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Ellipse

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The solid elliptical structural shape has a filled cross-section bounded by an ellipse. It exhibits anisotropic bending properties, with different strengths about its major and minor axes, which can be leveraged for specific loading conditions. While less common than rectangular or circular solid sections, its smooth, continuous curve can be aesthetically pleasing, leading to its use in architectural elements, custom furniture, and some specialised mechanical components where directional strength needs align with its geometry.

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Calculations

Description

Symbol Name

Value

Unit

Comment

Ellipse

figure wim9dcd1cee4c

Minor axis diameter

b_{t}

5.00

mm

Major axis diameter

a_{t}

10.00

mm

Area

A_{t}

39.27

{mm}^{2}

A_{t}

=

\frac{π}{4} \cdot a_{t} \cdot b_{t}

=

\frac{π}{4} \cdot 10.00 \cdot 5.00

Perimeter

P_{t}

24.22

mm

P_{t}

=

π \cdot (3 \cdot (\frac{a_{t}}{2} + \frac{b_{t}}{2}) - \sqrt{(\frac{3 \cdot a_{t}}{2} + \frac{b_{t}}{2}) \cdot (\frac{a_{t}}{2} + \frac{3 \cdot b_{t}}{2})})

=

π \cdot (3 \cdot (\frac{10.00}{2} + \frac{5.00}{2}) - \sqrt{(\frac{3 \cdot 10.00}{2} + \frac{5.00}{2}) \cdot (\frac{10.00}{2} + \frac{3 \cdot 5.00}{2})})

Distance to centroid (x-axis)

C_{x}

5.00

mm

C_{x}

=

\frac{a_{t}}{2}

=

\frac{10.00}{2}

Distance to centroid (y-axis)

C_{y}

2.50

mm

C_{y}

=

\frac{b_{t}}{2}

=

\frac{5.00}{2}

Second moment of area about x-axis

I_{x}

61.36

{mm}^{4}

I_{x}

=

\frac{π \cdot a_{t} \cdot {b_{t}}^{3}}{64}

=

\frac{π \cdot 10.00 \cdot {5.00}^{3}}{64}

Second moment of area about y-axis

I_{y}

245.44

{mm}^{4}

I_{y}

=

\frac{π \cdot {a_{t}}^{3} \cdot b_{t}}{64}

=

\frac{π \cdot {10.00}^{3} \cdot 5.00}{64}

Polar moment of inertia about centre

J_{z}

306.80

{mm}^{4}

J_{z}

=

I_{x} + I_{y}

=

61.36 + 245.44

Second moment of area about x1-axis

I_{x_{1}}

306.80

{mm}^{4}

I_{x_{1}}

=

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

=

61.36 + 39.27 \cdot {2.50}^{2}

Second moment of area about y1-axis

I_{y_{1}}

1227.18

{mm}^{4}

I_{y_{1}}

=

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

=

245.44 + 39.27 \cdot {5.00}^{2}

Polar moment of inertia about vertex

J_{z_{1}}

1533.98

{mm}^{4}

J_{z_{1}}

=

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

=

306.80 + 1227.18

Radius of gyration about x-axis

K_{x}

1.25

mm

K_{x}

=

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

=

\sqrt{\frac{61.36}{39.27}}

Radius of gyration about y-axis

K_{y}

2.50

mm

K_{y}

=

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

=

\sqrt{\frac{245.44}{39.27}}

Radius of gyration about z-axis

K_{z}

2.80

mm

K_{z}

=

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

=

\sqrt{{1.25}^{2} + {2.50}^{2}}

Radius of gyration about x1-axis

K_{x_{1}}

2.80

mm

K_{x_{1}}

=

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

=

\sqrt{\frac{306.80}{39.27}}

Radius of gyration about y1-axis

K_{y_{1}}

5.59

mm

K_{y_{1}}

=

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

=

\sqrt{\frac{1227.18}{39.27}}

Radius of gyration about z1-axis

K_{z_{1}}

6.25

mm

K_{z_{1}}

=

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

=

\sqrt{{2.80}^{2} + {5.59}^{2}}

Elastic section modulus about x-axis

S_{x}

24.54

{mm}^{3}

S_{x}

=

\frac{I_{x}}{(b_{t} \cdot 0.5)}

=

\frac{61.36}{(5.00 \cdot 0.5)}

Elastic section modulus about y-axis

S_{y}

49.09

{mm}^{3}

S_{y}

=

\frac{I_{y}}{(a_{t} \cdot 0.5)}

=

\frac{245.44}{(10.00 \cdot 0.5)}

Plastic section modulus around x-axis

Z_{x}

41.67

{mm}^{3}

Z_{x}

=

\frac{a_{t} \cdot {b_{t}}^{2}}{6}

=

\frac{10.00 \cdot {5.00}^{2}}{6}

Plastic section modulus around y-axis

Z_{y}

83.33

{mm}^{3}

Z_{y}

=

\frac{b_{t} \cdot {a_{t}}^{2}}{6}

=

\frac{5.00 \cdot {10.00}^{2}}{6}