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## Geometrical Properties of Structural Sections

Posted by Win Aung Cho on 27-Sept-2009

# Section Properties

## Torsional Properties of Structural Sections

Polar Moments of Inertia is a torsional property of structural section.

### Circular cross section:

`Jxx = Iyy + Izz `

### Solid rectangular cross section

(width=b, depth=d ( b < d ) ):
```Jxx =  Q d b3
where Q = 1.0/3 - 0.224435 / (d/b + 0.160705);
```

### Open sections made up of thin plates

(length=b, thickness=t):
```Jxx = Si [  bi ti3 / 3 ]
```

### Closed single-box sections made up of thin plates

(length=b, thickness=t):
``` Jxx = 4 A2 / Si [ bi / ti ]
where A is the area enclosed by the box.
```

## Bending Properties of Structural Sections

The bending moments of inertia, Iyy and Izz, are the principle bending moments of inertia for the cross section.

## Cross Sectional Properties

### Circular Tube

```
A = PI ( Ro2 - Ri2 )
Asy = Asz = A / ( 1.124235 + 0.055610(Ri/Ro) + 1.097134(Ri/Ro)2 - 0.630057(Ri/Ro)3 ) Ý 0.5%
Asy = Asz = A / ( 1.06124 + 0.59546(Ri/Ro) )  Ý 2%
Jxx = (1/2) PI ( Ro4 - Ri4 )
Izz = Iyy = (1/4) PI ( Ro4 - Ri4 )

```

### Square Tube

(outer dimension = b x b, wall thickness = t):
```
A = b2 - (b - 2t)2
Asy = Asz = A / ( 2.08334 - 0.70154(t/b) - 8.00313(t/b)2 + 12.22572(t/b)3 ) Ý 0.5%
Asy = Asz = A / ( 2.1186 - 1.9900(t/b) ) Ý 2%
Jxx =  (b - t)3 t
Izz = Iyy = (1/12) ( b4 - (b - 2t)4 )

```

### Rectangular Tube

(outer dimension = a x b, wall thickness = t):
```
A = ab -  (a - 2t)(b - 2t)
Asy = A / ( 1.14766  + 0.28187(t/b) + 0.96199(b/a) - 2.17742(t/a) ) Ý 1% ... (a > b)
Asy = A / ( 1.10498 - 1.98518(t/a) + 8.74762(t/a)3 + 0.99548(b/a) + 0.69146(tb/a2) - 5.36255(t2b/a3) ) Ý 1% ... (b > a)
Asz = A / ( 1.10498 - 1.98518(t/b) + 8.74762(t/b)3 + 0.99548(a/b) + 0.69146(ta/b2) - 5.36255(t2a/b3) ) Ý 1% ... (a > b)
Asz = A / ( 1.14766  + 0.28187(t/a) + 0.96199(a/b) - 2.17742(t/b) ) Ý 1% ... (b > a)
Jxx =  2 t (a - t)2(b - t)2 / (a + b - 2t)
Iyy = (1/12) ( ab3 - (a - 2t)(b - 2t)3 )
Izz = (1/12) ( a3b - (a - 2t)3(b - 2t) )

```

### I sections

(depth = d, width = b, flange thickness = t, web thickness = w):
```
A = bd  - (d-2t)(b-w)
Asy = 1.64 b t
Asz = d w
Jxx = (1/3) ( 2 b t3 + d w3 )
Iyy = (1/12) ( bd3 - (b-w)(d-2t)3 )
Izz = (1/12) ( 2 t b3 + (d-2t)w3 )

```

Note: Most of the Standard rolled sections have filleted corners. Manufacturer specifcations or standard steel section manul for such section should therefore be used whenever available.

## Cross Sectional Properties Of Some Dressed Wood Sections

```             A       Asy     Asz     Jxx     Iyy     Izz
in^2    in^2    in^2    in^4    in^4    in^4

2x3    3.750   2.500   2.500   1.776   1.953   0.708
2x4    5.250   3.500   3.500   2.875   6.359   0.984
2x5    6.750   4.500   4.500   3.984  11.390   1.266
2x6    8.250   5.500   5.500   5.099  20.800   1.547
2x8   10.850   7.233   7.233   7.057  47.630   2.039
2x10  13.880   9.253   9.253   9.299  98.930   2.602
2x12  16.880  11.253  11.253  11.544 178.000   3.164
2x14  19.880  13.253  13.253  13.790 290.800   3.727
```
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