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.

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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}