Tapered I Beam

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The tapered I-beam is an I-section where the depth of the web, and sometimes the width of the flanges, varies along its length. This design optimises material usage by providing greater depth and strength in regions of high bending moment or shear, and less material where stresses are lower. Common applications include crane girders, portal frames, and cantilever beams, allowing for efficient and economical structural solutions.

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Calculations

Description

Symbol Name

Value

Unit

Comment

Tapered I-beam

Section depth

d_{t}

100.00

mm

Section breadth

b_{t}

50.00

mm

Outer flange thickness

s_{t}

10.00

mm

Inner flange thickness

n_{t}

20.00

mm

Web thickness

t_{w}

20.00

mm

Web depth

L_{t}

60.00

mm

L_{t}

=

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

=

100.00 - 2 \cdot 20.00

Tapered flange segment

a_{trapz}

15.00

mm

a_{trapz}

=

\frac{(b_{t} - t_{w})}{2}

=

\frac{(50.00 - 20.00)}{2}

h_{trapz}

10.00

mm

h_{trapz}

=

n_{t} - s_{t}

=

20.00 - 10.00

A_{flange,rect}

500.00

A_{flange,rect}

=

b_{t} \cdot s_{t}

=

50.00 \cdot 10.00

A_{trapz}

350.00

{mm}^{2}

A_{trapz}

=

\frac{(t_{w} + b_{t})}{2} \cdot h_{trapz}

=

\frac{(20.00 + 50.00)}{2} \cdot 10.00

A_{web}

1200.00

{mm}^{2}

A_{web}

=

L_{t} \cdot t_{w}

=

60.00 \cdot 20.00

Total area

A_{t}

2900.00

{mm}^{2}

A_{t}

=

2 \cdot A_{flange,rect} + 2 \cdot A_{trapz} + A_{web}

=

2 \cdot 500.00 + 2 \cdot 350.00 + 1200.00

Perimeter

P_{t}

332.11

mm

P_{t}

=

2 \cdot b_{t} + 2 \cdot L_{t} + 4 \cdot s_{t} + 4 \cdot \sqrt{{a_{trapz}}^{2} + {h_{trapz}}^{2}}

=

2 \cdot 50.00 + 2 \cdot 60.00 + 4 \cdot 10.00 + 4 \cdot \sqrt{{15.00}^{2} + {10.00}^{2}}

Slope of tapered segment

g_{t}

0.66667

-

g_{t}

=

\frac{h_{trapz}}{a_{trapz}}

=

\frac{10.00}{15.00}

α_{degrees}

38.20

α_{degrees}

=

\deg (g_{t})

=

\deg (0.66667)

Distance to centroid (x-axis)

C_{x}

25.00

mm

C_{x}

=

\frac{b_{t}}{2}

=

\frac{50.00}{2}

Distance to centroid (y-axis)

C_{y}

50.00

mm

C_{y}

=

\frac{d_{t}}{2}

=

\frac{100.00}{2}

I_{local,trapz}

2738.10

{mm}^{2}

I_{local,trapz}

=

\frac{{h_{trapz}}^{3} \cdot ({t_{w}}^{2} + 4 \cdot t_{w} \cdot b_{t} + {b_{t}}^{2})}{(36 \cdot (t_{w} + b_{t}))}

=

\frac{{10.00}^{3} \cdot ({20.00}^{2} + 4 \cdot 20.00 \cdot 50.00 + {50.00}^{2})}{(36 \cdot (20.00 + 50.00))}

Distance from the bottom to the top trapezoid centroid

y_{trapz,top}

85.71

mm

y_{trapz,top}

=

d_{t} - s_{t} - \frac{h_{trapz} \cdot (2 \cdot t_{w} + b_{t})}{(3 \cdot (t_{w} + b_{t}))}

=

100.00 - 10.00 - \frac{10.00 \cdot (2 \cdot 20.00 + 50.00)}{(3 \cdot (20.00 + 50.00))}

Distance from the bottom to the bottom trapezoid centroid

y_{trapz,bot}

14.29

mm

y_{trapz,bot}

=

s_{t} + \frac{h_{trapz} \cdot (2 \cdot t_{w} + b_{t})}{(3 \cdot (t_{w} + b_{t}))}

=

10.00 + \frac{10.00 \cdot (2 \cdot 20.00 + 50.00)}{(3 \cdot (20.00 + 50.00))}

I_{xx,1}

(2.393e+6)

{mm}^{4}

I_{xx,1}

=

\frac{2 \cdot A_{flange,rect} \cdot {s_{t}}^{2}}{12} + 2 \cdot A_{flange,rect} \cdot {(0.5 \cdot (d_{t} - s_{t}))}^{2} + \frac{A_{web} \cdot {L_{t}}^{2}}{12}

=

\frac{2 \cdot 500.00 \cdot {10.00}^{2}}{12} + 2 \cdot 500.00 \cdot {(0.5 \cdot (100.00 - 10.00))}^{2} + \frac{1200.00 \cdot {60.00}^{2}}{12}

I_{xx,2}

898333

{mm}^{4}

I_{xx,2}

=

2 \cdot I_{local,trapz} + A_{trapz} \cdot {(y_{trapz,top} - C_{y})}^{2} + A_{trapz} \cdot {(y_{trapz,bot} - C_{y})}^{2}

=

2 \cdot 2738.10 + 350.00 \cdot {(85.71 - 50.00)}^{2} + 350.00 \cdot {(14.29 - 50.00)}^{2}

Total second moment of area about the x-axis

I_{xx}

(3.292e+6)

{mm}^{4}

I_{xx}

=

I_{xx,1} + I_{xx,2}

=

(2.393e+6) + 898333

Final second moment of area about the y-axis

I_{yy}

332917

{mm}^{4}

The local trapezoid formula is only suitable for Isosceles Trapezoids

I_{yy}

=

\frac{2 \cdot A_{flange,rect} \cdot {b_{t}}^{2}}{12} + \frac{2 \cdot h_{trapz} \cdot (t_{w} + b_{t}) \cdot ({t_{w}}^{2} + {b_{t}}^{2})}{48} + \frac{A_{web} \cdot {t_{w}}^{2}}{12}

=

\frac{2 \cdot 500.00 \cdot {50.00}^{2}}{12} + \frac{2 \cdot 10.00 \cdot (20.00 + 50.00) \cdot ({20.00}^{2} + {50.00}^{2})}{48} + \frac{1200.00 \cdot {20.00}^{2}}{12}

Polar moment of inertia about centre

J_{z}

(3.625e+6)

{mm}^{4}

J_{z}

=

I_{xx} + I_{yy}

=

(3.292e+6) + 332917

Second moment of area about the y1-axis

I_{y_{1}}

(2.145e+6)

{mm}^{4}

I_{y_{1}}

=

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

=

332917 + 2900.00 \cdot {25.00}^{2}

Second moment of area about the x1-axis

I_{x_{1}}

(1.054e+7)

{mm}^{4}

I_{x_{1}}

=

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

=

(3.292e+6) + 2900.00 \cdot {50.00}^{2}

Polar moment of inertia about vertex

J_{z_{1}}

(1.269e+7)

{mm}^{4}

J_{z_{1}}

=

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

=

(1.054e+7) + (2.145e+6)

Radius of gyration about x-axis

K_{x}

33.69

mm

K_{x}

=

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

=

\sqrt{\frac{(3.292e+6)}{2900.00}}

Radius of gyration about y-axis

K_{y}

10.71

mm

K_{y}

=

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

=

\sqrt{\frac{332917}{2900.00}}

Radius of gyration about z-axis

K_{z}

35.35

mm

K_{z}

=

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

=

\sqrt{{33.69}^{2} + {10.71}^{2}}

Radius of gyration about x1-axis

K_{x_{1}}

60.29

mm

K_{x_{1}}

=

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

=

\sqrt{\frac{(1.054e+7)}{2900.00}}

Radius of gyration about y1-axis

K_{y_{1}}

27.20

mm

K_{y_{1}}

=

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

=

\sqrt{\frac{(2.145e+6)}{2900.00}}

Radius of gyration about z1-axis

K_{z_{1}}

66.14

mm

K_{z_{1}}

=

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

=

\sqrt{{60.29}^{2} + {27.20}^{2}}

Elastic section modulus about x-axis

S_{x}

65833.3

{mm}^{3}

S_{x}

=

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

=

\frac{(3.292e+6)}{50.00}

Elastic section modulus about y-axis

S_{y}

13316.7

{mm}^{3}

S_{y}

=

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

=

\frac{332917}{25.00}

Distance between flange rectangles

h_{w,s}

80.00

mm

h_{w,s}

=

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

=

100.00 - 2 \cdot 10.00

Plastic section modulus about x-axis

Z_{x}

88000.0

{mm}^{3}

Z_{x}

=

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

=

\frac{20.00 \cdot {100.00}^{2}}{4} + 15.00 \cdot 100.00 \cdot (20.00 + 10.00) - \frac{2}{3} \cdot 15.00 \cdot ({20.00}^{2} + 20.00 \cdot 10.00 + {10.00}^{2})

Z_{y}

25000.0

Z_{y}

=

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

=

\frac{100.00 \cdot {20.00}^{2}}{4} + 20.00 \cdot 15.00 \cdot (20.00 + 10.00) + \frac{2}{3} \cdot {15.00}^{2} \cdot (20.00 + 2 \cdot 10.00)