CIV-31

Whitney rectangular stress block

a = As fy / (0.85 f′c b), Mn = As fy (d − a/2). Equivalent rectangular compression.

Reading speed

Reinforced concreteWhitneyACI 318

Governing equation

a=\dfrac{A_s f_y}{0.85 f'_c b},\quad M_n=A_s f_y\bigl(d-a/2\bigr)

$A_s \in \left[2, 60\right]$

where

A_s: Steel area (cm^2)
f_y: Steel yield (MPa)
f'_c: Concrete strength (MPa)
b: Width (mm)
d: Effective depth (mm)
a: Block depth (mm)
M_n: Nominal moment (kN·m)

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.

00:0 / 00:08

①

Governing law — red

a=\dfrac{A_s f_y}{0.85 f'_c b},\quad M_n=A_s f_y\bigl(d-a/2\bigr)

$A_s \in \left[2, 60\right]$

$A_s = 2$ $a = 37.34$

Along the curve

②

Domain of the variable — blue

A_s_{\min} = 2,\quad A_s_{\max} = 60

$A_s \in \left[2, 60\right]$

$A_s = 2$ $A_s = 2$

Along the curve

③

Response curve — green

a=\dfrac{A_s f_y}{0.85 f'_c b},\quad M_n=A_s f_y\bigl(d-a/2\bigr)

$A_s \in \left[2, 60\right]$

$A_s = 2$ $a = 37.34$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}a}{\mathrm{d}A_s}

$A_s \in \left[2, 60\right]$

$A_s = 2.365$ $a^{\prime} = 18.35$

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-31 — Whitney rectangular stress block) is the form associated with Whitney · ACI 318. Working symbols: $A_s$ , $f_y$ , $f'_c$ , $b$ , $d$ $\rightarrow$ $a$ , $M_n$ . Whitney replaced the curved concrete stress with a rectangle of 0.85 f′c and depth a = β1 c.

Purpose

Purpose: compute

a

M_n

from

A_s

f_y

f'_c

b

d

in Structural & civil via

a=\dfrac{A_s f_y}{0.85 f'_c b},\quad M_n=A_s f_y\bigl(d-a/2\bigr)

a = As fy / (0.85 f′c b), Mn = As fy (d − a/2). Equivalent rectangular compression. 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

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

f_y = 420.000\,\mathrm{MPa}

f'_c = 30.000\,\mathrm{MPa}

b = 300.000\,\mathrm{mm}

d = 450.000\,\mathrm{mm}

, the governing relation

a=\dfrac{A_s f_y}{0.85 f'_c b},\quad M_n=A_s f_y\bigl(d-a/2\bigr)

yields

a = 65.9\,\mathrm{mm}

M_n = 210.2\,\mathrm{kN·m}

. A beam, a steel bar, a compression block of depth a. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Steel area A_s12 cm^2Steel yield f_y420 MPaConcrete strength f'_c30 MPaWidth b300 mmEffective depth d450 mm

Outputs

Block depth a65.9 mm
Nominal moment M_n210.2 kN·m

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A beam, a steel bar, a compression block of depth a.

Whitney replaced the curved concrete stress with a rectangle of 0.85 f′c and depth a = β1 c.

Reading speed

Watch on YouTube