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Hollow Ellipse

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The hollow elliptical structural shape features an elliptical outer profile with a hollow interior. This closed section provides good torsional resistance and directional bending strength, similar to solid ellipses, but with improved material efficiency and a better strength-to-weight ratio. Its unique aesthetic qualities make it a choice for architecturally expressive columns, beams, and decorative structural elements where visual appeal is as important as performance.

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Calculations

Description

Symbol Name

Value

Unit

Comment

Hollow Ellipse

figure wim242b1ef908

Outer minor axis

b_{out}

5.00

mm

Inner minor axis

b_{i n}

2.00

mm

Outer major axis

a_{out}

10.00

mm

Inner major axis

a_{i n}

4.00

mm

Area

A_{t}

32.99

{mm}^{2}

A_{t}

=

\frac{π}{4} \cdot (a_{out} \cdot b_{out} - a_{i n} \cdot b_{i n})

=

\frac{π}{4} \cdot (10.00 \cdot 5.00 - 4.00 \cdot 2.00)

Outer perimeter

P_{t}

24.22

mm

P_{t}

=

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

Inner perimeter

P_{t,inner}

9.69

mm

P_{t,inner}

=

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

=

π \cdot (3 \cdot (\frac{4.00}{2} + \frac{2.00}{2}) - \sqrt{(\frac{3 \cdot 4.00}{2} + \frac{2.00}{2}) \cdot (\frac{4.00}{2} + \frac{3 \cdot 2.00}{2})})

Distance to centroid (x-axis)

C_{x}

5.00

mm

C_{x}

=

\frac{a_{out}}{2}

=

\frac{10.00}{2}

Distance to centroid (y-axis)

C_{y}

2.50

mm

C_{y}

=

\frac{b_{out}}{2}

=

\frac{5.00}{2}

Second moment of area about x-axis

I_{x}

59.79

{mm}^{4}

I_{x}

=

\frac{π \cdot (a_{out} \cdot {b_{out}}^{3} - a_{i n} \cdot {b_{i n}}^{3})}{64}

=

\frac{π \cdot (10.00 \cdot {5.00}^{3} - 4.00 \cdot {2.00}^{3})}{64}

Second moment of area about y-axis

I_{y}

239.15

{mm}^{4}

I_{y}

=

\frac{π \cdot ({a_{out}}^{3} \cdot b_{out} - {a_{i n}}^{3} \cdot b_{i n})}{64}

=

\frac{π \cdot ({10.00}^{3} \cdot 5.00 - {4.00}^{3} \cdot 2.00)}{64}

Polar moment of inertia about centre

J_{z}

298.94

{mm}^{4}

J_{z}

=

I_{x} + I_{y}

=

59.79 + 239.15

Second moment of area about x1-axis

I_{x_{1}}

265.96

{mm}^{4}

I_{x_{1}}

=

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

=

59.79 + 32.99 \cdot {2.50}^{2}

Second moment of area about y-axis

I_{y_{1}}

1063.82

{mm}^{4}

I_{y_{1}}

=

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

=

239.15 + 32.99 \cdot {5.00}^{2}

Polar moment of inertia about vertex

J_{z_{1}}

1329.78

{mm}^{4}

J_{z_{1}}

=

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

=

265.96 + 1063.82

Radius of gyration about x-axis

K_{x}

1.35

mm

K_{x}

=

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

=

\sqrt{\frac{59.79}{32.99}}

Radius of gyration about y-axis

K_{y}

2.69

mm

K_{y}

=

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

=

\sqrt{\frac{239.15}{32.99}}

Radius of gyration about z-axis

K_{z}

3.01

mm

K_{z}

=

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

=

\sqrt{{1.35}^{2} + {2.69}^{2}}

Radius of gyration about x1-axis

K_{x_{1}}

2.84

mm

K_{x_{1}}

=

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

=

\sqrt{\frac{265.96}{32.99}}

Radius of gyration about y1-axis

K_{y_{1}}

5.68

mm

K_{y_{1}}

=

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

=

\sqrt{\frac{1063.82}{32.99}}

Radius of gyration about z1-axis

K_{z_{1}}

6.35

mm

K_{z_{1}}

=

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

=

\sqrt{{2.84}^{2} + {5.68}^{2}}

Elastic section modulus about x-axis

S_{x}

23.92

{mm}^{3}

S_{x}

=

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

=

\frac{59.79}{2.50}

Elastic section modulus about y-axis

S_{y}

47.83

{mm}^{3}

S_{y}

=

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

=

\frac{239.15}{5.00}

Plastic section modulus about x-axis

Z_{x}

39.00

{mm}^{3}

Z_{x}

=

\frac{(a_{out} \cdot {b_{out}}^{2} - a_{i n} \cdot {b_{i n}}^{2})}{6}

=

\frac{(10.00 \cdot {5.00}^{2} - 4.00 \cdot {2.00}^{2})}{6}

Plastic section modulus about y-axis

Z_{y}

78.00

{mm}^{3}

Z_{y}

=

\frac{(b_{out} \cdot {a_{out}}^{2} - b_{i n} \cdot {a_{i n}}^{2})}{6}

=

\frac{(5.00 \cdot {10.00}^{2} - 2.00 \cdot {4.00}^{2})}{6}