Unequal I Beam

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The unequal I-beam features flanges of different widths or thicknesses. This specialised profile is designed for applications where stress distribution is asymmetrical, such as in composite beams where the top flange interacts with a concrete slab, effectively shifting the neutral axis. It can also be used for crane runway beams or other members subjected to eccentric loads or specific connection requirements, optimising material for unbalanced loading conditions.

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Calculations

Description

Symbol Name

Value

Unit

Comment

Unequal I-beam

Geometry input parameters

Section depth

d_{t}

100.00

mm

Web thickness

t_{w}

5.00

mm

Width of top flange

b_{f,top}

50.00

mm

Top flange thickness

t_{f,top}

5.00

mm

Width of bot flange

b_{f,bot}

30.00

mm

Bot flange thickness

t_{f,bot}

4.00

mm

Calculated geometry parameters

Web depth

h_{w}

91.00

mm

h_{w}

=

d_{t} - t_{f,top} - t_{f,bot}

=

100.00 - 5.00 - 4.00

Area

A_{t}

825.00

{mm}^{2}

A_{t}

=

b_{f,top} \cdot t_{f,top} + h_{w} \cdot t_{w} + b_{f,bot} \cdot t_{f,bot}

=

50.00 \cdot 5.00 + 91.00 \cdot 5.00 + 30.00 \cdot 4.00

Perimeter

P_{t}

350.00

mm

P_{t}

=

2 \cdot (b_{f,bot} + b_{f,top} + d_{t} - t_{w})

=

2 \cdot (30.00 + 50.00 + 100.00 - 5.00)

Distance to centroid (x-axis)

C_{x}

25.00

mm

C_{x}

=

\frac{\max (b_{f,bot}, b_{f,top})}{2}

=

\frac{\max (30.00, 50.00)}{2}

Distance to centroid (y-axis)

C_{y}

57.14

mm

C_{y}

=

d_{t} - \frac{1}{(2 \cdot A_{t})} \cdot (t_{w} \cdot {d_{t}}^{2} + {t_{f,top}}^{2} \cdot (b_{f,top} - t_{w}) + t_{f,bot} \cdot (b_{f,bot} - t_{w}) \cdot (2 \cdot d_{t} - t_{f,bot}))

=

100.00 - \frac{1}{(2 \cdot 825.00)} \cdot (5.00 \cdot {100.00}^{2} + {5.00}^{2} \cdot (50.00 - 5.00) + 4.00 \cdot (30.00 - 5.00) \cdot (2 \cdot 100.00 - 4.00))

Second moments of area

Second moment of area about the x-axis (1)

I_{1, x}

(3.178e+6)

{mm}^{4}

I_{1, x}

=

\frac{1}{3} \cdot (b_{f,top} \cdot {(d_{t} - C_{y})}^{3} + b_{f,bot} \cdot {C_{y}}^{3})

=

\frac{1}{3} \cdot (50.00 \cdot {(100.00 - 57.14)}^{3} + 30.00 \cdot {57.14}^{3})

Second moment of area about the x-axis (2)

I_{2, x}

(2.064e+6)

{mm}^{4}

I_{2, x}

=

\frac{1}{3} \cdot ((b_{f,top} - t_{w}) \cdot {(d_{t} - C_{y} - t_{f,top})}^{3} + (b_{f,bot} - t_{w}) \cdot {(C_{y} - t_{f,bot})}^{3})

=

\frac{1}{3} \cdot ((50.00 - 5.00) \cdot {(100.00 - 57.14 - 5.00)}^{3} + (30.00 - 5.00) \cdot {(57.14 - 4.00)}^{3})

Final second moment of area about the x-axis

I_{xx}

(1.113e+6)

{mm}^{4}

I_{xx}

=

I_{1, x} - I_{2, x}

=

(3.178e+6) - (2.064e+6)

Second moment of area about the y-axis

I_{yy}

62031.3

{mm}^{4}

I_{yy}

=

\frac{(t_{f,bot} \cdot {b_{f,bot}}^{3})}{12} + \frac{(h_{w} \cdot {t_{w}}^{3})}{12} + \frac{(t_{f,top} \cdot {b_{f,top}}^{3})}{12}

=

\frac{(4.00 \cdot {30.00}^{3})}{12} + \frac{(91.00 \cdot {5.00}^{3})}{12} + \frac{(5.00 \cdot {50.00}^{3})}{12}

Second moment of area about the z-axis

I_{zz}

(1.175e+6)

{mm}^{4}

I_{zz}

=

I_{xx} + I_{yy}

=

(1.113e+6) + 62031.3

Other properties

Radius of gyration about x-axis

K_{x}

36.74

mm

K_{x}

=

\sqrt{\frac{I_{xx}}{A_{t}}}

=

\sqrt{\frac{(1.113e+6)}{825.00}}

Radius of gyration about y-axis

K_{y}

8.67

mm

K_{y}

=

\sqrt{\frac{I_{yy}}{A_{t}}}

=

\sqrt{\frac{62031.3}{825.00}}

Elastic section modulus about x-axis

S_{x}

19485.1

{mm}^{3}

S_{x}

=

\frac{I_{xx}}{C_{y}}

=

\frac{(1.113e+6)}{57.14}

Elastic section modulus about y-axis

S_{y}

2481.25

{mm}^{3}

S_{y}

=

\frac{I_{yy}}{C_{x}}

=

\frac{62031.3}{25.00}

Plastic modulus

Upper flange area

A_{f}^{u}

250.00

mm

A_{f}^{u}

=

b_{f,top} \cdot t_{f,top}

=

50.00 \cdot 5.00

Lower flange area

A_{f}^{d}

120.00

mm

A_{f}^{d}

=

b_{f,bot} \cdot t_{f,bot}

=

30.00 \cdot 4.00

PNA Located Between Flanges

y_{pna, 1}

62.50

mm

y_{pna, 1}

=

t_{f,bot} + \frac{1}{(t_{w})} \cdot (\frac{A_{t}}{2} - A_{f}^{d})

=

4.00 + \frac{1}{(5.00)} \cdot (\frac{825.00}{2} - 120.00)

PNA Located in Lower Flange

y_{pna, 2}

13.75

mm

y_{pna, 2}

=

\frac{A_{t}}{(2 \cdot b_{f,bot})}

=

\frac{825.00}{(2 \cdot 30.00)}

PNA Located in Upper Flange

y_{pna, 3}

91.75

mm

y_{pna, 3}

=

d_{t} - \frac{A_{t}}{(2 \cdot b_{f,top})}

=

100.00 - \frac{825.00}{(2 \cdot 50.00)}

Z_{x, 1, temp}

2640.63

{mm}^{3}

Z_{x, 1, temp}

=

\frac{t_{w} \cdot {(d_{t} - y_{pna, 1} - t_{f,top})}^{2}}{2}

=

\frac{5.00 \cdot {(100.00 - 62.50 - 5.00)}^{2}}{2}

Z_{x, 1}

27206.3

{mm}^{3}

Z_{x, 1}

=

A_{f}^{d} \cdot (y_{pna, 1} - \frac{t_{f,bot}}{2}) + A_{f}^{u} \cdot (d_{t} - y_{pna, 1} - \frac{t_{f,top}}{2}) + \frac{t_{w} \cdot {(y_{pna, 1} - t_{f,bot})}^{2}}{2} + Z_{x, 1, temp}

=

120.00 \cdot (62.50 - \frac{4.00}{2}) + 250.00 \cdot (100.00 - 62.50 - \frac{5.00}{2}) + \frac{5.00 \cdot {(62.50 - 4.00)}^{2}}{2} + 2640.63

Z_{x, 2, temp}

20937.5

{mm}^{3}

Z_{x, 2, temp}

=

b_{f,top} \cdot t_{f,top} \cdot (d_{t} - y_{pna, 2} - \frac{t_{f,top}}{2})

=

50.00 \cdot 5.00 \cdot (100.00 - 13.75 - \frac{5.00}{2})

Z_{x, 2}

41465.6

{mm}^{3}

Z_{x, 2}

=

\frac{b_{f,bot} \cdot {y_{pna, 2}}^{2}}{2} + \frac{(b_{f,bot} \cdot {(t_{f,bot} - y_{pna, 2})}^{2})}{2} + \frac{t_{w} \cdot h_{w}}{2} \cdot (d_{t} + t_{f,bot} - t_{f,top} - 2 \cdot y_{pna, 2}) + Z_{x, 2, temp}

=

\frac{30.00 \cdot {13.75}^{2}}{2} + \frac{(30.00 \cdot {(4.00 - 13.75)}^{2})}{2} + \frac{5.00 \cdot 91.00}{2} \cdot (100.00 + 4.00 - 5.00 - 2 \cdot 13.75) + 20937.5

Z_{x, 3, temp}

19223.8

{mm}^{3}

Z_{x, 3, temp}

=

t_{w} \cdot h_{w} \cdot (y_{pna, 3} - t_{f,bot} - \frac{h_{w}}{2})

=

5.00 \cdot 91.00 \cdot (91.75 - 4.00 - \frac{91.00}{2})

Z_{x, 3}

31959.4

{mm}^{3}

Z_{x, 3}

=

b_{f,bot} \cdot t_{f,bot} \cdot (y_{pna, 3} - \frac{t_{f,bot}}{2}) + Z_{x, 3, temp} + \frac{b_{f,top} \cdot {(d_{t} - y_{pna, 3})}^{2}}{2} + \frac{b_{f,top} \cdot {(d_{t} - t_{f,top} - y_{pna, 3})}^{2}}{2}

=

30.00 \cdot 4.00 \cdot (91.75 - \frac{4.00}{2}) + 19223.8 + \frac{50.00 \cdot {(100.00 - 91.75)}^{2}}{2} + \frac{50.00 \cdot {(100.00 - 5.00 - 91.75)}^{2}}{2}

Z_{x, final}

27206.3

mm

Z_{x, final}

=

{\begin{cases} Z_{x, 1} & if and (A_{f}^{d} < \frac{A_{t}}{2}, A_{f}^{u} < \frac{A_{t}}{2}) \\ {\begin{cases} Z_{x, 2} & if A_{f}^{d} > \frac{A_{t}}{2} \\ Z_{x, 3} & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} 27206.3 & if and (120.00 < \frac{825.00}{2}, 250.00 < \frac{825.00}{2}) \\ {\begin{cases} 41465.6 & if 120.00 > \frac{825.00}{2} \\ 31959.4 & otherwise. \end{cases} & otherwise. \end{cases}

Z_{y}

4593.75

{mm}^{3}

Z_{y}

=

\frac{(t_{f,bot} \cdot {b_{f,bot}}^{2})}{4} + \frac{(h_{w} \cdot {t_{w}}^{2})}{4} + \frac{(t_{f,top} \cdot {b_{f,top}}^{2})}{4}

=

\frac{(4.00 \cdot {30.00}^{2})}{4} + \frac{(91.00 \cdot {5.00}^{2})}{4} + \frac{(5.00 \cdot {50.00}^{2})}{4}