In plastic analysis, what is the shape factor (M_p / M_y) for a triangular cross-section (about its centroidal axis in bending)?

Introduction / Context:
The shape factor indicates how much plastic moment capacity exceeds the elastic yield moment for a given section shape. Triangular sections have a larger shape factor than rectangles and circles because a greater proportion of the area lies away from the neutral axis, enabling more plastic redistribution after first yield.

Given Data / Assumptions:

• Elastic–perfectly plastic material in bending.
• Triangular cross-section, bending about a centroidal axis parallel to the base.
• Plane sections remain plane; full plasticity possible.

Concept / Approach:
M_y depends on elastic section modulus, while M_p is obtained by balancing equal tension and compression areas at yield stress. For a triangular section, the standard result is f = M_p / M_y ≈ 2.34, significantly higher than rectangular (1.5) or circular (≈1.7) sections.

Step-by-Step Solution:
Step 1: Recall definition of shape factor f = M_p / M_y.Step 2: Use known tabulated values for common shapes.Step 3: For triangle, f ≈ 2.34; select 2.34.

Verification / Alternative check:
Plastic design references consistently list triangular section shape factor close to 2.34, validating the choice.

Why Other Options Are Wrong:

• 1.5: Rectangular section value.
• 1.34: Not a standard shape factor for a triangle.
• 2.5: Higher than accepted value for a triangle.

Common Pitfalls:
Misremembering the ordering (triangle > circle > rectangle); using material-dependent values rather than geometric constants.

Final Answer:
2.34