CIV-04

Transformed cracked RC section

Modular ratio n = Es/Ec converts steel to equivalent concrete.

Reading speed

Reinforced concreteNavier transformed sectionACI 318

Governing equation

n=\dfrac{E_s}{E_c},\quad \tfrac12 b x^2 = n A_s(d-x)

$A_s \in \left[200, 8000\right]$

where

E_s: Steel modulus (GPa)
E_c: Concrete modulus (GPa)
b: Width (mm)
d: Effective depth (mm)
A_s: Tension steel (mm²)
n: Modular ratio (—)
x: Neutral axis (mm)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

n=\dfrac{E_s}{E_c},\quad \tfrac12 b x^2 = n A_s(d-x)

$A_s \in \left[200, 8000\right]$

$A_s = 200$ $n = 58.96$

Along the curve

②

Domain of the variable — blue

A_s_{\min} = 200,\quad A_s_{\max} = 8000

$A_s \in \left[200, 8000\right]$

$A_s = 200$ $A_s = 200$

Along the curve

③

Response curve — green

n=\dfrac{E_s}{E_c},\quad \tfrac12 b x^2 = n A_s(d-x)

$A_s \in \left[200, 8000\right]$

$A_s = 200$ $n = 58.96$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}n}{\mathrm{d}A_s}

$A_s \in \left[200, 8000\right]$

$A_s = 249.1$ $n^{\prime} = 0.121$

Along the curve

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-04 — Transformed cracked RC section) is the form associated with Navier transformed section · ACI 318. Working symbols: $E_s$ , $E_c$ , $b$ , $d$ , $A_s$ $\rightarrow$ $n$ , $x$ . Uncracked or cracked elastic RC is homogenised by replacing As with n As. The cracked centroid satisfies b x²/2 = n As (d−x).

Purpose

Purpose: compute

n

x

from

E_s

E_c

b

d

A_s

in Structural & civil via

n=\dfrac{E_s}{E_c},\quad \tfrac12 b x^2 = n A_s(d-x)

Modular ratio n = Es/Ec converts steel to equivalent concrete. 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}

b = 300.000\,\mathrm{mm}

d = 450.000\,\mathrm{mm}

A_s = 1200.000\,\mathrm{mm^{2}}

, the governing relation

n=\dfrac{E_s}{E_c},\quad \tfrac12 b x^2 = n A_s(d-x)

yields

n = 6.667\,\mathrm{—}

x = 130.531\,\mathrm{mm}

. Singly reinforced rectangle, cracked elastic, no compression steel. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Steel modulus E_s200 GPaConcrete modulus E_c30 GPaWidth b300 mmEffective depth d450 mmTension steel A_s1200 mm²

Outputs

Modular ratio n6.667 —
Neutral axis x130.531 mm

Reading speed

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Narration of this film

Singly reinforced rectangle, cracked elastic, no compression steel.

Uncracked or cracked elastic RC is homogenised by replacing As with n As. The cracked centroid satisfies b x²/2 = n As (d−x).

Reading speed

Watch on YouTube