Mechanics of Materials

R. C. Hibbeler

Chapter 7

Transverse Shear - all with Video Answers

Educators

Chapter Questions

Problem 1

If the wide-flange beam is subjected to a shear of $V=20 \mathrm{kN},$ determine the shear stress on the web at $A$ Indicate the shear-stress components on a volume element located at this point.

Chai Santi

Chai Santi

Numerade Educator

Problem 2

If the wide-flange beam is subjected to a shear of $V=20 \mathrm{kN},$ determine the maximum shear stress in the beam.

Chai Santi

Numerade Educator

Problem 3

If the wide-flange beam is subjected to a shear of $V=20 \mathrm{kN},$ determine the shear force resisted by the web of the beam.

Chai Santi

Numerade Educator

Problem 4

If the $\mathrm{T}$ -beam is subjected to a vertical shear of $V=12$ kip, determine the maximum shear stress in the beam. Also, compute the shear-stress jump at the flangeweb junction $A B .$ Sketch the variation of the shear-stress intensity over the entire cross section.

Anand Jangid

Numerade Educator

Problem 5

If the T-beam is subjected to a vertical shear of $V=12$ kip, determine the vertical shear force resisted by the flange.

Chai Santi

Numerade Educator

Problem 6

The wood beam has an allowable shear stress of $\tau_{\text {allow }}=7 \mathrm{MPa}$. Determine the maximum shear force $V$ that can be applied to the cross section.

Chai Santi

Numerade Educator

Problem 7

The shaft is supported by a smooth thrust bearing at $A$ and a smooth journal bearing at $B .$ If $P=20 \mathrm{kN}$, determine the absolute maximum shear stress in the shaft.

Chai Santi

Numerade Educator

Problem 8

The shaft is supported by a smooth thrust bearing at $A$ and a smooth journal bearing at $B$. If the shaft is made from a material having an allowable shear stress of $\tau_{\text {allow }}=75 \mathrm{MPa}$, determine the maximum value for $P$.

Chai Santi

Numerade Educator

Problem 9

Determine the largest shear force $V$ that the member can sustain if the allowable shear stress is $\tau_{\text {allow }}=8$ $\mathrm{ksi}$.

Chai Santi

Numerade Educator

Problem 10

If the applied shear force $V=18$ kip, determine the maximum shear stress in the member.

Chai Santi

Numerade Educator

Problem 11

The overhang beam is subjected to the uniform distributed load having an intensity of $w=50 \mathrm{kN} / \mathrm{m}$ Determine the maximum shear stress developed in the beam.

Chai Santi

Numerade Educator

Problem 12

The beam has a rectangular cross section and is made of wood having an allowable shear stress of $\tau_{\text {allow }}=200$ psi. Determine the maximum shear force $V$ that can be developed in the cross section of the beam. Also, plot the shear-stress variation over the cross section.

Chai Santi

Numerade Educator

Problem 13

Determine the maximum shear stress in the strut if it is subjected to a shear force of $V=20 \mathrm{kN}$.

Chai Santi

Numerade Educator

Problem 14

Determine the maximum shear force $V$ that the
strut can support if the allowable shear stress for the material is $\tau_{\text {allow }}=40 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 15

The strut is subjected to a vertical shear of $V=130 \mathrm{kN}$ Plot the intensity of the shear-stress distribution acting over the cross-sectional area, and compute the resultant shear force developed in the vertical segment $A B$.

Chai Santi

Numerade Educator

Problem 16

The steel rod has a radius of 1.25 in. If it is subjected to a shear of $V=5$ kip, determine the maximum shear stress.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

If the beam is subjected to a shear of $V=15 \mathrm{kN}$ determine the web's shear stress at $A$ and $B$. Indicate the shear-stress components on a volume element located at these points. Set $w=125 \mathrm{mm}$. Show that the neutral axis is located at $\bar{y}=0.1747 \mathrm{m}$ from the bottom and $I_{N A}=0.2182\left(10^{-3}\right) \mathrm{m}^{4}$.

Chai Santi

Numerade Educator

Problem 18

If the wide-flange beam is subjected to a shear of $V=30 \mathrm{kN},$ determine the maximum shear stress in the beam. Set $w=200 \mathrm{mm}$.

Chai Santi

Numerade Educator

Problem 19

If the wide-flange beam is subjected to a shear of $V=30 \mathrm{kN},$ determine the shear force resisted by the web of the beam. Set $w=200 \mathrm{mm}$.

Chai Santi

Numerade Educator

Problem 20

The steel rod is subjected to a shear of 30 kip. Determine the maximum shear stress in the rod.

Chai Santi

Numerade Educator

Problem 21

If the beam is made from wood having an allowable shear stress $\tau_{\text {allow }}=400$ psi, determine the maximum magnitude of $\mathbf{P}$. Set $d=4$ in.

Chai Santi

Numerade Educator

Problem 22

Determine the shear stress at point $B$ on the web of the cantilevered strut at section $a-a$.

Chai Santi

Numerade Educator

Problem 23

Determine the maximum shear stress acting at section $a-a$ of the cantilevered strut.

Chai Santi

Numerade Educator

Problem 24

Determine the maximum shear stress in the T-beam at the critical section where the internal shear force is maximum.

Chai Santi

Numerade Educator

Problem 25

Determine the maximum shear stress in the T-beam at section $C .$ Show the result on a volume element at this point.

Chai Santi

Numerade Educator

Problem 26

The beam has a square cross section and is made of wood having an allowable shear stress of $\tau_{\text {allow }}=1.4 \mathrm{ksi}$. If it is subjected to a shear of $V=1.5$ kip, determine the smallest dimension $a$ of its sides.

Chai Santi

Numerade Educator

Problem 27

The beam is slit longitudinally along both sides as shown. If it is subjected to an internal shear of $V=250 \mathrm{kN}$ compare the maximum shear stress developed in the beam before and after the cuts were made.

Chai Santi

Numerade Educator

Problem 28

The beam is to be cut longitudinally along both sides as shown. If it is made from a material having an allowable shear stress of $\tau_{\text {allow }}=75 \mathrm{MPa}$, determine the maximum allowable internal shear force $\mathbf{V}$ that can be
applied before and after the cut is made.

Chai Santi

Numerade Educator

Problem 29

Write a computer program that can be used to determine the maximum shear stress in the beam that has the cross section shown, and is subjected to a specified constant distributed load $w$ and concentrated force $P .$ Show an application of the program using the values $L=4 \mathrm{m}$ $a=2 \mathrm{m}, \quad P=1.5 \mathrm{kN}, \quad d_{1}=0, \quad d_{2}=2 \mathrm{m}, \quad w=400 \mathrm{N} / \mathrm{m}$
\[
t_{1}=15 \mathrm{mm}, t_{2}=20 \mathrm{mm}, b=50 \mathrm{mm}, \text { and } h=150 \mathrm{mm}.
\]

Hast Aggarwal

Numerade Educator

Problem 30

The beam has a rectangular cross section and is subjected to a load $P$ that is just large enough to develop a fully plastic moment $M_{p}=P L$ at the fixed support. If the material is elastic-plastic, then at a distance $x<L$ the moment $M=P x$ creates a region of plastic yielding with an associated elastic core having a height $2 y^{\prime}$. This situation has been described by Eq. $6-30$ and the moment $\mathbf{M}$ is distributed over the cross section as shown in Fig. $6-48 e$ Prove that the maximum shear stress developed in the beam is given by $\tau_{\max }=\frac{3}{2}\left(P / A^{\prime}\right),$ where $A^{\prime}=2 y^{\prime} b,$ the cross-sectional area of the elastic core.

Chai Santi

Numerade Educator

Problem 31

The beam in Fig. $6-48 f$ is subjected to a fully plastic moment $\mathbf{M}_{p} .$ Prove that the longitudinal and transverse shear stresses in the beam are zero. Hint: Consider an
element of the beam as shown in Fig. $7-4 c$.

Chai Santi

Numerade Educator

Problem 32

The beam is constructed from two boards fastened together at the top and bottom with two rows of nails spaced every 6 in. If each nail can support a 500 -Ib shear force, determine the maximum shear force $V$ that can be applied to beam.

Chai Santi

Numerade Educator

Problem 33

The beam is construced from two boards fastened together at the top and bottom with two rows of nails spaced every 6 in. If an internal shear force of $V=600$ lb is applied to the boards, determine the shear force resisted by each nail.

Chai Santi

Numerade Educator

Problem 34

The boards are glued together to form the built-up beam. If the wood has an allowable shear stress of $\tau_{\text {allow }}=3 \mathrm{MPa}$ and the glue seam at $B$ can withstand a maximum shear stress of 1.5 MPa, determine the maximum allowable internal shear $\mathbf{V}$ that can be developed in the beam.

Chai Santi

Numerade Educator

Problem 35

The boards are glued together to form the built-up beam. If the wood has an allowable shear stress of $\tau_{\text {allow }}=$ $3 \mathrm{MPa}$, and the glue seam at $D$ can withstand a maximum shear stress of $1.5 \mathrm{MPa}$, determine the maximum allowable shear $\mathbf{V}$ that can be developed in the beam.

Chai Santi

Numerade Educator

Problem 36

Three identical boards are bolted together to form the built-up beam. Each bolt has a shear strength of 1.5 kip and the bolts are spaced at a distance of $s=6$ in. If the wood has an allowable shear stress of $\tau_{\text {allow }}=450 \mathrm{psi}$, determine the maximum allowable internal shear $\mathbf{V}$ that can act on the beam.

Anand Jangid

Numerade Educator

Problem 37

Three identical boards are bolted together to form the built-up beam. If the wood has an allowable shear stress of $\tau_{\text {allow }}=450$ psi, determine the maximum allowable internal shear $\mathbf{V}$ that can act on the beam. Also, find the corresponding average shear stress in the $\frac{3}{8}$ in. diameter bolts which are spaced equally at $s=6$ in.

Anand Jangid

Numerade Educator

Problem 38

The beam is subjected to a shear of $V=2 \mathrm{kN}$ Determine the average shear stress developed in each nail if the nails are spaced $75 \mathrm{mm}$ apart on each side of the beam. Each nail has a diameter of $4 \mathrm{mm}$.

Chai Santi

Numerade Educator

Problem 39

A beam is constructed from three boards bolted together as shown. Determine the shear force developed in each bolt if the bolts are spaced $s=250 \mathrm{mm}$ apart and the applied shear is $V=35 \mathrm{kN}$.

Chai Santi

Numerade Educator

Problem 40

The simply-supported beam is built-up from three boards by nailing them together as shown. The wood has an allowable shear stress of $\tau_{\text {allow }}=1.5 \mathrm{MPa}$ and an allowable bending stress of $\sigma_{\text {allow }}=9$ MPa. The nails are spaced at $s=75 \mathrm{mm},$ and each has a shear strength of 1.5 kN. Determine the maximum allowable force $\mathbf{P}$ that can be applied to the beam.

Chai Santi

Numerade Educator

Problem 41

The simply-supported beam is built-up from three boards by nailing them together as shown. If $P=12 \mathrm{kN}$ determine the maximum allowable spacing $s$ of the nails to support that load, if each nail can resist a shear force of $1.5 \mathrm{kN}$.

Chai Santi

Numerade Educator

Problem 42

The T-beam is constructed as shown. If the nails can each support a shear force of $950 \mathrm{lb}$, determine the maximum shear force $V$ that the beam can support and the corresponding maximum nail spacing $s$ to the nearest $\frac{1}{8}$ in. The allowable shear stress for the wood is $\tau_{\text {allow }}=450 \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 43

The box beam is made from four pieces of plastic that are glued together as shown. If the glue has an allowable strength of $400 \mathrm{lb} / \mathrm{in}^{2}$, determine the maximum shear the beam will support.

Chai Santi

Numerade Educator

Problem 44

The box beam is made from four pieces of plastic that are glued together as shown. If $V=2$ kip, determine the shear stress resisted by the seam at each of the glued joints.

Chai Santi

Numerade Educator

Problem 45

A beam is constructed from four boards which are nailed together. If the nails are on both sides of the beam and each can resist a shear of $3 \mathrm{kN}$, determine the maximum load $P$ that can be applied to the end of the beam.

Chai Santi

Numerade Educator

Problem 46

The beam is subjected to a shear of $V=800 \mathrm{N}$ Determine the average shear stress developed in the nails along the sides $A$ and $B$ if the nails are spaced $s=100 \mathrm{mm}$ apart. Each nail has a diameter of $2$ $\mathrm{mm}$.

Chai Santi

Numerade Educator

Problem 47

The beam is made from four boards nailed together as shown. If the nails can each support a shear force of $100 \mathrm{Ib},$ determine their required spacing $s^{\prime}$ and $s$ if the beam is subjected to a shear of $V=700 \mathrm{lb}$.

Chai Santi

Numerade Educator

Problem 48

The beam is made from three polystyrenc strips that are glued together as shown. If the glue has a shear strength of $80 \mathrm{kPa}$, determine the maximum load $P$ that can be applied without causing the glue to lose its bond.

Chai Santi

Numerade Educator

Problem 49

The timber T-beam is subjected to a load consisting of $n$ concentrated forces, $P_{n-}$ If the allowable shear $V_{\text {nail for }}$ each of the nails is known, write a computer program that will specify the nail spacing between each load. Show an application of the program using the values $L=15 \mathrm{ft}$
\[
\begin{array}{l}
a_{1}=4 \mathrm{ft}, P_{1}=600 \mathrm{lb}, a_{2}=8 \mathrm{ft}, P_{2}=1500 \mathrm{lb}, b_{1}=1.5 \mathrm{in} . \\
h_{1}=10 \mathrm{in} ., b_{2}=8 \mathrm{in}, h_{2}=1 \mathrm{in} ., \text { and } V_{\mathrm{mil}}=200 \mathrm{lb}.
\end{array}
\]

Hast Aggarwal

Numerade Educator

Problem 50

A shear force of $V=300 \mathrm{kN}$ is applied to the box girder. Determine the shear flow at points $A$ and $B$.

Chai Santi

Numerade Educator

Problem 51

A shear force of $V=450 \mathrm{kN}$ is applied to the box girder. Determine the shear flow at points $C$ and $D$.

Chai Santi

Numerade Educator

Problem 52

A shear force of $V=18 \mathrm{kN}$ is applied to the symmetric box girder. Determine the shear flow at $A$ and $B$.

Chai Santi

Numerade Educator

Problem 53

A shear force of $V=18 \mathrm{kN}$ is applied to the box girder. Determine the shear flow at $C$.

Chai Santi

Numerade Educator

Problem 54

The aluminum strut is $10 \mathrm{mm}$ thick and has the cross section shown. If it is subjected to a shear of, $V=150 \mathrm{N}$ determine the shear flow at points $A$ and $B$.

Chai Santi

Numerade Educator

Problem 55

The aluminum strut is $10 \mathrm{mm}$ thick and has the cross section shown. If it is subjected to a shear of $V=150 \mathrm{N}$. determine the maximum shear flow in the strut.

Chai Santi

Numerade Educator

Problem 56

The beam is subjected to a shear force of $V=5$ kip. Determine the shear flow at points $A$ and $B$.

Chai Santi

Numerade Educator

Problem 57

The beam is constructed from four plates and is subjected to a shear force of $V=5$ kip. Determine the maximum shear flow in the cross section.

Chai Santi

Numerade Educator

Problem 58

The channel is subjected to a shear of $V=75 \mathrm{kN}$ Determine the shear flow developed at point $A$.

Chai Santi

Numerade Educator

Problem 59

The channel is subjected to a shear of $V=75 \mathrm{kN}$. Determine the maximum shear flow in the channel.

Chai Santi

Numerade Educator

Problem 60

The built-up beam is formed by welding together the thin plates of thickness $5 \mathrm{mm}$. Determine the location of the shear center $O$.

Chai Santi

Numerade Educator

Problem 61

The assembly is subjected to a vertical shear of $V=7$ kip. Determine the shear flow at points $A$ and $B$ and the maximum shear flow in the cross section.

Chai Santi

Numerade Educator

Problem 62

Determine the shear-stress variation over the cross section of the thin-walled tube as a function of elevation $y$ and show that $\tau_{\max }=2 V / A,$ where $A=2 \pi \pi .$ Hint: Choose a differential area element $d A=R t d \theta .$ Using $d Q=y d A$ formulate $Q$ for a circular section from $\theta$ to $(\pi-\theta)$ and show that $Q=2 R^{2} t \cos \theta,$ where $\cos \theta=\sqrt{R^{2}-y^{2}} / R$.

Chai Santi

Numerade Educator

Problem 63

Determine the location $e$ of the shear center, point $O,$ for the thin-walled member having the cross section shown. The member segments have the same thickness $t$.

Chai Santi

Numerade Educator

Problem 64

Determine the location $e$ of the shear center, point $O,$ for the thin-walled member having the cross section shown. The member segments have the same thickness $t$.

Chai Santi

Numerade Educator

Problem 65

The beam supports a vertical shear of $V=7$ kip. Determine the resultant force developed in segment $A B$ of the beam.

Chai Santi

Numerade Educator

Problem 66

The built-up beam is fabricated from the three thin plates having a thickness $t$. Determine the location of the shear center $O$.

Chai Santi

Numerade Educator

Problem 67

Determine the location $e$ of the shear center, point $O$ for the thin-walled member having the cross section shown. The member segments have the same thickness $t$.

Chai Santi

Numerade Educator

Problem 68

A thin plate of thickness $t$ is bent to form the beam having the cross section shown. Determine the location of the shear center $O$.

Chai Santi

Numerade Educator

Problem 69

A thin plate of thickness $t$ is bent to form the beam having the cross section shown. Determine the location of the shear center $O$.

Chai Santi

Numerade Educator

Problem 70

Determine the location $e$ of the shear center, point $O$ for the thin-walled member having the cross section shown.

Chai Santi

Numerade Educator

Problem 71

The beam is fabricated from four boards nailed together as shown. Determine the shear force each nail along the sides $C$ and the top $D$ must resist if the nails are uniformly spaced at $s=3$ in. The beam is subjected to a shear of $V=4.5$ kip.

Chai Santi

Numerade Educator

Problem 72

The T-beam is subjected to a shear of $V=150 \mathrm{kN}$ Determine the amount of this force that is supported by the web $B$.

Chai Santi

Numerade Educator

Problem 73

$7-73 .$ The member is subject to a shear force of $V=2 \mathrm{kN} .$ Determine the shear flow at points $A, B,$ and $C$. The thickness of each thin-walled segment in $15 \mathrm{mm}$.

Chai Santi

Numerade Educator

Problem 74

Determine the shear stress at points $B$ and $C$ on the web of the beam located at section $a-a$.

Chai Santi

Numerade Educator

Problem 75

Determine the maximum shear stress a section $a-a$ in the beam.

Chai Santi

Numerade Educator