Dimensional analysis of flexural rigidity (E * I) for a beam element Express the dimensions of flexural rigidity EI using mass (M), length (L), and time (T) base dimensions.

Difficulty: Easy

Correct Answer: ML^3T^-2

Explanation:


Introduction:
Flexural rigidity EI appears in beam deflection and vibration formulas. Dimensional analysis is a quick way to verify equations and to understand how physical quantities scale with geometry and material properties. This exercise asks for the base dimensions of EI in terms of M, L, and T.


Given Data / Assumptions:

  • E (Young’s modulus) has the dimensions of stress.
  • I (second moment of area) reflects cross-sectional geometry.
  • Standard SI base dimensions are used: force = M L T^-2.


Concept / Approach:

Young’s modulus E is stress/strain. Strain is dimensionless, so dimensions of E equal those of stress. Stress = force/area = (M L T^-2) / L^2 = M L^-1 T^-2. The second moment of area I has dimensions of L^4. Multiplying gives EI = (M L^-1 T^-2) * (L^4) = M L^3 T^-2.


Step-by-Step Solution:

1) E → M L^-1 T^-2.2) I → L^4.3) EI → (M L^-1 T^-2) * (L^4) = M L^3 T^-2.


Verification / Alternative check:

Check with common beam equation y'' = w / (E I): w (load per length) has dimensions of force/length = M T^-2; y'' has L^-3; therefore EI must be M L^3 T^-2 to balance dimensions.


Why Other Options Are Wrong:

Options A and C omit the L^3 factor; D is inverted and dimensionally inconsistent; E adds an incorrect exponent on L and omits M.


Common Pitfalls:

Confusing area moment of inertia (L^4) with mass moment (M L^2); mixing force and mass units; forgetting strain is dimensionless.


Final Answer:

ML^3T^-2

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