Chapter 6, Problem 6.1PFS

To determine

The selection of the lightest available W12 section to support the axial compression loads.

Answer to Problem 6.1PFS

W12 x 65

Explanation of Solution

Given data:

$\begin{array}{l}{P}_D=150kips\\ P_L=230kips\\ L_C=18ft\\ Steelused=A992,Grade50\end{array}$

LRFD method:

The factored load is given by

$\begin{array}{l}{P}_u=1.2P_D+1.6P_L\\ where,P_{u}isfactoredload\\ P_{D}isaxialcompressiondeadload\\ P_{L}isaxialcompressionliveload\end{array}$

Substituting the values,

$\begin{array}{l}{P}_u=1.2\left(150\right)+1.6\left(230\right)\\ P_u=548k\end{array}$

Assume slenderness ratio as

$\frac{KL}{r}=60$

From table 4-14 in the AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=60$ , the critical stress is

${\varphi }_c{f}_{cr}=34.6ksi$

The required area is

$\begin{array}{l}{A}_{req}=\frac{P_u}{{\varphi }_c{f}_{cr}}\\ where,A_{req}isrequiredarea\\ P_{u}isfactoredload\\ {\varphi }_c{f}_{cr}iscriticalstress\end{array}$

Substituting the values,

$\begin{array}{l}{A}_{req}=\frac{548}{34.6}\\ A_{req}=15.84in{.}^2\end{array}$

Try section $W12\times 58$ .

From table 1-1 in the AISC manual,

The properties of the trial section are

$\begin{array}{l}Area,A=17.0in{.}^2\\ Minimumradiusofgyration,r_{\min }=2.51in.\end{array}$

The slenderness ratio is

$\begin{array}{l}\lambda =\frac{L_c}{r_{\min }}\\ where,\lambda isslendernessratio\\ L_{c}iseffectivelengthofcolumn\\ r_{\min }is\min imumradiusofgyration\end{array}$

Substituting the values,

$\begin{array}{l}\lambda =\frac{18\times 12}{2.51}\\ \lambda =86.06\end{array}$

From table 4-14 in the AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=86$ , the critical stress is

${\varphi }_c{f}_{cr}=26.2ksi$

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=87$ , the critical stress is

${\varphi }_c{f}_{cr}=25.9ksi$

By interpolation, the critical stress for $F_y=50ksi$ and slenderness ratio $\frac{KL}{r}=86.06$ is

$\begin{array}{l}{\varphi }_c{f}_{cr}=26.2+\frac{25.9-26.2}{87-86}\left(86.06-86\right)\\ {\varphi }_c{f}_{cr}=26.182ksi\\ {\varphi }_c{f}_{cr}=26.18ksi\end{array}$

The load-carrying capacity of the section is

$\begin{array}{l}{\varphi }_c{P}_n=A\times {\varphi }_c{f}_{cr}\\ where,{\varphi }_c{P}_{n}isavailableloadcarryingcapacity\\ Aisareaof\sec tion\\ {\varphi }_c{f}_{cr}iscriticalstress\end{array}$

Substituting the values,

$\begin{array}{l}{\varphi }_c{P}_n=17\times 26.18\\ {\varphi }_c{P}_n=445.06k\\ {\varphi }_c{P}_n<P_u\\ 445.06k<548k\end{array}$

The load-carrying capacity of the trial section $W12\times 58$ is less than the required capacity. Hence, the section is not safe.

Try section $W12\times 65$ .

From table 1-1 in the AISC manual,

The properties of the trial section are

$\begin{array}{l}Area,A=19.1in{.}^2\\ Minimumradiusofgyration,r_{\min }=3.02in.\end{array}$

The slenderness ratio is

$\lambda =\frac{L_c}{r_{\min }}$

Substituting the values,

$\begin{array}{l}\lambda =\frac{18\times 12}{3.02}\\ \lambda =71.52\end{array}$

From table 4-14 in the AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=71$ , the critical stress is

${\varphi }_c{f}_{cr}=31.1ksi$

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=72$ , the critical stress is

${\varphi }_c{f}_{cr}=30.8ksi$

By interpolation, the critical stress for $F_y=50ksi$ and slenderness ratio $\frac{KL}{r}=71.52$ is

$\begin{array}{l}{\varphi }_c{f}_{cr}=31.1+\frac{30.8-31.1}{72-71}\left(71.52-71\right)\\ {\varphi }_c{f}_{cr}=30.944ksi\\ {\varphi }_c{f}_{cr}=30.94ksi\end{array}$

The load-carrying capacity of the section is

${\varphi }_c{P}_n=A\times {\varphi }_c{f}_{cr}$

Substituting the values,

$\begin{array}{l}{\varphi }_c{P}_n=19.1\times 30.94\\ {\varphi }_c{P}_n=590.954k\\ {\varphi }_c{P}_n=590.95k\\ {\varphi }_c{P}_n>P_u\\ 590.95k>548k\end{array}$

The load-carrying capacity of the trial section $W12\times 65$ is more than the required capacity. Hence, the section is safe.

Therefore, use section $W12\times 65$ in LRFD method.

ASD method:

The factored load is given by

$\begin{array}{l}{P}_a=P_D+P_L\\ where,P_{a}isfactoredload\\ P_{D}isaxialcompressiondeadload\\ P_{L}isaxialcompressionliveload\end{array}$

Substituting the values,

$\begin{array}{l}{P}_a=150+230\\ P_a=380k\end{array}$

Assume slenderness ratio as

$\frac{KL}{r}=60$

From table 4-14 in AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=60$ , the critical stress is

$\frac{f_{cr}}{{\Omega }_c}=23ksi$

The required area is

$\begin{array}{l}{A}_{req}=\frac{P_a}{\frac{f_{cr}}{{\Omega }_c}}\\ where,A_{req}isrequiredarea\\ P_{a}isfactoredload\\ \frac{f_{cr}}{{\Omega }_c}iscriticalstress\end{array}$

Substituting the values,

$\begin{array}{l}{A}_{req}=\frac{380}{23}\\ A_{req}=16.52in{.}^2\end{array}$

Try section $W12\times 58$ .

From table 1-1 in the AISC manual,

The properties of the trial section are

$\begin{array}{l}Area,A=17.0in{.}^2\\ Minimumradiusofgyration,r_{\min }=2.51in.\end{array}$

The slenderness ratio is

$\begin{array}{l}\lambda =\frac{L_c}{r_{\min }}\\ where,\lambda isslendernessratio\\ L_{c}iseffectivelengthofcolumn\\ r_{\min }is\min imumradiusofgyration\end{array}$

Substituting the values,

$\begin{array}{l}\lambda =\frac{18\times 12}{2.51}\\ \lambda =86.06\end{array}$

From table 4-14 in the AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=86$ , the critical stress is

$\frac{f_{cr}}{{\Omega }_c}=17.4ksi$

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=87$ , the critical stress is

$\frac{f_{cr}}{{\Omega }_c}=17.2ksi$

By interpolation, the critical stress for $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=86.06$ is

$\begin{array}{l}\frac{f_{cr}}{{\Omega }_c}=17.4+\frac{17.2-17.4}{87-86}\left(86.06-86\right)\\ \frac{f_{cr}}{{\Omega }_c}=17.388ksi\\ \frac{f_{cr}}{{\Omega }_c}=17.39ksi\end{array}$

The load-carrying capacity of the section is

$\begin{array}{l}\frac{P_n}{{\Omega }_c}=A\times \frac{f_{cr}}{{\Omega }_c}\\ where,\frac{P_n}{{\Omega }_c}isavailableloadcarryingcapacity\\ Aisareaof\sec tion\\ \frac{f_{cr}}{{\Omega }_c}iscriticalstress\end{array}$

Substituting the values,

$\begin{array}{l}\frac{P_n}{{\Omega }_c}=17\times 17.39\\ \frac{P_n}{{\Omega }_c}=295.63k\\ \frac{P_n}{{\Omega }_c}<P_a\\ 295.63k<380k\end{array}$

The load-carrying capacity of the trial section $W12\times 58$ is less than the required capacity. Hence, the section is not safe.

Try section $W12\times 65$ .

From table 1-1 in the AISC manual,

The properties of the trial section are

$\begin{array}{l}Area,A=19.1in{.}^2\\ Minimumradiusofgyration,r_{\min }=3.02in.\end{array}$

The slenderness ratio is

$\lambda =\frac{L_c}{r_{\min }}$

Substituting the values,

$\begin{array}{l}\lambda =\frac{18\times 12}{3.02}\\ \lambda =71.52\end{array}$

From table 4-14 in the AISC manual,

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=71$ , the critical stress is

$\frac{f_{cr}}{{\Omega }_c}=20.7ksi$

Corresponding to yield stress, $F_y=50ksi$ and slenderness ratio, $\frac{KL}{r}=72$ , the critical stress is

$\frac{f_{cr}}{{\Omega }_c}=20.5ksi$

By interpolation, the critical stress for $F_y=50ksi$ and slenderness ratio $\frac{KL}{r}=71.52$ is

$\begin{array}{l}\frac{f_{cr}}{{\Omega }_c}=20.7+\frac{20.5-20.7}{72-71}\left(71.52-86\right)71\\ \frac{f_{cr}}{{\Omega }_c}=20.596ksi\\ \frac{f_{cr}}{{\Omega }_c}=20.60ksi\end{array}$

The load-carrying capacity of the section is

$\frac{P_n}{{\Omega }_c}=A\times \frac{f_{cr}}{{\Omega }_c}$

Substituting the values,

$\begin{array}{l}\frac{P_n}{{\Omega }_c}=19.1\times 20.60\\ \frac{P_n}{{\Omega }_c}=393.46k\\ \frac{P_n}{{\Omega }_c}>P_a\\ 393.46k>380k\end{array}$

The load-carrying capacity of the trial section $W12\times 65$ is more than the required capacity. Hence, the section is safe.

Therefore, use section $W12\times 65$ in ASD method.

Conclusion:

Select the section $W12\times 65$ in both the LRFD method and ASD method.