# Step-by-Step Solution: Lightest Section Selection for Axial Load ## **Given Data** - **Dead Load (DL):** 600 kips - **Live Load (LL):** 300 kips - **Total Factored Load (Pu):** To be calculated - **Element Length (L):** 28 ft = 336 in - **Braced Length (Lb, y-direction):** 14 ft = 168 in - **End Conditions:** Fixed / Pinned - **Steel Grade:** ASTM A572 Grade 60 (Fy = 60 ksi) - **Verify:** Slenderness and local buckling --- ## **1. Calculate Required Strength** ### **Factored Axial Load (\( P_u \))** \[ P_u = 1.2 \times DL + 1.6 \times LL = 1.2 \times 600 + 1.6 \times 300 = 720 + 480 = 120 \text{ kips} \] --- ## **2. Effective Length Factor (K)** - **Fixed-Pinned End:** \( K \approx .7 \) (AISC Table C-C2.2) --- ## **3. Slenderness Ratio (\( KL/r \))** - **Unbraced Length (L):** 28 ft = 336 in - **\( K \) factor:** .7 - **\( KL \):** \( .7 \times 336 = 235.2 \) in \[ \frac{KL}{r_{min}} \leq 200 \] - We'll solve for required \( r_{min} \) after selecting a section. --- ## **4. Required Nominal Strength (\( P_n \))** ### **LRFD \(\phi_c = .9\) (Compression Member Strength):** \[ \phi_c P_n \geq P_u \implies P_n \geq \frac{P_u}{\phi_c} \] \[ P_n \geq \frac{120}{.9} = 1333 \text{ kips} \] --- ## **5. Required Section Selection** ### **A. Use AISC Equation for Axial Strength** \[ P_n = F_{cr} A_g \] - \( F_{cr} \): Critical stress (depends on slenderness) - \( A_g \): Gross area #### **i. Calculate \(\lambda = \frac{KL}{r}\) for trial sections** \[ \lambda = \frac{KL}{r} \] - Try W14 and W12 sections (wide-flange, compact, common for columns) #### **ii. Calculate \( F_{cr} \) (per AISC 360-16, Ch E3):** - For \( \lambda \leq 4.71\sqrt{\frac{E}{F_y}} \): \[ F_{cr} = .658^{F_y / F_e} F_y \] - For \( \lambda > 4.71\sqrt{\frac{E}{F_y}} \): \[ F_{cr} = .877 F_e \] Where: - \( F_e = \frac{\pi^2 E}{(KL/r)^2} \) - \( E = 29000 \) ksi - \( F_y = 60 \) ksi --- ### **B. Trial Section: W14x132** **From AISC Manual (14th Ed):** - \( A_g = 38.8 \) in² - \( r_x = 6.36 \) in, \( r_y = 4.09 \) in #### **i. Slenderness:** \[ \lambda = \frac{KL}{r_{min}} = \frac{235.2}{4.09} = 57.5 \] #### **ii. Calculate \( F_e \):** \[ F_e = \frac{\pi^2 \times 29000}{(57.5)^2} = \frac{2868707}{3306.25} \approx 868 \text{ ksi} \] \[ \frac{F_y}{F_e} = \frac{60}{868} = .069 \] \[ F_{cr} = .658^{.069} \times 60 = .969 \times 60 = 58.1 \text{ ksi} \] #### **iii. Axial Strength:** \[ P_n = F_{cr} \times A_g = 58.1 \times 38.8 = 2255 \text{ kips} \] \[ \phi_c P_n = .9 \times 2255 = 2029 \text{ kips} > 120 \text{ kips} \] #### **iv. Check Slenderness** \[ \lambda = 57.5 < 200 \quad \text{(OK)} \] #### **v. Local Buckling** - W14x132 is non-slender, per AISC tables. --- ### **C. Try Lighter Section: W14x99** - \( A_g = 29.1 \) in² - \( r_{min} = 3.78 \) in \[ \lambda = \frac{235.2}{3.78} = 62.2 \] \[ F_e = \frac{2868707}{(62.2)^2} = \frac{2868707}{3873.8} = 740 \text{ ksi} \] \[ F_{cr} = .658^{60/740} \times 60 = .658^{.081} \times 60 = .956 \times 60 = 57.4 \text{ ksi} \] \[ P_n = 57.4 \times 29.1 = 1672 \text{ kips} \] \[ \phi_c P_n = .9 \times 1672 = 1505 \text{ kips} > 120 \text{ kips} \] - **Slenderness \( \lambda = 62.2 < 200 \) — OK** - **Local buckling:** W14x99 is non-slender (per AISC) --- ### **D. Try Even Lighter Section: W14x90** - \( A_g = 26.5 \) in² - \( r_{min} = 3.72 \) in \[ \lambda = \frac{235.2}{3.72} = 63.2 \] \[ F_e = \frac{2868707}{(63.2)^2} = \frac{2868707}{3994.2} = 718.5 \text{ ksi} \] \[ F_{cr} = .658^{60/718.5} \times 60 = .658^{.0835} \times 60 = .954 \times 60 = 57.2 \text{ ksi} \] \[ P_n = 57.2 \times 26.5 = 1516 \text{ kips} \] \[ \phi_c P_n = .9 \times 1516 = 1364 \text{ kips} > 120 \text{ kips} \] - **Slenderness \( \lambda = 63.2 < 200 \) — OK** - **Local buckling:** W14x90 is non-slender --- ### **E. Try W14x82** - \( A_g = 24. \) in² - \( r_{min} = 3.67 \) in \[ \lambda = \frac{235.2}{3.67} = 64.1 \] \[ F_e = \frac{2868707}{(64.1)^2} = \frac{2868707}{411.8} = 698 \text{ ksi} \] \[ F_{cr} = .658^{60/698} \times 60 = .658^{.086} \times 60 = .952 \times 60 = 57.1 \text{ ksi} \] \[ P_n = 57.1 \times 24. = 137 \text{ kips} \] \[ \phi_c P_n = .9 \times 137 = 1233 \text{ kips} > 120 \text{ kips} \] - **Slenderness \( \lambda = 64.1 < 200 \) — OK** - **Local buckling:** W14x82 is non-slender --- ### **F. Try W14x74** - \( A_g = 21.7 \) in² - \( r_{min} = 3.60 \) in \[ \lambda = \frac{235.2}{3.60} = 65.3 \] \[ F_e = \frac{2868707}{(65.3)^2} = \frac{2868707}{4266.1} = 672 \text{ ksi} \] \[ F_{cr} = .658^{60/672} \times 60 = .658^{.0893} \times 60 = .950 \times 60 = 57. \text{ ksi} \] \[ P_n = 57. \times 21.7 = 1237 \text{ kips} \] \[ \phi_c P_n = .9 \times 1237 = 1113 \text{ kips} < 120 \text{ kips (NOT OK)} \] --- ## **6. Summary Table** | Section | \(A_g\) (in²) | \(r_{min}\) (in) | \(\phi_c P_n\) (kips) | \(KL/r\) | OK? | |----------|--------------|------------------|----------------------|---------|-----| | W14x99 | 29.1 | 3.78 | 1505 | 62.2 | Yes | | W14x90 | 26.5 | 3.72 | 1364 | 63.2 | Yes | | W14x82 | 24. | 3.67 | 1233 | 64.1 | Yes | | W14x74 | 21.7 | 3.60 | 1113 | 65.3 | No | --- ## **7. Final Answer** ### **Lightest Section That Meets All Requirements:** \[ \boxed{\text{W14x82}} \] - **Gross Area:** 24. in² - **\(\phi_c P_n = 1233 \, \text{kips} > 120 \, \text{kips}\)** - **Slenderness:** 64.1 (< 200) - **Local Buckling:** Non-slender --- ## **8. Short Checklist** - [x] Axial strength OK - [x] Slenderness OK - [x] Local buckling OK (non-slender section) --- ## **9. Sketch**  *Alt text: Cross-section of W14x82 wide-flange beam.* --- ## **10. Final Answer** **The lightest A572 Gr. 60 wide-flange section that supports the factored axial load, meets slenderness and local buckling criteria, and fits the bracing/support conditions is:** \[ \boxed{\text{W14x82}} \]