Area Moment of Inertia & Section Modulus: Calculator & Formulas

Diese Seite auf Deutsch: Flächenrägheits- und Widerstandsmoment-Rechner

This online calculator computes the axial and polar area moments of inertia (also known as second moment of area or second area moment), the section modulus, the outer-fibre distance and the cross sectional area of many beams. From many surfaces, the torsional moment of inertia and the torsionial section modulus can also be calculated.

In addition, the mass can be computed, too. Steel, aluminum and different types of wood are available as material. At the bottom of the page, the formulas for the axial area moment of inertia and section modulus are listed in a table.

Calculator for Area Moment of Inertia and Section Modulus

By default, one can calculate the moments, mass and cross section for an I-beam (I100).

cross section
height H	mm
width B	mm
material *

diameter d	mm
height h	mm
width b	mm
length of beam *	m

units: mm cm
can be changed later, too

2nd area moment			section modulus
in mm⁴			in mm³
I_y			W_y
I_z			W_z
I_t			W_t
I_p


cross section
	mm²
mass
	kg

outer-fibre distance:

* You have to fill in these fields only if the mass should be calculated, too. Only the smallest section modulus will be calculated!

Explanation of the abbreviations

Dm	diameter in mm
I_y, I_z	axial area moments of inertia
W_y, W_z	section modulus
I_t	torsional moment of inertia
W_t	torsional section modulus
I_p	polar area moments of inertia: I_p = I_y + I_z for circular cross sections: I_t = I_p
e_1-4	outer-fibre distance, see following section

Outer-fibre distance

The outer-fibre distance is the distance from the neutral fiber to the outer fibre. For homogeneous cross-sections, the neutral fibre always runs through the center of gravity SP of the surface which lies in the center of the coordinate system. The 4 outer fibres are the furthest away from the respective coordinate axes.

The sketch shows which lengths denote the four outer-fiber distances e₁, e₂, e₃ and e₄.

Therefore the lengths e₁ and e₂ are the vertical outer-fibre distances and the lengths e₃ and e₄ are to the horizontal outer-fibre distances.

Some Notes for the Use of the calculator (manual)
Background Knowledge and Formulas
- Formulas for axial Area Moments of Inertia & Section Modulus
- Correlation Section Modulus < > Area Moment of Inertia
Additional Information about the Calculator
- Comparison: idealized model and real I-beam I100
- Sketches of the available cross sectional areas

Manual

The following cross sectional areas are available, whereby profiles marked by * can have a clearance hole (bore), too:

circle, with slot too *
pipe / hollow circular
semi-circle
rectangle-section *
rectangle-pipe / hollow rectangular *
I/H-section (I/H-beam) *
U/C-section (U/C-beam) *

T-section (T-beam)
L-section (angle section), isosceles and not isosceles
L-section (isosceles) rotated through 45°
isosceles triangle
hexagon / six-sided figure
octagon / eight-sided figure

Below you will find sketches of all cross sectional areas. The cross sectional areas must always be symmetrical to the two coordinate axes.
All white fields have to be completed. Results are displayed on a green background.
Accuracy can not be guaranteed – for corrections or additions please use my contact form!

Background Knowledge and Formulas

In the following table you will find the formulas for the axial area moments of inertia and the section modulus. Then the mathematical relationship between these two quantities is explained.

Formulas for axial Area Moments of Inertia & Section Modulus

The following relationships apply to all the formulas listed in the table below:

b₃ = B – b
b₄ = B – 2·b
h₃ = H – h
h₄ = H – 2·h

Cross Section	Axial Area Moment of Inertia	Section Modulus
circle-section	$$I_y=I_z=\frac{\pi·D^4}{64}$$	$$W_y=W_z=\frac{\pi·D^3}{32}$$
pipe-section	$$I_y=I_z=\frac{\pi·(D^4-d^4)}{64}$$	$$W_y=W_z=\frac{\pi·(D^4-d^4)}{32·D}$$
square-section	$$I_y=I_z=\frac{B^4}{12}$$	$$W_y=W_z=\frac{B^3}{6}$$
square rotated	$$I_y=I_z=\frac{B^4}{12}$$	$$W_y=W_z=\frac{B^3}{6·\sqrt{2}}$$ $$e_{1,2,3,4}=\frac{\sqrt{2}}{2}·B$$
rectangle-section	$$I_y=\frac{B·H^3}{12}$$	$$W_y=\frac{B·H^2}{6}$$
rectangle-section	$$I_z=\frac{H·B^3}{12}$$	$$W_y=\frac{H·B^2}{6}$$
rectangle-pipe	$$I_y=\frac{B·H^3-b_4·(h_4)^3}{12}$$	$$W_y=\frac{B·H^3-b_4·(h_4)^3}{6·H}$$
	$$b_4=B-2·b\qquad h_4=H-2·h\qquad$$ (b₄ and h₄ = inside dimensions)
	$$I_z=\frac{H·B^3-h_4·(b_4)^3}{12}$$	$$W_z=\frac{H·B^3-h_4·(b_4)^3}{6·B}$$
I/H-section	$$I_y=\frac{B·H^3-b_3·(h_4)^3}{12}$$	$$W_y=\frac{B·H^3-b_3·(h_4)^3}{6·H}$$
	$$b_3=B-b\qquad h_4=H-2·h$$
	$$I_z=\frac{2·h·B^3+h_4·b^3}{12}$$	$$W_z=\frac{2·h·B^3+h_4·b^3}{6·B}$$
C/U-section $$b_3= B – b$$ $$h_4=H-2·h$$	$$I_y=\frac{B·H^3-b_3·(h_4)^3}{12}$$	$$W_y=\frac{B·H^3-b_3·(h_4)^3}{6·H}$$
	$$I_z=\frac{2·h·B^3+h_4·b^3}{3}-(2·h·B+h_4·b)e_3^2$$	$$W_{{z}_{1,2}}=\frac{I_z}{e_{3,4}}$$
	$$e_3=\frac{1}{2}·\frac{2·h·B^2+h_4·b^2}{2·h·B+h_4·b}$$	$$e_4=B-e_3$$
T-section $$b_3=B-b$$ $$h_3=H-h$$	$$I_y=\frac{b·H^3+b_3·h^3}{3}-(b·H+b_3·h)·e_2^2$$	$$W_{y_{1,2}}=\frac{I_y}{e_{1,2}}$$
	$$e_2=\frac{1}{2}·\frac{b·H^2+b_3·h^2}{b·H+b_3·h}$$	$$e_1=H-e_2$$
	$$I_z=\frac{h·B^3+h_3·b^3}{12}$$	$$W_z=\frac{h·B^3+h_3·b^3}{6·B}$$
L-section $$b_3=B-b$$ $$h_3=H-h$$	$$I_y=\frac{b·H^3+b_3·h^3}{3}-(b·H+b_3·h)·e_2^2$$	$$W_{y_{1,2}}=\frac{I_y}{e_{1,2}}$$
	$$e_2=\frac{1}{2}·\frac{b·H^2+b_3·h^2}{b·H+b_3·h}$$	$$e_1=H-e_2$$
	$$I_z=\frac{h·B^3+h_3·b^3}{3}-(h·B+h_3·b)·e_3^2$$	$$W_{z_{1,2}}=\frac{I_z}{e_{3,4}}$$
	$$e_3=\frac{1}{2}·\frac{h·B^2+h_3·b^2}{h·B+h_3·b}$$	$$e_4=B-e_3$$
isosceles triangle	$$I_y=\frac{B·H^3}{36}$$	$$W_{y_1}=\frac{B·H^2}{12}\quad e_1=\frac{H}{3}$$
	$$I_y=\frac{B·H^3}{36}$$	$$W_{y_2}=\frac{B·H^2}{24}\quad e_2=\frac{2·H}{3}$$
	$$I_z=\frac{H·B^3}{48}$$	$$W_{z_{1,2}}=\frac{H·B^2}{24}$$
semi-circle	$$I_y=0.10976·\left(\frac{D}{2}\right)^4=0.10976·r^4$$	$$W_{y_1}=0.25861·\left(\frac{D}{2}\right)^3$$
	$$I_y=0.10976·\left(\frac{D}{2}\right)^4=0.10976·r^4$$	$$W_{y_2}=0.19069·\left(\frac{D}{2}\right)^3$$
	$$I_z=\frac{1}{2}·\frac{\pi·D^4}{64}=\frac{\pi·D^4}{128}=\frac{\pi·R^4}{8}$$	$$W_{y_{1,2}}=\frac{1}{2}·\frac{\pi·D^3}{32}=\frac{\pi·D^3}{64}$$
hexagon	$$I_y=I_z=\frac{5·\sqrt{3}}{16}·B^4$$	$$W_y=\frac{5}{8}·B^3$$
hexagon	$$I_y=I_z=\frac{5·\sqrt{3}}{16}·B^4$$	$$W_z=\frac{5·\sqrt{3}}{16}·B^3$$
octagon	$$I_y=I_z=\frac{1+2·\sqrt{2}}{6}·B^4$$	$$W_y=W_z=0.6906·B^3$$

Correlation Section Modulus < > Area Moment of Inertia

The section modulus can be calculated by the following formulas if the area moment of inertia and the outer-fibre distance are known.

The section modulus W_y relative to the y-axis is:

$$W_{y_{1,2}}=\frac{I_y}{e_{1,2}}$$

The section modulus W_z relative to the z-axis is:

$$W_{z_{1,2}}=\frac{I_z}{e_{3,4}}$$

I_y	area moment of inertia relative to the y-axis
I_z	area moment of inertia relative to the z-axis
e₁	lower outer-fibre distance in the z-direction
e₂	upper outer-fibre distance in the z-direction
e₃	left outer-fibre distance in the y-direction
e₄	right outer-fibre distance in the y-direction

If the cross sectional area is not symmetrical to an axis (e₁ ≠ e₂ and/or e₃ ≠ e₄), there are two different section modulus relative to this axis, see figure above. Only the smallest section modulus will be calculated!

Additional Information about the Calculator

Comparison: idealized model and real I-beam I100

Figure 1: On the left a narrow I-beam I100, on the right the simplified model.

In figure 1 you can see a narrow I-beam I100 on the left, on the right you find a simplified model, as it used by the calculator.

The variations in the calculation arise from the fact that the real I-beam has oblique flange surfaces and the inner edges are rounded. This can be seen very well in figure 1.

All drawings were created by using the free programs FreeCAD and GIMP.

The following table compares the calculated values and the real values:

	I_y in cm⁴	W_y in cm³	I_z in cm⁴	W_z in cm³	A in mm²	m’ in kg/m
calculated values	172.1	34.4	14.2	5.7	1069	8.40
real values	171	34.2	12.2	4.88	1060	8.34
variations in %	0.64	0.58	16.4	16.8	0.85	0.72

As you can see, I_y and W_y, the cross sectional area A and the mass per meter match very well. The z-values differ slightly more, but are still useful as an estimate.