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.

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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

Outer top flange thickness

s_{f, top}

10.00

mm

Inner top flange thickness

n_{t, top}

15.00

mm

Width of bot flange

b_{f,bot}

30.00

mm

Outer bot flange thickness

s_{f, bot}

5.00

mm

Inner bot flange thickness

n_{t, bot}

10.00

mm

Calculated Geometry Parameters

Outer flange to flange depth

h_{central}

85.00

mm

h_{central}

=

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

=

100.00 - 5.00 - 10.00

Web depth

L_{t}

75.00

mm

L_{t}

=

d_{t} - n_{t, bot} - n_{t, top}

=

100.00 - 10.00 - 15.00

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}

5.00

mm

a_{v,top}

=

n_{t, top} - s_{f, top}

=

15.00 - 10.00

Tapered bot flange segment vertical length

a_{v,bot}

5.00

mm

a_{v,bot}

=

n_{t, bot} - s_{f, bot}

=

10.00 - 5.00

A_{top,flange}

500.00

{mm}^{2}

A_{top,flange}

=

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

=

50.00 \cdot 10.00

A_{top,trapz}

137.50

{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 5.00

A_{web}

375.00

{mm}^{2}

A_{web}

=

t_{w} \cdot L_{t}

=

5.00 \cdot 75.00

A_{bot,trapz}

87.50

{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 5.00

A_{bot,flange}

150.00

{mm}^{2}

A_{bot,flange}

=

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

=

30.00 \cdot 5.00

Total area of the section

A_{t}

1250.00

{mm}^{2}

A_{t}

=

A_{bot,flange} + A_{bot,trapz} + A_{web} + A_{top,trapz} + A_{top,flange}

=

150.00 + 87.50 + 375.00 + 137.50 + 500.00

Perimeter

P_{t}

333.02

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 (5.00 + 10.00 + 75.00 + \sqrt{{5.00}^{2} + {12.50}^{2}} + \sqrt{{5.00}^{2} + {22.50}^{2}})

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}

95.00

mm

y_{top,flange}

=

d_{t} - 0.5 \cdot s_{f, top}

=

100.00 - 0.5 \cdot 10.00

y_{top,trapz}

88.18

mm

y_{top,trapz}

=

d_{t} - s_{f, top} - \frac{a_{v,top}}{3} \cdot (\frac{(2 \cdot t_{w} + b_{f,top})}{(t_{w} + b_{f,top})})

=

100.00 - 10.00 - \frac{5.00}{3} \cdot (\frac{(2 \cdot 5.00 + 50.00)}{(5.00 + 50.00)})

y_{web}

47.50

mm

y_{web}

=

d_{t} - n_{t, top} - 0.5 \cdot L_{t}

=

100.00 - 15.00 - 0.5 \cdot 75.00

y_{bot,trapz}

6.90

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})

=

5.00 + \frac{5.00 \cdot (2 \cdot 5.00 + 30.00)}{(5.00 + 30.00)} \cdot (\frac{1}{3})

y_{bot,flange}

2.50

mm

y_{bot,flange}

=

0.5 \cdot s_{f, bot}

=

0.5 \cdot 5.00

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

C_{y}

62.73

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{(150.00 \cdot 2.50 + 87.50 \cdot 6.90 + 375.00 \cdot 47.50 + 137.50 \cdot 88.18 + 500.00 \cdot 95.00)}{1250.00}

Second moment of area around XX axis

Second moment of area about x axis for the top flange

I_{xx,top,flange}

524736

{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{500.00 \cdot {10.00}^{2}}{12} + 500.00 \cdot {(95.00 - 62.73)}^{2}

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

I_{xx,top,trapz}

89271.0

{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_{y})}^{2}

=

\frac{{5.00}^{3} \cdot ({5.00}^{2} + 4 \cdot 5.00 \cdot 50.00 + {50.00}^{2})}{(36 \cdot (5.00 + 50.00))} + 137.50 \cdot {(88.18 - 62.73)}^{2}

Second moment of area about x axis for the web

I_{xx,web}

170069

{mm}^{4}

I_{xx,web}

=

\frac{A_{web} \cdot {L_{t}}^{2}}{12} + A_{web} \cdot (y_{web} - C_{y})

=

\frac{375.00 \cdot {75.00}^{2}}{12} + 375.00 \cdot (47.50 - 62.73)

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

I_{xx,bot,trapz}

272874

{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_{y})}^{2}

=

\frac{{5.00}^{3} \cdot ({5.00}^{2} + 4 \cdot 5.00 \cdot 30.00 + {30.00}^{2})}{(36 \cdot (5.00 + 30.00))} + 87.50 \cdot {(6.90 - 62.73)}^{2}

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

I_{xx,bot,flange}

544521

{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{150.00 \cdot {5.00}^{2}}{12} + 150.00 \cdot {(2.50 - 62.73)}^{2}

Second moment of area about x-axis

I_{x x}

(1.601e+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}

=

524736 + 544521 + 170069 + 89271.0 + 272874

Second moment of area about the x1-axis

I_{xx, 1}

(6.521e+6)

{mm}^{4}

I_{xx, 1}

=

I_{x x} + A_{t} \cdot {C_{y}}^{2}

=

(1.601e+6) + 1250.00 \cdot {62.73}^{2}

Second moment of area around YY axis

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

I_{yy,top,flange}

104167

{mm}^{4}

I_{yy,top,flange}

=

\frac{1}{12} \cdot s_{f, top} \cdot {b_{f,top}}^{3}

=

\frac{1}{12} \cdot 10.00 \cdot {50.00}^{3}

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

I_{yy,bot,flange}

11250.0

{mm}^{4}

I_{yy,bot,flange}

=

\frac{1}{12} \cdot s_{f, bot} \cdot {b_{f,bot}}^{3}

=

\frac{1}{12} \cdot 5.00 \cdot {30.00}^{3}

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

I_{yy,web}

781.25

{mm}^{4}

I_{yy,web}

=

\frac{1}{12} \cdot L_{t} \cdot {t_{w}}^{3}

=

\frac{1}{12} \cdot 75.00 \cdot {5.00}^{3}

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

I_{yy,top,trapz}

7207.03

{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{a_{h,top}}{3})}^{2}

=

\frac{1}{36} \cdot (15.00 - 10.00) \cdot {22.50}^{3} + \frac{(22.50 \cdot (15.00 - 10.00))}{2} \cdot {(\frac{5.00}{2} + \frac{22.50}{3})}^{2}

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

I_{yy,bot,trapz}

1660.16

{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{a_{h,bot}}{3})}^{2}

=

\frac{1}{36} \cdot (10.00 - 5.00) \cdot {12.50}^{3} + \frac{(12.50 \cdot (10.00 - 5.00))}{2} \cdot {(\frac{5.00}{2} + \frac{12.50}{3})}^{2}

Second moment of area about y-axis

I_{y y}

125065

{mm}^{4}

I_{y y}

=

I_{yy,top,flange} + I_{yy,bot,flange} + I_{yy,web} + I_{yy,top,trapz} + I_{yy,bot,trapz}

=

104167 + 11250.0 + 781.25 + 7207.03 + 1660.16

Second moment of area about the y1-axis

I_{yy, 1}

906315

{mm}^{4}

I_{yy, 1}

=

I_{y y} + A_{t} \cdot {C_{x}}^{2}

=

125065 + 1250.00 \cdot {25.00}^{2}

Other properties

Polar second moment of area about the z-axis

J_{z}

(1.727e+6)

{mm}^{4}

J_{z}

=

I_{x x} + I_{y y}

=

(1.601e+6) + 125065

Polar second moment of area about the z1-axis

J_{z_{1}}

(7.427e+6)

{mm}^{4}

J_{z_{1}}

=

I_{xx, 1} + I_{yy, 1}

=

(6.521e+6) + 906315

Radius of gyration about x-axis

K_{x}

35.79

mm

K_{x}

=

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

=

\sqrt{\frac{(1.601e+6)}{1250.00}}

Radius of gyration about y-axis

K_{y}

10.00

mm

K_{y}

=

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

=

\sqrt{\frac{125065}{1250.00}}

Radius of gyration about the z-axis

K_{z}

37.16

mm

K_{z}

=

\sqrt{{K_{x}}^{2} + {K_{y}}^{2}}

=

\sqrt{{35.79}^{2} + {10.00}^{2}}

Radius of gyration about the x1-axis

K_{x_{1}}

72.23

mm

K_{x_{1}}

=

\sqrt{\frac{I_{xx, 1}}{A_{t}}}

=

\sqrt{\frac{(6.521e+6)}{1250.00}}

Radius of gyration about the y1-axis

K_{y_{1}}

26.93

mm

K_{y_{1}}

=

\sqrt{\frac{I_{yy, 1}}{A_{t}}}

=

\sqrt{\frac{906315}{1250.00}}

Radius of gyration about the z1-axis

K_{z_{1}}

77.08

mm

K_{z_{1}}

=

\sqrt{{K_{x_{1}}}^{2} + {K_{y_{1}}}^{2}}

=

\sqrt{{72.23}^{2} + {26.93}^{2}}

Elastic section modulus about x-axis

S_{x}

25528.2

{mm}^{3}

S_{x}

=

\frac{I_{x x}}{C_{y}}

=

\frac{(1.601e+6)}{62.73}

Elastic section modulus about y-axis

S_{y}

5002.60

{mm}^{3}

S_{y}

=

\frac{I_{y y}}{C_{x}}

=

\frac{125065}{25.00}

Plastic modulus about X axis

Half of section area

A_{half}

625.00

{mm}^{2}

A_{half}

=

\frac{A_{t}}{2}

=

\frac{1250.00}{2}

Lower section boundary

Y_{0}

mm

Lower trapezoid-flange boundary

Y_{1}

5.00

mm

Y_{1}

=

s_{f, bot}

=

5.00

Lower trapezoid-web boundary

Y_{2}

10.00

mm

Y_{2}

=

n_{t, bot}

=

10.00

Upper trapezoid-web boundary

Y_{3}

85.00

mm

Y_{3}

=

d_{t} - n_{t, top}

=

100.00 - 15.00

Upper trapezoid-flange boundary

Y_{4}

90.00

mm

Y_{4}

=

d_{t} - s_{f, top}

=

100.00 - 10.00

Upper section boundary

Y_{5}

100.00

mm

Y_{5}

=

d_{t}

=

100.00

Area boundary bottom flange

A_{L1}

150.00

{mm}^{2}

A_{L1}

=

A_{bot,flange}

=

150.00

Area boundary bottom trapezoid

A_{L2}

237.50

{mm}^{2}

A_{L2}

=

A_{L1} + A_{bot,trapz}

=

150.00 + 87.50

Area boundary web

A_{L3}

612.50

{mm}^{2}

A_{L3}

=

A_{L2} + A_{web}

=

237.50 + 375.00

Area boundary top trapezoid

A_{L4}

750.00

{mm}^{2}

A_{L4}

=

A_{L3} + A_{top,trapz}

=

612.50 + 137.50

Area boundary top flange

A_{L5}

1250.00

{mm}^{2}

A_{L5}

=

A_{L4} + A_{top,flange}

=

750.00 + 500.00

Bottom trapezoid gradient

k_{bot}

-5.00

rad

k_{bot}

=

\frac{(t_{w} - b_{f,bot})}{a_{v,bot}}

=

\frac{(5.00 - 30.00)}{5.00}

Top trapezoid gradient

k_{top}

9.00

k_{top}

=

\frac{(b_{f,top} - t_{w})}{a_{v,top}}

=

\frac{(50.00 - 5.00)}{5.00}

Bottom quadratic discriminant

Δ_{bot}

-3850.00

{mm}^{2}

Δ_{bot}

=

{b_{f,bot}}^{2} + 2 \cdot k_{bot} \cdot (A_{half} - A_{L1})

=

{30.00}^{2} + 2 \cdot -5.00 \cdot (625.00 - 150.00)

Top quadratic discriminant

Δ_{top}

250.00

{mm}^{2}

Δ_{top}

=

{t_{w}}^{2} + 2 \cdot k_{top} \cdot (A_{half} - A_{L3})

=

{5.00}^{2} + 2 \cdot 9.00 \cdot (625.00 - 612.50)

Candidate for bottom flange

y_{try,bot,flange}

20.83

mm

y_{try,bot,flange}

=

\frac{A_{half}}{b_{f,bot}}

=

\frac{625.00}{30.00}

Candidate for bottom trapezoid

y_{try,bot,trapz}

-1.00

mm

y_{try,bot,trapz}

=

{\begin{cases} Y_{1} + \frac{(- b_{f,bot} + \sqrt{Δ_{bot}})}{k_{bot}} & if Δ_{bot} \geq 0 \\ - 1 & otherwise. \end{cases}

=

{\begin{cases} 5.00 + \frac{(- 30.00 + \sqrt{-3850.00})}{-5.00} & if -3850.00 \geq 0 \\ - 1 & otherwise. \end{cases}

Candidate for web

y_{try,web}

87.50

mm

y_{try,web}

=

Y_{2} + \frac{(A_{half} - A_{L2})}{t_{w}}

=

10.00 + \frac{(625.00 - 237.50)}{5.00}

Candidate for top trapezoid

Y_{try,top,trapz}

86.20

mm

Y_{try,top,trapz}

=

{\begin{cases} Y_{3} + \frac{(- t_{w} + \sqrt{Δ_{top}})}{k_{top}} & if Δ_{top} \geq 0 \\ - 1 & otherwise. \end{cases}

=

{\begin{cases} 85.00 + \frac{(- 5.00 + \sqrt{250.00})}{9.00} & if 250.00 \geq 0 \\ - 1 & otherwise. \end{cases}

Candidate for top flange

y_{try,top,flange}

87.50

mm

y_{try,top,flange}

=

Y_{4} + \frac{(A_{half} - A_{L4})}{b_{f,top}}

=

90.00 + \frac{(625.00 - 750.00)}{50.00}

Plastic neutral axis location

y_{p}

86.20

mm

y_{p}

=

{\begin{cases} y_{try,bot,flange} & if A_{half} \leq A_{L1} \\ {\begin{cases} y_{try,bot,trapz} & if A_{half} \leq A_{L2} \\ {\begin{cases} y_{try,web} & if A_{half} \leq A_{L3} \\ {\begin{cases} Y_{try,top,trapz} & if A_{half} \leq A_{L4} \\ y_{try,top,flange} & otherwise. \end{cases} & otherwise. \end{cases} & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} 20.83 & if 625.00 \leq 150.00 \\ {\begin{cases} -1.00 & if 625.00 \leq 237.50 \\ {\begin{cases} 87.50 & if 625.00 \leq 612.50 \\ {\begin{cases} 86.20 & if 625.00 \leq 750.00 \\ 87.50 & otherwise. \end{cases} & otherwise. \end{cases} & otherwise. \end{cases} & otherwise. \end{cases}

Bottom flange term

Z_{bot,flange}

12555.2

{mm}^{3}

Z_{bot,flange}

=

{\begin{cases} \frac{b_{f,bot}}{2} \cdot ({y_{p}}^{2} + {(s_{f, bot} - y_{p})}^{2}) & if y_{p} < s_{f, bot} \\ A_{bot,flange} \cdot (y_{p} - \frac{s_{f, bot}}{2}) & otherwise. \end{cases}

=

{\begin{cases} \frac{30.00}{2} \cdot ({86.20}^{2} + {(5.00 - 86.20)}^{2}) & if 86.20 < 5.00 \\ 150.00 \cdot (86.20 - \frac{5.00}{2}) & otherwise. \end{cases}

The width at the PNA level when inside the bottom trapezoid

w_{pna,bot,trapz}

1348.73

mm

w_{pna,bot,trapz}

=

b_{f,bot} + (y_{p} - s_{f, bot}) \cdot (\frac{(y_{p} - s_{f, bot})}{a_{v,bot}})

=

30.00 + (86.20 - 5.00) \cdot (\frac{(86.20 - 5.00)}{5.00})

Area bottom wedge

A_{bw,bot,trapz}

55977.3

A_{bw,bot,trapz}

=

\frac{(b_{f,bot} + w_{pna,bot,trapz})}{2} \cdot (y_{p} - s_{f, bot})

=

\frac{(30.00 + 1348.73)}{2} \cdot (86.20 - 5.00)

Lever arm bottom wedge

y_{bw,bot,trapz}

27.66

mm

y_{bw,bot,trapz}

=

\frac{(y_{p} - s_{f, bot})}{3} \cdot (\frac{(w_{pna,bot,trapz} + 2 \cdot b_{f,bot})}{(w_{pna,bot,trapz} + b_{f,bot})})

=

\frac{(86.20 - 5.00)}{3} \cdot (\frac{(1348.73 + 2 \cdot 30.00)}{(1348.73 + 30.00)})

Bottom wedge term

Z_{bw,bot,trapz}

(1.548e+6)

{mm}^{3}

Z_{bw,bot,trapz}

=

A_{bw,bot,trapz} \cdot y_{bw,bot,trapz}

=

55977.3 \cdot 27.66

Area top wedge

A_{tw,bot,trapz}

-51577.9

{mm}^{2}

A_{tw,bot,trapz}

=

\frac{(w_{pna,bot,trapz} + t_{w})}{2} \cdot (n_{t, bot} - y_{p})

=

\frac{(1348.73 + 5.00)}{2} \cdot (10.00 - 86.20)

Lever arm top wedge

y_{tw,bot,trapz}

-25.49

mm

y_{tw,bot,trapz}

=

\frac{(n_{t, bot} - y_{p})}{3} \cdot (\frac{(w_{pna,bot,trapz} + 2 \cdot t_{w})}{(w_{pna,bot,trapz} + t_{w})})

=

\frac{(10.00 - 86.20)}{3} \cdot (\frac{(1348.73 + 2 \cdot 5.00)}{(1348.73 + 5.00)})

Top wedge term

Z_{tw,bot,trapz}

(1.315e+6)

{mm}^{3}

Z_{tw,bot,trapz}

=

A_{tw,bot,trapz} \cdot y_{tw,bot,trapz}

=

-51577.9 \cdot -25.49

Bottom trapezoid term

Z_{bot,trapz}

6938.44

{mm}^{3}

Z_{bot,trapz}

=

{\begin{cases} A_{bot,trapz} \cdot (y_{p} - y_{bot,trapz}) & if y_{p} > n_{t, bot} \\ {\begin{cases} A_{bot,trapz} \cdot (y_{bot,trapz} - y_{p}) & if y_{p} < s_{f, bot} \\ Z_{bw,bot,trapz} + Z_{tw,bot,trapz} & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} 87.50 \cdot (86.20 - 6.90) & if 86.20 > 10.00 \\ {\begin{cases} 87.50 \cdot (6.90 - 86.20) & if 86.20 < 5.00 \\ (1.548e+6) + (1.315e+6) & otherwise. \end{cases} & otherwise. \end{cases}

Web term

Z_{web}

14513.0

{mm}^{3}

Z_{web}

=

{\begin{cases} A_{web} \cdot (y_{p} - y_{web}) & if y_{p} > d_{t} - n_{t, top} \\ {\begin{cases} A_{web} \cdot (y_{web} - y_{p}) & if y_{p} < n_{t, bot} \\ \frac{t_{w}}{2} \cdot ({(y_{p} - n_{t, bot})}^{2} + {((d_{t} - n_{t, top}) - y_{p})}^{2}) & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} 375.00 \cdot (86.20 - 47.50) & if 86.20 > 100.00 - 15.00 \\ {\begin{cases} 375.00 \cdot (47.50 - 86.20) & if 86.20 < 10.00 \\ \frac{5.00}{2} \cdot ({(86.20 - 10.00)}^{2} + {((100.00 - 15.00) - 86.20)}^{2}) & otherwise. \end{cases} & otherwise. \end{cases}

The width at the PNA level when inside the top trapezoid

w_{pna,top,trapz}

15.81

mm

w_{pna,top,trapz}

=

t_{w} + (b_{f,top} - t_{w}) \cdot (\frac{(y_{p} - (d_{t} - n_{t, top}))}{a_{v,top}})

=

5.00 + (50.00 - 5.00) \cdot (\frac{(86.20 - (100.00 - 15.00))}{5.00})

Area bottom wedge

A_{bw,top,trapz}

12.50

{mm}^{2}

A_{bw,top,trapz}

=

\frac{(t_{w} + w_{pna,top,trapz})}{2} \cdot (y_{p} - (d_{t} - n_{t, top}))

=

\frac{(5.00 + 15.81)}{2} \cdot (86.20 - (100.00 - 15.00))

Lever arm bottom wedge

y_{bw,top,trapz}

0.49662

mm

y_{bw,top,trapz}

=

\frac{(y_{p} - (d_{t} - n_{t, top}))}{3} \cdot (\frac{(w_{pna,top,trapz} + 2 \cdot t_{w})}{(w_{pna,top,trapz} + t_{w})})

=

\frac{(86.20 - (100.00 - 15.00))}{3} \cdot (\frac{(15.81 + 2 \cdot 5.00)}{(15.81 + 5.00)})

Bottom wedge term

Z_{bw,top,trapz}

6.21

{mm}^{3}

Z_{bw,top,trapz}

=

A_{bw,top,trapz} \cdot y_{bw,top,trapz}

=

12.50 \cdot 0.49662

Area top wedge

A_{tw,top,trapz}

125.00

{mm}^{2}

A_{tw,top,trapz}

=

\frac{(w_{pna,top,trapz} + b_{f,top})}{2} \cdot ((d_{t} - s_{f, top}) - y_{p})

=

\frac{(15.81 + 50.00)}{2} \cdot ((100.00 - 10.00) - 86.20)

Lever arm top wedge

y_{tw,top,trapz}

2.23

mm

y_{tw,top,trapz}

=

\frac{((d_{t} - s_{f, top}) - y_{p})}{3} \cdot (\frac{(w_{pna,top,trapz} + 2 \cdot b_{f,top})}{(w_{pna,top,trapz} + b_{f,top})})

=

\frac{((100.00 - 10.00) - 86.20)}{3} \cdot (\frac{(15.81 + 2 \cdot 50.00)}{(15.81 + 50.00)})

Top wedge term

Z_{tw,top,trapz}

278.53

{mm}^{3}

Z_{tw,top,trapz}

=

A_{tw,top,trapz} \cdot y_{tw,top,trapz}

=

125.00 \cdot 2.23

Top trapezoid term

Z_{top,trapz}

284.74

{mm}^{3}

Z_{top,trapz}

=

{\begin{cases} A_{top,trapz} \cdot (y_{p} - y_{top,trapz}) & if y_{p} > d_{t} - s_{f, top} \\ {\begin{cases} A_{top,trapz} \cdot (y_{top,trapz} - y_{p}) & if y_{p} < d_{t} - n_{t, top} \\ Z_{bw,top,trapz} + Z_{tw,top,trapz} & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} 137.50 \cdot (86.20 - 88.18) & if 86.20 > 100.00 - 10.00 \\ {\begin{cases} 137.50 \cdot (88.18 - 86.20) & if 86.20 < 100.00 - 15.00 \\ 6.21 + 278.53 & otherwise. \end{cases} & otherwise. \end{cases}

Top flange term

Z_{top,flange}

4399.37

{mm}^{3}

Z_{top,flange}

=

{\begin{cases} A_{top,flange} \cdot (y_{top,flange} - y_{p}) & if y_{p} < d_{t} - s_{f, top} \\ \frac{b_{f,top}}{2} \cdot ({(d_{t} - y_{p})}^{2} + {(s_{f, top} - (d_{t} - y_{p}))}^{2}) & otherwise. \end{cases}

=

{\begin{cases} 500.00 \cdot (95.00 - 86.20) & if 86.20 < 100.00 - 10.00 \\ \frac{50.00}{2} \cdot ({(100.00 - 86.20)}^{2} + {(10.00 - (100.00 - 86.20))}^{2}) & otherwise. \end{cases}

Plastic section modulus about x-axis

Z_{x}

38690.7

{mm}^{3}

Z_{x}

=

Z_{bot,flange} + Z_{bot,trapz} + Z_{web} + Z_{top,trapz} + Z_{top,flange}

=

12555.2 + 6938.44 + 14513.0 + 284.74 + 4399.37

Plastic modulus about Y axis

Flanges contribution term

Z_{flanges}

7375.00

{mm}^{3}

Z_{flanges}

=

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

=

\frac{(10.00 \cdot {50.00}^{2})}{4} + \frac{(5.00 \cdot {30.00}^{2})}{4}

Central web term

Z_{web}

531.25

{mm}^{3}

Z_{web}

=

\frac{(h_{central} \cdot {t_{w}}^{2})}{4}

=

\frac{(85.00 \cdot {5.00}^{2})}{4}

Top wings term

Z_{wings,top}

1125.00

{mm}^{3}

Z_{wings,top}

=

2 \cdot (0.5 \cdot a_{v,top} \cdot a_{h,top}) \cdot (\frac{t_{w}}{2} + \frac{a_{h,top}}{3})

=

2 \cdot (0.5 \cdot 5.00 \cdot 22.50) \cdot (\frac{5.00}{2} + \frac{22.50}{3})

Bottom wings term

Z_{wings,bot}

416.67

{mm}^{3}

Z_{wings,bot}

=

2 \cdot (0.5 \cdot a_{h,bot} \cdot a_{v,bot}) \cdot (\frac{t_{w}}{2} + \frac{a_{h,bot}}{3})

=

2 \cdot (0.5 \cdot 12.50 \cdot 5.00) \cdot (\frac{5.00}{2} + \frac{12.50}{3})

Plastic section mosulus about y-axis

Z_{y}

9447.92

{mm}^{3}

Z_{y}

=

Z_{flanges} + Z_{web} + Z_{wings,top} + Z_{wings,bot}

=

7375.00 + 531.25 + 1125.00 + 416.67