Cold And Hot Formed Rectangular Hollow Sections

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Rectangular Hollow Sections include both cold-formed and hot-finished rectangular box profiles. Cold-formed sections are manufactured at ambient temperatures, while hot-finished sections are formed at elevated temperatures, providing lower residual stresses. Both types are widely used for structural frames, beams, columns, and trusses.

Calculations

Description

Symbol Name

Value

Unit

Comment

Cold Formed Rectangular Hollow Sections

Section depth

h_{t}

100.00

mm

Section breadth

b_{t}

50.00

mm

Section thickness

t_{t}

6.00

mm

Material properties

Young's modulus

E_{t}

210000

\frac{N}{{mm}^{2}}

Poisson's ratio

v_{t}

0.30000

-

Shear modulus

G_{t}

80769.2

\frac{N}{{mm}^{2}}

G_{t} = \frac{E_{t}}{(2 \cdot (1 + v_{t}))} = \frac{210000}{(2 \cdot (1 + 0.30000))}

Density

ρ_{t}

7850.00

\frac{kg}{m^{3}}

Calculations

Internal corner radii

r_{i n}

6.00

mm

BS EN 10219-2

r_{i n} = {\begin{cases} t_{t} \cdot 1 & if t_{t} \leq 6 \\ ({\begin{cases} t_{t} \cdot 1.5 & if and (t_{t} > 6, t_{t} \leq 10) \\ t_{t} \cdot 2 & otherwise. \end{cases}) & otherwise. \end{cases} = {\begin{cases} 6.00 \cdot 1 & if 6.00 \leq 6 \\ ({\begin{cases} 6.00 \cdot 1.5 & if and (6.00 > 6, 6.00 \leq 10) \\ 6.00 \cdot 2 & otherwise. \end{cases}) & otherwise. \end{cases}

External corner radii

r_{out}

12.00

mm

BS EN 10219-2

r_{out} = {\begin{cases} t_{t} \cdot 2 & if t_{t} \leq 6 \\ {\begin{cases} t_{t} \cdot 2.5 & if and (t_{t} > 6, t_{t} \leq 10) \\ t_{t} \cdot 3 & otherwise. \end{cases} & otherwise. \end{cases} = {\begin{cases} 6.00 \cdot 2 & if 6.00 \leq 6 \\ {\begin{cases} 6.00 \cdot 2.5 & if and (6.00 > 6, 6.00 \leq 10) \\ 6.00 \cdot 3 & otherwise. \end{cases} & otherwise. \end{cases}

Average of internal and external corner radii

R_{c}

9.00

mm

R_{c} = \frac{(r_{i n} + r_{out})}{2} = \frac{(6.00 + 12.00)}{2}

Area of section

A_{t}

1563.29

{mm}^{2}

A_{t} = 2 \cdot t_{t} \cdot (b_{t} + h_{t} - 2 \cdot t_{t}) - (4 - π) \cdot (2 \cdot t_{t} \cdot R_{c}) = 2 \cdot 6.00 \cdot (50.00 + 100.00 - 2 \cdot 6.00) - (4 - π) \cdot (2 \cdot 6.00 \cdot 9.00)

Mass per meter length

m_{m}

12.27

\frac{kg}{m}

m_{m} = \frac{A_{t}}{10^{6}} \cdot ρ_{t} = \frac{1563.29}{10^{6}} \cdot 7850.00

Web local buckling ratio

\frac{c_{w}}{t_{s}}

13.67

-

\frac{c_{w}}{t_{s}} = \frac{(h_{t} - 3 \cdot t_{t})}{t_{t}} = \frac{(100.00 - 3 \cdot 6.00)}{6.00}

Flange local buckling ratio

\frac{c_{f}}{t_{s}}

5.33

-

\frac{c_{f}}{t_{s}} = \frac{(b_{t} - 3 \cdot t_{t})}{t_{t}} = \frac{(50.00 - 3 \cdot 6.00)}{6.00}

D_{y,i}

38.00

mm

D_{y,i} = \frac{h_{t}}{2} - r_{out} = \frac{100.00}{2} - 12.00

I_{yy,corners}

658422

{mm}^{4}

I_{yy,corners} = 4 \cdot (\frac{π \cdot ({r_{out}}^{4} - {r_{i n}}^{4})}{16} + \frac{π \cdot ({r_{out}}^{2} - {r_{i n}}^{2})}{4} \cdot {D_{y,i}}^{2} + \frac{2 \cdot D_{y,i} \cdot ({r_{out}}^{3} - {r_{i n}}^{3})}{3}) = 4 \cdot (\frac{π \cdot ({12.00}^{4} - {6.00}^{4})}{16} + \frac{π \cdot ({12.00}^{2} - {6.00}^{2})}{4} \cdot {38.00}^{2} + \frac{2 \cdot 38.00 \cdot ({12.00}^{3} - {6.00}^{3})}{3})

I_{yy,web}

438976

{mm}^{4}

I_{yy,web} = \frac{t_{t} \cdot {(h_{t} - 2 \cdot r_{out})}^{3}}{6} = \frac{6.00 \cdot {(100.00 - 2 \cdot 12.00)}^{3}}{6}

I_{yy}

(1.788e+6)

{mm}^{4}

I_{yy} = I_{yy,web} + 2 \cdot (\frac{(b_{t} - 2 \cdot r_{out}) \cdot {t_{t}}^{3}}{12} + (b_{t} - 2 \cdot r_{out}) \cdot t_{t} \cdot {(\frac{(h_{t} - t_{t})}{2})}^{2}) + I_{yy,corners} = 438976 + 2 \cdot (\frac{(50.00 - 2 \cdot 12.00) \cdot {6.00}^{3}}{12} + (50.00 - 2 \cdot 12.00) \cdot 6.00 \cdot {(\frac{(100.00 - 6.00)}{2})}^{2}) + 658422

D_{z,i}

13.00

mm

D_{z,i} = \frac{b_{t}}{2} - r_{out} = \frac{50.00}{2} - 12.00

I_{zz,corners}

125024

{mm}^{4}

I_{zz,corners} = 4 \cdot (\frac{π \cdot ({r_{out}}^{4} - {r_{i n}}^{4})}{16} + \frac{π \cdot ({r_{out}}^{2} - {r_{i n}}^{2})}{4} \cdot {D_{z,i}}^{2} + \frac{2 \cdot D_{z,i} \cdot ({r_{out}}^{3} - {r_{i n}}^{3})}{3}) = 4 \cdot (\frac{π \cdot ({12.00}^{4} - {6.00}^{4})}{16} + \frac{π \cdot ({12.00}^{2} - {6.00}^{2})}{4} \cdot {13.00}^{2} + \frac{2 \cdot 13.00 \cdot ({12.00}^{3} - {6.00}^{3})}{3})

I_{zz,web}

17576.0

{mm}^{4}

I_{zz,web} = \frac{t_{t} \cdot {(b_{t} - 2 \cdot r_{out})}^{3}}{6} = \frac{6.00 \cdot {(50.00 - 2 \cdot 12.00)}^{3}}{6}

I_{zz}

586744

{mm}^{4}

I_{zz} = I_{zz,web} + 2 \cdot (\frac{(h_{t} - 2 \cdot r_{out}) \cdot {t_{t}}^{3}}{12} + (h_{t} - 2 \cdot r_{out}) \cdot t_{t} \cdot {(\frac{(b_{t} - t_{t})}{2})}^{2}) + I_{zz,corners} = 17576.0 + 2 \cdot (\frac{(100.00 - 2 \cdot 12.00) \cdot {6.00}^{3}}{12} + (100.00 - 2 \cdot 12.00) \cdot 6.00 \cdot {(\frac{(50.00 - 6.00)}{2})}^{2}) + 125024

Radius of gyration about major axis

K_{yy}

33.81

mm

K_{yy} = \sqrt{\frac{I_{yy}}{A_{t}}} = \sqrt{\frac{(1.788e+6)}{1563.29}}

Radius of gyration about minor axis

K_{zz}

19.37

mm

K_{zz} = \sqrt{\frac{I_{zz}}{A_{t}}} = \sqrt{\frac{586744}{1563.29}}

Elastic section modulus about major axis

S_{yy}

35750.8

{mm}^{3}

S_{yy} = \frac{I_{yy}}{(\frac{h_{t}}{2})} = \frac{(1.788e+6)}{(\frac{100.00}{2})}

Elastic section modulus about the minor axis

S_{zz}

23469.8

{mm}^{3}

S_{zz} = \frac{I_{zz}}{(\frac{b_{t}}{2})} = \frac{586744}{(\frac{50.00}{2})}

Plastic section modulus about YY axis (corners)

Z_{yy,corners}

7454.55

{mm}^{3}

Z_{yy,corners} = 2 \cdot (\frac{π \cdot ({r_{out}}^{2} - {r_{i n}}^{2})}{4} \cdot (\frac{h_{t}}{2} - r_{out}) + \frac{({r_{out}}^{3} - {r_{i n}}^{3})}{3}) = 2 \cdot (\frac{π \cdot ({12.00}^{2} - {6.00}^{2})}{4} \cdot (\frac{100.00}{2} - 12.00) + \frac{({12.00}^{3} - {6.00}^{3})}{3})

Plastic section modulus about YY axis

Z_{yy}

46901.1

Z_{yy} = 2 \cdot (\frac{(b_{t} - 2 \cdot r_{out}) \cdot t_{t} \cdot (h_{t} - t_{t})}{2} + t_{t} \cdot {(\frac{(h_{t} - 2 \cdot r_{out})}{2})}^{2} + Z_{yy,corners}) = 2 \cdot (\frac{(50.00 - 2 \cdot 12.00) \cdot 6.00 \cdot (100.00 - 6.00)}{2} + 6.00 \cdot {(\frac{(100.00 - 2 \cdot 12.00)}{2})}^{2} + 7454.55)

Plastic section modulus about ZZ axis (corners)

Z_{zz,corners}

3213.40

{mm}^{3}

Z_{zz,corners} = 2 \cdot (\frac{π \cdot ({r_{out}}^{2} - {r_{i n}}^{2})}{4} \cdot (\frac{b_{t}}{2} - r_{out}) + \frac{({r_{out}}^{3} - {r_{i n}}^{3})}{3}) = 2 \cdot (\frac{π \cdot ({12.00}^{2} - {6.00}^{2})}{4} \cdot (\frac{50.00}{2} - 12.00) + \frac{({12.00}^{3} - {6.00}^{3})}{3})

Plastic section modulus about ZZ axis

Z_{zz}

28518.8

{mm}^{3}

Z_{zz} = 2 \cdot (\frac{(h_{t} - 2 \cdot r_{out}) \cdot t_{t} \cdot (b_{t} - t_{t})}{2} + t_{t} \cdot {(\frac{(b_{t} - 2 \cdot r_{out})}{2})}^{2} + Z_{zz,corners}) = 2 \cdot (\frac{(100.00 - 2 \cdot 12.00) \cdot 6.00 \cdot (50.00 - 6.00)}{2} + 6.00 \cdot {(\frac{(50.00 - 2 \cdot 12.00)}{2})}^{2} + 3213.40)

Mean perimeter length

P

260.55

mm

P = 2 \cdot ((b_{t} - t_{t}) + (h_{t} - t_{t})) - 2 \cdot R_{c} \cdot (4 - π) = 2 \cdot ((50.00 - 6.00) + (100.00 - 6.00)) - 2 \cdot 9.00 \cdot (4 - π)

Area enclosed by mean parameter

A_{p}

4066.47

{mm}^{2}

A_{p} = (b_{t} - t_{t}) \cdot (h_{t} - t_{t}) - {R_{c}}^{2} \cdot (4 - π) = (50.00 - 6.00) \cdot (100.00 - 6.00) - {9.00}^{2} \cdot (4 - π)

Torsional Constant

I_{t}

(1.542e+6)

{mm}^{4}

I_{t} = \frac{4 \cdot {A_{p}}^{2} \cdot t_{t}}{P} + \frac{{t_{t}}^{3} \cdot P}{3} = \frac{4 \cdot {4066.47}^{2} \cdot 6.00}{260.55} + \frac{{6.00}^{3} \cdot 260.55}{3}

Torsional section modulus

W_{t}

41434.2

{mm}^{3}

W_{t} = \frac{I_{t}}{(t_{t} + \frac{2 \cdot A_{p}}{P})} = \frac{(1.542e+6)}{(6.00 + \frac{2 \cdot 4066.47}{260.55})}

Surface area per metre

A_{s}

0.27940

\frac{m^{2}}{m}

A_{s} = \frac{(2 \cdot (b_{t} - 2 \cdot r_{out}) + 2 \cdot (h_{t} - 2 \cdot r_{out}) + 2 \cdot π \cdot r_{out})}{1000} = \frac{(2 \cdot (50.00 - 2 \cdot 12.00) + 2 \cdot (100.00 - 2 \cdot 12.00) + 2 \cdot π \cdot 12.00)}{1000}