👋 Introduction

Steel Properties is one of the first essential design in most of the steel design, as no matter structural analysis or capacity design, the steel properties is the major parameter to compute. While there's steel section properties table available, it's also always a good idea to understand how every value comes from. This article aims to demonstrate the steps involved in calculating the Area, Second Moment of Inertia, Elastic Modulus and Plastic Modulus of a I section through worked numerical examples.

The concept behind this and its corresponding structural meaning can be found in another article: 4 Steel Section Properties Every Steel Designer Should Know.

ℹ️ Symbols Difference between Eurocode and AISC

Please note that this is first principle logic about the section properties, so it can be applied to all codes of standard worldwide. However, symbols may be a bit different. Below is a table showing the symbol difference between eurocode and AISC:

Item	Eurocode	AISC
Area	A	A
Second Moment of Area(Major)	$I_{y}$	$I_{x}$
Second Moment of Area(Minor)	$I_{z}$	$I_{y}$
Elastic Section Modulus(Major)	$W_{el,y}$	$S_{x}$
Elastic Section Modulus(Minor)	$W_{el,z}$	$S_{y}$
Plastic Section Modulus(Major)	$W_{pl,y}$	$Z_{x}$
Plastic Section Modulus(Minor)	$W_{pl,z}$	$Z_{y}$

📐 Design Data

Below are the design information of the I steel section (457x191x67).

Steel Section
	Type	I-section
	Standard	457x191x67
	Breath, b	190mm
	Height, h	453mm
	Flange Thickness, t_f	12.7mm
	Web Thickness, t_w	8.5mm
	Radius, r	10mm

Other Geometry
Height of web, h_w	= h - 2 x t_f =	428mm

🔢 Calculation Steps

*Below is using Eurocode symbol to calculate. Also, the calculation will consider not only the flange and web, but also the fillet part (the 4 small corner connecting web and flange).

Area of I section (no fillet) ^ⓘ
$A_{I,no_fillet}$	=	$(h - 2 \times t_{f}) \times t_{w} + 2 \times b \times t_{f}$
	=	$(453mm - 2 \times 13mm) \times 8.5mm + 2 \times 190mm \times 13.5$
	=	$8461 mm^2$

Area of fillet ^ⓘ
$A_{fillet}$	=	$4 \times (r)^2 \times (1 - \frac{\pi}{4})$
	=	$4 \times (10mm)^2 \times (1 - \frac{\pi}{4})$
	=	$89 mm^2$

Total Area of I section ^ⓘ
$A_{I}$	=	$A_{I, no fillet} + A_{fillet}$
	=	$8461 mm^2 + 89 mm^2$
	=	$8551 mm^2$

Second Moment of area (Major Axis) ^ⓘ
$I_{y,main}$	=	$2 \times (\frac{b \times (t_{f})^3}{12} + b \times t_{f} \times (h_{w})^2)$
		$+ \frac{t_{w} \times (h_{w})^3}{12}$
	=	$2 \times (\frac{190mm \times (12.7mm)^3}{12} + 190mm \times 12.7mm \times (428mm)^2)$
		$+ \frac{8.5mm \times (428mm)^3}{12}$
	=	$29000 cm^4$

Second Moment of area, fillet (Major Axis) ^ⓘ
$I_{y,fillet}$	=	$4 \times (\frac{r^4}{12} + r^2 \times (\frac{h_{w}}{2} - \frac{r}{2})^2$
		$- (\frac{\pi}{16} - \frac{4}{9\pi}) \times r^4 - \frac{\pi \times (r^2}{4} \times (\frac{h_{w}}{2}$
		$- r + \frac{4 \times r}{3 \times \pi})^2)$
	=	$4 \times (\frac{(10mm)^4}{12} + (10mm)^2 \times (\frac{428mm}{2} - \frac{10mm}{2})^2$
		$- (\frac{\pi}{16} - \frac{4}{9\pi}) \times (10mm)^4 - \frac{\pi \times ((10mm)^2}{4} \times (\frac{428mm}{2}$
		$- r + \frac{4 \times 10mm}{3 \times \pi})^2)$
	=	$400 cm^4$

Total Second Moment of area (Major Axis) ^ⓘ
$I_{y}$	=	$I_{y,main} + I_{y,fillet}$
	=	$29000 cm^4 + 400 cm^4$
	=	$29400 cm^4$

Second Moment of area, main (Minor Axis) ^ⓘ
$I_{z,main}$	=	$2 \times (\frac{(b)^3 \times t_{f}}{12}) + \frac{(t_{w})^3 \times h_{w}}{12}$
	=	$2 \times (\frac{(190mm)^3 \times 12.7mm}{12}) + \frac{(8.5mm)^3 \times 428mm}{12}$
	=	$1452 cm^4$

Second Moment of area, fillet (Minor Axis) ^ⓘ
$I_{z,fillet}$	=	$4 \times (\frac{(r)^4}{12} + (r)^2 \times (\frac{t_w + 2 //times r}{2} - \frac{r}{2})^2$
		$- (\frac{\pi}{16} - \frac{4}{9\pi}) \times (r)^4 - \frac{\pi \times (r)^2}{4} \times (\frac{t_w + 2 //times r}{2} - r + \frac{4 //times r}{3\pi})^2)$
	=	$4 \times (\frac{(10mm)^4}{12} + (10mm)^2 \times (\frac{8.5mm + 2 //times 10mm}{2} - \frac{10mm}{2})^2$
		$- (\frac{\pi}{16} - \frac{4}{9\pi}) \times (10mm)^4 - \frac{\pi \times (10mm)^2}{4} \times (\frac{12.7mm + 2 //times 10mm}{2} - r + \frac{4 //times 10mm}{3\pi})^2)$
	=	$1.36 cm^4$

Total Second Moment of area (Minor Axis) ^ⓘ
$I_{z}$	=	$I_{z,main} + I_{z,fillet}$
	=	$1452 cm^4 + 1.36 cm^4$
	=	$1453 cm^4$

Elastic Section Modulus (major axis) ^ⓘ
$W_{el,y}$	=	$\frac{I_{y}}{h / 2}$
	=	$\frac{29400 cm^4}{453mm / 2}$
	=	$1296 cm^3$

Elastic Section Modulus (minor axis) ^ⓘ
$W_{el,z}$	=	$\frac{I_{z}}{b / 2}$
	=	$\frac{1453}{190mm / 2}$
	=	$153 cm^3$

Plastic Section Modulus (main part) ^ⓘ
$W_{pl,y0}$	=	$b \times t_{f} \times (h_{w} + t_{f}) + h_{w} \times t_{w} \times \frac{h_{w}}{4}$
	=	$190mm \times 12.7mm \times (428mm + 12.7mm) + 428mm \times 12.7mm \times \frac{428mm}{4}$
	=	$1452 cm^3$

Plastic Section Modulus(fillet) ^ⓘ
$W_{pl,y,fillet}$	=	$4 \times ((r)^2 \times (\frac{h_{w}}{2} - \frac{r}{2}) - \frac{\pi \times (r)^2}{4}(\frac{h_{w}}{2} - r + \frac{4 \times r}{3 \times \pi}))$
	=	$4 \times ((10mm)^2 \times (\frac{428mm}{2} - \frac{10mm}{2}) - \frac{\pi \times (10mm)^2}{4}(\frac{428mm}{2} - r + \frac{4 \times 10mm}{3 \times \pi}))$
	=	$19 cm^3$

Plastic Section Modulus (major axis) ^ⓘ
$W_{pl,y}$	=	$W_{pl,y0} + W_{pl,fillet}$
	=	$1452 cm^3 + 19 cm^3$
	=	$1471 cm^3$

Plastic Section Modulus (minor axis) ^ⓘ
$W_{pl,z}$	=	$2 \times \frac{(b)^{2} \times t_{f}}{4} + \frac{(t_{w})^{2} \times h_{w}}{4}$
	=	$2 \times \frac{(190mm)^{2} \times 12.7mm}{4} + \frac{(8.5mm)^{2} \times 428mm}{4}$
	=	$237 cm^3$

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CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.