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

Difficulty: Easy

Correct Answer: 2.34

Explanation:


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

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