# Step-by-Step Solution ## 1. **Given Data** - **Axial tensile load, \( P = 24,500 \) lb** - **Material:** ASTM A36 Structural Steel - **Member type:** Standard equal-leg angle (single angle) - **Code:** AISC (American Institute of Steel Construction) --- ## 2. **Material Properties (ASTM A36)** - **Yield strength, \( F_y = 36 \) ksi** - **Ultimate strength, \( F_u = 58 \) ksi** --- ## 3. **AISC Tensile Strength Check** ### **AISC Specification Section D1 (Tension Members)** AISC requires the design strength to be the lesser of: - **Yielding on the gross section:** \[ \phi_t P_y = \phi_t F_y A_g \] - **Fracture on the effective net section:** \[ \phi_t P_u = \phi_t F_u A_{e} \] Where: - \( \phi_t \) = .90 (resistance factor for tension) - \( A_g \) = gross area - \( A_{e} \) = effective net area (for single angle, can often use \( A_n \), the net area) Assume bolted end connection with one line of bolts (typical for angles). --- ## 4. **Required Strength Calculation** \[ P_{required} = 24,500~\text{lb} \] --- ## 5. **Required Area Calculation** ### **(a) Yielding Limit State** \[ \phi_t F_y A_g \geq P_{required} \] \[ .90 \times 36,000 \times A_g \geq 24,500 \] \[ A_g \geq \frac{24,500}{.9 \times 36,000} \] \[ A_g \geq \frac{24,500}{32,400} \] \[ A_g \geq .756~\text{in}^2 \] ### **(b) Fracture Limit State** Assume one line of bolts, so net area is gross area minus one bolt hole. \[ \phi_t F_u A_n \geq P_{required} \] \[ .90 \times 58,000 \times A_n \geq 24,500 \] \[ A_n \geq \frac{24,500}{.9 \times 58,000} \] \[ A_n \geq \frac{24,500}{52,200} \] \[ A_n \geq .469~\text{in}^2 \] --- ## 6. **Select an Angle Section** Check standard equal-leg angles in the **AISC Manual**. Let's try **L 2 × 2 × 1/4** (from AISC tables): - **\( A_g = 1.19~\text{in}^2 \)** - **Legs:** 2 in × 2 in - **Thickness:** .25 in #### **Net Area Calculation** Assume 7/8" bolt (standard for this size), and oversized hole: \[ d_{hole} = 7/8" + 1/16" = .9375" \] Assume one bolt line, net width per leg: \[ b_n = b - d_{hole} \] But for angles, use the table value for net area for one bolt. From tables: - **Net area, \( A_n \) (one bolt):** \( A_n \approx .98~\text{in}^2 \) --- ## 7. **Strength Check** ### **(a) Yielding on Gross Area** \[ \phi_t F_y A_g = .9 \times 36,000 \times 1.19 = 38,484~\text{lb} \] ### **(b) Fracture on Net Area** \[ \phi_t F_u A_n = .9 \times 58,000 \times .98 = 51,156~\text{lb} \] Both are **greater than 24,500 lb**. --- ## 8. **Final Answer** ### **Suitable Angle Section** \[ \boxed{ \text{Use L 2} \times \text{2} \times \frac{1}{4}~\text{(A36 steel, ASTM A36)},~A_g = 1.19~\text{in}^2 } \] - **Gross area, \( A_g \):** 1.19 in² - **Net area, \( A_n \):** .98 in² (after one bolt hole) - **Design strength (yielding):** 38,484 lb - **Design strength (fracture):** 51,156 lb **Both are >24,500 lb, so this angle is suitable.** --- ## 9. **Summary Table** | Section | \( A_g \) (in²) | \( A_n \) (in²) | \( \phi_t F_y A_g \) (lb) | \( \phi_t F_u A_n \) (lb) | OK? | |------------------|-----------------|-----------------|--------------------------|--------------------------|------| | L 2 × 2 × 1/4 | 1.19 | .98 | 38,484 | 51,156 | Yes | --- ## 10. **Note** - For final design, check connection details, length, slenderness, and other AISC requirements. - If a smaller section is desired, you may check L 1-1/2 × 1-1/2 × 1/4 (but its net area may be marginal). --- **Image:**  *Equal-leg steel angle section*