Tapered Channel

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The tapered channel section is a C-shaped profile where the web depth or flange dimensions vary along its length. This allows for material optimisation, concentrating material in areas of higher stress, similar to tapered I-beams but with the open geometry of a channel. It is used in applications like vehicle frames, support brackets, and structural members where loading varies and weight reduction is a design goal.

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Calculations

Description

Symbol Name

Value

Unit

Comment

Tapered Channel

Basic geometry input parameters

Section depth

d_{t}

100.00

mm

Section breadth

b_{t}

50.00

mm

Outer flange thickness

s_{t}

8.00

mm

Inner flange thickness

n_{t}

10.00

mm

Web thickness

t_{w}

10.00

mm

Calculated geometry parameters

Flange to flange depth

h_{t}

84.00

mm

h_{t}

=

d_{t} - 2 \cdot s_{t}

=

100.00 - 2 \cdot 8.00

Web depth

L_{t}

80.00

mm

L_{t}

=

d_{t} - 2 \cdot n_{t}

=

100.00 - 2 \cdot 10.00

Tapered horizontal flange segment

a_{h,t}

40.00

mm

a_{h,t}

=

b_{t} - t_{w}

=

50.00 - 10.00

Tapered vertical flange segment

a_{v,t}

2.00

mm

a_{v,t}

=

n_{t} - s_{t}

=

10.00 - 8.00

Area of the trapezoidal segment

A_{trapz}

360.00

{mm}^{2}

A_{trapz}

=

\frac{(s_{t} + n_{t})}{2} \cdot a_{h,t}

=

\frac{(8.00 + 10.00)}{2} \cdot 40.00

Area of the rectangular web

A_{w}

1000.00

{mm}^{2}

A_{w}

=

d_{t} \cdot t_{w}

=

100.00 \cdot 10.00

Area

A_{t}

1720.00

{mm}^{2}

A_{t}

=

2 \cdot (A_{trapz}) + A_{w}

=

2 \cdot (360.00) + 1000.00

Perimeter

P_{t}

372.10

mm

P_{t}

=

2 \cdot (b_{t} + L_{t} + 2 \cdot s_{t} + \sqrt{({a_{h,t}}^{2} + {(n_{t} - s_{t})}^{2})})

=

2 \cdot (50.00 + 80.00 + 2 \cdot 8.00 + \sqrt{({40.00}^{2} + {(10.00 - 8.00)}^{2})})

Slope of tapered segment

g_{t}

0.10000

-

g_{t}

=

\frac{(h_{t} - L_{t})}{(b_{t} - t_{w})}

=

\frac{(84.00 - 80.00)}{(50.00 - 10.00)}

y_{trapz,top}

95.48

mm

y_{trapz,top}

=

d_{t} - \frac{({s_{t}}^{2} + s_{t} \cdot n_{t} + {n_{t}}^{2})}{(3 \cdot (s_{t} + n_{t}))}

=

100.00 - \frac{({8.00}^{2} + 8.00 \cdot 10.00 + {10.00}^{2})}{(3 \cdot (8.00 + 10.00))}

y_{w}

50.00

mm

y_{w}

=

\frac{d_{t}}{2}

=

\frac{100.00}{2}

y_{trapz,bot}

5.48

mm

y_{trapz,bot}

=

n_{t} - \frac{({s_{t}}^{2} + s_{t} \cdot n_{t} + {n_{t}}^{2})}{(3 \cdot (s_{t} + n_{t}))}

=

10.00 - \frac{({8.00}^{2} + 8.00 \cdot 10.00 + {10.00}^{2})}{(3 \cdot (8.00 + 10.00))}

Distance to centroid (y-axis)

C_{y}

50.00

mm

C_{y}

=

\frac{d_{t}}{2}

=

\frac{100.00}{2}

x_{trapz}

24.07

mm

x_{trapz}

=

\frac{b_{t}}{3} \cdot (\frac{(2 \cdot s_{t} + n_{t})}{(s_{t} + n_{t})})

=

\frac{50.00}{3} \cdot (\frac{(2 \cdot 8.00 + 10.00)}{(8.00 + 10.00)})

x_{w}

5.00

mm

x_{w}

=

\frac{t_{w}}{2}

=

\frac{10.00}{2}

x_{f}

29.26

mm

x_{f}

=

t_{w} + \frac{(a_{h,t} \cdot (n_{t} + 2 \cdot s_{t}))}{(3 \cdot (s_{t} + n_{t}))}

=

10.00 + \frac{(40.00 \cdot (10.00 + 2 \cdot 8.00))}{(3 \cdot (8.00 + 10.00))}

Distance to centroid (x-axis)

C_{x}

15.16

mm

C_{x}

=

\frac{(A_{w} \cdot x_{w} + 2 \cdot A_{trapz} \cdot x_{f})}{A_{t}}

=

\frac{(1000.00 \cdot 5.00 + 2 \cdot 360.00 \cdot 29.26)}{1720.00}

Second moments of area

y_{0}

4.52

mm

y_{0}

=

\frac{({s_{t}}^{2} + s_{t} \cdot n_{t} + {n_{t}}^{2})}{(3 \cdot (s_{t} + n_{t}))}

=

\frac{({8.00}^{2} + 8.00 \cdot 10.00 + {10.00}^{2})}{(3 \cdot (8.00 + 10.00))}

δ_{y}

45.48

mm

δ_{y}

=

\frac{d_{t}}{2} - y_{0}

=

\frac{100.00}{2} - 4.52

Web term

I_{x,w}

833333

{mm}^{4}

I_{x,w}

=

\frac{t_{w} \cdot {d_{t}}^{3}}{12}

=

\frac{10.00 \cdot {100.00}^{3}}{12}

Flange term

I_{x,f}

1706.67

{mm}^{4}

I_{x,f}

=

\frac{a_{h,t} \cdot {s_{t}}^{3}}{12}

=

\frac{40.00 \cdot {8.00}^{3}}{12}

Second moment of area about x-axis

I_{x}

(2.326e+6)

{mm}^{4}

I_{x}

=

I_{x,w} + 2 \cdot (I_{x,f} + A_{trapz} \cdot {δ_{y}}^{2})

=

833333 + 2 \cdot (1706.67 + 360.00 \cdot {45.48}^{2})

Web term

I_{y,w}

8333.33

{mm}^{4}

I_{y,w}

=

\frac{d_{t} \cdot {t_{w}}^{3}}{12}

=

\frac{100.00 \cdot {10.00}^{3}}{12}

Flange term

I_{y,f}

42666.7

I_{y,f}

=

\frac{s_{t} \cdot {a_{h,t}}^{3}}{12}

=

\frac{8.00 \cdot {40.00}^{3}}{12}

Second moment of area about y-axis

I_{y}

340020

{mm}^{4}

I_{y}

=

I_{y,w} + A_{w} \cdot {(x_{w} - C_{x})}^{2} + 2 \cdot (I_{y,f} + A_{trapz} \cdot {(x_{f} - C_{x})}^{2})

=

8333.33 + 1000.00 \cdot {(5.00 - 15.16)}^{2} + 2 \cdot (42666.7 + 360.00 \cdot {(29.26 - 15.16)}^{2})

Second moment of area about x1-axis

I_{x_{1}}

(6.626e+6)

{mm}^{4}

I_{x_{1}}

=

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

=

(2.326e+6) + 1720.00 \cdot {50.00}^{2}

Second moment of area about y1-axis

I_{y_{1}}

735062

{mm}^{4}

I_{y_{1}}

=

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

=

340020 + 1720.00 \cdot {15.16}^{2}

Polar moment of inertia about centre

J_{z}

(2.666e+6)

{mm}^{4}

J_{z}

=

I_{x} + I_{y}

=

(2.326e+6) + 340020

Polar moment of inertia about vertex

J_{z_{1}}

(7.361e+6)

{mm}^{4}

J_{z_{1}}

=

I_{x_{1}} + I_{y_{1}}

=

(6.626e+6) + 735062

Radius of gyration about x-axis

K_{x}

36.77

mm

K_{x}

=

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

=

\sqrt{\frac{(2.326e+6)}{1720.00}}

Radius of gyration about y-axis

K_{y}

14.06

mm

K_{y}

=

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

=

\sqrt{\frac{340020}{1720.00}}

Radius of gyration about z-axis

K_{z}

39.37

mm

K_{z}

=

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

=

\sqrt{{36.77}^{2} + {14.06}^{2}}

Radius of gyration about x1-axis

K_{x_{1}}

62.07

mm

K_{x_{1}}

=

\sqrt{\frac{I_{x_{1}}}{A_{t}}}

=

\sqrt{\frac{(6.626e+6)}{1720.00}}

Radius of gyration about y1-axis

K_{y_{1}}

20.67

mm

K_{y_{1}}

=

\sqrt{\frac{I_{y_{1}}}{A_{t}}}

=

\sqrt{\frac{735062}{1720.00}}

Radius of gyration about z1-axis

K_{z_{1}}

65.42

mm

K_{z_{1}}

=

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

=

\sqrt{{62.07}^{2} + {20.67}^{2}}

Elastic section modulus about x-axis

S_{x}

46522.3

{mm}^{3}

S_{x}

=

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

=

\frac{(2.326e+6)}{50.00}

Elastic section modulus about y-axis

S_{y}

9758.09

{mm}^{3}

Minimum value located at the tips of the flanges

S_{y}

=

\frac{I_{y}}{(b_{t} - C_{x})}

=

\frac{340020}{(50.00 - 15.16)}

Plastic section modulus around X-axis

Taper

k_{t}

-0.05000

k_{t}

=

\frac{(s_{t} - n_{t})}{a_{h,t}}

=

\frac{(8.00 - 10.00)}{40.00}

u_{quad,term}

-206.88

mm

u_{quad,term}

=

\frac{(\sqrt{{n_{t}}^{2} - (k_{t}) \cdot (t_{w} \cdot d_{t} - \frac{A_{t}}{2})})}{k_{t}}

=

\frac{(\sqrt{{10.00}^{2} - (-0.05000) \cdot (10.00 \cdot 100.00 - \frac{1720.00}{2})})}{-0.05000}

Plastic neutral axis about y-axis

x_{PNA}

8.60

mm

x_{PNA}

=

{\begin{cases} \frac{A_{t}}{(2 \cdot d_{t})} & if \frac{A_{t}}{2} \leq A_{w} \\ {\begin{cases} \frac{(\frac{A_{t}}{2} - t_{w} \cdot d_{t})}{(2 \cdot n_{t})} & if k_{t} = 0 \\ t_{w} + u_{quad,term} & otherwise. \end{cases} & otherwise. \end{cases}

=

{\begin{cases} \frac{1720.00}{(2 \cdot 100.00)} & if \frac{1720.00}{2} \leq 1000.00 \\ {\begin{cases} \frac{(\frac{1720.00}{2} - 10.00 \cdot 100.00)}{(2 \cdot 10.00)} & if -0.05000 = 0 \\ 10.00 + -206.88 & otherwise. \end{cases} & otherwise. \end{cases}

Z_{w,pna,web}

3796.00

{mm}^{3}

Z_{w,pna,web}

=

d_{t} \cdot (\frac{{x_{PNA}}^{2}}{2} + \frac{{(t_{w} - x_{PNA})}^{2}}{2})

=

100.00 \cdot (\frac{{8.60}^{2}}{2} + \frac{{(10.00 - 8.60)}^{2}}{2})

Z_{f,pna,web}

14874.7

{mm}^{3}

Z_{f,pna,web}

=

(t_{w} - x_{PNA}) \cdot a_{h,t} \cdot (s_{t} + n_{t}) + \frac{{a_{h,t}}^{2}}{3} \cdot (n_{t} + 2 \cdot s_{t})

=

(10.00 - 8.60) \cdot 40.00 \cdot (8.00 + 10.00) + \frac{{40.00}^{2}}{3} \cdot (10.00 + 2 \cdot 8.00)

Z_{w,pna,flange}

3600.00

{mm}^{3}

Z_{w,pna,flange}

=

d_{t} \cdot (x_{PNA} \cdot t_{w} - \frac{{t_{w}}^{2}}{2})

=

100.00 \cdot (8.60 \cdot 10.00 - \frac{{10.00}^{2}}{2})

e_{t}

-1.40

mm

e_{t}

=

x_{PNA} - t_{w}

=

8.60 - 10.00

Z_{f,pna,flange}

14914.0

{mm}^{3}

Z_{f,pna,flange}

=

2 \cdot (n_{t} \cdot (\frac{{e_{t}}^{2}}{2} + \frac{{(a_{h,t} - e_{t})}^{2}}{2}) + k_{t} \cdot (\frac{{e_{t}}^{3}}{6} + \frac{({a_{h,t}}^{3} - {e_{t}}^{3})}{3} - \frac{e_{t} \cdot ({a_{h,t}}^{2} - {e_{t}}^{2})}{2}))

=

2 \cdot (10.00 \cdot (\frac{{-1.40}^{2}}{2} + \frac{{(40.00 - -1.40)}^{2}}{2}) + -0.05000 \cdot (\frac{{-1.40}^{3}}{6} + \frac{({40.00}^{3} - {-1.40}^{3})}{3} - \frac{-1.40 \cdot ({40.00}^{2} - {-1.40}^{2})}{2}))

Final plastic section modulus around X-axis

Plastic section modulus about the y-axis

Z_{x}

18670.7

{mm}^{3}

Z_{x}

=

{\begin{cases} Z_{w,pna,web} + Z_{f,pna,web} & if x_{PNA} \leq t_{w} \\ Z_{w,pna,flange} + Z_{f,pna,flange} & otherwise. \end{cases}

=

{\begin{cases} 3796.00 + 14874.7 & if 8.60 \leq 10.00 \\ 3600.00 + 14914.0 & otherwise. \end{cases}

Calculations for the plastic modulus Y-axis

Plastic neutral axis about x-axis

y_{PNA}

50.00

mm

y_{PNA}

=

\frac{d_{t}}{2}

=

\frac{100.00}{2}

Plastic sextion modulus about x-axis

Z_{y}

57746.7

{mm}^{3}

Z_{y}

=

\frac{t_{w} \cdot {d_{t}}^{2}}{4} + \frac{(d_{t} \cdot a_{h,t} \cdot (s_{t} + n_{t}))}{2} - \frac{a_{h,t}}{3} \cdot ({s_{t}}^{2} + s_{t} \cdot n_{t} + {n_{t}}^{2})

=

\frac{10.00 \cdot {100.00}^{2}}{4} + \frac{(100.00 \cdot 40.00 \cdot (8.00 + 10.00))}{2} - \frac{40.00}{3} \cdot ({8.00}^{2} + 8.00 \cdot 10.00 + {10.00}^{2})