Equilibrium of Rigid Bodies

Why it matters

Understanding the equilibrium of rigid bodies is essential for designing stable structures and mechanical systems. Equilibrium is necessary but does not by itself establish strength, stiffness, buckling resistance or stability.

Key ideas

• Rigid Body: An idealization where deformation is negligible, and the body maintains a constant shape.
• Equilibrium: A state where the sum of forces and the sum of moments acting on a body are zero.
• Conditions for Equilibrium:
  1. The vector sum of all external forces acting on the body must be zero (ΣF = 0).
  2. The vector sum of all external moments about any point must be zero (ΣM = 0).
• Types of Supports and Reactions in a planar model:
  • Fixed Support: Restrains all movements, providing three reactions (two forces and a moment).
  • Pinned Support: Allows rotation, providing two reactions (two forces).
  • Roller Support: Allows translation tangent to its supporting surface and rotation, providing one normal reaction for frictionless contact.

Formulas

• ΣF_x = 0: Sum of horizontal forces must be zero.
• ΣF_y = 0: Sum of vertical forces must be zero.
• ΣM = 0: Sum of moments about any point must be zero.

Worked example

Given: A simply supported beam of length 6 m with a uniform load of 2 kN/m.

1. Calculate the total load:

   • Total load, W = 2 kN/m × 6 m = 12 kN

2. Determine reactions at supports:

   • Assume reactions at supports A and B are R_A and R_B.
   • By symmetry, R_A = R_B = W/2 = 12 kN / 2 = 6 kN

3. Verify equilibrium:

   • ΣF_y = R_A + R_B - W = 6 kN + 6 kN - 12 kN = 0
   • ΣM_A = 0: Taking moments about A, R_B × 6 m - W × 3 m = 0
   • 6 kN × 6 m - 12 kN × 3 m = 0

Answer: Reactions at supports A and B are 6 kN each.

Common mistakes

• Neglecting to consider all forces and moments, especially those at supports.
• Incorrectly assuming symmetry in non-uniform load distributions.
• Forgetting to convert units consistently.

For GATE CE

Questions often involve calculating reactions at supports, analyzing equilibrium under various loading conditions, and determining the effects of different support types. Practice problems with varying complexity and load types.

Quick check

1. What are the conditions for equilibrium of a rigid body?
2. How many reactions does a fixed support provide?
3. What is the total load on a beam with a uniform load of 3 kN/m over 4 m?

Answers: 1. ΣF = 0 and ΣM = 0; 2. Three in a planar model (six in a fully fixed spatial model); 3. 12 kN.