Modular ratio of two materials: choose the correct definition\n\nIn composite members (e.g., reinforced concrete or bimetal bars), the modular ratio between two materials is defined as the ratio of:

Introduction / Context:
When different materials act together, designers often transform one material into an equivalent section of another using the modular ratio. This is standard for cracked transformed sections in reinforced concrete and laminated members.

Given Data / Assumptions:

• Two linearly elastic, isotropic materials with moduli E1 and E2.
• Perfect bond so that strains are compatible at the interface.
• Axial or bending behavior within elastic range.

Concept / Approach:
The modular ratio n is defined as n = E1 / E2. It scales the area (or moment of inertia) of one material to an equivalent amount of the other so that identical strains produce proportionate stresses via Hooke’s law.

Step-by-Step Solution:
Hooke’s law: σ = E * ε.For equal ε in bonded materials, stress ratio equals modulus ratio: σ1 / σ2 = E1 / E2 = n.Transformed section method: replace area A2 of material 2 by n * A2 in material 1’s “units.”

Verification / Alternative check:
Check bending of a composite section: internal force resultants balance when areas are transformed by n, producing correct stress distribution proportional to E.

Why Other Options Are Wrong:
Options (a) and (b) describe definitions of E and G individually, not a ratio between two materials. (d) refers to the ratio of shear moduli, not the standard modular ratio used in transformed sections. (e) density is unrelated.

Common Pitfalls:
Using temperature-dependent or nonlinear moduli without care; mixing tangent and secant moduli; applying wrong n for sustained vs. short-term (as in RC design).

Final Answer:
their modulus of elasticities (E1 / E2)