Mechanics of Materials in SI Units

Russell C. Hibbeler

Chapter 13

Buckling of Columns - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine the critical buckling load for the column. The material can be assumed rigid.

Satpal Satpal

Satpal Satpal

Numerade Educator

Problem 2

The column consists of a rigid member that is pinned at its bottom and attached to a spring at its top. If the spring is unstretched when the column is in the vertical position, determine the critical load that can be placed on the column.

Surendra Kumar

Numerade Educator

Problem 3

The leg in (a) acts as a column and can be modeled (b) by the two pin-connected members that are attached to a torsional spring having a stiffness $k$ (torque $/ \mathrm{rad}$ ). Determine the critical buckling load. Assume the bone material is rigid.
(a)
(b)

Satpal Satpal

Numerade Educator

Problem 3

Determine if the frame can support a load of $P=20 \mathrm{kN}$ if the factor of safety with respect to buckling of member $A B$ is F.S. $=3$. Assume that $A B$ is made of steel and is pinned at its ends for $x-x$ axis buckling and fixed at its ends for $y-y$ axis buckling. $E_{\mathrm{zt}}=200 \mathrm{GPa}, \sigma_Y=360 \mathrm{MPa}$.

Prob. 13-33

Surendra Kumar

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Problem 4

Rigid bars $A B$ and $B C$ are pin connected at $B$. If the spring at $D$ has a stiffness $k$, determine the critical load $P_c$ that can be applied to the bars.

Surendra Kumar

Numerade Educator

Problem 5

A 2014-T6 aluminum alloy column has a length of 6 m and is fixed at one end and pinned at the other. If the cross-sectional area has the dimensions shown, determine the critical load. $\sigma_y=250 \mathrm{MPa}$.

Surendra Kumar

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Problem 6

Solve Prob. 13-5 if the column is pinned at its top and bottom.

Chai Santi

Numerade Educator

Problem 7

The W360 $\times 57$ columin is made of $\mathrm{A}-36$ steel and is fixed supported at its base. If it is subjected to an axial load of $P=75 \mathrm{kN}$, determine the factor of safety with respect to buckling.

Surendra Kumar

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Problem 8

The W360 $\times 57$ column is made of A-36 steel. Determine the critical load if its bottom end is fixed supported and its top is free to move about the strong axis and is pinned about the weak axis.

Satpal Satpal

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Problem 9

A steel column has a length of 9 m and is fixed at both ends If the cross-sectional area has the dimensions shown, determine the critical load. $E_{2 t}=200 \mathrm{GPa}, \sigma_Y=250 \mathrm{MPa}$.

Anand Jangid

Numerade Educator

Problem 9

A steel column has a length of 9 m and is pinned at its top and bottom. If the cross-sectional area has the dimensions shown, determine the critical load. $E_{\mathrm{mt}}=200 \mathrm{GPa}$, $\sigma_Y=250 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 11

The A-36 steel angle has a cross-sectional area of $A=1550 \mathrm{~mm}^2$ and a radius of gyration about the $x$ axis of $r_x=31.5 \mathrm{~mm}$ and about the $y$ axis of $r_y=21.975 \mathrm{~mm}$. The smallest radius of gyration occurs about the $z$ axis and is $r_z=16.1 \mathrm{~mm}$. If the angle is to be used as a pin-connected 3-m-long column, determine the largest axial load that can be applied through its centroid $C$ without causing it to buckle.

Satpal Satpal

Numerade Educator

Problem 12

The deck is supported by the two $40-\mathrm{mm}$-square columns. Column $A B$ is pinned at $A$ and fixed at $B$, whereas $C D$ is pinned at $C$ and $D$. If the deck is prevented from sidesway, determine the greatest weight of the load that can be applied without causing the deck to collapse. The center of gravity of the load is located at $d-2 \mathrm{~m}$. Both columns are made from Douglas Fir.

Surendra Kumar

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Problem 13

The deck is supported by the two 40 -mm-square columns Column $A B$ is pinned at $A$ and fixed at $B$, whereas $C D$ is pinned at $C$ and $D$. If the deck is prevented from sidesway, determine the position $d$ of the center of gravity of the load and the load's greatest magnitude without causing the deck to collapse. Both columns are made from Douglas Fir.

Surendra Kumar

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Problem 14

Determine the maximum force $P$ that can be applied to the handle so that the A - 36 steel control rod $B C$ does not buckle. The rod has a diameter of 25 mm .

Satpal Satpal

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Problem 15

Determine the maximum load $P$ the frame can support without buckling member $A B$. Assume that $A B$ is made of steel and is pinned at its ends for $y$ - $y$ axis buckling and fixed at its ends for $x-x$ axis buckling. $E_{\mathrm{xi}}=200 \mathrm{GPa}$, $\sigma_y=360 \mathrm{MPa}$.

Surendra Kumar

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Problem 16

The two steel channels are to be laced together to form a 9 -m-long bridge column assumed to be pin connected at its ends. Each channel has a cross-sectional area of $A=1950 \mathrm{~mm}^2$ and moments of inertia $I_x=21.60\left(10^6\right) \mathrm{mm}^4, l_y=0.15\left(10^6\right) \mathrm{mm}^4$. The centroid $C$ of its area is located in the figure. Determine the proper distance $d$ between the centroids of the channels so that buckling occurs about the $x-x$ and $y^{\prime}-y^{\prime}$ axes due to the same load. What is the value of this critical load? Neglect the effect of the lacing. $E_{\mathrm{m}}=200 \mathrm{GPa}, \sigma_Y=350 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 17

The W250 $\times 67$ is made of A992 steel and is used as a column that has a length of 4.55 m . If its ends are assumed pin supported, and it is subjected to an axial load of 500 kN , determine the factor of safety with respect to buckling.

Surendra Kumar

Numerade Educator

Problem 18

The W250 $\times 67$ is made of A992 steel and is used as a column that has a length of 4.55 m . If the ends of the column are fixed supported, can the column support the critical load without yielding?

Probs. 13-17/18

Surendra Kumar

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Problem 19

The 50 -mm-diameter C 86100 bronze rod is fixed supported at $A$ and has a gap of 2 mm from the wall at $B$. Determine the increase in temperature $\Delta T$ that will cause the rod to buckle. Assume that the contact at $B$ acts as a pin.

Surendra Kumar

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Problem 20

An A992 steel W200 $\times 46$ column of length 9 m is fixed at one end and free at its other end. Determine the allowable axial load the column can support if F.S. -2 against buckling.

Satpal Satpal

Numerade Educator

Problem 21

The $3-\mathrm{m}$ wooden rectangular column has the dimensions shown. Determine the critical load if the ends are assumed to be pin connected. $E_w=12 \mathrm{GPa}$, $\sigma_Y=35 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 22

The 3 -m column has the dimensions shown. Determine the critical load if the bottom is fixed and the top is pinned. $E_m=12 \mathrm{GPa}_2, \sigma_Y=35 \mathrm{MPa}$.

Probs. 13-21/22

Surendra Kumar

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Problem 23

If load $C$ has a mass of 500 kg , determine the required minimum diameter of the solid $L 2$-steel rod $A B$ to the nearest mm so that it will not buckle. Use F.S. -2 against buckling.

Surendra Kumar

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Problem 24

If the diameter of the solid L2-steel rod $A B$ is 50 mm , determine the maximum mass $C$ that the rod can support without buckling. Use F.S. $=2$ against buckling.

Chai Santi

Numerade Educator

Problem 25

The members of the truss are assumed to be pin connected. If member GF is an A-36 steel rod having a diameter of 50 mm , determine the greatest magnitude of load $\mathbf{P}$ that can be supported by the truss without causing this member to buckle.

Satpal Satpal

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Problem 26

The members of the truss are assumed to be pin connected. If member $A G$ is an $\mathrm{A}-36$ steel rod having a diameter of 50 mm , determine the greatest magnitude of load $\mathbf{P}$ that can be supported by the truss without causing this member to buckle.

Probs. 13-25/26

Satpal Satpal

Numerade Educator

Problem 27

Determine the maximum allowable intensity $w$ of the distributed load that can be applied to member $B C$ without causing member $A B$ to buckle. Assume that $A B$ is made of steel and is pinned at its ends for $x-x$ axis buckling and fixed at its ends for $y-y$ axis buckling. Use a factor of safety with respect to buckling of 3 . $E_z=200 \mathrm{GPa}^2, \sigma_Y=360 \mathrm{MPa}$

Chai Santi

Numerade Educator

Problem 28

Determine if the frame can support a load of $w=6 \mathrm{kN} / \mathrm{m}$ if the factor of safety with respect to buckling of member $A B$ is 3 . Assume that $A B$ is made of steel and is pinned at its ends for $x-x$ axis buckling and fixed at its ends for $y-y$ axis buckling, $E_{\mathrm{st}}=200 \mathrm{GPa}, \sigma_Y=360 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 29

A 6061-T6 aluminum alloy solid circular rod of length 4 m is pinned at both of its ends If it is subjected to an axial load of 15 kN and F. -2 against buckling, determine the minimum required diameter of the rod to the nearest mm.

Satpal Satpal

Numerade Educator

Problem 30

A 6061-T6 aluminum alloy solid circular rod of length 4 m is pinned at one end while fixed at the other end. If it is subjected to an axial load of 15 kN and F.S. -2 against buckling, determine the minimum required diameter of the rod to the nearest mm.

Satpal Satpal

Numerade Educator

Problem 31

The $\mathrm{A}-36$ steel bar $A B$ has a square cross section. If it is pin connected at its ends, determine the maximum allowable load $P$ that can be applied to the frame. Use a factor of safety with respect to buckling of 2 .

Surendra Kumar

Numerade Educator

Problem 32

Determine the maximum allowable load $P$ that can be applied to member $B C$ without causing member $A B$ to buckle. Assume that $A B$ is made of steel and is pinned at its ends for $x-x$ axis buckling and fixed at its ends for $y-y$ axis buckling. Use a factor of safety with respect to buckling of F.S. $=3 . E_{\mathrm{at}}=200 \mathrm{GPa}, \sigma_Y=360 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 34

The steel bar $A B$ has a rectangular cross section. If it is pin connected at its ends, determine the maximum allowable intensity $w$ of the distributed load that can be applied to $B C$ without causing $A B$ to buckle. Use a factor of safety with respect to buckling of 1.5 . $E_w=200 \mathrm{GPia}, \sigma_Y=$ 360 MPa

Chai Santi

Numerade Educator

Problem 35

The W360 $\times 45$ is used as a structural A-36 steel column that can be assumed pinned at both of its ends. Determine the largest axial force $P$ that can be applied without causing it to buckle.

Surendra Kumar

Numerade Educator

Problem 36

The beam supports the load of $P=30 \mathrm{kN}$. As a result, the A-36 steel member $B C$ is subjected to a compressive load. Due to the forked ends on the member, consider the supports at $B$ and $C$ to act as pins for $x-x$ axis buckling and as fixed supports for $y$-y axis buckling Determine the factor of safety with respect to buckling about each of these axes.

Chai Santi

Numerade Educator

Problem 37

Determine the greatest load $P$ the frame will support without causing the $\mathrm{A}-36$ steel member $B C$ to buckle. Due to the forked ends on the member, consider the supports at $B$ and $C$ to act as pins for $x-x$ axis buckling and as fixed supports for $y$ - $y$ axis buckling.

Satpal Satpal

Numerade Educator

Problem 38

The members of the truss are assumed to be pin connected. If member $A B$ is an $A-36$ steel rod of 40 mm diameter, determine the maximum force $P$ that can be supported by the truss without causing the member to buckle.

Satpal Satpal

Numerade Educator

Problem 39

The members of the truss are assumed to be pin connected. If member $C B$ is an $A-36$ steel rod of 40 mm diameter, determine the maximum load $P$ that can be supported by the truss without causing the member to buckle.

Probs. 13-38/39

Satpal Satpal

Numerade Educator

Problem 40

The steel bar $A B$ of the frame is assumed to be pin connected at its ends for $y-y$ axis buckling. If $P=18 \mathrm{kN}$, determine the factor of safety with respect to buckling about the $y-y$ axis. $E_{z t}=200 \mathrm{GPa}, \sigma y=360 \mathrm{MPa}$.

Prob. 13-40

Satpal Satpal

Numerade Educator

Problem 41

The ideal column has a weight $w$ (force/length) and is subjected to the axial load P. Determine the maximum moment in the column at midspan. El is constant.

Surendra Kumar

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Problem 42

The ideal column is subjected to the force $\mathbf{F}$ at its midpoint and the axial load $\mathbf{P}$. Determine the maximum moment in the column at midspan. EI is constant.

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Problem 43

The column with constant EI has the end constraints shown. Determine the critical load for the column.

Satpal Satpal

Numerade Educator

Problem 44

Consider an ideal column as in Fig. 13-10c, having both ends fixed. Show that the critical load on the column is $P_{\mathrm{c}}=4 \pi^2 E I / L^2$. Hinr. Due to the vertical deflection of the top of the column, a constant moment $\mathbf{M}^{\prime}$ will be developed at the supports. Show that $d^2 v / d x^2+(P / E I) v-M^{\prime} / E I$. The solution is of the form $v=C_1 \sin (\sqrt{P / E l} x)+$ $C_2 \cos (\sqrt{P / E I} x)+M^{\prime} / P$.

Surendra Kumar

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Problem 45

Consider an ideal column as in Fig. 13-10.d, having one end fixed and the other pinned. Show that the critical load on the column is $P_{\mathrm{Gr}}=20.19 \mathrm{EI} / \mathrm{L}^2$.

Surendra Kumar

Numerade Educator

Problem 46

The W360 $\times 39$ structural A-36 steel member is used as a 6 -m-long column that is assumed to be fixed at its top and fixed at its bottom. If the $75-\mathrm{kN}$ load is applied at an eccentric distance of 250 mm , determine the maximum stress in the column.

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Problem 47

The W360 $\times 39$ structural A-36 steel member is used as a column that is assumed to be fixed at its top and pinned at its bottom. If the $75-\mathrm{kN}$ load is applied at an eccentric distance of 250 mm , determine the maximum stress in the column.

Check back soon!

Problem 48

The aluminum column is fixed at the bottom and free at the top. Determine the maximum force $P$ that can be applied at $A$ without causing it to buckle or yield. Use a factor of safety of 3 with respect to buckling and yielding. $E_{31}=70 \mathrm{GPa}, \sigma_\gamma=95 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 49

The aluminum rod is fixed at its base and free at its top. If the eccentric load $P=200 \mathrm{kN}$ is applied, determine the greatest allowable length $L$ of the rod so that it does not buckle or yield. $E_{\mathrm{al}}=72 \mathrm{GPa}, \sigma_Y=410 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 50

The aluminum rod is fixed at its base and free and at its top. If the length of the rod is $L=2 \mathrm{~m}$, determine the greatest allowable load $P$ that can be applied so that the rod does not buckle or yield. Also, determine the largest sidesway deflection of the rod due to the loading. $E_{\mathrm{a}}=72 \mathrm{GPa}, \sigma_Y=410 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 51

. The wood column is fixed at its base and free at its top Determine the load $P$ that can be applied to the edge of the column without causing the column to fail either by buckling or by yielding. $E_x=12 \mathrm{GPa}, \sigma_Y=55 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 52

The tube is made of copper and has an outer diameter of 35 mm and a wall thickness of 7 mm . Determine the eccentric load $P$ that it can support without failure. The tube is pin supported at its ends. $E_{\mathrm{cu}}=120 \mathrm{GPa}, \sigma y=750 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 53

The W250 $\times 45 \mathrm{~A}-36$-steel column is pinned at its top and fixed at its base. Also, the column is braced along its weak axis at mid-height. If $P=250 \mathrm{kN}$, investigate whether the column is adequate to support this loading. Use F.S. -2 against buckling and F.S. $=1.5$ against yielding.

Surendra Kumar

Numerade Educator

Problem 54

The W250 $\times 45 \mathrm{~A}$-36-steel column is pinned at its top and fixed at its base. Also, the column is braced along its weak axis at mid-height. Determine the allowable force $P$ that the column can support without causing it either to buckle or yield. Use F.S. -2 against buckling and F.S. -1.5 against yielding.

Surendra Kumar

Numerade Educator

Problem 55

The wood column is pinned at its base and top. If the eccentric force $P=10 \mathrm{kN}$ is applied to the column, investigate whether the column is adequate to support this loading without buckling or yielding. Take $E=10 \mathrm{GPa}$ and $\sigma_Y=15 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 56

The wood column is pinned at its base and top. Determine the maximum eccentric force $P$ the column can support without causing it to either buckle or yield. Take $E=10 \mathrm{GPa}$ and $\sigma_y=15 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 57

The wood column is fixed at its base and can be assumed pin connected at its top. Determine the maximum eccentric load $P$ that can be applied without causing the column to buckle or yield. $E_n=12 \mathrm{GPa}, \sigma_Y=56 \mathrm{MPa}$.

Satpal Satpal

Numerade Educator

Problem 58

The wood column is fixed at its base and can be assumed fixed connected at its top. Determine the maximum eccentric load $P$ that can be applied without causing the column to buckle or yield. $E_w=12 \mathrm{GPa}, \omega_Y=56 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 59

Determine the maximum eccentric load $P$ the 2014-T6-aluminum-alloy strut can support without causing it either to buckle or yield. The ends of the strut are pin connected.

Satpal Satpal

Numerade Educator

Problem 60

The W200 $\times 22$ A-36-steel column is fixed at its base. Its top is constrained to rotate about the $y-y$ axis and free to move along the $y-y$ axis. Also, the column is braced along the $x-x$ axis at its mid-height. Determine the allowable eccentric force $P$ that can be applied without causing the column either to buckle or yield. Use F.S. -2 against buckling and F.S. $=1.5$ against yielding.

Chai Santi

Numerade Educator

Problem 61

The W200 $\times 22$ A-36-steel column is fixed at its base. Its top is constrained to rotate about the $y-y$ axis and free to move along the $y-y$ axis. Also, the column is braced along the $x-x$ axis at its mid-height. If $P=25 \mathrm{kN}$, determine the maximum normal stress developed in the column.

Chai Santi

Numerade Educator

Problem 62

The brass rod is fixed at one end and free at the other end. If the eccentric load $P=200 \mathrm{kN}$ is applied, determine the greatest allowable length $L$ of the rod so that it does not buckle or yield. $E_{\mathrm{Er}}=101 \mathrm{GPa}, \sigma_Y=69 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 63

The brass rod is fixed at one end and free at the other end. If the length of the rod is $L=2 \mathrm{~m}$, determine the greatest allowable load $P$ that can be applied so that the rod does not backle or yield. Also, determine the largest sidesway deflection of the rod due to the loading $E_{\mathrm{br}}=101 \mathrm{GPa}$, $\sigma_Y=69 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 64

Determine the load $P$ required to cause the W310 $\times 74$ structural A-36 steel column to fail either by buckling or by yielding. The column is fixed at its bottom and the cables at its top act as a pin to hold it.

Surendra Kumar

Numerade Educator

Problem 65

The W250 $\times 28$ A-36-steel column is fixed at its base. Its top is constrained to rotate about the $y-y$ axis and free to move along the $y-y$ axis. If $e=350 \mathrm{~mm}$, determine the allowable eccentric force $P$ that can be applied without causing the column either to buckle or yield. Use F.S. -2 against buckling and F.S. -1.5 against yielding

Chai Santi

Numerade Educator

Problem 66

The W250 $\times 28 \mathrm{~A}$-36-steel column is fixed at its base. Its top is constrained to rotate about the $y-y$ axis and free to move along the $y$-y axis. Determine the force $\mathbf{P}$ and its eccentricity e so that the column will yield and buckle simultaneously.

Chai Santi

Numerade Educator

Problem 67

The 6061-T6 aluminum alloy solid shaft is fixed at one end but free at the other end. If the shaft has a diameter of 100 mm , determine its maximum allowable length $L$ if it is subjected to the eccentric force $P=80 \mathrm{kN}$.

Satpal Satpal

Numerade Educator

Problem 68

The 6061-T6 aluminum alloy solid shaft is fixed at one end but free at the other end. If the length is $L=3 \mathrm{~m}$, determine its minimum required diameter if it is subjected to the eccentric force $P=60 \mathrm{kN}$.

Satpal Satpal

Numerade Educator

Problem 69

A column of intermediate length buckles when the compressive stress is 280 MPa . If the slenderness ratio is 60 , determine the tangent modulus

Satpal Satpal

Numerade Educator

Problem 70

The stress-strain diagram for the material of a column can be approximated as shown. Plot $P / A$ vs. $K L / r$ for the column.

Chai Santi

Numerade Educator

Problem 71

The stress-strain diagram for a material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are pinned. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 72

The stress-strain diagram for a material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are fixed. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 73

The stress-strain diagram for a material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and length of 1.5 m is made from this material, determine the critical load provided one end is pinned and the other is fixed. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 74

Construct the buckling curve, $P / A$ versus $L / r$, for a column that has a bilinear stress-strain curve in compression as shown. The column is pinned at its ends.

Chai Santi

Numerade Educator

Problem 75

The stress-strain diagram of the material can be approximated by the two line segments. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are pinned. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 76

The stress-strain diagram of the material can be approximated by the two line segments. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are fixed. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 77

The stress-strain diagram of the material can be approximated by the two line segments. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided one end is pinned and the other is fixed. Assume that the load acts through the axis of the bar. Use Engesser's equation.

Satpal Satpal

Numerade Educator

Problem 78

Determine the largest length of a structural A-36 steel rod if it is fixed supported and subjected to an axial load of 100 kN . The rod has a diameter of 50 mm . Use the AISC equations.

Chai Santi

Numerade Educator

Problem 79

Check if a W250 $\times 58$ column can safely support an axial force of $P=1150 \mathrm{kN}$. The column is 6 m long and is pinned at both ends and braced against its weak axis at mid-height. It is made of steel having $E=200 \mathrm{GPa}$ and $\sigma_Y=350 \mathrm{MPa}$. Use the AISC column design formulas.

Surendra Kumar

Numerade Educator

Problem 80

A W200 $\times 36$ A-36-steel column of $9-\mathrm{m}$ length is pinned at both ends and braced against its weak axis at mid-height. Determine the allowable axial force $P$ that can be safely supported by the column. Use the AISC column design formulas.

Surendra Kumar

Numerade Educator

Problem 81

Using the AISC equations, select from Appendix B the lightest-weight structural A-36 steel column that is 9 m long and supports an axial load of 1000 kN . The ends are fixed.

Chai Santi

Numerade Educator

Problem 82

Using the AISC equations, select from Appendix B the lightest-weight structural A-36 steel column that is 72 m long and supports an axial load of 450 kN . The ends are fixed.

Satpal Satpal

Numerade Educator

Problem 83

Determine the largest length of a W250 $\times 67$ A992 structural steel column if it is pin supported and subjected to an axial load of 1450 kN . Use the AISC equations.

Chai Santi

Numerade Educator

Problem 84

Determine the largest length of a W250 $\times 18$ structural A- 36 steel section if it is pin supported and is subjected to an axial load of 140 kN . Use the AISC equations.

Satpal Satpal

Numerade Educator

Problem 85

Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 4.2 m long and supports an axial load of 200 kN . The ends are pinned.

Chai Santi

Numerade Educator

Problem 86

Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 3.6 m long and supports an axial load of 200 kN . The ends are fixed.

Chai Santi

Numerade Educator

Problem 87

Check if a W250 $\times 67$ column can safely support an axial force of $P=1000 \mathrm{kN}$. The column is 4.5 m long and is pinned at both of its ends It is made of steel having $E=200 \mathrm{GPa}$ and $\sigma_Y=350 \mathrm{MPa}$. Use the AISC column design formulas

Surendra Kumar

Numerade Educator

Problem 88

A 1.5 -m-long rod is used in a machine to transmit an axial compressive load of 15 kN . Determine its smallest diameter if it is pin connected at its ends and is made of a 2014-T6 aluminum alloy.

Chai Santi

Numerade Educator

Problem 89

Using the AISC equations, check if a column having the cross section shown can support an axial force of 1500 kN . The column has a length of 4 m , is made from A992 steel, and its ends are pinned.

Chai Santi

Numerade Educator

Problem 90

The beam and column arrangement is used in a railroad yard for loading and unloading cars. If the maximum anticipated hoist load is 560 kN , determine if the $\mathbf{W} 200 \times 46$ wide-flange $\mathrm{A}-\mathbf{3}$ steel column is adequate for supporting the load. The hoist travels along the bottom flange of the beam, $0.4 \mathrm{~m} \leq x \leq 7.5 \mathrm{~m}$, and has negligible size. Assume the beam is pinned to the column at $B$ and roller supported at $A$. The column is also pinned at $C$.

Surendra Kumar

Numerade Educator

Problem 91

The bar is made of a 2014-T6 aluminum alloy. Determine its smallest thickness $b$ if its width is $5 b$. Assume that it is pin connected at its ends.

Satpal Satpal

Numerade Educator

Problem 92

The bar is made of a 2014-T6 aluminum alloy. Determine its smallest thickness $b$ if its width is $5 b$. Assume that it is fixed connected at its ends.

Satpal Satpal

Numerade Educator

Problem 93

The 2014-T6 aluminum hollow section has the cross section shown. If the column is 3 m long and is fixed at both ends, determine the allowable axial force $P$ that can be safely supported by the column.

Satpal Satpal

Numerade Educator

Problem 94

The 2014-T6 aluminum hollow section has the cross section shown. If the column is fixed at its base and pinned at its top, and is subjected to the axial force $P=500 \mathrm{kN}$ determine the maximum length of the column for it to safely support the load.

Satpal Satpal

Numerade Educator

Problem 95

The 2014-T6 aluminum column of 3-m length has the cross section shown. If the column is pinned at both ends and braced against the weak axis at its mid-height, determine the allowable axial force $P$ that can be safely supported by the column.

Satpal Satpal

Numerade Educator

Problem 96

The 2014-T6 aluminum column has the cross section shown. If the column is pinned at both ends and subjected to an axial force $P=100 \mathrm{kN}$, determine the maximum length the column can have to safely support the loading.

Satpal Satpal

Numerade Educator

Problem 97

The tube is 6 mm thick, is made of a 2014 -T6 aluminum alloy, and is fixed at its bottom and pinned at its top. Determine the largest axial load that it can support.

Satpal Satpal

Numerade Educator

Problem 98

The tube is 6 mm thick, is made of a 2014-T6 aluminum alloy, and is fixed connected at its ends. Determine the largest axial load that it can support.

Satpal Satpal

Numerade Educator

Problem 99

The tube is 6 mm thick, is made of 2014 -T6 aluminum alloy and is pin connected at its ends. Determine the largest axial load it can support.

Satpal Satpal

Numerade Educator

Problem 100

The column is made of wood. It is fixed at its bottom and free at its top. Use the NFPA formulas to determine its greatest allowable length if it supports an axial load of $P=10 \mathrm{kN}$.

Chai Santi

Numerade Educator

Problem 101

The column is made of wood. It is fixed at its bottom and free at its top. Use the NFPA formulas to determine the largest allowable axial load $P$ that it can support if it has a length $L=1.2 \mathrm{~m}$.

Satpal Satpal

Numerade Educator

Problem 102

The wooden column shown is formed by gluing together the $150 \mathrm{~mm} \times 12 \mathrm{~mm}$ boards. If the column is pinned at both ends and is subjected to an axial load $P=100 \mathrm{kN}$ determine the required number of boards needed to form the column in order to safely support the loading.

Hast Aggarwal

Numerade Educator

Problem 103

The timber column has a square cross section and is assumed to be pin connected at its top and bottom. If it supports an axial load of 250 kN , determine its smallest side dimension a to the nearest multiples of $\$ \mathrm{~mm}$. Use the NFPA formulas.

Chai Santi

Numerade Educator

Problem 104

A rectangular wooden column has the cross section shown. If the column is 1.8 m long and subjected to an axial force of $P=75 \mathrm{kN}$, determine the required minimum dimension $a$ of its cross-sectional area to the nearest multiples of 5 mm so that the column can safely support the loading. The column is pinned at both ends.

Satpal Satpal

Numerade Educator

Problem 105

A rectangular wooden column has the cross section shown. If $a=75 \mathrm{~mm}$ and the column is 3.6 m long, determine the allowable axial force $P$ that can be safely supported by the column if it is pinned at its top and fixed at its base.

Satpal Satpal

Numerade Educator

Problem 106

A rectangular wooden column has the cross section shown. If $a=75 \mathrm{~mm}$ and the column is subjected to an axial force of $P=75 \mathrm{kN}$ determine the maximum length the column can have to safely support the load. The column is pinned at its top and fixed at its base.

Satpal Satpal

Numerade Educator

Problem 107

The W360 $\times 33$ structural A-36 steel column is fixed at its top and bottom. If a horizontal load (not shown) causes it to support end moments of $M=15 \mathrm{kN} \cdot \mathrm{m}$, determine the mavimum allowahle avial foren $P$ that can he applied. Bending is about the $x-x$ axis. Use the AISC equations of Sec. 13.6 and Eq. 13-30.

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Problem 108

The W360 $\times 33$ structural A-36 steel column is fixed at its top and bottom. If a horizontal load (not shown) causes it to support end moments of $M=70 \mathrm{kN} \cdot \mathrm{m}$, determine the maximum allowable axial force $P$ that can be applied. Bending is about the $x-x$ axis. Use the interaction formula with $\left(\sigma_{\mathrm{h}}\right)_{\text {allow }}=168 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator

Problem 109

The W360 $\times 79$ structural A-36 steel column supports an axial load of 400 kN in addition to an eccentric load $P$. Determine the maximum allowable value of $P$ based on the AISC equations of Sec. 13.6 and Eq. 13-30. Assume the column is fixed at its base, and at its top it is free to sway in the $x-z$ plane while it is pinned in the $y-z$ plane.

Surendra Kumar

Numerade Educator

Problem 110

The W310 $\times 67$ structural $A-36$ steel column supports an axial load of 400 kN in addition to an eccentric load of $P=30 \mathrm{kN}$. Determine if the column fails based on the AISC equations of Sec. 13.6 and Eq. 13-30.Assume that the column is fixed at its base, and at its top it is free to sway in the $x-z$ plane while it is pinned in the $y$ - $z$ plane.

Chai Santi

Numerade Educator

Problem 111

The W360 $\times 57$ structural A-36 steel column is fixed at its bottom and free at its top. Determine the greatest eccentric load $P$ that can be applied using Eq. 13-30 and the AISC equations of Sec. 13.6.

Surendra Kumar

Numerade Educator

Problem 112

The W250 $\times 67$ structural A-36 steel column is fixed at its bottom and free at its top. If it is subjected to a load of $P=10 \mathrm{kN}$, determine if it is safe based on the AISC equations of Sec. 13.6 and Eq. 13-30.

Surendra Kumar

Numerade Educator

Problem 113

The A-36-steel W250 $\times 67$ column is fixed at its base. Its top is constrained to move along the $x-x$ axis but free to rotate about and move along the $y$ - $y$ axis. Determine the maximum eccentric force $P$ that can be safely supported by the column using an interaction formula. The allowable bending stress is $\left(\sigma_b\right)_{\mathrm{albw}}-100 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 114

The A-36-steel W310 $\times 74$ column is fixed at its base. Its top is constrained to move along the $x-x$ axis but free to rotate about and move along the $y-y$ axis. If the eccentric force $P=75 \mathrm{kN}$ is applied to the column, investigate if the column is adequate to support the loading. Use the allowable stress method.

Chai Santi

Numerade Educator

Problem 115

The A-36-steel W250 $\times 67$ column is fixed at its base. Its top is constrained to move along the $x-x$ axis but free to rotate about and move along the $y-y$ axis. Determine the maximum eccentric force $P$ that can be safely supported by the column using the allowable stress method.

Chai Santi

Numerade Educator

Problem 116

The A-36-steel W310 $\times 74$ column is fixed at its base. Its top is constrained to move along the $x-x$ axis but free to rotate about and move along the $y-y$ axis. If the eccentric force $P=65 \mathrm{kN}$ is applied to the column, investigate if the column is adequate to support the loading. Use the interaction formula. The allowable bending stress is $\left(\sigma_b\right)_{\text {alurw }}=100 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 117

A 4.8-m-long column is made of aluminum alloy 2014-T6. If it is fixed at its top and bottom, and a compressive load $\mathbf{P}$ is applied at point $A$, determine the maximum allowable magnitude of $\mathbf{P}$ using the equations of Sec. 13.6 and Eq. 13-30.

Chai Santi

Numerade Educator

Problem 118

A 4.8 -m-long column is made of aluminum alloy 2014-T6. If it is fixed at its top and bottom, and a compressive load $\mathbf{P}$ is applied at point $A$, determine the maximum allowable magnitude of $\mathbf{P}$ using the equations of Sec. 13.6 and the interaction formula with $\left(\sigma_2\right)$ alow $=140 \mathrm{MPa}$.

Chai Santi

Numerade Educator

Problem 119

The 2014-T6 hollow column is fixed at its base and free at its top. Determine the maximum eccentric force $P$ that can be safely supported by the column. Use the allowable stress method. The thickness of the wall for the section is $t=12 \mathrm{~mm}$.

Satpal Satpal

Numerade Educator

Problem 120

The 2014-T6 hollow column is fixed at its base and free at its top. Determine the maximum eccentric force $P$ that can be safely supported by the column. Use the interaction formula. The allowable bending stress is $\left(\omega_2\right)_{\text {allow }}-200 \mathrm{MP}$. The thickness of the wall for the section is $t=12 \mathrm{~mm}$.

Surendra Kumar

Numerade Educator

Problem 121

Determine if the column can support the eccentric compressive load of $P=7.5 \mathrm{kN}$. Assume that the ends are pin connected. Use the NFPA equations in Sec. 13.6 and Eq. 13-30.

Satpal Satpal

Numerade Educator

Problem 122

Determine if the column can support the eccentric compressive load of $P=7.5 \mathrm{kN}$. Assume that the bottom is fixed and the top is pinned. Use the NFPA equations in Sec. 13.6 and Eq. 13-30.

Prohs. 13-121/122

Satpal Satpal

Numerade Educator

Problem 123

The 250 -mm-diameter utility pole supports the transformer that has a weight of 3 kN and center of gravity at $G$. If the pole is fixed to the ground and free at its top, determine if it is adequate according to the NFPA equations of Sec. 13.6 and Eq. 13-30.

Satpal Satpal

Numerade Educator

Problem 124

Using the NFPA equations of Sec. 13.6 and Eq. 13-30, determine the maximum allowable eccentric load $P$ that can be applied to the wood column. Assume that the column is pinned at both its top and bottom.

Surendra Kumar

Numerade Educator

Problem 125

Using the NFPA equations of Sec. 13.6 and Eq. 13-30, determine the maximum allowable eccentric load $P$ that can be applied to the wood column. Assume that the column is pinned at the top and fixed at the bottom.

Surendra Kumar

Numerade Educator

Problem 126

The 3-m-long bar is made of aluminum alloy 2014-T6. If it is fixed at its bottom and pinned at the top, determine the maximum allowable eccentric load $\mathbf{P}$ that can be applied using the formulas in Sec. 13.6 and Eq. 13-30.

Satpal Satpal

Numerade Educator

Problem 127

The 3-m-long bar is made of aluminum alloy 2014 T6. If it is fixed at its bottom and pinned at the top, determine the maximum allowable eccentric load $\mathbf{P}$ that can be applied using the equations of Sec. 13.6 and the interaction formula with $\left(\sigma_b\right)_{3 l b w}=126 \mathrm{MPa}$.

Surendra Kumar

Numerade Educator