Mechanics of Materials

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

Chapter 10

Columns - all with Video Answers

Educators

Chapter Questions

Problem 1

Knowing that the spring at $A$ is of constant $k$ and that the bar $A B$ is rigid, determine the critical load $P_{c r}$.

Chai Santi

Chai Santi

Numerade Educator

Problem 2

Two rigid bars $A C$ and $B C$ are connected by a pin at $C$ as shown. Knowing that the torsional spring at $B$ is of constant $K$, determine the critical load $P_{\alpha}$ for the system.

Chai Santi

Numerade Educator

Problem 3

Two rigid bars $A C$ and $B C$ are connected as shown to a spring of constant $k .$ Knowing that the spring can act in either tension or compression, determine the critical load $P_{c x}$ for the system.

Chai Santi

Numerade Educator

Problem 4

Two rigid bars $A C$ and $B C$ are connected as shown to a spring of constant $k .$ Knowing that the spring can act in either tension or compression, determine the critical load $P_{c x}$ for the system.

Chai Santi

Numerade Educator

Problem 5

The steel rod $B C$ is attached to the rigid bar $A B$ and to the fixed support at $C .$ Knowing that $G=11.2 \times 10^{6}$ psi, determine the diameter of rod $B C$ for which the critical load $P_{\mathrm{cr}}$ of the system is $80 \mathrm{lb}$.

Narayan Hari

Numerade Educator

Problem 6

The rigid rod $A B$ is attached to a hinge at $A$ and to two springs, each of constant $k=2$ kips/in., that can act in either tension or compression. Knowing that $h=2 \mathrm{ft}$, determine the critical load.

Surendra Kumar

Numerade Educator

Problem 7

The rigid bar $A D$ is attached to two springs of constant $k$ and is in equilibrium in the position shown. Knowing that the equal and opposite loads $\mathrm{P}$ and $\mathrm{P}^{\prime}$ remain horizontal, determine the magnitude $P_{c r}$ of the critical load for the system.

Chai Santi

Numerade Educator

Problem 8

A frame consists of four L-shaped members connected by four torsional springs, each of constant $K .$ Knowing that equal loads $\mathbf{P}$ are applied at points $A$ and $D$ as shown, determine the critical value $P_{c r}$ of the loads applied to the frame.

Chai Santi

Numerade Educator

Problem 9

Determine the critical load of a pin-ended steel tube that is $5 \mathrm{m}$ long and has a 100 -mm outer diameter and a 16 -mm wall thick. ness. Use $E=200 \mathrm{GPa}$.

Satpal Satpal

Numerade Educator

Problem 10

Determine the critical load of a pin-ended wooden stick that is
3 ft long and has a $\frac{3}{16} \times 1 \frac{1}{4}$ -in. rectangular cross section. Use $E=1.6 \times 10^{6} \mathrm{psi}$.

Satpal Satpal

Numerade Educator

Problem 11

A column of effective length $L$ can be made by gluing together identical planks in either of the arrangements shown. Determine the ratio of the critical load using the arrangement $a$ to the critical load using the arrangement $b$.

Satpal Satpal

Numerade Educator

Problem 12

A compression member of 1.5 -m effective length consists of a solid 30 -mm-diameter brass rod. In order to reduce the weight of the member by $25 \%$, the solid rod is replaced by a hollow rod of the cross section shown. Determine $(a)$ the percent reduction in the critical load, ( $b$ ) the value of the critical load for the hollow rod. Use $E=200 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 13

Determine the radius of the round strut so that the round and square struts have the same cross-sectional area and compute the critical load of each strut. Use $E=200 \mathrm{GPa}$.

Surendra Kumar

Numerade Educator

Problem 14

Determine $(a)$ the critical load for the square strut, $(b)$ the radius of the round strut for which both struts have the same critical $\frac{1}{4}$ load. (c) Express the cross-sectional area of the square strut as a percentage of the cross-sectional area of the round strut. Use $E=200 \mathrm{GPa}$.

Surendra Kumar

Numerade Educator

Problem 15

A column is made from half of a $\mathrm{W} 360 \times 216$ rolled-steel shape, with the geometric properties as shown. Using a factor of safety equal to $2.6,$ determine the allowable centric load if the effective length of the column is $6.5 \mathrm{m}$. Use $E=200 \mathrm{GPa}$.

Satpal Satpal

Numerade Educator

Problem 16

A column is made from half of a $\mathrm{W} 360 \times 216$ rolled-steel shape, with the geometric properties as shown. Using a factor of safety equal to $2.6,$ determine the allowable centric load if the effective length of the column is $6.5 \mathrm{m}$. Use $E=200 \mathrm{GPa}$.

Satpal Satpal

Numerade Educator

Problem 17

A column of 22 -ft effective length is made by welding two $9 \times 0.5$ -in. plates to a $\mathrm{W} 8 \times 35$ as shown. Determine the allowable centric load if a factor of safety of 2.3 is required. Use $E=29 \times 10^{6}$ psi.

Penny Riley

Numerade Educator

Problem 18

A single compression member of 8.2 -m effective length is obtained by connecting two $\mathrm{C} 200 \times 17.1$ steel channels with lacing bars as shown. Knowing that the factor of safety is $1.85,$ determine the allowable centric load for the member. Use $E=200 \mathrm{GPa}$ and $d=100 \mathrm{mm}$.

Anand Jangid

Numerade Educator

Problem 19

Knowing that $P=5.2 \mathrm{kN}$, determine the factor of safety for the structure shown. Use $E=200 \mathrm{GPa}$ and consider only buckling in the plane of the structure.

Surendra Kumar

Numerade Educator

Problem 20

Members $A B$ and $C D$ are 30 -mm-diameter steel rods, and members $B C$ and $A D$ are 22 -mm-diameter steel rods. When the turnbuckle is tightened, the diagonal member $A C$ is put in tension. Knowing that a factor of safety with respect to buckling of 2.75 is required, determine the largest allowable tension in $A C$. Use $E=200 \mathrm{GPa}$ and consider only buckling in the plane of the structure.

Anand Jangid

Numerade Educator

Problem 21

The uniform brass bar $A B$ has a rectangular cross section and is supported by pins and brackets as shown. Each end of the bar can rotate freely about a horizontal axis through the pin, but rotation about a vertical axis is prevented by the brackets. (a) Determine the ratio $b / d$ for which the factor of safety is the same about the horizontal and vertical axes.
(b) Determine the factor of safety if $P=1.8$ kips, $L=7 \mathrm{ft}, d=1.5$ in., and $E=29 \times 10^{6} \mathrm{psi}$.

Anand Jangid

Numerade Educator

Problem 22

A 1 -in.-square aluminum strut is maintained in the position shown by a pin support at $A$ and by sets of rollers at $B$ and $C$ that prevent rotation of the strut in the plane of the figure. Knowing that $L_{A B}=3 \mathrm{ft},$ determine $(a)$ the largest values of $L_{B C}$ and $L_{C D}$ that can be used if the allowable load $\mathbf{P}$ is to be as large as possible,
(b) the magnitude of the corresponding allowable load. Consider only buckling in the plane of the figure and use $E=10.4 \times 10^{6} \mathrm{psi}$.

Rashmi Sinha

Numerade Educator

Problem 23

A 1 -in.-square aluminum strut is maintained in the position shown by a pin support at $A$ and by sets of rollers at $B$ and $C$ that prevent rotation of the strut in the plane of the figure. Knowing that $L_{A B}=3 \mathrm{ft}, L_{B C}=4 \mathrm{ft},$ and $L_{C D}=1 \mathrm{ft},$ determine the allowable load $\mathbf{P}$ using a factor of safety with respect to buckling of 3.2 Consider only buckling in the plane of the figure and use $E=10.4 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 24

Column $A B C$ has a uniform rectangular cross section with $b=12 \mathrm{mm}$ and $d=22 \mathrm{mm} .$ The column is braced in the $x z$ plane at its midpoint $C$ and carries a centric load $\mathbf{P}$ of magnitude $3.8 \mathrm{kN} .$ Knowing that a factor of safety of 3.2 is required, determine the largest allowable length $L$. Use $E=200$ GPa.

Chai Santi

Numerade Educator

Problem 25

Column $A B C$ has a uniform rectangular cross section and is braced in the $x z$ plane at its midpoint $C$. (a) Determine the ratio $b / d$ for which the factor of safety is the same with respect to buckling in the $x z$ and $y z$ planes.
(b) Using the ratio found in part $a$ design the cross section of the column so that the factor of safety will be 3.0 when $P=4.4 \mathrm{kN}, L=1 \mathrm{m},$ and $E=200 \mathrm{GPa}$.

Anand Jangid

Numerade Educator

Problem 26

Column $A B$ carries a centric load $\mathbf{P}$ of magnitude 15 kips. Cables $B C$ and $B D$ are taut and prevent motion of point $B$ in the $x z$ plane. Using Euler's formula and a factor of safety of $2.2,$ and neglecting the tension in the cables, determine the maximum allowable length $L$. Use $E=29 \times 10^{6}$ psi.

Vipender Yadav

Numerade Educator

Problem 27

Each of the five struts shown consists of a solid steel rod.
(a) Knowing that the strut of Fig. (1) is of a 20 -mm diameter, determine the factor of safety with respect to buckling for the loading shown. ( $b$ ) Determine the diameter of each of the other struts for which the factor of safety is the same as the factor of safety obtained in part $a .$ Use $E=200 \mathrm{GPa}$.

Satpal Satpal

Numerade Educator

Problem 28

A rigid block of mass $m$ can be supported in each of the four ways shown. Each column consists of an aluminum tube that has a $44-\mathrm{mm}$ outer diameter and a $4-\mathrm{mm}$ wall thickness. Using $E=70 \mathrm{GPa}$ and a factor of safety of $2.8,$ determine the allowable mass for each support condition.

Penny Riley

Numerade Educator

Problem 29

An axial load $\mathbf{P}=15 \mathrm{kN}$ is applied at point $D$ that is $4 \mathrm{mm}$ from the geometric axis of the square aluminum bar $B C$. Using $E=70 \mathrm{GPa},$ determine $(a)$ the horizontal deflection of end $C$
(b) the maximum stress in the column.

Chai Santi

Numerade Educator

Problem 30

An axial load $\mathbf{P}$ is applied to the 32 -mm-diameter steel rod $A B$ as shown. For $P=37 \mathrm{kN}$ and $e=1.2 \mathrm{mm},$ determine $(a)$ the deflection at the midpoint $C$ of the rod, $(b)$ the maximum stress in the rod. Use $E=200 \mathrm{GPa}$.

Narayan Hari

Numerade Educator

Problem 31

The line of action of the 310 -kN axial load is parallel to the geometric axis of the column $A B$ and intersects the $x$ axis at $x=e$ Using $E=200 \mathrm{GPa}$, determine (a) the eccentricity $e$ when the deflection of the midpoint $C$ of the column is $9 \mathrm{mm}$, ( $b$ ) the maximum stress in the column.

Chai Santi

Numerade Educator

Problem 32

An axial load $\mathbf{P}$ is applied to the 1.375 -in. diameter steel rod $A B$ as shown. When $P=21$ kips, it is observed that the horizontal deflection at midpoint $C$ is 0.03 in. Using $E=29 \times 10^{6} \mathrm{psi}$, determine $(a)$ the eccentricity $e$ of the load, $(b)$ the maximum stress in the rod.

Narayan Hari

Numerade Educator

Problem 33

An axial load $\mathbf{P}$ is applied to the 32 -mm-square aluminum bar $B C$ as shown. When $P=24 \mathrm{kN}$, the horizontal deflection at end $C$ is $4 \mathrm{mm} .$ Using $E=70 \mathrm{GPa}$, determine $(a)$ the eccentricity $e$ of the load,
(b) the maximum stress in the bar.

Chai Santi

Numerade Educator

Problem 34

The axial load $\mathbf{P}$ is applied at a point located on the $x$ axis at a distance $e$ from the geometric axis of the rolled-steel column $B C$ When $P=82$ kips, the horizontal deflection of the top of the column is 0.20 in. Using $E=29 \times 10^{6}$ psi, determine ( $a$ ) the eccentricity $e$ of the load, $(b)$ the maximum stress in the column.

Chai Santi

Numerade Educator

Problem 35

An axial load $\mathbf{P}$ is applied at point $D$ that is 0.25 in. from the geometric axis of the square aluminum bar $B C$. Using $E=10.1 \times$ $10^{6} \mathrm{psi}$, determine $(a)$ the load $\mathbf{P}$ for which the horizontal deflection of end $C$ is 0.50 in.
(b) the corresponding maximum stress in the column.

Chai Santi

Numerade Educator

Problem 36

A brass pipe having the cross section shown has an axial load $\mathbf{P}$ applied $5 \mathrm{mm}$ from its geometric axis. Using $E=120 \mathrm{GPa}$, determine $(a)$ the load $\mathbf{P}$ for which the horizontal deflection at the midpoint $C$ is $5 \mathrm{mm},(b)$ the corresponding maximum stress in the column.

Chai Santi

Numerade Educator

Problem 37

Solve Prob. $10.36,$ assuming that the axial load $\mathbf{P}$ is applied $10 \mathrm{mm}$ from the geometric axis of the column.

Shoukat Ali

Other Schools

Problem 38

The line of action of the axial load $\mathbf{P}$ is parallel to the geometric axis of the column $A B$ and intersects the $x$ axis at $x=0.8$ in. Using $E=29 \times 10^{6} \mathrm{psi}$, determine (a) the load $\mathbf{P}$ for which the horizontal deflection at the end $C$ is 0.5 in.
(b) the corresponding maximum stress in the column.

Naman Kumar

Numerade Educator

Problem 39

The line of action of the axial load $\mathbf{P}$ is parallel to the geometric axis of the column and applied at a point located on the $x$ axis at a distance $e=12 \mathrm{mm}$ from the geometric axis of the $\mathrm{W} 310 \times 60$ rolled-steel column $B C$. Assuming that $L=7.0 \mathrm{m}$ and using $E=200 \mathrm{GPa},$ determine $(a)$ the load $\mathbf{P}$ for which the horizontal deflection of the midpoint $C$ of the column is $15 \mathrm{mm}$
$(b)$ the corresponding maximum stress in the column.

Chai Santi

Numerade Educator

Problem 40

Solve Prob. 10.39 , assuming that $L$ is $9.0 \mathrm{m}$.

Sheila Akinleye

Sheila Akinleye

Numerade Educator

Problem 41

The steel bar $A B$ has a $\frac{3}{4} \times \frac{3}{6}$ -in. square cross section and is held by pins that are a fixed distance apart and are located at a distance $e=0.03$ in. from the geometric axis of the bar. Knowing that at temperature $T_{0}$ the pins are in contact with the bar and that the force in the bar is zero, determine the increase in temperature for which the bar will just make contact with point $C$ if $d=0.01$ in. Use $E=29 \times 10^{6} \mathrm{psi}$ and a coefficient of thermal expansion $\alpha=6.5 \times 10^{-6} /^{\circ} \mathrm{F}$.

Chai Santi

Numerade Educator

Problem 42

For the bar of Prob. 10.41 , determine the required distance $d$ for which the bar will just make contact with point $C$ when the temperature increases by $120^{\circ} \mathrm{F}$.

Manik Pulyani

Numerade Educator

Problem 43

A 3.5 -m-long steel tube having the cross section and properties shown is used as a column. For the grade of steel used $\sigma_{Y}=$ $250 \mathrm{MPa}$ and $E=200 \mathrm{GPa} .$ Knowing that a factor of safety of 2.6 with respect to permanent deformation is required, determine the allowable load $\mathbf{P}$ when the eccentricity $e$ is $(a) 15 \mathrm{mm},(b)$ $7.5 \mathrm{mm}$. (Hint: since the factor of safety must be applied to the load $\mathbf{P},$ not to the stress, use Fig. 10.24 to determine $P_{Y}$ ).

Ameer Said

Numerade Educator

Problem 44

Solve Prob. 10.43 , assuming that the length of the tube is increased to $5 \mathrm{m}$.

James Kiss

Numerade Educator

Problem 45

An axial load $\mathbf{P}$ is applied to the $\mathrm{W} 8 \times 28$ rolled-steel column $B C$ that is free at its top $C$ and fixed at its base $B$. Knowing that the eccentricity of the load is $e=0.6$ in. and that for the grade of steel used $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$, determine (a) the magnitude of $P$ of the allowable load when a factor of safety of 2.5 with respect to permanent deformation is required, ( $b$ ) the ratio of the load found in part $a$ to the magnitude of the allowable centric load for the column. (See hint of Prob. $10.43 .$ )

Chai Santi

Numerade Educator

Problem 46

An axial load $\mathbf{P}$ of magnitude 50 kips is applied at a point located on the $x$ axis at a distance $e=0.25$ in. from the geometric axis of the $\mathrm{W} 8 \times 28$ rolled-steel column $B C$. Knowing that the column is free at its top $C$ and fixed at its base $B$ and that $\sigma_{Y}=36$ ksi and $E=29 \times 10^{6} \mathrm{psi},$ determine the factor of safety with respect to yield. (See hint of Prob. $10.43 .)$

Chai Santi

Numerade Educator

Problem 47

A 100 -kN axial load $\mathbf{P}$ is applied to the $\mathrm{W} 150 \times 18$ rolled-steel column $B C$ that is free at its top $C$ and fixed at its base $B$. Knowing that the eccentricity of the load is $e=6 \mathrm{mm}$, determine the largest permissible length $L$ if the allowable stress in the column is 80 MPa. Use $E=200 \mathrm{GPa}$.

Surendra Kumar

Numerade Educator

Problem 48

A $26-$ kip axial load $\mathbf{P}$ is applied to a $\mathrm{W} 6 \times 12$ rolled-steel column $B C$ that is free at its top $C$ and fixed at its base $B$. Knowing that the eccentricity of the load is $e=0.25$ in., determine the largest permissible length $L$ if the allowable stress in the column is 14 ksi. Use $E=29 \times 10^{6}$ psi.

Surendra Kumar

Numerade Educator

Problem 49

Axial loads of magnitude $P=135$ kips are applied parallel to the geometric axis of the $\mathrm{W} 10 \times 54$ rolled-steel column $A B$ and intersect the $x$ axis at a distance $e$ from the geometric axis. Knowing that $\sigma_{\text {all }}=12$ ksi and $E=29 \times 10^{6}$ psi, determine the largest permissible length $L$ when $(a) e=0.25$ in.
$(b) e=0.5$ in.

Narayan Hari

Numerade Educator

Problem 50

Axial loads of magnitude $P=84 \mathrm{kN}$ are applied parallel to the geometric axis of the $\mathrm{W} 200 \times 22.5$ rolled-steel column $A B$ and intersect the $x$ axis at a distance $e$ from the geometric axis. Knowing that $\sigma_{\text {all }}=75$ MPa and $E=200$ GPa, determine the largest permissible length $L$ when $(a) e=5 \mathrm{mm}$
(b) $e=12 \mathrm{mm}$.

Naman Kumar

Numerade Educator

Problem 51

An axial load of magnitude $P=220 \mathrm{kN}$ is applied at a point located on the $x$ axis at a distance $e=6 \mathrm{mm}$ from the geometric axis of the wide-flange column $B C$. Knowing that $E=200 \mathrm{GPa}$ choose the lightest $W 200$ shape that $\operatorname{can}$ be used if $\sigma_{\text {all }}=$ $120 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 52

Solve Prob. 10.51 , assuming that the magnitude of the axial load is $P=345 \mathrm{kN}$.

Vidhi Bhatt

Numerade Educator

Problem 53

A 12 -kip axial load is applied with an eccentricity $e=0.375$ in. to the circular steel rod $B C$ that is free at its top $C$ and fixed at its base
$B$. Knowing that the stock of rods available for use have diameters in increments of $\frac{1}{8}$ in. from 1.5 in. to 3.0 in., determine the lightest rod that can be used if $\sigma_{\mathrm{all}}=15 \mathrm{ksi}$. Use $E=29 \times 10^{6} \mathrm{psi}$.

Anand Jangid

Numerade Educator

Problem 54

Solve Prob. $10.53,$ assuming that the 12 -kip axial load will be applied to the rod with an eccentricity $e=\frac{1}{2} d$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 55

Axial loads of magnitude $P=175 \mathrm{kN}$ are applied parallel to the geometric axis of a $\mathrm{W} 250 \times 44.8$ rolled-steel column $A B$ and intersect the $x$ axis at a distance $e=12 \mathrm{mm}$ from its geometric axis. Knowing that $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$, determine the factor of safety with respect to yield. (Hint: since the factor of safety must be applied to the load $\mathbf{P},$ not to the stresses, use Fig. 10.24 to determine $P_{Y \cdot}$)

Satpal Satpal

Numerade Educator

Problem 56

Solve Prob. $10.55,$ assuming that $e=16 \mathrm{mm}$ and $P=155 \mathrm{kN}$.

Ashley High

Numerade Educator

Problem 57

Using allowable stress design, determine the allowable centric load for a column of 6 -m effective length that is made from the following rolled-steel shape: $(a) \mathrm{W} 200 \times 35.9,(b) \mathrm{W} 200 \times 86 .$ Use $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

A $\mathrm{W} 8 \times 31$ rolled-steel shape is used for a column of 21 -ft effective length. Using allowable stress design, determine the allowable centric load if the yield strength of the grade of steel used is
$(a) \sigma_{Y}=36 \mathrm{ksi}$
$(b) \sigma_{Y}=50$ ksi. Use $E=29 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 59

A rectangular structural tube having the cross section shown is used as a column of 5 -m effective length. Knowing that $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$, use allowable stress design to determine the largest centric load that can be applied to the steel colum.

Satpal Satpal

Numerade Educator

Problem 60

A column having a 3.5 -m effective length is made of sawn lumber with a $114 \times 140$ -mm cross section. Knowing that for the grade of wood used the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=7.6 \mathrm{MPa}$ and the adjusted modulus $E=2.8 \mathrm{GPa},$ determine the maximum allowable centric load for the column.

Satpal Satpal

Numerade Educator

Problem 61

A sawn lumber column with a $7.5 \times 5.5$ -in. cross section has an 18-ft effective length. Knowing that for the grade of wood used the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=1200$ psi and that the adjusted modulus $E=470 \times 10^{3}$ psi, determine the maximum allowable centric load for the column.

Hast Aggarwal

Numerade Educator

Problem 62

Bar $A B$ is free at its end $A$ and fixed at its base $B$. Determine the allowable centric load $\mathbf{P}$ if the aluminum alloy is $(a) 6061-\mathrm{T} 6$
(b) $2014-\mathrm{T} 6$.

Satpal Satpal

Numerade Educator

Problem 63

A compression member has the cross section shown and an effective length of $5 \mathrm{ft}$. Knowing that the aluminum alloy used is $2014-\mathrm{T} 6,$ determine the allowable centric load.

Surendra Kumar

Numerade Educator

Problem 64

A compression member has the cross section shown and an effective length of 5 ft. Knowing that the aluminum alloy used is $6061-\mathrm{T} 6,$ determine the allowable centric load.

Surendra Kumar

Numerade Educator

Problem 65

A compression member of 8.2 -ft effective length is obtained by bolting together two $\mathrm{L} 5 \times 3 \times \frac{1}{2}$ -in. steel angles as shown. Using allowable stress design, determine the allowable centric load for the column. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 66

A compression member of $9-\mathrm{m}$ effective length is obtained by welding two 10 -mm-thick steel plates to a $\mathrm{W} 250 \times 80$ rolled-steel shape as shown. Knowing that $\sigma_{Y}=345 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$ and using allowable stress design, determine the allowable centric load for the compression member.

Satpal Satpal

Numerade Educator

Problem 67

A column of 6.4 -m effective length is obtained by connecting four L89 $\times 89 \times 9.5$ -mm steel angles with lacing bars as shown. Using allowable stress design, determine the allowable centric load for the column. Use $\sigma_{Y}=345 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Anand Jangid

Numerade Educator

Problem 68

A column of 21 -ft effective length is obtained by connecting $\mathrm{C} 10 \times 20$ steel channels with lacing bars as shown. Using allowable stress design, determine the allowable centric load for the column. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Surendra Kumar

Numerade Educator

Problem 69

The glued laminated column shown is made from four planks, each of $38 \times 190$ -mm cross section. Knowing that for the grade of wood used the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=10 \mathrm{MPa}$ and $E=12 \mathrm{GPa}$, determine the maximum allowable centric load if the effective length of the column is $(a) 7 \mathrm{m},(b) 3 \mathrm{m}$.

Satpal Satpal

Numerade Educator

Problem 70

An aluminum structural tube is reinforced by bolting two plates to it as shown for use as a column of 1.7 -m effective length. Knowing that all material is aluminum alloy $2014-\mathrm{T} 6,$ determine the maximum allowable centric load.

Surendra Kumar

Numerade Educator

Problem 71

The glued laminated column shown is free at its top $A$ and fixed at its base
B. Using wood that has an adjusted allowable stress for compression parallel to the grain $\sigma_{C}=9.2 \mathrm{MPa}$ and an adjusted modulus of elasticity $E=5.7 \mathrm{GPa}$, determine the smallest cross section that can support a centric load of $62 \mathrm{kN}$.

Satpal Satpal

Numerade Educator

Problem 72

An 18 -kip centric load is applied to a rectangular sawn lumber column of 22 -ft effective length. Using lumber for which the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=1050$ psi and the adjusted modulus is $E=440 \times 10^{3} \mathrm{psi}$ determine the smallest cross section that can be used. Use $b=2 d$.

Anand Jangid

Numerade Educator

Problem 73

A laminated column of 2.1 -m effective length is to be made by gluing together wood pieces of $25 \times 150-\mathrm{mm}$ cross section. Knowing that for the grade of wood used the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=7.7 \mathrm{MPa}$ and the adjusted modulus is $E=5.4 \mathrm{GPa}$, determine the number of wood pieces that must be used to support the concentric load shown when
$(a) P=52 \mathrm{kN},(b) P=108 \mathrm{kN}$.

Satpal Satpal

Numerade Educator

Problem 74

For a rod made of aluminum alloy $2014-\mathrm{T} 6,$ select the smallest square cross section that can be used if the rod is to carry a 55 -kip centric load.

Surendra Kumar

Numerade Educator

Problem 75

A 72 -kN centric load must be supported by an aluminum column as shown. Using the aluminum alloy $6061-$ Th, determine the minimum dimension $b$ that can be used.

Satpal Satpal

Numerade Educator

Problem 76

An aluminum tube of $90-$ mm outer diameter is to carry a centric load of $120 \mathrm{kN} .$ Knowing that the stock of tubes available for use are made of alloy $2014-\mathrm{T} 6$ and with wall thicknesses in increments of $3 \mathrm{mm}$ from $6 \mathrm{mm}$ to $15 \mathrm{mm}$, determine the lightest tube that can be used.

Victor Salazar

Numerade Educator

Problem 77

A column of $4.6-\mathrm{m}$ effective length must carry a centric load of $525 \mathrm{kN} .$ Knowing that $\sigma_{Y}=345 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$, use allowable stress design to select the wide-flange shape of $200-\mathrm{mm}$ nominal depth that should be used.

Satpal Satpal

Numerade Educator

Problem 78

A column of 22.5 -ft effective length must carry a centric load of 288 kips. Using allowable stress design, select the wide-flange shape of 14 -in. nominal depth that should be used. Use $\sigma_{Y}=$ $50 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 79

A column of 17 -ft effective length must carry a centric load of 235 kips. Using allowable stress design, select the wide-flange shape of 10 -in. nominal depth that should be used. Use $\sigma_{Y}=$ $36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Hast Aggarwal

Numerade Educator

Problem 80

A centric load $\mathbf{P}$ must be supported by the steel bar $A B$. Using allowable stress design, determine the smallest dimension $d$ of the cross section that can be used when $(a) P=108 \mathrm{kN},(b) P=$ $166 \mathrm{kN} .$ Use $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 81

A square steel tube having the cross section shown is used as a column of 26 -ft effective length to carry a centric load of 65 kips. Knowing that the tubes available for use are made with wall thicknesses ranging from $\frac{1}{4}$ in. to $\frac{3}{4}$ in. in increments of $\frac{1}{16}$ in., use allowable stress design to determine the lightest tube that can be used. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator

Problem 82

Solve Prob. $10.81,$ assuming that the effective length of the column is decreased to $20 \mathrm{ft}$.

Chai Santi

Numerade Educator

Problem 83

Two $89 \times 64-\mathrm{mm}$ angles are bolted together as shown for use as a column of 2.4 -m effective length to carry a centric load of $180 \mathrm{kN} .$ Knowing that the angles available have thicknesses of $6.4 \mathrm{mm}, 9.5 \mathrm{mm},$ and $12.7 \mathrm{mm},$ use allowable stress design to determine the lightest angles that can be used. Use $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Chai Santi

Numerade Educator

Problem 84

Two $89 \times 64$ -mm angles are bolted together as shown for use as a column of 2.4 -m effective length to carry a centric load of $325 \mathrm{kN} .$ Knowing that the angles available have thicknesses of $6.4 \mathrm{mm}, 9.5 \mathrm{mm},$ and $12.7 \mathrm{mm},$ use allowable stress design to determine the lightest angles that can be used. Use $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 85

A rectangular steel tube having the cross section shown is used as a column of 14.5 -ft effective length. Knowing that $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$, use load and resistance factor design to determine the largest centric live load that can be applied if the centric dead load is 54 kips. Use a dead load factor $\gamma_{D}=1.2,$ a live load factor $\gamma_{L}=1.6$ and the resistance factor $\phi=0.90$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 86

A column with a 5.8 -m effective length supports a centric load, with ratio of dead to live load equal to 1.35 . The dead load factor is $\gamma_{D}=1.2,$ the live load factor $\gamma_{L}=1.6,$ and the resistance factor $\phi=0.90 .$ Use load and resistance factor design to determine the allowable centric dead and live loads if the column is made of the following rolled-steel shape: $(a) \mathrm{W} 250 \times 67,(b) \mathrm{W} 360 \times 101$ Use $\sigma_{Y}=345 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 87

A steel column of 5.5 -m effective length must carry a centric dead load of $310 \mathrm{kN}$ and a centric live load of $375 \mathrm{kN}$. Knowing that $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$, use load and resistance factor design to select the wide-flange shape of 310 -mm nominal depth that should be used. The dead load factor $\gamma_{D}=1.2,$ the live load factor $\gamma_{L}=1.6,$ and the resistance factor $\phi=0.90$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 88

The steel tube having the cross section shown is used as a column of 15 -ft effective length to carry a centric dead load of 51 kips and a centric live load of 58 kips. Knowing that the tubes available for use are made with wall thicknesses in increments of $\frac{1}{16}$ in. from $\frac{3}{16}$ in. to $\frac{3}{8}$ in., use load and resistance factor design to determine the lightest tube that can be used. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6}$ psi. The dead load factor $\gamma_{D}=1.2,$ the live load factor $\gamma_{L}=1.6,$ and the resistance factor $\phi=0.90$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 89

An eccentric load is applied at a point $22 \mathrm{mm}$ from the geometric axis of a 60 -mm-diameter rod made of a steel for which $\sigma_{Y}=$ $250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$. Using the allowable-stress method, determine the allowable load $\mathbf{P}$.

Anand Jangid

Numerade Educator

Problem 90

Solve Prob, $10.89,$ assuming that the load is applied at a point $40 \mathrm{mm}$ from the geometric axis and that the effective length is $0.9 \mathrm{m}$.

Vidhi Bhatt

Numerade Educator

Problem 91

A sawn-lumber column of $5.0 \times 7.5$ -in. cross section has an effective length of $8.5 \mathrm{ft}$. The grade of wood used has an adjusted allowable stress for compression parallel to the grain $\sigma_{C}=$ 1180 psi and an adjusted modulus $E=440 \times 10^{3}$ psi. Using the allowable-stress method, determine the largest eccentric load $\mathbf{P}$ that can be applied when $(a) e=0.5$ in., $(b) e=1.0$ in.

Hast Aggarwal

Numerade Educator

Problem 92

Solve Prob. 10.91 using the interaction method and an allowable stress in bending of 1300 psi.

Ameer Said

Numerade Educator

Problem 93

A column of 5.5 -m effective length is made of the aluminum alloy $2014-\mathrm{T} 6$ for which the allowable stress in bending is $220 \mathrm{MPa}$ Using the interaction method, determine the allowable load $\mathbf{P}$ knowing that the eccentricity is $(a) e=0,(b) e=40 \mathrm{mm}$.

Surendra Kumar

Numerade Educator

Problem 94

Solve Prob. $10.93,$ assuming that the effective length of the column is $3.0 \mathrm{m}$.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 95

A steel compression member of 9 -ft effective length supports an eccentric load as shown. Using the allowable-stress method, determine the maximum allowable eccentricity $e$ if $(a) P=$ 30 kips,
$(b) P=18$ kips. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Anand Jangid

Numerade Educator

Problem 96

Solve Prob. $10.95,$ assuming that the effective length of the column is increased to $12 \mathrm{ft}$ and that
$(a) P=20$ kips
$(b) P=$
15 kips.

Satpal Satpal

Numerade Educator

Problem 97

Two $\mathrm{L} 4 \times 3 \times \frac{3}{8}$ -in. steel angles are welded together to form the column $A B .$ An axial load $\mathbf{P}$ of magnitude 14 kips is applied at point $D .$ Using the allowable-stress method, determine the largest allowable length $L$. Assume $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Naman Kumar

Numerade Educator

Problem 98

Solve Prob. 10.97 using the interaction method with $P=18$ kips and an allowable stress in bending of 22 ksi.

Hast Aggarwal

Numerade Educator

Problem 99

A rectangular column is made of a grade of sawn wood that has an adjusted allowable stress for compression parallel to the grain $\sigma_{C}=8.3 \mathrm{MPa}$ and an adjusted modulus of elasticity $E=11.1 \mathrm{GPa}$ Using the allowable-stress method, determine the largest allow able effective length $L$ that can be used.

Satpal Satpal

Numerade Educator

Problem 100

Solve Prob. 10.99 , assuming that $P=105 \mathrm{kN}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 101

An eccentric load $P=48 \mathrm{kN}$ is applied at a point $20 \mathrm{mm}$ from the geometric axis of a 50 -mm-diameter rod made of the aluminum alloy $6061-$ T6. Using the interaction method and an allowable stress in bending of 145 MPa, determine the largest allowable effective length $L$ that can be used.

Narayan Hari

Numerade Educator

Problem 102

Solve Prob. 10.101 , assuming that the aluminum alloy used is $2014-76$ and that the allowable stress in bending is $180 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 103

A compression member made of steel has a 720 -mm effective length and must support the 198 -kN load $\mathbf{P}$ as shown. For the material used $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$. Using the interaction method with an allowable bending stress equal to $150 \mathrm{MPa}$ determine the smallest dimension $d$ of the cross section that can be used.

Satpal Satpal

Numerade Educator

Problem 104

Solve Prob. $10.103,$ assuming that the effective length is $1.62 \mathrm{m}$ and that the magnitude of $P$ of the eccentric load is $128 \mathrm{kN}$.

Vidhi Bhatt

Numerade Educator

Problem 105

A steel tube of $80-\mathrm{mm}$ outer diameter is to carry a 93 -kN load $\mathbf{P}$ with an eccentricity of $20 \mathrm{mm}$. The tubes available for use are made with wall thicknesses in increments of 3 mm from 6 mm to $15 \mathrm{mm} .$ Using the allowable-stress method, determine the lightest tube that can be used. Assume $E=200 \mathrm{GPa}$ and $\sigma_{Y}=$ $250 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 106

Solve Prob. $10.105,$ using the interaction method with $P=165 \mathrm{kN}$ $e=15 \mathrm{mm},$ and an allowable stress in bending of $150 \mathrm{MPa}$.

Narayan Hari

Numerade Educator

Problem 107

A sawn lumber column of rectangular cross section has a $2.2-\mathrm{m}$ effective length and supports a 41 -kN load as shown. The sizes available for use have $b$ equal to $90 \mathrm{mm}, 140 \mathrm{mm}, 190 \mathrm{mm},$ and $240 \mathrm{mm} .$ The grade of wood has an adjusted allowable stress for compression parallel to the grain $\sigma_{C}=8.1 \mathrm{MPa}$ and an adjusted modulus $E=8.3$ GPa. Using the allowable-stress method, determine the lightest section that can be used.

Satpal Satpal

Numerade Educator

Problem 108

Solve Prob. $10.107,$ assuming that $e=40 \mathrm{mm}$.

Erica Bartos

Numerade Educator

Problem 109

A compression member of rectangular cross section has an effective length of 36 in. and is made of the aluminum alloy $2014-\mathrm{T} 6$ for which the allowable stress in bending is 24 ksi. Using the interaction method, determine the smallest dimension $d$ of the cross section that can be used when $e=0.4$ in.

Surendra Kumar

Numerade Educator

Problem 110

Solve Prob. 10.109 , assuming that $e=0.2$ in.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 111

An aluminum tube of 3 -in. outside diameter is to carry a load of 10 kips having an eccentricity $e=0.6$ in. Knowing that the stock of tubes available for use are made of alloy 2014 - 76 and have wall thicknesses in increments of $\frac{1}{16}$ in. up to $\frac{1}{2}$ in., determine the lightest tube that can be used. Use the allowable-stress method.

Victor Salazar

Numerade Educator

Problem 112

Solve Prob. 10.111 , using the interaction method of design with an allowable stress in bending of 25 ksi.

Hast Aggarwal

Numerade Educator

Problem 113

A steel column having a 24 -ft effective length is loaded eccentrically as shown. Using the allowable-stress method, select the wide-flange shape of 14 -in. nominal depth that should be used. Use $\sigma_{Y}=36$ ksi and $E=29 \times 10^{6}$ psi.

Chai Santi

Numerade Educator

Problem 114

Solve Prob. 10.113 using the interaction method, assuming that $\sigma_{Y}=50 \mathrm{ksi}$ and the allowable stress in bending is $30 \mathrm{ksi}$.

Chai Santi

Numerade Educator

Problem 115

A steel compression member of 5.8 -m effective length is to support a $296-\mathrm{kN}$ eccentric load $\mathbf{P}$. Using the interaction method, select the wide-flange shape of $200-\mathrm{mm}$ nominal depth that should be used. Use $E=200 \mathrm{GPa}, \sigma_{Y}=250 \mathrm{MPa},$ and $\sigma_{\mathrm{all}}=$
150 MPa in bending.

Satpal Satpal

Numerade Educator

Problem 116

A steel column of 7.2 -m effective length is to support an $83-\mathrm{kN}$ eccentric load $\mathbf{P}$ at a point $D,$ located on the $x$ axis as shown. Using the allowable-stress method, select the wide-flange shape of $250-\mathrm{mm}$ nominal depth that should be used. Use $E=200 \mathrm{GPa}$ and $\sigma_{Y}=250 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 117

Determine $(a)$ the critical load for the steel strut, $(b)$ the dimension $d$ for which the aluminum strut will have the same critical load.
(c) Express the weight of the aluminum strut as a percent of the weight of the steel strut.

Rashmi Sinha

Numerade Educator

Problem 118

The rigid rod $A B$ is attached to a hinge at $A$ and to two springs, each of constant
$k .$ If $h=450 \mathrm{mm}, d=300 \mathrm{mm},$ and $m=200 \mathrm{kg}$
determine the range of values of $k$ for which the equilibrium of rod $A B$ is stable in the position shown. Each spring can act in either tension or compression.

John Palmer

Numerade Educator

Problem 119

A column of $3-\mathrm{m}$ effective length is to be made by welding together two $\mathrm{C} 130 \times 13$ rolled-steel channels. Using $E=200 \mathrm{GPa}$ determine for each arrangement shown the allowable centric load if a factor of safety of 2.4 is required.

Satpal Satpal

Numerade Educator

Problem 120

(a) Considering only buckling in the plane of the structure shown and using Euler's formula, determine the value of $\theta$ between
0 and $90^{\circ}$ for which the allowable magnitude of the load $\mathbf{P}$ is maximum.
(b) Determine the corresponding maximum value of $P$ knowing that a factor of safety of 3.2 is required. Use $E=29 \times$ $10^{6}$ psi.

Surendra Kumar

Numerade Educator

Problem 121

Member $A B$ consists of a single $\mathrm{C} 130 \times 10.4$ steel channel of length $2.5 \mathrm{m} .$ Knowing that the pins $A$ and $B$ pass through the centroid of the cross section of the channel, determine the factor of safety for the load shown with respect to buckling in the plane of the figure when $\theta=30^{\circ} .$ Use $E=200 \mathrm{GPa}$.

Anand Jangid

Numerade Educator

Problem 122

The line of action of the 75 -kip axial load is parallel to the geometric axis of the column $A B$ and intersects the $x$ axis at $x=0.6$ in. Using $E=29 \times 10^{6} \mathrm{psi}$, determine $(a)$ the horizontal deflection of the midpoint $C$ of the column,
(b) the maximum stress in the column.

Satpal Satpal

Numerade Educator

Problem 123

Supports $A$ and $B$ of the pin-ended column shown are at a fixed distance $L$ from each other. Knowing that at a temperature $T_{0}$ the force in the column is zero and that buckling occurs when the temperature is $T_{1}=T_{0}+\Delta T$, express $\Delta T$ in terms of $b, L$ and the coefficient of thermal expansion $\alpha$.

Farnaz Mohseni

Numerade Educator

Problem 124

A column is made from half of a $\mathrm{W} 360 \times 216$ rolled-steel shape, with the geometric properties as shown. Using allowable stress design, determine the allowable centric load if the effective length of the column is $(a) 4.0 \mathrm{m},(b) 6.5 \mathrm{m} .$ Use $\sigma_{Y}=345 \mathrm{MPa}$
and $E=200 \mathrm{GPa}$.

Satpal Satpal

Numerade Educator

Problem 125

A rectangular column with a 4.4 -m effective length is made of glued laminated wood. Knowing that for the grade of wood used the adjusted allowable-stress for compression parallel to the grain is $\sigma_{C}=8.3 \mathrm{MPa}$ and the adjusted modulus $E=4.6 \mathrm{GPa}$ determine the maximum allowable centric load for the column.

Satpal Satpal

Numerade Educator

Problem 126

A column of 4.5 -m effective length must carry a centric load of $900 \mathrm{kN} .$ Knowing that $\sigma_{\mathrm{Y}}=345 \mathrm{MPa}$ and $E=200 \mathrm{GPa},$ use
allowable-stress design to select the wide-flange shape of $250-m m$ nominal depth that should be used.

Satpal Satpal

Numerade Educator

Problem 127

An 11 -kip vertical load $\mathbf{P}$ is applied at the midpoint of one edge of the square cross section of the steel compression member $A B$ which is free at its top $A$ and fixed at its base $B$. Knowing that for the grade of steel used $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$ and using the allowable-stress method, determine the smallest allowable dimension $d$.

Anand Jangid

Numerade Educator

Problem 128

A column of 14 -ft effective length consists of a section of steel tubing having the cross section shown. Using the allowablestress method, determine the maximum allowable eccentricity $e$ if $(a) P=55$ kips,
(b) $P=35$ kips. Use $\sigma_{Y}=36 \mathrm{ksi}$ and $E=29 \times 10^{6} \mathrm{psi}$.

Chai Santi

Numerade Educator