# Step-by-Step Solution ## **Given Data** -Dead Load (DL):** 600 kips - **Live Load (LL):** 300 kips - **Total Axial Load (P):** \( P = DL + LL = 600 + 300 = 900 \) kips - **Element Length (L):** 28 ft (\( 336 \) in) - **Braced in y-direction:** 14 ft (\( 168 \) in) from base - **End Conditions:** Fixed at base, pinned at top - **Steel Grade:** ASTM A572 Grade60 (\( F_y = 60 \) ksi **Check:** Slenderness and local buckling --- **Step 1: Required Strength (\( P \** Apply \( \phi = .9 \) LRFD: \[ P_u = 1.2 \times DL + 1.6 \times LL = 1.2 \times 600 + 1.6 \times 300 = 720 + 480 = 120 \text{ kips} \] --- ## **Step 2: Effective Length Factor (\( K \))** For **fixed-pinned** condition (AISC Table): \[ K \approx .7 \] --- ## **Step 3: Slenderness Ratio (\( \lambda \))** - **Effective Length:** \( K \cdot L = .7 \times 28\,\text{ft} = 19.6\,\text{ft} = 235.2\,\text{in} \) - **Maximum Unbraced Length:** \( L_{u} = 14\,\text{ft} = 168\,\text{in} \) (in y-direction) We must check both axes: - **x-axis (strong axis):** \( K \cdot L = 235.2\,\text{in} \) - **y-axis (weak axis):** \( L = 168\,\text{in} \) (braced at 14 ft) --- ## **Step 4: Select a Trial Section** For a heavy axial load, start with a **W-shape**. Assume **W14x176** as a trial (from AISC Manual): - \( A_g = 51.8\,\text{in}^2 \) - \( r_x = 5.98\,\text{in} \) - \( r_y = 4.02\,\text{in} \) - \( b_f = 14.74\,\text{in} \) - \( t_f = 1.19\,\text{in} \) - \( d = 16.50\,\text{in} \) - \( t_w = .59\,\text{in} \) --- ## **Step 5: Slenderness Check** \[ \lambda = \frac{K L}{r} \] - **x-axis:** \( \frac{235.2}{5.98} = 39.33 \) - **y-axis:** \( \frac{168}{4.02} = 41.79 \) **Use the larger:** \( \lambda = 41.8 \) *Limit for columns (AISC):* \( \lambda \leq 200 \) → **OK** --- ## **Step 6: Axial Strength** ### **Nominal Strength (\( P_n \))** \[ F_y = 60\,\text{ksi} \] \[ P_n = F_{cr} \times A_g \] Calculate slenderness parameter: \[ \frac{K L}{r} = 41.8 \] \[ F_{cr} = .658^{(F_y / F_{e})} F_y \] Where: \[ F_e = \frac{\pi^2 E}{(K L/r)^2} \] \[ E = 29000\,\text{ksi} \] \[ F_e = \frac{\pi^2 \times 29000}{(41.8)^2} = \frac{286,870}{1747} = 164.2\,\text{ksi} \] \[ \frac{F_y}{F_e} = \frac{60}{164.2} = .365 \] \[ F_{cr} = .658^{.365} \times 60 = .853 \times 60 = 51.2\,\text{ksi} \] \[ P_n = 51.2 \times 51.8 = 2652\,\text{kips} \] \[ \phi_c = .9 \] \[ \phi_c P_n = .9 \times 2652 = 2387\,\text{kips} \] --- ## **Step 7: Compare with Required Strength** \[ \phi_c P_n = 2387\,\text{kips} > P_u = 120\,\text{kips} \] **OK!** --- ## **Step 8: Try Lighter Section** Try **W14x132**: - \( A_g = 38.8\,\text{in}^2 \) - \( r_x = 5.45\,\text{in} \) - \( r_y = 3.65\,\text{in} \) \[ \frac{K L}{r_x} = \frac{235.2}{5.45} = 43.2 \] \[ \frac{L}{r_y} = \frac{168}{3.65} = 46. \] \[ F_e = \frac{\pi^2 \times 29000}{(46.)^2} = \frac{286870}{2116} = 135.7\,\text{ksi} \] \[ \frac{F_y}{F_e} = \frac{60}{135.7} = .442 \] \[ F_{cr} = .658^{.442} \times 60 = .827 \times 60 = 49.6\,\text{ksi} \] \[ P_n = 49.6 \times 38.8 = 1926\,\text{kips} \] \[ \phi_c P_n = .9 \times 1926 = 1733\,\text{kips} \] \[ 1733\,\text{kips} > 120\,\text{kips} \qquad \text{OK} \] --- ## **Step 9: Try Even Lighter Section** Try **W14x99**: - \( A_g = 29.1\,\text{in}^2 \) - \( r_x = 4.87\,\text{in} \) - \( r_y = 3.33\,\text{in} \) \[ \frac{K L}{r_x} = \frac{235.2}{4.87} = 48.3 \] \[ \frac{L}{r_y} = \frac{168}{3.33} = 50.5 \] \[ F_e = \frac{286870}{(50.5)^2} = \frac{286870}{255} = 112.6\,\text{ksi} \] \[ \frac{F_y}{F_e} = \frac{60}{112.6} = .533 \] \[ F_{cr} = .658^{.533} \times 60 = .800 \times 60 = 48.\,\text{ksi} \] \[ P_n = 48. \times 29.1 = 1397\,\text{kips} \] \[ \phi_c P_n = .9 \times 1397 = 1257\,\text{kips} \] \[ 1257\,\text{kips} > 120\,\text{kips} \qquad \text{OK} \] --- ## **Step 10: Try Lighter (W14x90)** - \( A_g = 26.5\,\text{in}^2 \) - \( r_x = 4.75\,\text{in} \) - \( r_y = 3.22\,\text{in} \) \[ \frac{K L}{r_x} = \frac{235.2}{4.75} = 49.5 \] \[ \frac{L}{r_y} = \frac{168}{3.22} = 52.2 \] \[ F_e = \frac{286870}{(52.2)^2} = \frac{286870}{2725} = 105.3\,\text{ksi} \] \[ \frac{F_y}{F_e} = \frac{60}{105.3} = .57 \] \[ F_{cr} = .658^{.57} \times 60 = .787 \times 60 = 47.2\,\text{ksi} \] \[ P_n = 47.2 \times 26.5 = 125\,\text{kips} \] \[ \phi_c P_n = .9 \times 125 = 1125\,\text{kips} \] \[ 1125\,\text{kips} < 120\,\text{kips} \qquad \text{NOT OK} \] --- ## **Step 11: Local Buckling Check (W14x99)** - **Web slenderness:** \( h/t_w \) for W14x99 (\( h = d - 2k = 16.73 - 2 \times 1.45 = 13.83\,\text{in} \)), \( t_w = .395\,\text{in} \) - \( h/t_w = 13.83 / .395 = 35. \) AISC limit for non-compact web (Table B4.1b): \( \lambda_{rw} = 3.76 \sqrt{E/F_y} = 3.76 \sqrt{29000/60} = 3.76 \times 21.99 = 82.7 \) \( 35. < 82.7 \) → **Web is non-slender**. - **Flange slenderness:** \( b_f/(2 t_f) = 10.065/(2 \times .645) = 10.065 / 1.29 = 7.8 \) AISC limit for flange: \( \lambda_{rf} = .56 \sqrt{E/F_y} = .56 \times 21.99 = 12.3 \) \( 7.8 < 12.3 \) → **Flange is non-slender**. --- # **Final Answer** ## **The lightest W-shape section that supports the load is:** ## **W14x99 (A572 Gr. 60)** - \( \phi_c P_n = 1257\,\text{kips} > 120\,\text{kips} \) - **Slenderness and local buckling OK** --- ### **Summary Table** | Section | \( \phi_c P_n \) (kips) | Slenderness OK | Local Buckling OK | |---------|:-----------------------:|:--------------:|:-----------------:| | W14x176 | 2387 | Yes | Yes | | W14x132 | 1733 | Yes | Yes | | W14x99 | 1257 | Yes | Yes | | W14x90 | 1125 | No | -- | --- **Select:** \[ \boxed{\text{W14x99 (A572 Gr. 60)}} \] --- **References:** - AISC Steel Construction Manual, 15th Edition - ASTM A572 Specification --- **Note:** Always check with the latest code and local requirements before finalizing design. If you need calculations for a different section or more details on bracing, let me know!