A C7 × 9.8 tension member is connected to a 3⁄8-in.-thick gusset plate as shown in Figure P3.5-5. Both the member and the gusset plate are A36 steel.
a. Compute the available block shear strength of the tension member for both LRFD and ASD.
b. Compute the available block shear strength of the gusset plate for both LRFD and ASD.
The Correct Answer and Explanation is :
To compute the available block shear strength of the tension member and the gusset plate, we will use the Load and Resistance Factor Design (LRFD) and Allowable Stress Design (ASD) methods, as specified by the American Institute of Steel Construction (AISC) for A36 steel.
Given:
- Tension Member: C7 × 9.8 channel section
- Gusset Plate Thickness: 3/8 inches
- Material: A36 steel (Fy = 36 ksi, Fu = 58 ksi)
a. Available Block Shear Strength of the Tension Member
LRFD Method:
- Calculate the Gross Area in Shear (Agv):
- Agv = 2 × (height of the channel) × (thickness of the flange)
- Agv = 2 × (7 inches) × (0.375 inches) = 5.25 in²
- Calculate the Net Area in Shear (Anv):
- Anv = Agv – (number of bolt holes × area of one bolt hole)
- Assuming 2 bolt holes of 0.75 inches diameter:
- Area of one bolt hole = π × (0.75/2)² ≈ 0.4418 in²
- Anv = 5.25 in² – 2 × 0.4418 in² ≈ 4.3664 in²
- Calculate the Net Area in Tension (Ant):
- Ant = (width of the flange – (number of bolt holes × diameter of bolt)) × thickness of the flange
- Ant = (7 inches – 2 × 0.75 inches) × 0.375 inches = 4.5 inches × 0.375 inches = 1.6875 in²
- Calculate the Nominal Block Shear Strength (Rn):
- Rn = 0.6 × Fu × Anv + Ubs × Fu × Ant
- Assuming Ubs = 1.0 for uniform tension stress:
- Rn = 0.6 × 58 ksi × 4.3664 in² + 1.0 × 58 ksi × 1.6875 in² ≈ 151.5 kips + 97.8 kips = 249.3 kips
- Calculate the Design Strength (φRn):
- φ = 0.75 (for tension members)
- φRn = 0.75 × 249.3 kips ≈ 187 kips
ASD Method:
- Calculate the Allowable Block Shear Strength (Rn):
- Rn = 0.6 × Fy × Anv + Ubs × Fu × Ant
- Rn = 0.6 × 36 ksi × 4.3664 in² + 1.0 × 58 ksi × 1.6875 in² ≈ 94.5 kips + 97.8 kips = 192.3 kips
- Calculate the Allowable Strength (Rn/Ω):
- Ω = 2.0 (for ASD)
- Rn/Ω = 192.3 kips / 2.0 ≈ 96.15 kips
b. Available Block Shear Strength of the Gusset Plate
LRFD Method:
- Calculate the Gross Area in Shear (Agv):
- Agv = (number of bolt lines) × (thickness of the gusset plate) × (distance between bolt lines)
- Agv = 2 × 0.375 inches × 3 inches = 2.25 in²
- Calculate the Net Area in Shear (Anv):
- Anv = Agv – (number of bolt holes × area of one bolt hole)
- Anv = 2.25 in² – 2 × 0.4418 in² ≈ 1.3664 in²
- Calculate the Nominal Block Shear Strength (Rn):
- Rn = 0.6 × Fu × Anv + Ubs × Fu × Ant
- Assuming Ubs = 1.0 for uniform tension stress:
- Rn = 0.6 × 58 ksi × 1.3664 in² + 1.0 × 58 ksi × 1.6875 in² ≈ 47.6 kips + 97.8 kips = 145.4 kips
- Calculate the Design Strength (φRn):
- φ = 0.75 (for tension members)
- φRn = 0.75 × 145.4 kips ≈ 109.05 kips
ASD Method:
- Calculate the Allowable Block Shear Strength (Rn):
- Rn = 0.6 × Fy × Anv + Ubs × Fu × Ant
- Rn = 0.6 × 36 ksi × 1.3664 in² + 1.0 × 58 ksi × 1.6875 in² ≈ 29.5 kips + 97.8 kips = 127.3 kips
- Calculate the Allowable Strength (Rn/Ω):
- Ω = 2.0 (for ASD)
- Rn/Ω = 127.3 kips / 2.0 ≈ 63.65 kips