CIV-23

Transformed RC section

n = Es/Ec. Steel area becomes n As in the concrete modular ratio.

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Reinforced concreteTransformed section

Governing equation

n=E_s/E_c,\quad A_{s,tr}=n A_s

where

E_s: Steel modulus (GPa)
E_c: Concrete modulus (GPa)
A_s: Steel area (cm^2)
n: Modular ratio (—)
A_{s,tr}: Transformed steel (cm^2)

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-23 — Transformed RC section) is the form associated with Transformed section. Working symbols: $E_s$ , $E_c$ , $A_s$ $\rightarrow$ $n$ , $A_{s,tr}$ . Uncracked or cracked elastic analysis. Strain is plane; stress in steel is n times the concrete fibre.

Purpose

Purpose: compute

n

A_{s,tr}

from

E_s

E_c

A_s

in Structural & civil via

n=E_s/E_c,\quad A_{s,tr}=n A_s

n = Es/Ec. Steel area becomes n As in the concrete modular ratio. Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E_s = 200.000\,\mathrm{GPa}

E_c = 30.000\,\mathrm{GPa}

A_s = 12.000\,\mathrm{cm^2}

, the governing relation

n=E_s/E_c,\quad A_{s,tr}=n A_s

yields

n = 6.67\,\mathrm{—}

A_{s,tr} = 80.00\,\mathrm{cm^2}

. A concrete rectangle, a steel bar widened n times. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Steel modulus E_s200 GPaConcrete modulus E_c30 GPaSteel area A_s12 cm^2

Outputs

Modular ratio n6.67 —
Transformed steel A_{s,tr}80.00 cm^2

Reading speed

Watch on YouTube

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Narration of this film

A concrete rectangle, a steel bar widened n times.

Uncracked or cracked elastic analysis. Strain is plane; stress in steel is n times the concrete fibre.

Reading speed

Watch on YouTube