Let's solve the problem **step by step**: --- ## **Step 1: Problem Summary** - **Compressive load, \( P_u \) = 20 kips** - **Yield strength, \( F_y \) = 50 ksi** - **Modulus of elasticity, \( E \) = 29,000 ksi** - **Ω (Omega, safety factor) = 1.25** - **Diagonal brace length: \( L_d \) (not numerically specified, but marked as \( 14' \) in the image)** We are to **select an economical W-section (wide flange) steel member** that can resist this compressive load per AISC specifications. --- ## **Step 2: Calculate Required Nominal Strength** The required **nominal compressive strength** (\( P_n \)) considering the safety factor: \[ P_n = \Omega \cdot P_u = 1.25 \times 20~\text{kips} = 25~\text{kips} \] --- ## **Step 3: Determine Required Member Properties** ### **3.1. Bracing Length** The diagonal's length (\( L \)) is given as **14 ft** (from the image). Convert to inches: \[ L = 14 \times 12 = 168~\text{in} \] ### **3.2. Effective Length Factor (\( K \))** Assume both ends are pinned (typical for bracing): \[ K = 1. \] ### **3.3. Required Compressive Strength** The required **design compressive strength** (\( \phi P_n \)) must be at least \( P_u = 20~\text{kips} \). But with ASD (using Ω), we've found \( P_n = 25~\text{kips} \). --- ## **Step 4: AISC Compressive Strength Calculation** ### **4.1. Slenderness Ratio (\( \lambda \))** \[ \lambda = \frac{KL}{r} \] where \( r \) = radius of gyration (select the weaker axis for W-shapes, usually \( r_y \)). Let's start with a **trial section**. We'll check a few common economical W-sections from the AISC tables. --- ## **Step 5: Try a Trial Section** Let's try **W8x18** (a common small W-section): From AISC tables: - Area \( A = 5.29~\text{in}^2 \) - \( r_y = 1.38~\text{in} \) \[ \lambda = \frac{KL}{r_y} = \frac{168}{1.38} = 121.74 \] --- ## **Step 6: Calculate Critical Stress \( F_{cr} \)** The AISC column strength is calculated using: \[ F_{cr} = \begin{cases} .658^{F_y/F_e} F_y & \text{if } \lambda_c \leq 1.5 \\ .877 F_e & \text{if } \lambda_c > 1.5 \end{cases} \] Where: \[ F_e = \frac{\pi^2 E}{(KL/r)^2} \] So, \[ F_e = \frac{\pi^2 \times 29000}{(121.74)^2} \] \[ = \frac{286,870}{14,827} \] \[ = 19.35~\text{ksi} \] Now, \[ \lambda_c = \sqrt{\frac{F_y}{F_e}} = \sqrt{\frac{50}{19.35}} = 1.61 \] Since \( \lambda_c > 1.5 \): \[ F_{cr} = .877 \times 19.35 = 16.97~\text{ksi} \] --- ## **Step 7: Calculate Nominal Compressive Strength (\( P_n \))** \[ P_n = F_{cr} \times A = 16.97~\text{ksi} \times 5.29~\text{in}^2 = 89.87~\text{kips} \] --- ## **Step 8: Compare to Required Strength** \[ P_n = 89.87~\text{kips} > 25~\text{kips} \] So, **W8x18** is more than adequate (even much more than needed). Let's try a lighter section: **W8x10** (smallest W8): - \( A = 2.94~\text{in}^2 \) - \( r_y = .84~\text{in} \) \[ \lambda = \frac{168}{.84} = 200 \] \[ F_e = \frac{286,870}{(200)^2} = \frac{286,870}{40,000} = 7.17~\text{ksi} \] \[ \lambda_c = \sqrt{\frac{50}{7.17}} = 2.64 \] \[ F_{cr} = .877 \times 7.17 = 6.29~\text{ksi} \] \[ P_n = 6.29 \times 2.94 = 18.48~\text{kips} \] This is **less than 25 kips**; so, W8x10 is NOT sufficient. --- ## **Step 9: Try W8x15** - \( A = 4.41~\text{in}^2 \) - \( r_y = 1.11~\text{in} \) \[ \lambda = \frac{168}{1.11} = 151.35 \] \[ F_e = \frac{286,870}{(151.35)^2} = \frac{286,870}{22,915} = 12.52~\text{ksi} \] \[ \lambda_c = \sqrt{\frac{50}{12.52}} = 2.00 \] \[ F_{cr} = .877 \times 12.52 = 10.97~\text{ksi} \] \[ P_n = 10.97 \times 4.41 = 48.39~\text{kips} \] This is **well above 25 kips**. --- ## **Step 10: Select the Most Economical Section** - **W8x10:** Not enough - **W8x15:** Adequate (48.39 kips > 25 kips) - **W8x18:** More than enough (over-designed) **W8x15 is the most economical standard section that meets the requirement.** --- ## **Step 11: Final Answer** --- ### **Final Answer** \[ \boxed{ \text{Select } \mathbf{W8 \times 15} \text{ as the diagonal brace.} } \] - **It has a nominal compressive strength of 48.4 kips, which is greater than the required 25 kips.** - **It is the lightest W-shape that satisfies the strength requirement for this bracing member.** --- If you need to check a lighter section or a different W-shape series, the same steps apply!