Chapter 4, Problem 4.3.1P

To determine

(a)

Nominal compressive strength of column

Answer to Problem 4.3.1P

$1267.1\text{kips}$

Explanation of Solution

Calculation:

Obtain the effective length factor for the support condition

$L_{eff}=L$

Calculate the effective slenderness ratio using the formula as follows

$\begin{array}{l}\text{Slenderness ratio =}\frac{KL}{r}\\ \text{Consider the properties of the steel from the AISC stell table }\\ \text{Substitute 1 for k,(10}\times \text{12)in for L,and 2}{\text{.65 for r}}_y\text{.}\\ \text{Slenderness ratio=}\frac{KL}{r}\\ \text{ =}\frac{1\left(10\times 12\right)}{2.65}\\ \text{ =45}\text{.2}\end{array}$

Calculate the buckling stress by using the formula as follows:

$F_e=\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}$

$\text{substitute }29000\text{ for E,and 45}\text{.2 for KL/r}\text{.}$

$F_e=\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}$

$\begin{array}{l}\text{ =}\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}\\ \text{ =}\frac{{\pi }^2(29000)}{{45.2}^2}\\ \text{ =140}\text{.1ksi}\end{array}$

Check for slenderness ratio by using the relation as follows:

$\begin{array}{l}\text{Slenderness ratio=4}\text{.71}\sqrt{\frac{E}{F_y}}\\ F_y\text{ is the yield strength}\text{.}\\ \text{Substitute 50ksi for }{F}_y,\text{and }29000\text{ for E}\text{.}\\ \text{Slenderness ratio=4}\text{.71}\sqrt{\frac{29000}{50}}\\ \text{ =113}\text{.4}\\ \text{Since the value 45}\text{.28 is less than 113}\text{.4, so caluculate critical buckling stress using the formula}\text{.}\\ F_{cr}={0.658}^{\left(\frac{F_y}{F_e}\right)}{F}_y\\ \text{ =0}{\text{.658}}^{\left(\frac{50}{140.1}\right)}\left(50\right)\\ \text{ =43}\text{.1ksi}\end{array}$

Calculate the nominal compressive strength of column by using the formula as follows:

$\begin{array}{l}{P}_n=F_{cr}{A}_g\\ \text{Substitute 29}\text{.40 for }{A}_g,\text{and 43}{\text{.03 for F}}_{cr}\\ P_n=F_{cr}{A}_g\\ \text{ =43}\text{.1}\left(29.40\right)\\ \text{ =1267}\text{.1kips}\end{array}$

Conclusion:

Therefore, the nominal compressive strength of column $1267.1\text{kips}$.

To determine

(b)

Nominal compressive strength of column

Answer to Problem 4.3.1P

$\text{400kips}$

Explanation of Solution

Calculation:

Calculate the effective slenderness ratio using the formula as follows

$\begin{array}{l}\text{Slenderness ratio =}\frac{KL}{r}\\ \text{Consider the properties of the steel from the AISC steel table }\\ \text{Substitute 1 for k,(30}\times \text{12)in for L,and 2}{\text{.65 for r}}_y\text{.}\\ \text{Slenderness ratio=}\frac{KL}{r}\\ \text{ =}\frac{1\left(30\times 12\right)}{2.65}\\ \text{ =135}\text{.84}\end{array}$

Calculate the buckling stress by using the formula as follows:

$F_e=\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}$

$\text{substitute }29000\text{ for E,and 135}\text{.84 for KL/r}\text{.}$

$F_e=\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}$

$\begin{array}{l}\text{ =}\frac{{\pi }^{2}E}{{\left(KL/r\right)}^2}\\ \text{ =}\frac{{\pi }^2(29000)}{{135.84}^2}\\ \text{ =15}\text{.50ksi}\end{array}$

Check for slenderness ratio by using the relation as follows:

$\begin{array}{l}\text{Slenderness ratio=4}\text{.71}\sqrt{\frac{E}{F_y}}\\ F_y\text{ is the yield strength}\text{.}\\ \text{Substitute 50ksi for }{F}_y,\text{and }29000\text{ for E}\text{.}\\ \text{Slenderness ratio=4}\text{.71}\sqrt{\frac{29000}{50}}\\ \text{ =113}\text{.4}\\ \text{Since the value 135}\text{.84 is greater than 113}\text{.4, so caluculate critical buckling stress using the formula}\text{.}\\ F_{cr}=0.877F_e\\ \text{ =0}\text{.877(15}\text{.50)}\\ \text{ =13}\text{.60ksi}\end{array}$

Calculate the nominal compressive strength of column by using the formula as follows:

$\begin{array}{l}{P}_n=F_{er}{A}_g\\ \text{Substitute 29}\text{.40 for }{A}_g,\text{and 13}{\text{.60 for F}}_{cr}\\ P_n=F_{cr}{A}_g\\ \text{ =13}\text{.60}\left(29.40\right)\\ \text{ =400kips}\end{array}$

Conclusion:

Therefore, the nominal compressive strength of column $\text{400kips}$.