Solve Prob. 10.97 using the interaction method with P 5 18

Knowing that the spring at A is of constant k and that the bar AB is rigid, determine the critical load \(P_{\mathrm{cr}}\).

Two rigid bars AC and BC are connected by a pin at C as shown. Knowing that the torsional spring at B is of constant K, determine the critical load \(P_{\mathrm{cr}}\) for the system.

Two rigid bars AC and BC 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 \(P_{\mathrm{cr}}\) for the system.

Two rigid bars AC and BC 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 \(P_{\mathrm{cr}}\) for the system.

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

The rigid rod AB 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 ft , determine the critical load.

The rigid bar AD is attached to two springs of constant k and is in equilibrium in the position shown. Knowing that the equal and opposite loads P and \(\mathrm{P}^{\prime}\) remain horizontal, determine the magnitude \(\mathrm{P}_{\mathrm{cr}}\) of the critical load for the system.

A frame consists of four L-shaped members connected by four torsional springs, each of constant K . Knowing that equal loads P are applied at points A and D as shown, determine the critical value \(\mathrm{P}_{\mathrm{cr}\) of the loads applied to the frame.

Determine the critical load of a pin-ended steel tube that is 5 m long and has a 100-mm outer diameter and a 16-mm wall thickness. Use E = 200 GPa.

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

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.

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 GPa.

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 GPa.

Determine (a) the critical load for the square strut, (b) the radius of the round strut for which both struts have the same critical 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 GPa.

A column with the cross section shown has a 13.5-ft effective length. Using a factor of safety equal to 2.8, determine the allowable centric load that can be applied to the column. Use \(E=29 \times 10^{6} \mathrm{psi}\).

A column is made from half of a W360 X 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 m. Use E = 200 GPa.

A column of 22-ft effective length is made by welding two 9 3 0.5-in. plates to a W8 X 35 as shown. Determine the allowable centric load if a factor of safety of 2.3 is required. Use \(E=29 \times 10^{6} \mathrm{psi}\).

A single compression member of 8.2-m effective length is obtained by connecting two C200 X 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 GPa and d = 100 mm.

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

Members AB and CD are 30-mm-diameter steel rods, and members BC and AD are 22-mm-diameter steel rods. When the turnbuckle is tightened, the diagonal member AC 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 AC. Use E = 200 GPa and consider only buckling in the plane of the structure.

The uniform brass bar AB 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 ft, d = 1.5 in., and \(E=29 \times 10^{6} \mathrm{psi}\).

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 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 \(E=10.4 \times 10^{6} \mathrm{psi}\).

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 \(L_{A B}=3 \mathrm{ft}, L_{B C}=4 \mathrm{ft} \text {, and } L_{C D}=1 \mathrm{ft}\), determine the allowable load P using a factor of safety with respect to buckling of 3.2. Consider only buckling in the plane of the figure and use \(E=10.4 \times 10^{6} \mathrm{psi}\).

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

Column ABC has a uniform rectangular cross section and is braced in the xz 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 xz and yz 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 kN , L = 1 m, and E = 200 GPa .

Column AB carries a centric load P of magnitude 15 kips. Cables BC and BD are taut and prevent motion of point B in the xz 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 \(E=29 \times 10^{6} \mathrm{psi}\).

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 5 200 GPa.

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-mm outer diameter and a 4-mm wall thickness. Using E = 70 GPa and a factor of safety of 2.8, determine the allowable mass for each support condition.

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

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

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

An axial load P is applied to the 1.375-in. diameter steel rod AB 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.

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

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

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

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

Solve Prob. 10.36, assuming that the axial load P is applied 10 mm from the geometric axis of the column. A brass pipe having the cross section shown has an axial load P applied 5 mm from its geometric axis. Using E 5 120 GPa, determine (a) the load P for which the horizontal deflection at the midpoint C is 5 mm, (b) the corresponding maximum stress in the column.

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

The line of action of the axial load 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 mm from the geometric axis of the W310 X 60 rolled-steel column BC. Assuming that L = 7.0 m and using E = 200 GPa, determine (a) the load P for which the horizontal deflection of the midpoint C of the column is 15 mm, (b) the corresponding maximum stress in the column.

Solve Prob. 10.39, assuming that L is 9.0 m. The line of action of the axial load 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 mm from the geometric axis of the W310 X 60 rolled-steel column BC. Assuming that L = 7.0 m and using E = 200 GPa, determine (a) the load P for which the horizontal deflection of the midpoint C of the column is 15 mm, (b) the corresponding maximum stress in the column.

The steel bar AB has a \(\frac{3}{8} \times \frac{3}{8}-\mathrm{in}\) 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 \(\alpha=6.5 \times 10^{-6} /{ }^{\circ} \mathrm{F}\).

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 \(120^{\circ} \mathrm{F}\).

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 GPa. Knowing that a factor of safety of 2.6 with respect to permanent deformation is required, determine the allowable load P when the eccentricity e is (a) 15 mm, (b) 7.5 mm. (Hint: Since the factor of safety must be applied to the load P, not to the stress, use Fig. 10.24 to determine \(P_{Y}\)).

Solve Prob. 10.43, assuming that the length of the tube is increased to 5 m. 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 GPa. Knowing that a factor of safety of 2.6 with respect to permanent deformation is required, determine the allowable load P when the eccentricity e is (a) 15 mm, (b) 7.5 mm. (Hint: Since the factor of safety must be applied to the load P, not to the stress, use Fig. 10.24 to determine \(P_{Y}\)).

An axial load P is applied to the W8 X 28 rolled-steel column BC 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 \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\), 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.)

An axial load 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 W8 X 28 rolled-steel column BC. Knowing that the column is free at its top C and fixed at its base B and that \(\sigma_{Y}=36 \mathrm{ksi}\) and \(E=29 \times 10^{6} \mathrm{psi}\), determine the factor of safety with respect to yield. (See hint of Prob. 10.43.)

A 100-kN axial load P is applied to the W150 X 18 rolled-steel column BC that is free at its top C and fixed at its base B. Knowing that the eccentricity of the load is e = 6 mm, determine the largest permissible length L if the allowable stress in the column is 80 MPa. Use E = 200 GPa.

A 26-kip axial load P is applied to a W6 X 12 rolled-steel column BC 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 \(E=29 \times 10^{6} \mathrm{psi}\).

Axial loads of magnitude P = 135 kips are applied parallel to the geometric axis of the W10 X 54 rolled-steel column AB and intersect the x axis at a distance e from the geometric axis. Knowing that \(\sigma_{\text {all }}=12 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\), determine the largest permissible length L when (a) e = 0.25 in., (b) e = 0.5 in.

Axial loads of magnitude P = 84 kN are applied parallel to the geometric axis of the W200 X 22.5 rolled-steel column AB and intersect the x axis at a distance e from the geometric axis. Knowing that \(\sigma_{\text {all }}=75 \mathrm{MPa}\) and E = 200 GPa, determine the largest permissible length L when (a) e = 5 mm, (b) e = 12 mm.

An axial load of magnitude P = 220 kN is applied at a point located on the x axis at a distance e = 6 mm from the geometric axis of the wide-flange column BC. Knowing that E = 200 GPa, choose the lightest W200 shape that can be used if \(\sigma_{\text {all }}=120 \mathrm{MPa}\).

Solve Prob. 10.51, assuming that the magnitude of the axial load is P = 345 kN. An axial load of magnitude P = 220 kN is applied at a point located on the x axis at a distance e = 6 mm from the geometric axis of the wide-flange column BC. Knowing that E = 200 GPa, choose the lightest W200 shape that can be used if \(\sigma_{\text {all }}=120 \mathrm{MPa}\).

A 12-kip axial load is applied with an eccentricity e = 0.375 in. to the circular steel rod BC 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 \(\frac{1}{8} \mathrm{in} .\) from 1.5 in. to 3.0 in., determine the lightest rod that can be used if \(\sigma_{\text {all }}=15 \mathrm{ksi}\). Use \(E=29 \times 10^{6} \mathrm{psi}\).

Solve Prob. 10.53, assuming that the 12-kip axial load will be applied to the rod with an eccentricity \(e=\frac{1}{2} d\). A 12-kip axial load is applied with an eccentricity e = 0.375 in. to the circular steel rod BC 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 \(\frac{1}{8} \mathrm{in} .\) from 1.5 in. to 3.0 in., determine the lightest rod that can be used if \(\sigma_{\text {all }}=15 \mathrm{ksi}\). Use \(E=29 \times 10^{6} \mathrm{psi}\).

Axial loads of magnitude P = 175 kN are applied parallel to the geometric axis of a W250 X 44.8 rolled-steel column AB and intersect the x axis at a distance e = 12 mm from its geometric axis. Knowing that \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa, determine the factor of safety with respect to yield. (Hint: Since the factor of safety must be applied to the load P, not to the stresses, use Fig. 10.24 to determine \(P_{Y}\).)

Solve Prob. 10.55, assuming that e = 16 mm and P = 155 kN. Axial loads of magnitude P = 175 kN are applied parallel to the geometric axis of a W250 X 44.8 rolled-steel column AB and intersect the x axis at a distance e = 12 mm from its geometric axis. Knowing that \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa, determine the factor of safety with respect to yield. (Hint: Since the factor of safety must be applied to the load P, not to the stresses, use Fig. 10.24 to determine \(P_{Y}\).)

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) W200 X 35.9, (b) W200 X 86. Use \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa.

A W8 X 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) \(\sigma_{Y}=50 \mathrm{ksi}\). Use \(E=29 \times 10^{6} \mathrm{psi}\).

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 GPa, use allowable stress design to determine the largest centric load that can be applied to the steel column.

A column having a 3.5-m effective length is made of sawn lumber with a 114 X 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 GPa, determine the maximum allowable centric load for the column.

A sawn lumber column with a 7.5 X 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 \(\sigma_{C}=1200 \mathrm{psi}\) and that the adjusted modulus \(E=470 \times 10^{3}\) psi, determine the maximum allowable centric load for the column.

Bar AB is free at its end A and fixed at its base B. Determine the allowable centric load P if the aluminum alloy is (a) 6061-T6, (b) 2014-T6.

A compression member has the cross section shown and an effective length of 5 ft. Knowing that the aluminum alloy used is 2014-T6, determine the allowable centric load.

A compression member has the cross section shown and an effective length of 5 ft. Knowing that the aluminum alloy used is 6061-T6, determine the allowable centric load.

A compression member of 8.2-ft effective length is obtained by bolting together two \(\text { L5 } \times 3 \times \frac{1}{2} \text {-in }\). steel angles as shown. Using allowable stress design, determine the allowable centric load for the column. Use \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

A compression member of 9-m effective length is obtained by welding two 10-mm-thick steel plates to a W250 X 80 rolled-steel shape as shown. Knowing that \(\sigma_{Y}=345 \mathrm{MPa}\) and E = 200 GPa and using allowable stress design, determine the allowable centric load for the compression member.

A column of 6.4-m effective length is obtained by connecting four L89 X 89 X 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 GPa.

A column of 21-ft effective length is obtained by connecting C10 X 20 steel channels with lacing bars as shown. Using allowable stress design, determine the allowable centric load for the column. Use \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

The glued laminated column shown is made from four planks, each of 38 X 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 GPa, determine the maximum allowable centric load if the effective length of the column is (a) 7 m, (b) 3 m.

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-T6, determine the maximum allowable centric load.

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 GPa, determine the smallest cross section that can support a centric load of 62 kN.

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 \(\sigma_{C}=1050 \mathrm{psi}\) and the adjusted modulus is \(E=440 \times 10^{3} \mathrm{psi}\), determine the smallest cross section that can be used. Use b = 2d.

A laminated column of 2.1-m effective length is to be made by gluing together wood pieces of 25 X 150-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 GPa, determine the number of wood pieces that must be used to support the concentric load shown when (a) P = 52 kN, (b) P = 108 kN.

For a rod made of aluminum alloy 2014-T6, select the smallest square cross section that can be used if the rod is to carry a 55-kip centric load.

A 72-kN centric load must be supported by an aluminum column as shown. Using the aluminum alloy 6061-T6, determine the minimum dimension b that can be used.

An aluminum tube of 90-mm outer diameter is to carry a centric load of 120 kN. Knowing that the stock of tubes available for use are made of alloy 2014-T6 and with wall thicknesses in increments of 3 mm from 6 mm to 15 mm, determine the lightest tube that can be used.

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

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} \text { and } E=29 \times 10^{6} \mathrm{psi}.

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} \text { and } E=29 \times 10^{6} \mathrm{psi}.

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

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 \(\frac{1}{4} \text { in. to } \frac{3}{4} \text { in. }\) in increments of \(\frac{1}{16} \text { in. }\), use allowable stress design to determine the lightest tube that can be used. Use \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

Solve Prob. 10.81, assuming that the effective length of the column is decreased to 20 ft. 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 \(\frac{1}{4} \text { in. to } \frac{3}{4} \text { in. }\) in increments of \(\frac{1}{16} \text { in. }\), use allowable stress design to determine the lightest tube that can be used. Use \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

Two 89 X 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 180 kN. Knowing that the angles available have thicknesses of 6.4 mm, 9.5 mm, and 12.7 mm, use allowable stress design to determine the lightest angles that can be used. Use \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa.

Two 89 3 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 kN. Knowing that the angles available have thicknesses of 6.4 mm, 9.5 mm, and 12.7 mm, use allowable stress design to determine the lightest angles that can be used. Use \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa.

A rectangular steel tube having the cross section shown is used as a column of 14.5-ft effective length. Knowing that \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\), 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 \(\phi=0.90\).

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 f 5 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) W250 X 67, (b) W360 X 101. Use \(\sigma_{Y}=345 \mathrm{MPa}\) and E = 200 GPa.

A steel column of 5.5-m effective length must carry a centric dead load of 310 kN and a centric live load of 375 kN. Knowing that \(\sigma_{Y}=250 \mathrm{MPa}\) and E = 200 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 \(\phi=0.90\).

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 116 in. from 316 in. to 38 in., use load and resistance factor design to determine the lightest tube that can be used. Use \(\sigma_{Y}=36 \mathrm{ksi}\) and \(E=29 \times 10^{6} \mathrm{psi}\). The dead load factor \(\gamma_{D}=1.2\), the live load factor \(\gamma_{L}=1.6\), and the resistance factor \(\phi=0.90\).

An eccentric load is applied at a point 22 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 GPa. Using the allowable-stress method, determine the allowable load P.

Solve Prob, 10.89, assuming that the load is applied at a point 40 mm from the geometric axis and that the effective length is 0.9 m. An eccentric load is applied at a point 22 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 GPa. Using the allowable-stress method, determine the allowable load P.

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

Solve Prob. 10.91 using the interaction method and an allowable stress in bending of 1300 psi. A sawn-lumber column of 5.0 X 7.5-in. cross section has an effective length of 8.5 ft. The grade of wood used has an adjusted allowable stress for compression parallel to the grain \(\sigma_{C}= 1180 \mathrm{psi}\) and an adjusted modulus \. Using the allowable-stress method, determine the largest eccentric load P that can be applied when (a) e 5 0.5 in., (b) e 5 1.0 in.

A column of 5.5-m effective length is made of the aluminum alloy 2014-T6 for which the allowable stress in bending is 220 MPa. Using the interaction method, determine the allowable load P, knowing that the eccentricity is (a) e = 0, (b) e = 40 mm.

Solve Prob. 10.93, assuming that the effective length of the column is 3.0 m. A column of 5.5-m effective length is made of the aluminum alloy 2014-T6 for which the allowable stress in bending is 220 MPa. Using the interaction method, determine the allowable load P, knowing that the eccentricity is (a) e = 0, (b) e = 40 mm.

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 \(E=29 \times 10^{6} \mathrm{psi}\).

Solve Prob. 10.95, assuming that the effective length of the column is increased to 12 ft and that (a) P = 20 kips, (b) P = 15 kips. 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 \(E=29 \times 10^{6} \mathrm{psi}\).

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

Solve Prob. 10.97 using the interaction method with P = 18 kips and an allowable stress in bending of 22 ksi. Two \(\mathrm{L} 4 \times 3 \times \frac{3}{8} \text {-in }\). steel angles are welded together to form the column AB. An axial load P of magnitude 14 kips is applied at point D. Using the allowable-stress method, determine the largest allowable length L. Assume \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

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

Solve Prob. 10.99, assuming that P = 105 kN. A rectangular column is made of a grade of sawn wood that has an adjusted allowable stress for compression parallel to the grain \(\sigma_{Y}=8.3 \mathrm{MPa}\) and an adjusted modulus of elasticity E = 11.1 GPa. Using the allowable-stress method, determine the largest allowable effective length L that can be used.

An eccentric load P = 48 kN is applied at a point 20 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.

Solve Prob. 10.101, assuming that the aluminum alloy used is 2014-T6 and that the allowable stress in bending is 180 MPa. An eccentric load P = 48 kN is applied at a point 20 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.

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

Solve Prob. 10.103, assuming that the effective length is 1.62 m and that the magnitude of P of the eccentric load is 128 kN. A compression member made of steel has a 720-mm effective length and must support the 198-kN load P as shown. For the material used \sigma_{Y}=250 \mathrm{MPa}\) and E = 200 GPa. Using the interaction method with an allowable bending stress equal to 150 MPa, determine the smallest dimension d of the cross section that can be used.

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

Solve Prob. 10.105, using the interaction method with P = 165 kN, e = 15 mm, and an allowable stress in bending of 150 MPa. A steel tube of 80-mm outer diameter is to carry a 93-kN load P with an eccentricity of 20 mm. The tubes available for use are made with wall thicknesses in increments of 3 mm from 6 mm to 15 mm. Using the allowable-stress method, determine the lightest tube that can be used. Assume E = 200 GPa and \(\sigma_{Y}=250 \mathrm{MPa}\).

A sawn lumber column of rectangular cross section has a 2.2-m effective length and supports a 41-kN load as shown. The sizes available for use have b equal to 90 mm, 140 mm, 190 mm, and 240 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.

Solve Prob. 10.107, assuming that e = 40 mm. A sawn lumber column of rectangular cross section has a 2.2-m effective length and supports a 41-kN load as shown. The sizes available for use have b equal to 90 mm, 140 mm, 190 mm, and 240 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.

A compression member of rectangular cross section has an effective length of 36 in. and is made of the aluminum alloy 2014-T6 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.

Solve Prob. 10.109, assuming that e = 0.2 in. A compression member of rectangular cross section has an effective length of 36 in. and is made of the aluminum alloy 2014-T6 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.

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-T6 and have wall thicknesses in increments of \(\frac{1}{16} \mathrm{in} . \operatorname{up} \text { to } \frac{1}{2} \mathrm{in} .\), determine the lightest tube that can be used. Use the allowable-stress method.

Solve Prob. 10.111, using the interaction method of design with an allowable stress in bending of 25 ksi. 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-T6 and have wall thicknesses in increments of \(\frac{1}{16} \mathrm{in} . \operatorname{up} \text { to } \frac{1}{2} \mathrm{in} .\), determine the lightest tube that can be used. Use the allowable-stress method.

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 \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

Solve Prob. 10.113 using the interaction method, assuming that \(\sigma_{Y}=50 \mathrm{ksi}\) and the allowable stress in bending is 30 ksi. 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 \(\sigma_{Y}=36 \mathrm{ksi} \text { and } E=29 \times 10^{6} \mathrm{psi}\).

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

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