b$. If it is simply supported and carries the distributed loading shown, determine the variation of its depth $h$ as a function of $x$ so that it maintains a constant maximum bending stress $\sigma_{\text {allow throughout its length. }}$

Hast Aggarwal

Numerade Educator

Problem 35

Determine the variation in the depth $d$ of a cantilevered beam that supports a concentrated force $\mathbf{P}$ at its end so that it has a constant maximum bending stress $\sigma_{\text {allow }}$ throughout its length. The beam has a constant width $b_{0}$.

Chai Santi

Numerade Educator

Problem 36

Determine the variation of the radius $r$ of the cantilevered beam that supports the uniform distributed load so that it has a constant maximum bending stress $\sigma_{\max }$ throughout its length.

Chai Santi

Numerade Educator

Problem 37

The tapered beam supports a uniform distributed load $w$. If it is made from a plate that has a constant width $b_{0}$ determine the absolute maximum bending stress in the beam.

Hast Aggarwal

Numerade Educator

Problem 38

Determine the variation in the width $b$ as a function of $x$ for the cantilevered beam that supports a uniform distributed load along its centerline so that it has the same maximum bending stress $\sigma_{\text {allow }}$ throughout its length. The beam has a constant depth $t$.

Chai Santi

Numerade Educator

Problem 39

The tubular shaft has an inner diameter of $15 \mathrm{mm}$. Determine to the nearest millimeter its minimum outer
diameter if it is subjected to the gear loading. The bearings at $A$ and $B$ exert force components only in the $y$ and $z$ directions on the shaft. Use an allowable shear stress of $\tau_{\text {allow }}=70 \mathrm{MPa},$ and base the design on the maximum shear stress theory of failure.

Hast Aggarwal

Numerade Educator

Problem 40

Determine to the nearest millimeter the minimum diameter of the solid shaft if it is subjected to the gear loading. The bearings at $A$ and $B$ exert force components only in the $y$ and $z$ directions on the shaft. Base the design on the maximum distortion energy theory of failure with $\sigma_{\text {allow }}=150 \mathrm{MPa}$.

Hast Aggarwal

Numerade Educator

Problem 41

The 50 -mm-diameter shaft is supported by journal bearings at $A$ and $B$. If the pulleys $C$ and $D$ are subjected to the loadings shown, determine the absolute maximum bending stress in the shaft.

Hast Aggarwal

Numerade Educator

Problem 42

The pulleys fixed to the shaft are loaded as shown. If the journal bearings at $A$ and $B$ exert only horizontal and vertical forces on the shaft, determine the required diameter of the shaft to the nearest $\frac{1}{8}$ in. using the maximum shear stress theory of failure, $\tau_{\text {allow }}=12 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator

Problem 43

The pulleys fixed to the shaft are loaded as shown. If the journal bearings at $A$ and $B$ exert only horizontal and vertical forces on the shaft, determine the required diameter of the shaft to the nearest $\frac{1}{8}$ in. Use the maximum distortion energy theory of failure, $\sigma_{\text {allow }}=20 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator

Problem 44

The two pulleys fixed to the shaft are loaded as shown. If the journal bearings at $A$ and $B$ exert only vertical forces on the shaft, determine the required diameter of the shaft to the nearest $\frac{1}{8}$ in. using the maximum distortion energy theory. $\sigma_{\text {allow }}=67 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator

Problem 45

The shaft is supported by journal bearings at $A$ and $B$ that exert force components only in the $x$ and $z$ directions If the allowable normal stress for the shaft is $\sigma_{\text {allow }}=15 \mathrm{ksi}$ determine the smallest diameter of the shaft to the nearest $\frac{1}{8}$ in. Use the maximum distortion energy theory of failure.

Hast Aggarwal

Numerade Educator

Problem 46

The shaft is supported by journal bearings at $A$ and $B$ that exert force components only in the $x$ and $z$ directions. Determine the smallest diameter of the shaft to the nearest In. Use the maximum shear stress theory of failure with $\tau_{\text {allow }}=6 \mathrm{ksi}$.

Hast Aggarwal

Numerade Educator