To solve for the block shear capacity of the gusset plate in this structure, we will need to follow the block shear design principles according to the relevant codes (such as AISC or similar).

Block Shear Failure Mechanism

In this scenario, block shear occurs when part of the gusset plate experiences a combination of:

• Tension along one line of bolts (parallel to the applied force)
• Shear along the edge of the plate (perpendicular to the applied force).

Key Data from the Diagram and Problem Statement:

• Material: ASTM A572 Grade 50

  • Yield strength, $f_y = 345 \, \text{MPa}$
  • Ultimate tensile strength, $f_u = 450 \, \text{MPa}$

• Bolt Information:

  • Bolt diameter = 20 mm
  • Gusset plate thickness = 25 mm

• Channel Properties (MC460X63.5):

  • Cross-sectional area $A = 8130 \, \text{mm}^2$
  • Other geometric properties are also provided, but are primarily relevant for the channel.

Step-by-Step Calculation:

1. Determine the Area in Tension and Shear:

   • Tension Area, $A_t$: This is the area subjected to tension, which will be along the bolt line. $A_t = (n_{\text{bolt\_holes}} \times \text{bolt\_hole\_diameter} + \text{edge\_distance}) \times \text{thickness}$

   • Shear Area, $A_s$: This is the area subjected to shear along the edge of the gusset plate. $A_s = \text{edge\_length} \times \text{thickness}$

2. Shear Yielding and Tension Rupture:

   • Shear Strength: For shear, the capacity is governed by yielding: $P_s = 0.6 \times f_y \times A_s$

   • Tension Rupture: The capacity in tension is governed by rupture, and it can be calculated as: $P_t = f_u \times A_t$

3. Block Shear Capacity, $P_b$: Using the block shear formula: $P_b = 0.6 f_y A_s + 0.75 f_u A_t$ Here, the shear yielding and tension rupture terms are combined. The 0.75 factor is a safety factor applied to the tension rupture capacity.

4. Performing the Calculation: To complete the calculation, I would need the exact bolt hole spacing and edge distances from the plate to calculate the areas accurately. If you can provide that information, I can compute the final block shear capacity.

Would you like to proceed with any clarifications or provide more information regarding dimensions? Let me know!

Here are some related questions to consider:

1. What is the exact bolt hole spacing and edge distance?
2. Is there any specific standard or code you are following for this design?
3. Should we factor in any reduction for the bolt holes in calculating the shear area?
4. Do you have the full stress distribution for the gusset plate under load?
5. Is any particular safety factor specified in this design?

Tip: Always ensure to account for both shear and tension failure modes in a block shear calculation to avoid underestimating capacity.