Unequal Tapered I Beam

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The unequal-tapered I-beam combines the characteristics of varying depth (tapered) and dissimilar flanges (unequal). This highly specialised section is engineered for complex loading scenarios where both stress distribution along the member's length and stress asymmetry across the section need to be addressed efficiently. Such beams are found in custom-designed structures like complex bridge elements or specialised industrial equipment requiring maximal material optimisation.

Calculations

Description

Symbol Name

Value

Unit

Comment

Unequal Tapered I-beam

Geometry Input

Section depth

d_{t}

100.00

mm

Web thickness

t_{w}

5.00

mm

Width of top flange

b_{f,top}

50.00

mm

Width of bot flange

b_{f,bot}

30.00

mm

Outer top flange thickness

s_{f, top}

5.00

mm

Inner top flange thickness

n_{t, top}

8.00

mm

Outer bot flange thickness

s_{f, bot}

4.00

mm

Inner bot flange thickness

n_{t, bot}

6.00

mm

Calculated Geometry Parameters

Outer flange to flange depth

h_{t}

91.00

mm

h_{t} = d_{t} - s_{f, bot} - s_{f, top} = 100.00 - 4.00 - 5.00

Web depth

L_{t}

86.00

mm

L_{t} = d_{t} - n_{t, bot} - n_{t, top} = 100.00 - 6.00 - 8.00

Tapered top flange segment horizontal length

a_{h,top}

22.50

mm

a_{h,top} = 0.5 \cdot (b_{f,top} - t_{w}) = 0.5 \cdot (50.00 - 5.00)

Tapered bottom flange segment horizontal length

a_{h,bot}

12.50

mm

a_{h,bot} = 0.5 \cdot (b_{f,bot} - t_{w}) = 0.5 \cdot (30.00 - 5.00)

Tapered top flange segment vertical length

a_{v,top}

3.00

mm

a_{v,top} = n_{t, top} - s_{f, top} = 8.00 - 5.00

Tapered bot flange segment vertical length

a_{v,bot}

2.00

mm

a_{v,bot} = n_{t, bot} - s_{f, bot} = 6.00 - 4.00

A_{top,flange}

250.00

{mm}^{2}

A_{top,flange} = b_{f,top} \cdot s_{f, top} = 50.00 \cdot 5.00

A_{top,trapz}

82.50

{mm}^{2}

A_{top,trapz} = 0.5 \cdot (b_{f,top} + t_{w}) \cdot a_{v,top} = 0.5 \cdot (50.00 + 5.00) \cdot 3.00

A_{web}

430.00

{mm}^{2}

A_{web} = t_{w} \cdot L_{t} = 5.00 \cdot 86.00

A_{bot,trapz}

35.00

{mm}^{2}

A_{bot,trapz} = 0.5 \cdot (b_{f,bot} + t_{w}) \cdot a_{v,bot} = 0.5 \cdot (30.00 + 5.00) \cdot 2.00

A_{bot,flange}

120.00

{mm}^{2}

A_{bot,flange} = b_{f,bot} \cdot s_{f, bot} = 30.00 \cdot 4.00

Total area of the section

A_{t}

917.50

{mm}^{2}

A_{t} = A_{bot,flange} + A_{bot,trapz} + A_{web} + A_{top,trapz} + A_{top,flange} = 120.00 + 35.00 + 430.00 + 82.50 + 250.00

Perimeter

P_{t}

340.72

mm

P_{t} = b_{f,bot} + b_{f,top} + 2 \cdot (s_{f, bot} + s_{f, top} + L_{t} + \sqrt{{a_{v,bot}}^{2} + {a_{h,bot}}^{2}} + \sqrt{{a_{v,top}}^{2} + {a_{h,top}}^{2}}) = 30.00 + 50.00 + 2 \cdot (

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}

y_{top,flange}

97.50

mm

y_{top,flange} = d_{t} - 0.5 \cdot s_{f, top} = 100.00 - 0.5 \cdot 5.00

y_{top,trapz}

93.09

mm

y_{top,trapz} = d_{t} - s_{f, top} - a_{v,top} + \frac{a_{v,top}}{3} \cdot (\frac{(2 \cdot t_{w} + b_{f,top})}{(t_{w} + b_{f,top})}) = 100.00 - 5.00 - 3.00 + \frac{3.00}{3} \cdot (\frac{(2}{}

y_{web}

49.00

mm

y_{web} = d_{t} - n_{t, top} - 0.5 \cdot L_{t} = 100.00 - 8.00 - 0.5 \cdot 86.00

y_{bot,trapz}

4.76

mm

y_{bot,trapz} = s_{f, bot} + \frac{a_{v,bot} \cdot (2 \cdot t_{w} + b_{f,bot})}{(t_{w} + b_{f,bot})} \cdot (\frac{1}{3}) = 4.00 + \frac{2.00 \cdot (2 \cdot 5.00 + 30.00)}{(5.00 + 30.00)} \cdot (\frac{1}{}

y_{bot,flange}

2.00

mm

y_{bot,flange} = 0.5 \cdot s_{f, bot} = 0.5 \cdot 4.00

Distance to centroid (y-axis) from the bottom of the section

C_{y}

58.35

mm

C_{y} = \frac{(A_{bot,flange} \cdot y_{bot,flange} + A_{bot,trapz} \cdot y_{bot,trapz} + A_{web} \cdot y_{web} + A_{top,trapz} \cdot y_{top,trapz} + A_{top,flange} \cdot y_{top,flange})}{A_{t}} = \frac{(120.00 \cdot 2.00 +}{}

Second moment of area around XX axis

Second moment of area about x axis for the top flange

I_{xx,top,flange}

383797

{mm}^{4}

I_{xx,top,flange} = \frac{A_{top,flange} \cdot {s_{f, top}}^{2}}{12} + A_{top,flange} \cdot {(y_{top,flange} - C_{y})}^{2} = \frac{250.00 \cdot {5.00}^{2}}{12} + 250.00 \cdot {(97.50 - 58.35)}^{2}

Second moment of area about the x-axis for the top tapered sections

I_{xx,top,trapz}

99647.7

{mm}^{4}

I_{xx,top,trapz} = \frac{{a_{v,top}}^{3} \cdot ({t_{w}}^{2} + 4 \cdot t_{w} \cdot b_{f,top} + {b_{f,top}}^{2})}{(36 \cdot (t_{w} + b_{f,top}))} + A_{top,trapz} \cdot {(y_{top,trapz} - C_{}}^{}

Second moment of area about x axis for the web

I_{xx,web}

261005

{mm}^{4}

I_{xx,web} = \frac{A_{web} \cdot {L_{t}}^{2}}{12} + A_{web} \cdot (y_{web} - C_{y}) = \frac{430.00 \cdot {86.00}^{2}}{12} + 430.00 \cdot (49.00 - 58.35)

Second moment of area about the x-axis for the bottom tapered sections

I_{xx,bot,trapz}

100500

{mm}^{4}

I_{xx,bot,trapz} = \frac{{a_{v,bot}}^{3} \cdot ({t_{w}}^{2} + 4 \cdot t_{w} \cdot b_{f,bot} + {b_{f,bot}}^{2})}{(36 \cdot (t_{w} + b_{f,bot}))} + A_{bot,trapz} \cdot {(y_{bot,trapz} - C_{}}^{}

Second moment of area about x-axis for the bot flange

I_{xx,bot,flange}

381133

{mm}^{4}

I_{xx,bot,flange} = \frac{A_{bot,flange} \cdot {s_{f, bot}}^{2}}{12} + A_{bot,flange} \cdot {(y_{bot,flange} - C_{y})}^{2} = \frac{120.00 \cdot {4.00}^{2}}{12} + 120.00 \cdot {(2.00 - 58.35)}^{2}

Second moment of area about x-axis

I_{x x}

(1.2e+6)

{mm}^{4}

I_{x x} = I_{xx,top,flange} + I_{xx,bot,flange} + I_{xx,web} + I_{xx,top,trapz} + I_{xx,bot,trapz} = 383797 + 381133 + 261005 + 99647.7 + 100500

Second moment of area about the x1-axis

I_{xx, 1}

(4.3e+6)

{mm}^{4}

I_{xx, 1} = I_{x x} + A_{t} \cdot {C_{y}}^{2} = (1.2e+6) + 917.50 \cdot {58.35}^{2}

Second moment of area around YY axis

Second moment of area about the y-axis for the major flange

I_{yy,top,flange}

52083.3

{mm}^{4}

I_{yy,top,flange} = \frac{1}{12} \cdot s_{f, top} \cdot {b_{f,top}}^{3} = \frac{1}{12} \cdot 5.00 \cdot {50.00}^{3}

Second moment of area about the y-axis for the minor flange

I_{yy,bot,flange}

9000.00

{mm}^{4}

I_{yy,bot,flange} = \frac{1}{12} \cdot s_{f, bot} \cdot {b_{f,bot}}^{3} = \frac{1}{12} \cdot 4.00 \cdot {30.00}^{3}

Second moment if area about the y-axis for the web

I_{yy,web}

895.83

{mm}^{4}

I_{yy,web} = \frac{1}{12} \cdot L_{t} \cdot {t_{w}}^{3} = \frac{1}{12} \cdot 86.00 \cdot {5.00}^{3}

Second moment of area about the y-axis for the top tapered section

I_{yy,top,trapz}

4324.22

{mm}^{4}

I_{yy,top,trapz} = \frac{1}{36} \cdot (n_{t, top} - s_{f, top}) \cdot {a_{h,top}}^{3} + \frac{(a_{h,top} \cdot (n_{t, top} - s_{f, top}))}{2} \cdot {(\frac{t_{w}}{2} + \frac{_{}}{}}^{}

Second moment of area about the y-axis for the bottom tapered section

I_{yy,bot,trapz}

664.06

{mm}^{4}

I_{yy,bot,trapz} = \frac{1}{36} \cdot (n_{t, bot} - s_{f, bot}) \cdot {a_{h,bot}}^{3} + \frac{(a_{h,bot} \cdot (n_{t, bot} - s_{f, bot}))}{2} \cdot {(\frac{t_{w}}{2} + \frac{_{}}{}}^{}

Second moment of area about y-axis

I_{y y}

66967.4

{mm}^{4}

I_{y y} = I_{yy,top,flange} + I_{yy,bot,flange} + I_{yy,web} + I_{yy,top,trapz} + I_{yy,bot,trapz} = 52083.3 + 9000.00 + 895.83 + 4324.22 + 664.06

Second moment of area about the y1-axis

I_{yy, 1}

640405

{mm}^{4}

I_{yy, 1} = I_{y y} + A_{t} \cdot {C_{x}}^{2} = 66967.4 + 917.50 \cdot {25.00}^{2}

Other properties

Polar second moment of area about the z-axis

J_{z}

(1.3e+6)

{mm}^{4}

J_{z} = I_{x x} + I_{y y} = (1.2e+6) + 66967.4

Polar second moment of area about the z1-axis

J_{z_{1}}

(5.0e+6)

{mm}^{4}

J_{z_{1}} = I_{xx, 1} + I_{yy, 1} = (4.3e+6) + 640405

Radius of gyration about x-axis

K_{x}

36.56

mm

K_{x} = \sqrt{\frac{I_{x x}}{A_{t}}} = \sqrt{\frac{(1.2e+6)}{917.50}}

Radius of gyration about y-axis

K_{y}

8.54

mm

K_{y} = \sqrt{\frac{I_{y y}}{A_{t}}} = \sqrt{\frac{66967.4}{917.50}}

Radius of gyration about the z-axis

K_{z}

37.54

mm

K_{z} = \sqrt{{K_{x}}^{2} + {K_{y}}^{2}} = \sqrt{{36.56}^{2} + {8.54}^{2}}

Radius of gyration about the x1-axis

K_{x_{1}}

68.85

mm

K_{x_{1}} = \sqrt{\frac{I_{xx, 1}}{A_{t}}} = \sqrt{\frac{(4.3e+6)}{917.50}}

Radius of gyration about the y1-axis

K_{y_{1}}

26.42

mm

K_{y_{1}} = \sqrt{\frac{I_{yy, 1}}{A_{t}}} = \sqrt{\frac{640405}{917.50}}

Radius of gyration about the z1-axis

K_{z_{1}}

73.75

mm

K_{z_{1}} = \sqrt{{K_{x_{1}}}^{2} + {K_{y_{1}}}^{2}} = \sqrt{{68.85}^{2} + {26.42}^{2}}

Elastic section modulus about x-axis

S_{x}

21014.3

{mm}^{3}

S_{x} = \frac{I_{x x}}{C_{y}} = \frac{(1.2e+6)}{58.35}

Elastic section modulus about y-axis

S_{y}

2678.70

{mm}^{3}

S_{y} = \frac{I_{y y}}{C_{x}} = \frac{66967.4}{25.00}

PLASTIC SECTION CALCS TO BE DONE