Mechanics of Materials

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

Chapter 6

Shearing Stresses in Beams and Thin-Walled Members - all with Video Answers

Educators

Chapter Questions

Problem 1

Three full-size $50 \times 100-\mathrm{mm}$ boards are nailed together to form a beam that is subjected to a vertical shear of 1500 N. Knowing that the allowable shearing force in each nail is $400 \mathrm{N}$, determine the largest longitudinal spacing $s$ that can be used between each pair of nails.

Hast Aggarwal

Hast Aggarwal

Numerade Educator

Problem 2

For the built-up beam of Prob. $6.1,$ determine the allowable shear if the spacing between each pair of nails is $s=45 \mathrm{mm}$

Chai Santi

Numerade Educator

Problem 3

Three boards, each 2 in. thick, are nailed together to form a beam that is subjected to a vertical shear. Knowing that the allowable shearing force in each nail is $150 \mathrm{lb}$, determine the allowable shear if the spacing $s$ between the nails is 3 in.

Hast Aggarwal

Numerade Educator

Problem 4

A square box beam is made of two $20 \times 80-\mathrm{mm}$ planks and two $20 \times 120-\mathrm{mm}$ planks nailed together as shown. Knowing that the spacing between the nails is $s=30 \mathrm{mm}$ and that the vertical shear in the beam is $V=1200 \mathrm{N}$, determine ( $a$ ) the shearing force in each nail, $(b)$ the maximum shearing stress in the beam.

Rashmi Sinha

Numerade Educator

Problem 5

The American Standard rolled-steel beam shown has been reinforced by attaching to it two $16 \times 200$ -mm plates, using 18 -mmdiameter bolts spaced longitudinally every $120 \mathrm{mm}$. Knowing that the average allowable shearing stress in the bolts is $90 \mathrm{MPa}$ determine the largest permissible vertical shearing force.

Rashmi Sinha

Numerade Educator

Problem 6

The beam shown is fabricated by connecting two channel shapes and two plates, using bolts of $\frac{3}{4}$ -in. diameter spaced longitudinally every 7.5 in. Determine the average shearing stress in the bolts caused by a shearing force of 25 kips parallel to the $y$ axis.

Ajay Singhal

Numerade Educator

Problem 7

A column is fabricated by connecting the rolled-steel members shown by bolts of $\frac{3}{4}$ -in. diameter spaced longitudinally every 5 in. Determine the average shearing stress in the bolts caused by a shearing force of 30 kips parallel to the $y$ axis.

Ajay Singhal

Numerade Educator

Problem 8

The composite beam shown is fabricated by connecting two $\mathrm{W} 6 \times 20$ rolled-steel members, using bolts of $\frac{5}{8}$ -in. diameter spaced longitudinally every 6 in. Knowing that the average allowable shearing stress in the bolts is 10.5 ksi, determine the largest allowable vertical shear in the beam.

Rashmi Sinha

Numerade Educator

Problem 9

For beam and loading shown, consider section $n-n$ and determine $(a)$ the largest shearing stress in that section, $(b)$ the shearing stress at point $a$

Chai Santi

Numerade Educator

Problem 10

For beam and loading shown, consider section $n-n$ and determine $(a)$ the largest shearing stress in that section, $(b)$ the shearing stress at point $a$

Chai Santi

Numerade Educator

Problem 11

For beam and loading shown, consider section $n-n$ and determine $(a)$ the largest shearing stress in that section, $(b)$ the shearing stress at point $a$

Chai Santi

Numerade Educator

Problem 12

For beam and loading shown, consider section $n-n$ and determine $(a)$ the largest shearing stress in that section, $(b)$ the shearing stress at point $a$

Chai Santi

Numerade Educator

Problem 13

For a beam having the cross section shown, determine the largest allowable vertical shear if the shearing stress is not to exceed 60 MPa.

Chai Santi

Numerade Educator

Problem 14

For a beam having the cross section shown, determine the largest allowable vertical shear if the shearing stress is not to exceed 60 MPa.

Chai Santi

Numerade Educator

Problem 15

For a timber beam having the cross section shown, determine the largest allowable vertical shear if the shearing stress is not to exceed 150 psi.

Ajay Singhal

Numerade Educator

Problem 16

Two steel plates of $12 \times 220$ -mm rectangular cross section are welded to the $\mathrm{W} 250 \times 58$ beam as shown. Determine the largest allowable vertical shear if the shearing stress in the beam is not to exceed $90 \mathrm{MPa}$

Rashmi Sinha

Numerade Educator

Problem 17

Two $\mathrm{W} 8 \times 31$ rolled sections may be welded at $A$ and $B$ in either of the two ways shown in order to form a composite beam. Knowing that for each weld the allowable shearing force is 3000 lb per inch of weld, determine for each arrangement the maximum allowable vertical shear in the composite beam.

Hast Aggarwal

Numerade Educator

Problem 18

For the beam and loading shown, determine the minimum required width $b,$ knowing that for the grade of timber used, $\sigma_{\text {all }}=12$ MPa and $\tau_{\text {all }}=825$ kPa.

Hast Aggarwal

Numerade Educator

Problem 19

A timber beam $A B$ of length $L$ and rectangular cross section carries a single concentrated load $\mathbf{P}$ at its midpoint $C .(a)$ Show that the ratio $\tau_{m} / \sigma_{m}$ of the maximum values of the shearing and normal stresses in the beam is equal to $h / 2 L,$ where $h$ and $L$ are, respectively, the depth and the length of the beam.
(b) Determine the depth $h$ and the width $b$ of the beam, knowing that $L=2 \mathrm{m}$ $P=40 \mathrm{kN}, \tau_{m}=960 \mathrm{kPa},$ and $\sigma_{m}=12 \mathrm{MPa}$

Rashmi Sinha

Numerade Educator

Problem 20

A timber beam $A B$ of length $L$ and rectangular cross section carries a uniformly distributed load $w$ and is supported as shown.
(a) Show that the ratio $\tau_{m} / \sigma_{m}$ of the maximum values of the shearing and normal stresses in the beam is equal to $2 h / L$, where $h$ and $L$ are, respectively, the depth and the length of the beam.
(b) Determine the depth $h$ and the width $b$ of the beam, knowing that $L=5 \mathrm{m}, w=8 \mathrm{kN} / \mathrm{m}, \tau_{m}=1.08 \mathrm{MPa},$ and $\sigma_{m}=12 \mathrm{MPa}$

Rashmi Sinha

Numerade Educator

Problem 21

For the beam and loading shown, consider section $n-n$ and determine the shearing stress at $(a)$ point $a$ (b) point $b$

Chai Santi

Numerade Educator

Problem 22

For the beam and loading shown, consider section $n-n$ and determine the shearing stress at $(a)$ point $a$ (b) point $b$

Chai Santi

Numerade Educator

Problem 23

For the beam and loading shown, determine the largest shearing stress in section $n-n$

Surendra Kumar

Numerade Educator

Problem 24

For the beam and loading shown, determine the largest shearing stress in section $n-n$

Surendra Kumar

Numerade Educator

Problem 25

A beam having the cross section shown is subjected to a vertical shear $\mathbf{V}$. Determine $(a)$ the horizontal line along which the shearing stress is maximum, ( $b$ ) the constant $k$ in the following expression for the maximum shearing stress
\[
\tau_{\max }=k \frac{V}{A}
\]
where $A$ is the cross-sectional area of the beam.

Chai Santi

Numerade Educator

Problem 26

A beam having the cross section shown is subjected to a vertical shear $\mathbf{V}$. Determine $(a)$ the horizontal line along which the shearing stress is maximum, ( $b$ ) the constant $k$ in the following expression for the maximum shearing stress
\[
\tau_{\max }=k \frac{V}{A}
\]
where $A$ is the cross-sectional area of the beam.

Chai Santi

Numerade Educator

Problem 27

A beam having the cross section shown is subjected to a vertical shear $\mathbf{V}$. Determine $(a)$ the horizontal line along which the shearing stress is maximum, ( $b$ ) the constant $k$ in the following expression for the maximum shearing stress
\[
\tau_{\max }=k \frac{V}{A}
\]
where $A$ is the cross-sectional area of the beam.

Chai Santi

Numerade Educator

Problem 28

A beam having the cross section shown is subjected to a vertical shear $\mathbf{V}$. Determine $(a)$ the horizontal line along which the shearing stress is maximum, ( $b$ ) the constant $k$ in the following expression for the maximum shearing stress
\[
\tau_{\max }=k \frac{V}{A}
\]
where $A$ is the cross-sectional area of the beam.

Chai Santi

Numerade Educator

Problem 29

The built-up timber beam shown is subjected to a vertical shear of 1200 Ib. Knowing that the allowable shearing force in the nails is 75 lb, determine the largest permissible spacing $s$ of the nails.

Ajay Singhal

Numerade Educator

Problem 30

The built-up beam shown is made by gluing together two $20 \times 250-\mathrm{mm}$ plywood strips and two $50 \times 100-\mathrm{mm}$ planks. Knowing that the allowable average shearing stress in the glued joints is 350 kPa, determine the largest permissible vertical shear in the beam.

Rashmi Sinha

Numerade Educator

Problem 31

The built-up beam was made by gluing together several wooden planks. Knowing that the beam is subjected to a 1200 -lb vertical shear, determine the average shearing stress in the glued joint $(a)$ at $A,(b)$ at $B$

Rashmi Sinha

Numerade Educator

Problem 32

Several wooden planks are glued together to form the box beam shown. Knowing that the beam is subjected to a vertical shear of $3 \mathrm{kN},$ determine the average shearing stress in the glued joint $(a)$ at $A,(b)$ at $B$

Rashmi Sinha

Numerade Educator

Problem 33

The built-up wooden beam shown is subjected to a vertical shear of $8 \mathrm{kN} .$ Knowing that the nails are spaced longitudinally every $60 \mathrm{mm}$ at $A$ and every $25 \mathrm{mm}$ at $B,$ determine the shearing force in the nails $(a)$ at $A,(b)$ at B. (Given: $I_{x}=1.504 \times 10^{9} \mathrm{mm}^{4} .$ )

Rashmi Sinha

Numerade Educator

Problem 34

Knowing that a $\mathrm{W} 360 \times 122$ rolled-steel beam is subjected to a $250-\mathrm{kN}$ vertical shear, determine the shearing stress $(a)$ at point $A$
(b) at the centroid $C$ of the section.

Anand Jangid

Numerade Educator

Problem 35

An extruded aluminum beam has the cross section shown. Knowing that the vertical shear in the beam is $150 \mathrm{kN}$, determine the shearing stress at $(a)$ point $a,(b)$ point $b$

Chai Santi

Numerade Educator

Problem 36

An extruded aluminum beam has the cross section shown. Knowing that the vertical shear in the beam is $150 \mathrm{kN}$, determine the shearing stress at $(a)$ point $a,(b)$ point $b$

Chai Santi

Numerade Educator

Problem 37

Knowing that a given vertical shear $\mathbf{V}$ causes a maximum shearing stress of $75 \mathrm{MPa}$ in an extruded beam having the cross section shown, determine the shearing stress at the three points indicated.

Ajay Singhal

Numerade Educator

Problem 38

The vertical shear is $1200 \mathrm{lb}$ in a beam having the cross section shown. Knowing that $d=4$ in., determine the shearing stress at
$(a)$ point $a$
$(b)$ point $b$

Anand Jangid

Numerade Educator

Problem 39

The vertical shear is $1200 \mathrm{lb}$ in a beam having the cross section shown. Determine $(a)$ the distance $d$ for which $\tau_{a}=\tau_{b},(b)$ the corresponding shearing stress at points $a$ and $b$

Anand Jangid

Numerade Educator

Problem 40

The extruded aluminum beam has a uniform wall thickness of $\frac{1}{8}$ in. Knowing that the vertical shear in the beam is 2 kips, determine the corresponding shearing stress at each of the five points indicated.

Chai Santi

Numerade Educator

Problem 41

The extruded aluminum beam has a uniform wall thickness of $\frac{1}{8}$ in. Knowing that the vertical shear in the beam is 2 kips, determine the corresponding shearing stress at each of the five points indicated.

Chai Santi

Numerade Educator

Problem 42

Knowing that a given vertical shear $\mathbf{V}$ causes a maximum shearing stress of $50 \mathrm{MPa}$ in a thin-walled member having the cross section shown, determine the corresponding shearing stress at $(a)$ point $a$ (b) point $b,(c)$ point $c$

Chai Santi

Numerade Educator

Problem 43

Three planks are connected as shown by bolts of $\frac{3}{8}$ -in. diameter spaced every 6 in. along the longitudinal axis of the beam. For a vertical shear of 2.5 kips, determine the average shearing stress in the bolts.

Ajay Singhal

Numerade Educator

Problem 44

A beam consists of three planks connected as shown by steel bolts with a longitudinal spacing of $225 \mathrm{mm}$. Knowing that the shear in the beam is vertical and equal to $6 \mathrm{kN}$ and that the allowable average shearing stress in each bolt is $60 \mathrm{MPa}$, determine the smallest permissible bolt diameter that can be used.

Victor Salazar

Numerade Educator

Problem 45

A beam consists of five planks of $1.5 \times 6$ -in. cross section connected by steel bolts with a longitudinal spacing of 9 in. Knowing that the shear in the beam is vertical and equal to $2000 \mathrm{lb}$ and that the allowable average shearing stress in each bolt is 7500 psi, determine the smallest permissible bolt diameter that can be used.

Victor Salazar

Numerade Educator

Problem 46

Four $\mathrm{L} 102 \times 102 \times 9.5$ steel angle shapes and a $12 \times 400-\mathrm{mm}$ plate are bolted together to form a beam with the cross section shown. The bolts are of 22 -mm diameter and are spaced longitudinally every $120 \mathrm{mm} .$ Knowing that the beam is subjected to a vertical shear of $240 \mathrm{kN}$, determine the average shearing stress in each bolt.

Prashant Bana

Numerade Educator

Problem 47

A plate of $\frac{1}{4}$ -in. thickness is corrugated as shown and then used as a beam. For a vertical shear of 1.2 kips, determine $(a)$ the maximum shearing stress in the section, ( $b$ ) the shearing stress at point $B$. Also sketch the shear flow in the cross section.

Anand Jangid

Numerade Educator

Problem 48

A plate of $2-\mathrm{mm}$ thickness is bent as shown and then used as a beam. For a vertical shear of $5 \mathrm{kN}$, determine the shearing stress at the five points indicated and sketch the shear flow in the cross section.

Chai Santi

Numerade Educator

Problem 49

An extruded beam has the cross section shown and a uniform wall thickness of $3 \mathrm{mm}$. For a vertical shear of $10 \mathrm{kN}$, determine $(a)$ the shearing stress at point $A,(b)$ the maximum shearing stress in the beam. Also sketch the shear flow in the cross section.

Anand Jangid

Numerade Educator

Problem 50

A plate of thickness $t$ is bent as shown and then used as a beam. For a vertical shear of $600 \mathrm{lb}$, determine (a) the thickness $t$ for which the maximum shearing stress is 300 psi, $(b)$ the corresponding shearing stress at point $E$. Also sketch the shear flow in the cross section.

Anand Jangid

Numerade Educator

Problem 51

The design of a beam calls for connecting two vertical rectangular $\frac{3}{8} \times 4-$ in. plates by welding them to two horizontal $\frac{1}{2} \times 2$ -in. plates as shown. For a vertical shear $\mathbf{V}$, determine the dimension a for which the shear flow through the welded surfaces is maximum.

Julie Silva

Numerade Educator

Problem 52

The cross section of an extruded beam is a hollow square of side $a=3$ in. and thickness $t=0.25$ in. For a vertical shear of 15 kips, determine the maximum shearing stress in the beam and sketch the shear flow in the cross section.

Ajay Singhal

Numerade Educator

Problem 53

An extruded beam has a uniform wall thickness $t .$ Denoting by $\mathbf{V}$ the vertical shear and by $A$ the cross-sectional area of the beam, express the maximum shearing stress as $\tau_{\max }=k(V / A)$ and determine the constant $k$ for each of the two orientations shown.

Chai Santi

Numerade Educator

Problem 54

(a) Determine the shearing stress at point $P$ of a thin-walled pipe of the cross section shown caused by a vertical shear $\mathbf{V}$. ( $b$ ) Show that the maximum shearing stress occurs for $\theta=90^{\circ}$ and is equal to $2 V / A$, where $A$ is the cross-sectional area of the pipe.

Anand Jangid

Numerade Educator

Problem 55

For a beam made of two or more materials with different moduli of elasticity, show that Eq. (6.6)
\[
\tau_{\mathrm{ave}}=\frac{V Q}{I t}
\]
remains valid provided that both $Q$ and $I$ are computed by using the transformed section of the beam (see Sec. 4.4 ) and provided further that $t$ is the actual width of the beam where $\tau_{\text {ave }}$ is computed.

Hast Aggarwal

Numerade Educator

Problem 56

A composite beam is made by attaching the timber and steel portions shown with bolts of 12 -mm diameter spaced longitudinally every $200 \mathrm{mm}$. The modulus of elasticity is 10 GPa for the wood and 200 GPa for the steel. For a vertical shear of $4 \mathrm{kN}$ determine $(a)$ the average shearing stress in the bolts, $(b)$ the shearing stress at the center of the cross section. (Hint: Use the method indicated in Prob. $6.55 .$

Hast Aggarwal

Numerade Educator

Problem 57

A composite beam is made by attaching the timber and steel portions shown with bolts of 12 -mm diameter spaced longitudinally every $200 \mathrm{mm}$. The modulus of elasticity is 10 GPa for the wood and 200 GPa for the steel. For a vertical shear of $4 \mathrm{kN}$ determine $(a)$ the average shearing stress in the bolts, $(b)$ the shearing stress at the center of the cross section. (Hint: Use the method indicated in Prob. $6.55 .$

Hast Aggarwal

Numerade Educator

Problem 58

A steel bar and an aluminum bar are bonded together as shown to form a composite beam. Knowing that the vertical shear in the beam is 4 kips and that the modulus of elasticity is $29 \times 10^{6}$ psi for the steel and $10.6 \times 10^{6}$ psi for the aluminum, determine ( $a$ ) the average shearing stress at the bonded surface, (b) the maximum shearing stress in the beam. (Hint: Use the method indicated in Prob. $6.55 .$

Surendra Kumar

Numerade Educator

Problem 59

A steel bar and an aluminum bar are bonded together as shown to form a composite beam. Knowing that the vertical shear in the beam is 4 kips and that the modulus of elasticity is $29 \times 10^{6}$ psi for the steel and $10.6 \times 10^{6}$ psi for the aluminum, determine ( $a$ ) the average shearing stress at the bonded surface, (b) the maximum shearing stress in the beam. (Hint: Use the method indicated in Prob. $6.55 .$

Surendra Kumar

Numerade Educator

Problem 60

Consider the cantilever beam $A B$ discussed in Sec. 6.5 and the portion $A C K J$ of the beam that is located to the left of the transverse section $C C^{\prime}$ and above the horizontal plane $J K$, where $K$ is a point at a distance $y<y_{Y}$ above the neutral axis (Fig. P6.60)
$(a)$ Recalling that $\sigma_{x}=\sigma_{Y}$ between $C$ and $E$ and $\sigma_{x}=\left(\sigma_{y} / y_{y}\right) y$ between $E$ and $K,$ show that the magnitude of the horizontal shearing force $\mathbf{H}$ exerted on the lower face of the portion of beam $A C K J$ is
\[
H=\frac{1}{2} b \sigma_{Y}\left(2 c-y_{Y}-\frac{y^{2}}{y_{Y}}\right)
\]
(b) Observing that the shearing stress at $K$ is
\[
\tau_{x y}=\lim _{\Delta A \rightarrow 0} \frac{\Delta H}{\Delta A}=\lim _{\Delta x \rightarrow 0} \frac{1}{b} \frac{\Delta H}{\Delta x}=\frac{1}{b} \frac{\partial H}{\partial x}
\]
and recalling that $y_{Y}$ is a function of $x$ defined by Eq. $(6.14),$ derive Eq. (6.15)

Surendra Kumar

Numerade Educator

Problem 61

Determine the location of the shear center $O$ of a thinwalled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 62

Determine the location of the shear center $O$ of a thinwalled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 63

Determine the location of the shear center $O$ of a thinwalled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 64

Determine the location of the shear center $O$ of a thinwalled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 65

An extruded beam has the cross section shown. Determine $(a)$ the location of the shear center $O,(b)$ the distribution of the shearing stresses caused by the vertical shearing force $\mathbf{V}$ shown applied at $O$

Chai Santi

Numerade Educator

Problem 66

An extruded beam has the cross section shown. Determine $(a)$ the location of the shear center $O,(b)$ the distribution of the shearing stresses caused by the vertical shearing force $\mathbf{V}$ shown applied at $O$

Chai Santi

Numerade Educator

Problem 67

An extruded beam has the cross section shown. Determine $(a)$ the location of the shear center $O,(b)$ the distribution of the shearing stresses caused by the vertical shearing force $\mathbf{V}$ shown applied at $O$

Chai Santi

Numerade Educator

Problem 68

An extruded beam has the cross section shown. Determine $(a)$ the location of the shear center $O,(b)$ the distribution of the shearing stresses caused by the vertical shearing force $\mathbf{V}$ shown applied at $O$

Chai Santi

Numerade Educator

Problem 69

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 70

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 71

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 72

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 73

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 74

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator

Problem 75

A thin-walled beam has the cross section shown. Determine the location of the shear center $O$ of the cross section.

Chai Santi

Numerade Educator

Problem 76

A thin-walled beam has the cross section shown. Determine the location of the shear center $O$ of the cross section.

Chai Santi

Numerade Educator

Problem 77

A thin-walled beam of uniform thickness has the cross section shown. Determine the dimension $b$ for which the shear center $O$ of the cross section is located at the point indicated.

Chai Santi

Numerade Educator

Problem 78

A thin-walled beam of uniform thickness has the cross section shown. Determine the dimension $b$ for which the shear center $O$ of the cross section is located at the point indicated.

Chai Santi

Numerade Educator

Problem 79

For the angle shape and loading of Sample Prob. $6.6,$ check that $\int q d z=0$ along the horizontal leg of the angle and $\int q d y=P$ along its vertical leg.

Chris Trentman

Numerade Educator

Problem 80

For the angle shape and loading of Sample Prob. $6.6,(a)$ determine the points where the shearing stress is maximum and the corresponding values of the stress, $(b)$ verify that the points obtained are located on the neutral axis corresponding to the given loading.

Bret Rosen

Numerade Educator

Problem 81

Determine the distribution of the shearing stresses along line $D^{\prime} B^{\prime}$ in the horizontal leg of the angle shape for the loading shown. The $x^{\prime}$ and $y^{\prime}$ axes are the principal centroidal axes of the cross section.

Surendra Kumar

Numerade Educator

Problem 82

For the angle shape and loading of Prob. 6.81 , determine the distribution of the shearing stresses along line $D^{\prime} A^{\prime}$ in the vertical leg.

Chai Santi

Numerade Educator

Problem 83

A steel plate, $160 \mathrm{mm}$ wide and $8 \mathrm{mm}$ thick, is bent to form the channel shown. Knowing that the vertical load $\mathbf{P}$ acts at a point in the midplane of the web of the channel, determine ( $a$ ) the torque $\mathbf{T}$ that would cause the channel to twist in the same way that it does under the load $\mathbf{P},(b)$ the maximum shearing stress in the channel caused by the load $\mathbf{P}$.

Hast Aggarwal

Numerade Educator

Problem 84

Solve Prob. $6.83,$ assuming that a 6 -mm-thick plate is bent to form the channel shown.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 85

The cantilever beam $A B,$ consisting of half of a thin-walled pipe of 1.25 -in. mean radius and $\frac{3}{8}$ -in. wall thickness, is subjected to a $500-1 b$ vertical load. Knowing that the line of action of the load passes through the centroid $C$ of the cross section of the beam, determine $(a)$ the equivalent force-couple system at the shear center of the cross section, $(b)$ the maximum shearing stress in the beam. (Hint: The shear center $O$ of this cross section was shown in Prob. 6.74 to be located twice as far from its vertical diameter as its centroid $C .)$

Naman Kumar

Numerade Educator

Problem 86

Solve Prob. $6.85,$ assuming that the thickness of the beam is reduced to $\frac{1}{4}$ in.

Vipender Yadav

Numerade Educator

Problem 87

The cantilever beam shown consists of a Z shape of $\frac{1}{4}$ -in. thickness. For the given loading, determine the distribution of the shearing stresses along line $A^{\prime} B^{\prime}$ in the upper horizontal leg of the Z shape. The $x^{\prime}$ and $y^{\prime}$ axes are the principal centroidal axes of the cross section, and the corresponding moments of inertia $\operatorname{are} I_{x^{\prime}}=166.3$ in $^{4}$ and $I_{y^{\prime}}=13.61$ in $^{4}$

Hast Aggarwal

Numerade Educator

Problem 88

For the cantilever beam and loading of Prob. 6.87 , determine the distribution of the shearing stresses along line $B^{\prime} D^{\prime}$ in the vertical web of the Z shape.

Surendra Kumar

Numerade Educator

Problem 89

Three boards are nailed together to form the beam shown, which is subjected to a vertical shear. Knowing that the spacing between the nails is $s=75 \mathrm{mm}$ and that the allowable shearing force in each nail is $400 \mathrm{N}$, determine the allowable shear when $w=120 \mathrm{mm}$

Hast Aggarwal

Numerade Educator

Problem 90

For the beam and loading shown, consider section $n-n$ and determine ( $a$ ) the largest shearing stress in that section, ( $b$ ) the shearing stress at point $a$

Chai Santi

Numerade Educator

Problem 91

For the wide-flange beam with the loading shown, determine the largest $\mathbf{P}$ that can be applied, knowing that the maximum normal stress is 24 ksi and the largest shearing stress, using the approximation $\tau_{m}=V / A_{\text {web }},$ is 14.5 ksi.

Chai Santi

Numerade Educator

Problem 92

For the beam and loading shown, consider section $n-n$ and determine the shearing stress at $(a)$ point $a$ (b) point $b$

Chai Santi

Numerade Educator

Problem 93

The built-up timber beam is subjected to a 1500 -lb vertical shear. Knowing that the longitudinal spacing of the nails is $s=2.5$ in. and that each nail is 3.5 in. long, determine the shearing force in each nail.

Ajay Singhal

Numerade Educator

Problem 94

Knowing that a given vertical shear $\mathbf{V}$ causes a maximum shearing stress of 75 MPa in the hat-shaped extrusion shown, determine the corresponding shearing stress at $(a)$ point $a,(b)$ point $b$

Chai Santi

Numerade Educator

Problem 95

Three planks are connected as shown by bolts of $14-\mathrm{mm}$ diameter spaced every $150 \mathrm{mm}$ along the longitudinal axis of the beam. For a vertical shear of $10 \mathrm{kN}$, determine the average shearing stress in the bolts.

Ajay Singhal

Numerade Educator

Problem 96

Three $1 \times 18$ -in. steel plates are bolted to four $\mathrm{L} 6 \times 6 \times 1$ angles to form a beam with the cross section shown. The bolts have a $\frac{7}{8}-$ in. diameter and are spaced longitudinally every 5 in. Knowing that the allowable average shearing stress in the bolts is 12 ksi, determine the largest permissible vertical shear in the beam. (Given: $I_{x}=6123$ in $^{4}$.)

Rashmi Sinha

Numerade Educator

Problem 97

The composite beam shown is made by welding $\mathrm{C} 200 \times 17.1$ rolled-steel channels to the flanges of a $\mathrm{W} 250 \times 80$ wide-flange rolled-steel shape. Knowing that the beam is subjected to a vertical shear of $200 \mathrm{kN}$, determine $(a)$ the horizontal shearing force per meter at each weld, ( $b$ ) the shearing stress at point $a$ of the flange of the wide-flange shape.

Hast Aggarwal

Numerade Educator

Problem 98

The design of a beam requires welding four horizontal plates to a vertical $0.5 \times 5$ -in. plate as shown. For a vertical shear $\mathbf{V}$, determine the dimension $h$ for which the shear flow through the welded surfaces is maximum.

Chai Santi

Numerade Educator

Problem 99

A thin-walled beam of uniform thickness has the cross section shown. Determine the dimension $b$ for which the shear center $O$ of the cross section is located at the point indicated.

Chai Santi

Numerade Educator

Problem 100

Determine the location of the shear center $O$ of a thin-walled beam of uniform thickness having the cross section shown.

Chai Santi

Numerade Educator