Plastic Analysis of Structures

Why it matters

Plastic analysis of structures is crucial for understanding how structures behave beyond their elastic limit until failure. This knowledge allows engineers to design structures that are both efficient and safe, optimizing material use and ensuring stability under extreme conditions.

Key ideas

• Plastic Hinge: A point in a structure where the moment reaches the plastic moment capacity, allowing rotation without an increase in moment.
• Plastic Moment (Mp): The maximum moment a section can withstand when the cross-section has fully yielded in the idealized model.
• Load Factor: The ratio of the collapse load to the working load, indicating the safety margin.
• Mechanism: A condition where the structure forms enough plastic hinges to become a mechanism, leading to collapse.
• Shape Factor: The ratio of the plastic moment to the yield moment, indicating the reserve strength of a section.

Formulas

• Mp = Fy · Z
  • Mp: Plastic moment (Nm)
  • Fy: Yield strength of the material (N/m²)
  • Z: Plastic section modulus (m³)
• Load Factor = Collapse Load / Working Load
• Shape Factor = Mp / My
  • My: Yield moment (Nm)

Assume sufficient ductility and rotation capacity, with local buckling, lateral-torsional buckling, shear failure and axial-force interaction excluded. A plastic hinge rotates at M_p in the ideal perfectly plastic model; it does not require M > M_p.

Worked example

Given: A simply supported beam with a span of 6 m, subjected to a uniform load of 20 kN/m. The yield strength of the material is 250 MPa, and the plastic section modulus is 0.0002 m³.

1. Calculate the plastic moment (Mp):

   • Formula: Mp = Fy · Z
   • Calculation: Mp = 250 × 10^6 N/m² · 0.0002 m³ = 50,000 Nm

2. Determine the collapse load (Wc):

   • For a simply supported beam, Wc = 8 · Mp / L
   • Calculation: Wc = 8 · 50,000 Nm / 6 m = 66,667 N = 66.667 kN

3. Calculate the load factor:

   • Formula: Load Factor = Collapse Load / Working Load
   • Working Load = 20 kN/m · 6 m = 120 kN
   • Calculation: Load Factor = 66.667 kN / 120 kN = 0.556

Final Answer: The load factor is 0.556. The specified 20 kN/m exceeds the ideal collapse UDL of 8M_p/L² = 11.11 kN/m; this is not an acceptable working-load condition.

Common mistakes

• Confusing the plastic moment with the yield moment.
• Incorrectly calculating the number of plastic hinges required for a mechanism.
• Neglecting the effect of axial forces in beams during plastic analysis.

For GATE CE

Questions often involve calculating the plastic moment, determining the collapse load, and understanding the formation of mechanisms. Practice problems on identifying plastic hinges and calculating load factors are essential.

Quick check

1. What is a plastic hinge?
2. How is the plastic moment calculated?
3. What does the load factor indicate?

Answers: 1. A point where the moment exceeds the plastic moment capacity. 2. Mp = Fy · Z. 3. The safety margin between collapse load and working load.