CIV-08

Parallel-axis (Steiner) theorem

I = Icm + A d² about a parallel axis.

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Section propertiesHuygens–SteinerEN 1993-1-1

Governing equation

I = I_{\mathrm{cm}} + A d^2

$d \in \left[0, 800\right]$

where

b: Width (mm)
h: Height (mm)
d: Offset of axis (mm)
I_{\mathrm{cm}}: Centroidal I (cm⁴)
I: I about axis (cm⁴)

The figure, part by part

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①

Governing law — red

I = I_{\mathrm{cm}} + A d^2

$d \in \left[0, 800\right]$

$d = 0$ $I_{\mathrm{cm}} = 1.07\times 10^{5}$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0,\quad d_{\max} = 800

$d \in \left[0, 800\right]$

$d = 0$ $d = 0$

Along the curve

③

Response curve — green

I = I_{\mathrm{cm}} + A d^2

$d \in \left[0, 800\right]$

$d = 0$ $I_{\mathrm{cm}} = 1.07\times 10^{5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I_{\mathrm{cm}}}{\mathrm{d}d}

$d \in \left[0, 800\right]$

$d = 5.031$ $I_{\mathrm{cm}}^{\prime} = 80.5$

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-08 — Parallel-axis (Steiner) theorem) is the form associated with Huygens–Steiner · EN 1993-1-1. Working symbols: $b$ , $h$ , $d$ $\rightarrow$ $I_{\mathrm{cm}}$ , $I$ . Shifting the origin of the second-moment integral by d adds the Steiner term A d²; it is the workhorse of composite-section design.

Purpose

Purpose: compute

I_{\mathrm{cm}}

I

from

b

h

d

in Structural & civil via

I = I_{\mathrm{cm}} + A d^2

I = Icm + A d² about a parallel axis. 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

b = 200.000\,\mathrm{mm}

h = 400.000\,\mathrm{mm}

d = 250.000\,\mathrm{mm}

, the governing relation

I = I_{\mathrm{cm}} + A d^2

yields

I_{\mathrm{cm}} = 106666.667\,\mathrm{cm⁴}

I = 606666.667\,\mathrm{cm⁴}

. One rectangle shifted from its centroid. Units mm. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Width b200 mmHeight h400 mmOffset of axis d250 mm

Outputs

Centroidal I I_{\mathrm{cm}}106666.667 cm⁴
I about axis I606666.667 cm⁴

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Narration of this film

One rectangle shifted from its centroid. Units mm.

Shifting the origin of the second-moment integral by d adds the Steiner term A d²; it is the workhorse of composite-section design.

Reading speed

Watch on YouTube