Mathematical Representation of Modular Ratio:

Let the total axial force applied on the composite member P be shared by the different members as P₁, P₂ and P₃. If the deformations produced are δ₁, δ₂ and δ₃, then we get,

δ₁   =   P₁L₁/A₁E₁

δ₂ =   P₂L₂/A₂E₂

δ₃ = P₃ L₃/A₃ E₃

The compatibility condition requires that

P₁L₁/A₁E₁ = P₂L₂/A₂E₂ = P₃ L₃/A₃ E₃                                                                                      

Let the elastic modulii of the materials be redefined as E₁, m₂ E₁ and m₃ E₁, where

m₂ =   E₂/E₁

Eq. (7) may be rewritten as follows

L₁/E₁ × P₁/A₁  = L₂/m₂E₁ × P₂/A₂  = L₃/m₃E₁× P₃/A₃                                               

Recognising that L₁ = L₂ = L₃, we may rewrite Eq. (8) as

P₁/ A₁  = P₂/m₂ A₂ = P₃/m₃ A₃       

from which, we get P₂ = (P₁/A₁) m₂A₂ and

P₃ = (P₁/A₁) m₃ A₃

The equilibrium condition required to be satisfied is

P₁ + P₂ + P₃ = P

Or P₁+ ( P₁/A₁)m₂A₂  + (P₁/A₁)m₃A₃ =P

P₁(1+(m₂A₂/A₁)+m₃A₃/A₁) =P

P₁ = P/(1+(m₂A₂/A₁) +(m₃A₃/A₁))

After computing P₁ using Eq. (11), the loads shared through other members P₂ and P₃ may be calculated using the Eq. (10).