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Mechanics of Materials 7th

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

Chapter 5

Analysis and Design of Beams for Bending

Educators


Problem 1

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 2

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 3

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 4

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 5

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 6

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, ( $b$ ) determine the equations of the shear and bending-moment curves.

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Problem 7

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 8

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 9

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 10

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 11

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 12

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 13

Assuming that the reaction of the ground is uniformly dis tributed, draw the shear and bending-moment diagrams for the beam $A B$ and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 14

Assuming that the reaction of the ground is uniformly dis tributed, draw the shear and bending-moment diagrams for the beam $A B$ and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 15

For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at $C$.

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Problem 16

For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at $C$.

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Problem 17

For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at $C$.

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Problem 18

For the beam and loading shown, determine the maximum normal stress due to bending on section $a$ -a.

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Problem 19

For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at $C$.

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Problem 20

For the beam and loading shown, determine the maximum normal stress due to bending on a transverse section at $C$.

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Problem 21

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 22

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 23

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 24

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 25

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 26

Knowing that $W=12 \mathrm{kN},$ draw the shear and bending-moment diagrams for beam $A B$ and determine the maximum normal stress due to bending.

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Problem 27

Determine $(a)$ the magnitude of the counterweight $W$ for which the maximum absolute value of the bending moment in the beam is as small as possible, ( $b$ ) the corresponding maximum normal stress due to bending. (Hint: Draw the bending-moment diagram and equate the absolute values of the largest positive and negative bending moments obtained.)

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Problem 28

Determine $(a)$ the distance $a$ for which the absolute value of the bending moment in the beam is as small as possible, $(b)$ the corresponding maximum normal stress due to bending. (See hint of Prob. $5.27 .$

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Problem 29

Knowing that $P=Q=480 \mathrm{N},$ determine $(a)$ the distance $a$ for which the absolute value of the bending moment in the beam is as small as possible,
$(b)$ the corresponding maximum normal stress due to bending. (See hint of Prob. 5.27.)

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Problem 30

Solve Prob. $5.29,$ assuming that $P=480 N$ and $Q=320 N$

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Problem 31

Determine $(a)$ the distance $a$ for which the absolute value of the bending moment in the beam is as small as possible, $(b)$ the corresponding maximum normal stress due to bending. (See hint of Prob. $5.27 .$

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Problem 32

A solid steel rod of diameter $d$ is supported as shown. Knowing that for steel $\gamma=490 \mathrm{lb} / \mathrm{ft}^{3},$ determine the smallest diameter $d$ that can be used if the normal stress due to bending is not to exceed 4 ksi.

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Problem 33

A solid steel bar has a square cross section of side $b$ and is supported as shown. Knowing that for steel $\rho=7860 \mathrm{kg} / \mathrm{m}^{3},$ determine the dimension $b$ for which the maximum normal stress due to bending is $(a) 10 \mathrm{MPa}$ $(b) 50 \mathrm{MPa}$.

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Problem 34

Using the method of Sec. 5.2, solve Prob. 5.1 $a$.

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Problem 35

Using the method of Sec. $5.2,$ solve Prob. $5.2 a .$

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Problem 36

Using the method of Sec. 5.2, solve Prob. 5.3 a.

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Problem 37

Using the method of Sec. $5.2,$ solve Prob. $5.4 a$

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Problem 38

Using the method of Sec. $5.2,$ solve Prob. $5.5 a$

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Problem 39

Using the method of Sec. $5.2,$ solve Prob. $5.6 a$

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Problem 40

Using the method of Sec. $5.2,$ solve Prob. 5.7

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Problem 41

Using the method of Sec. 5.2, solve Prob. 5.8.

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Problem 42

Using the method of Sec. $5.2,$ solve Prob. 5.9

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Problem 43

Using the method of Sec. $5.2,$ solve Prob. 5.10

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Problem 44

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 45

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 46

Using the method of Sec. 5.2, solve Prob. 5.15.

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Problem 47

Using the method of Sec. $5.2,$ solve Prob. 5.16

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Problem 48

Using the method of Sec. $5.2,$ solve Prob. 5.18

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Problem 49

Using the method of Sec. 5.2, solve Prob. 5.20.

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Problem 50

Determine $(a)$ the equations of the shear and bendingmoment curves for the beam and loading shown, (b) the maximum absolute value of the bending moment in the beam.

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Problem 51

Determine $(a)$ the equations of the shear and bendingmoment curves for the beam and loading shown, (b) the maximum absolute value of the bending moment in the beam.

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Problem 52

Determine $(a)$ the equations of the shear and bendingmoment curves for the beam and loading shown, ( $b$ ) the maximum absolute value of the bending moment in the beam.

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Problem 53

Determine $(a)$ the equations of the shear and bendingmoment curves for the beam and loading shown, ( $b$ ) the maximum absolute value of the bending moment in the beam.

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Problem 54

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 55

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 56

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 57

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 58

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 59

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 60

Knowing that beam $A B$ is in equilibrium under the loading shown, draw the shear and bending-moment diagrams and determine the maximum normal stress due to bending.

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Problem 61

Knowing that beam $A B$ is in equilibrium under the loading shown, draw the shear and bending-moment diagrams and determine the maximum normal stress due to bending.

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Problem 62

The beam $A B$ supports two concentrated loads $\mathbf{P}$ and $\mathbf{Q}$. The normal stress due to bending on the bottom edge of the beam is +55 MPa at $D$ and +37.5 MPa at $F$. (a) Draw the shear and bending-moment diagrams for the beam. (b) Determine the maximum normal stress due to bending that occurs in the beam.

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Problem 63

The beam $A B$ supports a uniformly distributed load of $480 \mathrm{lb} / \mathrm{ft}$ and two concentrated loads $\mathbf{P}$ and $\mathbf{Q}$. The normal stress due to bending on the bottom edge of the lower flange is +14.85 ksi at $D$ and +10.65 ksi at $E .(a)$ Draw the shear and bending-moment diagrams for the beam. (b) Determine the maximum normal stress due to bending that occurs in the beam.

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Problem 64

Beam $A B$ supports a uniformly distributed load of $2 \mathrm{kN} / \mathrm{m}$ and two concentrated loads $\mathbf{P}$ and $\mathbf{Q} .$ It has been experimentally determined that the normal stress due to bending in the bottom edge of the beam is $-56.9 \mathrm{MPa}$ at $A$ and $-29.9 \mathrm{MPa}$ at $C .$ Draw the shear and bending-moment diagrams for the beam and determine the magnitudes of the loads $\mathbf{P}$ and $\mathbf{Q}$.

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Problem 65

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of $12 \mathrm{MPa}$.

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Problem 66

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of $12 \mathrm{MPa}$.

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Problem 67

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of 1750 psi.

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Problem 68

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of 1750 psi.

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Problem 69

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of $12 \mathrm{MPa}$.

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Problem 70

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of $12 \mathrm{MPa}$.

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Problem 71

Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select the most economical wide-flange beam to support the loading shown.

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Problem 72

Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select the most economical wide-flange beam to support the loading shown.

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Problem 73

Knowing that the allowable normal stress for the steel used is $160 \mathrm{MPa}$, select the most economical wide-flange beam to support the loading shown.

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Problem 74

Knowing that the allowable normal stress for the steel used is $160 \mathrm{MPa}$, select the most economical wide-flange beam to support the loading shown.

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Problem 75

Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select the most economical S-shape beam to support the loading shown.

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Problem 76

Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select the most economical S-shape beam to support the loading shown.

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Problem 77

Knowing that the allowable normal stress for the steel used is $160 \mathrm{MPa}$, select the most economical S-shape beam to support the loading shown.

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Problem 78

Knowing that the allowable normal stress for the steel used is $160 \mathrm{MPa}$, select the most economical S-shape beam to support the loading shown.

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Problem 79

A steel pipe of 100 -mm diameter is to support the loading shown. Knowing that the stock of pipes available has thicknesses varying from $6 \mathrm{mm}$ to $24 \mathrm{mm}$ in 3 -mm increments, and that the allowable normal stress for the steel used is $150 \mathrm{MPa}$, determine the minimum wall thickness $t$ that can be used.

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Problem 80

Two metric rolled-steel channels are to be welded along their edges and used to support the loading shown. Knowing that the allowable normal stress for the steel used is $150 \mathrm{MPa}$, determine the most economical channels that can be used.

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Problem 81

Two rolled-steel channels are to be welded back to back and used to support the loading shown. Knowing that the allowable normal stress for the steel used is 30 ksi, determine the most economical channels that can be used.

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Problem 82

Two L4 $\times 3$ rolled-steel angles are bolted together and used to support the loading shown. Knowing that the allowable normal stress for the steel used is 24 ksi, determine the minimum angle thickness that can be used.

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Problem 83

Assuming the upward reaction of the ground to be uniformly distributed and knowing that the allowable normal stress for the steel used is $170 \mathrm{MPa}$, select the most economical wide-flange beam to support the loading shown.

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Problem 84

Assuming the upward reaction of the ground to be uniformly distributed and knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select the most economical wide-flange beam to support the loading shown.

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Problem 85

Determine the largest permissible distributed load $w$ for the beam shown, knowing that the allowable normal stress is +80 MPa in tension and -130 MPa in compression.

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Problem 86

Solve Prob. $5.85,$ assuming that the cross section of the beam is inverted, with the flange of the beam resting on the supports at $B$ and $C$.

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Problem 87

Determine the largest permissible value of $\mathbf{P}$ for the beam and loading shown, knowing that the allowable normal stress is $+8 \mathrm{ksi}$ in tension and $-18 \mathrm{ksi}$ in compression.

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Problem 88

Solve Prob. $5.87,$ assuming that the T-shaped beam is inverted.

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Problem 89

Beams $A B, B C,$ and $C D$ have the cross section shown and are pin-connected at $B$ and $C .$ Knowing that the allowable normal stress is +110 MPa in tension and -150 MPa in compression, determine $(a)$ the largest permissible value of $w$ if beam $B C$ is not to be overstressed, $(b)$ the corresponding maximum distance $a$ for which the cantilever beams $A B$ and $C D$ are not overstressed.

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Problem 90

Beams $A B, B C,$ and $C D$ have the cross section shown and are pin-connected at $B$ and $C .$ Knowing that the allowable normal stress is +110 MPa in tension and -150 MPa in compression, determine $(a)$ the largest permissible value of $\mathbf{P}$ if beam $B C$ is not to be overstressed, (b) the corresponding maximum distance $a$ for which the cantilever beams $A B$ and $C D$ are not overstressed.

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Problem 91

Each of the three rolled-steel beams shown (numbered $1,2, \text { and } 3)$ is to carry a 64 -kip load uniformly distributed over the beam. Each of these beams has a 12 -ft span and is to be supported by the two 24 -ft rolled-steel girders $A C$ and $B D .$ Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi}$, select $(a)$ the most economical S shape for the three beams, $(b)$ the most economical W shape for the two girders.

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Problem 92

A 54 -kip load is to be supported at the center of the 16 -ft span shown. Knowing that the allowable normal stress for the steel used is 24 ksi, determine $(a)$ the smallest allowable length $l$ of beam $C D$ if the $\mathrm{W} 12 \times 50$ beam $A B$ is not to be overstressed, (b) the most economical W shape that can be used for beam $C D$ Neglect the weight of both beams.

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Problem 93

A uniformly distributed load of $66 \mathrm{kN} / \mathrm{m}$ is to be supported over the $6-\mathrm{m}$ span shown. Knowing that the allowable normal stress for the steel used is $140 \mathrm{MPa}$, determine $(a)$ the smallest allowable length $l$ of beam $C D$ if the $\mathrm{W} 460 \times 74$ beam $A B$ is not to be overstressed, $(b)$ the most economical $\mathrm{W}$ shape that can be used for beam $C D .$ Neglect the weight of both beams.

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Problem 94

A roof structure consists of plywood and roofing material supported by several timber beams of length $L=16 \mathrm{m} .$ The dead load carried by each beam, including the estimated weight of the beam, can be represented by a uniformly distributed load $w_{D}=350 \mathrm{N} / \mathrm{m} .$ The live load consists of a snow load, represented by a uniformly distributed load $w_{L}=600 \mathrm{N} / \mathrm{m},$ and a $6-\mathrm{kN}$ concentrated load $\mathbf{P}$ applied at the midpoint $C$ of each beam. Knowing that the ultimate strength for the timber used is $\sigma_{U}=50 \mathrm{MPa}$ and that the width of the beam is $b=75 \mathrm{mm},$ determine the minimum allowable depth $h$ of the beams, using LRFD with the load factors $\gamma_{D}=1.2, \gamma_{L}=1.6$ and the resistance factor $\phi=0.9$.

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Problem 95

Solve Prob. $5.94,$ assuming that the 6 -kN concentrated load $\mathbf{P}$ applied to each beam is replaced by 3 -kN concentrated loads $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ applied at a distance of $4 \mathrm{m}$ from each end of the beams.

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Problem 96

A bridge of length $L=48 \mathrm{ft}$ is to be built on a secondary road whose access to trucks is limited to two-axle vehicles of medium weight. It will consist of a concrete slab and of simply supported steel beams with an ultimate strength $\sigma_{U}=60$ ksi. The combined weight of the slab and beams can be approximated by a uniformly distributed load $w=0.75$ kips/ft on each beam. For the purpose of the design, it is assumed that a truck with axles located at a distance $a=14 \mathrm{ft}$ from each other will be driven across the bridge and that the resulting concentrated loads $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ exerted on each beam could be as large as 24 kips and 6 kips, respectively. Determine the most economical wide-flange shape for the beams, using LRFD with the load factors $\gamma_{D}=1.25, \gamma_{L}=1.75$ and the resistance factor $\phi=0.9 .[$ Hint: It can be shown that the maximum value of $\left|M_{L}\right|$ occurs under the larger load when that load is located to the left of the center of the beam at a distance equal to $\left.a P_{2} / 2\left(P_{1}+P_{2}\right) .\right]$

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Problem 97

Assuming that the front and rear axle loads remain in the same ratio as for the truck of Prob. 5.96, determine how much heavier a truck could safely cross the bridge designed in that problem.

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Problem 98

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. $(b)$ Use the equation obtained for $M$ to determine the bending moment at point $C,$ and check your answer by drawing the free-body diagram of the entire beam.

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Problem 99

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. $(b)$ Use the equation obtained for $M$ to determine the bending moment at point $C,$ and check your answer by drawing the free-body diagram of the entire beam.

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Problem 100

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. $(b)$ Use the equation obtained for $M$ to determine the bending moment at point $C,$ and check your answer by drawing the free-body diagram of the entire beam.

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Problem 101

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. (b) Use the equation obtained for $M$ to determine the bending moment at point $E,$ and check your answer by drawing the free-body diagram of the portion of the beam to the right of $E$.

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Problem 102

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. (b) Use the equation obtained for $M$ to determine the bending moment at point $E,$ and check your answer by drawing the free-body diagram of the portion of the beam to the right of $E$.

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Problem 103

(a) Using singularity functions, write the equations defining the shear and bending moment for the beam and loading shown. (b) Use the equation obtained for $M$ to determine the bending moment at point $E,$ and check your answer by drawing the free-body diagram of the portion of the beam to the right of $E$.

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Problem 104

(a) Using singularity functions, write the equations for the shear and bending moment for beam $A B C$ under the loading shown.
(b) Use the equation obtained for $M$ to determine the bending moment just to the right of point $B$

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Problem 105

(a) Using singularity functions, write the equations for the shear and bending moment for beam $A B C$ under the loading shown.
(b) Use the equation obtained for $M$ to determine the bending moment just to the right of point $B$

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Problem 106

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum value of the bending moment in the beam.

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Problem 107

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum value of the bending moment in the beam.

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Problem 108

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum value of the bending moment in the beam.

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Problem 109

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum value of the bending moment in the beam.

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Problem 110

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum normal stress due to bending.

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Problem 111

(a) Using singularity functions, write the equations for the shear and bending moment for the beam and loading shown.
(b) Determine the maximum normal stress due to bending.

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Problem 112

(a) Using singularity functions, find the magnitude and location of the maximum bending moment for the beam and loading shown.
(b) Determine the maximum normal stress due to bending.

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Problem 113

(a) Using singularity functions, find the magnitude and location of the maximum bending moment for the beam and loading shown.
(b) Determine the maximum normal stress due to bending.

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Problem 114

A beam is being designed to be supported and loaded as shown.
(a) Using singularity functions, find the magnitude and location of the maximum bending moment in the beam.
(b) Knowing that the allowable normal stress for the steel to be used is $24 \mathrm{ksi}$ find the most economical wide-flange shape that can be used.

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Problem 115

A beam is being designed to be supported and loaded as shown.
(a) Using singularity functions, find the magnitude and location of the maximum bending moment in the beam.
(b) Knowing that the allowable normal stress for the steel to be used is $24 \mathrm{ksi}$ find the most economical wide-flange shape that can be used.

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Problem 116

A timber beam is being designed to be supported and loaded as shown. $(a)$ Using singularity functions, find the magnitude and location of the maximum bending moment in the beam. (b) Knowing that the available stock consists of beams with an allowable normal stress of $12 \mathrm{MPa}$ and a rectangular cross section of $30-\mathrm{mm}$ width and depth $h$ varying from $80 \mathrm{mm}$ to $160 \mathrm{mm}$ in $10-m m$ increments, determine the most economical cross section that can be used.

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Problem 117

A timber beam is being designed to be supported and loaded as shown. $(a)$ Using singularity functions, find the magnitude and location of the maximum bending moment in the beam. (b) Knowing that the available stock consists of beams with an allowable normal stress of $12 \mathrm{MPa}$ and a rectangular cross section of $30-\mathrm{mm}$ width and depth $h$ varying from $80 \mathrm{mm}$ to $160 \mathrm{mm}$ in $10-m m$ increments, determine the most economical cross section that can be used.

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Problem 118

Using a computer and step functions, calculate the shear and bending moment for the beam and loading shown. Use the specified increment $\Delta L$, starting at point $A$ and ending at the right-hand support.

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Problem 119

Using a computer and step functions, calculate the shear and bending moment for the beam and loading shown. Use the specified increment $\Delta L$, starting at point $A$ and ending at the right-hand support.

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Problem 120

Using a computer and step functions, calculate the shear and bending moment for the beam and loading shown. Use the specified increment $\Delta L$, starting at point $A$ and ending at the right-hand support.

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Problem 121

Using a computer and step functions, calculate the shear and bending moment for the beam and loading shown. Use the specified increment $\Delta L$, starting at point $A$ and ending at the right-hand support.

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Problem 122

For the beam and loading shown and using a computer and step functions, $(a)$ tabulate the shear, bending moment, and maximum normal stress in sections of the beam from $x=0$ to $x=L,$ using the increments $\Delta L$ indicated, $(b)$ using smaller increments if necessary, determine with a $2 \%$ accuracy the maximum normal stress in the beam. Place the origin of the $x$ axis at end $A$ of the beam.

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Problem 123

For the beam and loading shown and using a computer and step functions, $(a)$ tabulate the shear, bending moment, and maximum normal stress in sections of the beam from $x=0$ to $x=L,$ using the increments $\Delta L$ indicated, $(b)$ using smaller increments if necessary, determine with a $2 \%$ accuracy the maximum normal stress in the beam. Place the origin of the $x$ axis at end $A$ of the beam.

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Problem 124

For the beam and loading shown and using a computer and step functions, $(a)$ tabulate the shear, bending moment, and maximum normal stress in sections of the beam from $x=0$ to $x=L,$ using the increments $\Delta L$ indicated, $(b)$ using smaller increments if necessary, determine with a $2 \%$ accuracy the maximum normal stress in the beam. Place the origin of the $x$ axis at end $A$ of the beam.

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Problem 125

For the beam and loading shown and using a computer and step functions, $(a)$ tabulate the shear, bending moment, and maximum normal stress in sections of the beam from $x=0$ to $x=L,$ using the increments $\Delta L$ indicated, $(b)$ using smaller increments if necessary, determine with a $2 \%$ accuracy the maximum normal stress in the beam. Place the origin of the $x$ axis at end $A$ of the beam.

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Problem 126

The beam $A B,$ consisting of a cast-iron plate of uniform thickness $b$ and length $L$, is to support the load shown. $(a)$ Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0} .(b)$ Determine the maximum allowable load if $L=36$ in., $h_{0}=12$ in., $b=1.25$ in., and $\sigma_{\text {all }}=24$ ksi.

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Problem 127

The beam $A B,$ consisting of a cast-iron plate of uniform thickness $b$ and length $L$, is to support the load shown. $(a)$ Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0} .(b)$ Determine the maximum allowable load if $L=36$ in., $h_{0}=12$ in., $b=1.25$ in., and $\sigma_{\text {all }}=24$ ksi.

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Problem 128

The beam $A B,$ consisting of a cast-iron plate of uniform thickness $b$ and length $L,$ is to support the distributed load $w(x)$ shown.
(a) Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0}$
(b) Determine the smallest value of $h_{0}$ if $L=750 \mathrm{mm}, b=30 \mathrm{mm}, w_{0}=300 \mathrm{kN} / \mathrm{m},$ and $\sigma_{\text {all }}=200 \mathrm{MPa}$

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Problem 129

The beam $A B,$ consisting of a cast-iron plate of uniform thickness $b$ and length $L,$ is to support the distributed load $w(x)$ shown.
(a) Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0}$
(b) Determine the smallest value of $h_{0}$ if $L=750 \mathrm{mm}, b=30 \mathrm{mm}, w_{0}=300 \mathrm{kN} / \mathrm{m},$ and $\sigma_{\text {all }}=200 \mathrm{MPa}$

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Problem 130

The beam $A B,$ consisting of an aluminum plate of uniform thickness $b$ and length $L,$ is to support the load shown.
(a) Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0}$ for portion $A C$ of the beam.
(b) Determine the maximum allowable load if $L=800 \mathrm{mm}, h_{0}=200 \mathrm{mm}, b=$ $25 \mathrm{mm},$ and $\sigma_{\text {all }}=72 \mathrm{MPa}$

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Problem 131

The beam $A B,$ consisting of an aluminum plate of uniform thickness $b$ and length $L,$ is to support the load shown.
(a) Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0}$ for portion $A C$ of the beam.
(b) Determine the maximum allowable load if $L=800 \mathrm{mm}, h_{0}=200 \mathrm{mm}, b=$ $25 \mathrm{mm},$ and $\sigma_{\text {all }}=72 \mathrm{MPa}$

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Problem 132

A preliminary design on the use of a cantilever prismatic timber beam indicated that a beam with a rectangular cross section 2 in. wide and 10 in. deep would be required to safely support the load shown in part $a$ of the figure. It was then decided to replace that beam with a built-up beam obtained by gluing together, as shown in part $b$ of the figure, five pieces of the same timber as the original beam and of $2 \times 2$ -in. cross section. Determine the respective lengths $l_{1}$ and $l_{2}$ of the two inner and outer pieces of timber that will yield the same factor of safety as the original design.

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Problem 133

A preliminary design on the use of a cantilever prismatic timber beam indicated that a beam with a rectangular cross section 2 in. wide and 10 in. deep would be required to safely support the load shown in part $a$ of the figure. It was then decided to replace that beam with a built-up beam obtained by gluing together, as shown in part $b$ of the figure, five pieces of the same timber as the original beam and of $2 \times 2$ -in. cross section. Determine the respective lengths $l_{1}$ and $l_{2}$ of the two inner and outer pieces of timber that will yield the same factor of safety as the original design.

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Problem 134

A preliminary design on the use of a simply supported prismatic timber beam indicated that a beam with a rectangular cross section $50 \mathrm{mm}$ wide and $200 \mathrm{mm}$ deep would be required to safely support the load shown in part $a$ of the figure. It was then decided to replace that beam with a built-up beam obtained by gluing together, as shown in part $b$ of the figure, four pieces of the same timber as the original beam and of $50 \times 50-\mathrm{mm}$ cross section. Determine the length $l$ of the two outer pieces of timber that will yield the same factor of safety as the original design.

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Problem 135

A preliminary design on the use of a simply supported prismatic timber beam indicated that a beam with a rectangular cross section $50 \mathrm{mm}$ wide and $200 \mathrm{mm}$ deep would be required to safely support the load shown in part $a$ of the figure. It was then decided to replace that beam with a built-up beam obtained by gluing together, as shown in part $b$ of the figure, four pieces of the same timber as the original beam and of $50 \times 50-\mathrm{mm}$ cross section. Determine the length $l$ of the two outer pieces of timber that will yield the same factor of safety as the original design.

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Problem 136

A machine element of cast aluminum and in the shape of a solid of revolution of variable diameter $d$ is being designed to support the load shown. Knowing that the machine element is to be of constant strength, express $d$ in terms of $x, L,$ and $d_{0}$.

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Problem 137

A machine element of cast aluminum and in the shape of a solid of revolution of variable diameter $d$ is being designed to support the load shown. Knowing that the machine element is to be of constant strength, express $d$ in terms of $x, L,$ and $d_{0}$.

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Problem 138

A transverse force $\mathbf{P}$ is applied as shown at end $A$ of the conical $\operatorname{taper} A B .$ Denoting by $d_{0}$ the diameter of the taper at $A,$ show that the maximum normal stress occurs at point $H,$ which is contained in a transverse section of diameter $d=1.5 d_{0}$.

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Problem 139

A cantilever beam $A B$ consisting of a steel plate of uniform depth $h$ and variable width $b$ is to support the distributed load $w$ along its centerline $A B .(a)$ Knowing that the beam is to be of constant strength, express $b$ in terms of $x, L,$ and $b_{0} .(b)$ Determine the maximum allowable value of $w$ if $L=15$ in., $b_{0}=8$ in., $h=0.75$ in. and $\sigma_{\text {all }}=24$ ksi.

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Problem 140

Assuming that the length and width of the cover plates used with the beam of Sample Prob. 5.12 are, respectively, $l=4 \mathrm{m}$ and $b=$ $285 \mathrm{mm},$ and recalling that the thickness of each plate is $16 \mathrm{mm}$ determine the maximum normal stress on a transverse section
$(a)$ through the center of the beam,
$(b)$ just to the left of $D$

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Problem 141

Two cover plates, each $\frac{1}{2}$ in. thick, are welded to a $\mathrm{W} 27 \times 84$ beam as shown. Knowing that $l=10 \mathrm{ft}$ and $b=10.5$ in., determine the maximum normal stress on a transverse section $(a)$ through the center of the beam, $(b)$ just to the left of $D$.

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Problem 142

Two cover plates, each $\frac{1}{2}$ in. thick, are welded to a $\mathrm{W} 27 \times 84$ beam as shown. Knowing that $\sigma_{\text {all }}=24 \mathrm{ksi}$ for both the beam and the plates, determine the required value of $(a)$ the length of the plates, $(b)$ the width of the plates.

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Problem 143

Knowing that $\sigma_{\text {all }}=150 \mathrm{MPa}$, determine the largest concentrated load $\mathbf{P}$ that can be applied at end $E$ of the beam shown.

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Problem 144

Two cover plates, each $7.5 \mathrm{mm}$ thick, are welded to a $\mathrm{W} 460 \times 74$ beam as shown. Knowing that $l=5 \mathrm{m}$ and $b=200 \mathrm{mm},$ determine the maximum normal stress on a transverse section
$(a)$ through the center of the beam,
$(b)$ just to the left of $D$

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Problem 145

Two cover plates, each $7.5 \mathrm{mm}$ thick, are welded to a $\mathrm{W} 460 \times 74$ beam as shown. Knowing that $\sigma_{\text {all }}=150$ MPa for both the beam and the plates, determine the required value of $(a)$ the length of the plates, $(b)$ the width of the plates.

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Problem 146

Two cover plates, each $\frac{5}{8}$ in. thick, are welded to a $\mathrm{W} 30 \times 99$ beam as shown. Knowing that $l=9 \mathrm{ft}$ and $b=12$ in., determine the maximum normal stress on a transverse section $(a)$ through the center of the beam, $(b)$ just to the left of $D$.

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Problem 147

Two cover plates, each $\frac{5}{8}$ in. thick, are welded to a $\mathrm{W} 30 \times 99$ beam as shown. Knowing that $\sigma_{\text {all }}=22 \mathrm{ksi}$ for both the beam and the plates, determine the required value of $(a)$ the length of the plates (b) the width of the plates.

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Problem 148

For the tapered beam shown, determine $(a)$ the transverse section in which the maximum normal stress occurs, (b) the largest distributed load $w$ that can be applied, knowing that $\sigma_{\text {all }}=140 \mathrm{MPa}$.

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Problem 149

For the tapered beam shown, knowing that $w=160 \mathrm{kN} / \mathrm{m},$ determine $(a)$ the transverse section in which the maximum normal stress occurs,
$(b)$ the corresponding value of the normal stress.

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Problem 150

For the tapered beam shown, determine $(a)$ the transverse section in which the maximum normal stress occurs, $(b)$ the largest distributed load $w$ that can be applied, knowing that $\sigma_{\text {all }}=24$ ksi.

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Problem 151

For the tapered beam shown, determine ( $a$ ) the transverse section in which the maximum normal stress occurs, (b) the largest concentrated load $\mathbf{P}$ that can be applied, knowing that $\sigma_{\text {all }}=24 \mathrm{ksi}$.

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Problem 152

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value $(a)$ of the shear, $(b)$ of the bending moment.

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Problem 153

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 154

Determine $(a)$ the distance $a$ for which the absolute value of the bending moment in the beam is as small as possible, $(b)$ the corresponding maximum normal stress due to bending. (See hint of Prob. $5.27 .$

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Problem 155

For the beam and loading shown, determine the equations of the shear and bending-moment curves and the maximum absolute value of the bending moment in the beam, knowing that $(a) k=1,(b) k=0.5$.

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Problem 156

Draw the shear and bending-moment diagrams for the beam and loading shown and determine the maximum normal stress due to bending.

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Problem 157

Beam $A B,$ of length $L$ and square cross section of side $a,$ is supported by a pivot at $C$ and loaded as shown. $(a)$ Check that the beam is in equilibrium.
(b) Show that the maximum normal stress due to bending occurs at $C$ and is equal to $w_{0} L^{2} /(1.5 a)^{3}$.

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Problem 158

For the beam and loading shown, design the cross section of the beam, knowing that the grade of timber used has an allowable normal stress of 1750 psi.

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Problem 159

Knowing that the allowable normal stress for the steel used is $24 \mathrm{ksi},$ select the most economical wide-flange beam to support the loading shown.

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Problem 160

Three steel plates are welded together to form the beam shown. Knowing that the allowable normal stress for the steel used is $22 \mathrm{ksi},$ determine the minimum flange width $b$ that can be used.

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Problem 161

(a) Using singularity functions, find the magnitude and location of the maximum bending moment for the beam and loading shown.
(b) Determine the maximum normal stress due to bending.

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Problem 162

The beam $A B,$ consisting of an aluminum plate of uniform thickness $b$ and length $L,$ is to support the load shown. $(a)$ Knowing that the beam is to be of constant strength, express $h$ in terms of $x, L,$ and $h_{0}$ for portion $A C$ of the beam. (b) Determine the maximum allowable load if $L=800 \mathrm{mm}, h_{0}=200 \mathrm{mm}, b=25 \mathrm{mm}$ and $\sigma_{\text {all }}=72 \mathrm{MPa}$.

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Problem 163

A cantilever beam $A B$ consisting of a steel plate of uniform depth $h$ and variable width $b$ is to support the concentrated load $\mathbf{P}$ at point $A .(a)$ Knowing that the beam is to be of constant strength, express $b$ in terms of $x, L,$ and $b_{0}$ (b) Determine the smallest allowable value of $h$ if $L=300 \mathrm{mm}, b_{0}=375 \mathrm{mm}$ $P=14.4 \mathrm{kN},$ and $\sigma_{\text {all }}=160 \mathrm{MPa}$.

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