# Geometric Properties of an Ellipse

Aim: To check the accuracy of the geometric properties calculation for a solid elliptical cross-section of a rod.

Name of a file with the initial data: Ellipse_Solid.cns

Formulation: Check the accuracy of the geometric properties calculation for a solid elliptical cross-section of a rod.

References:    Demidov S. P., Theory of Elasticity, M., Vysshaya Shkola, 1979.
Lurie A. I., Theory of Elasticity, M., Nauka, 1970.

Initial data:

 ν = 0.30 - Poisson’s ratio; a = 50 cm - length of the semi-major axis of the elliptical cross-section (along Y axis); b = 30 cm - length of the semi-minor axis of the elliptical cross-section (along Z axis).

Design model: The design model is created by triangulation (the number of triangles ≈ 3000) on the basis of a model of the external contour imported from the AutoCad graphic editor. The model of an external contour is a polygon inscribed in an ellipse with given properties and built in polar coordinates with an angle step of 3°. The number of vertices of a polygon in a model is 120.

## Results Obtained in Consul

Design model, coordinate and principal axes, center of mass, ellipse of inertia, core of the section

Comparison of results:

Parameter

Theory

CONSUL

Deviation, %

Cross-sectional area, A cm2

4712.389

4709.319

0.07

Conventional shear area along the principal U-axis, Av,y cm2

3724.143

3702.975

0.57

Conventional shear area along the principal V-axis, Av,z cm2

4147.170

4161.672

0.35

Angle of the principal axes of inertia α rad

1.5708

1.5708

0.00

Moment of inertia about the centroidal Y1 axis parallel to the coordinate Y axis, Iy cm4

1060287.521

1059491.143

0.08

Moment of inertia about the centroidal Z1 axis parallel to the coordinate Z axis, Iz cm4

2945243.113

2939784.432

0.19

Torsional moment of inertia, It cm4

3118492.708

3064969.367

1.72

Sectorial moment of inertia, Iw cm6

97835065.337

95561910.155

2.32

15.000

14.980

0.13

25.000

24.991

0.04

Maximum section modulus about U-axis, Wu+ cm3

58904.862

58795.689

0.19

Minimum section modulus about U-axis, Wu‒ cm3

58904.862

58795.689

0.19

Maximum section modulus about V-axis, Wv+ cm3

35342.917

35316.371

0.08

Minimum section modulus about V-axis, Wv‒ cm3

35342.917

35316.371

0.08

Plastic section modulus about U-axis, Wpl,u cm3

100000.000

99796.050

0.20

Plastic section modulus about V-axis, Wpl,v cm3

60000.000

59820.326

0.30

Maximum moment of inertia, Iu cm4

2945243.113

2939784.432

0.19

Minimum moment of inertia, Iv cm4

1060287.521

1059491.143

0.08

Maximum radius of gyration, iu cm

25.000

24.985

0.06

Minimum radius of gyration, iv cm

15.000

14.999

0.01

Core size along positive Y(U)-axis, a u+ cm

7.500

7.494

0.08

Core size along negative Y(U)-axis, a u‒ cm

7.500

7.480

0.27

Core size along positive Z(V)-axis, a v+ cm

12.500

12.491

0.07

Core size along negative Z(V)-axis, a v‒ cm

12.500

12.491

0.07

Y-coordinate of the center of mass, ym cm

0.000

0.000

Z-coordinate of the center of mass, zm cm

0.000

0.000

Y-coordinate of the shear center, yb cm

0.000

0.013

Z-coordinate of the shear center, zb cm

0.000

0.040

Perimeter, P cm

255.180

255.180

0.00

Internal perimeter, Pi cm

0.000

0.000

External perimeter, Pe cm

255.180

255.180

0.00

Polar moment of inertia, Ip cm4

4005530.633

3993669.583

0.30

Polar radius of gyration, ip cm

29.155

29.136

0.07

Polar section modulus, Wp cm3

80110.800

79872.926

0.30

Notes: Geometric properties of the solid elliptical cross-section of the rod can be determined analytically by the following formulas:

$A=\pi \cdot a\cdot b;$ $A_{v,y} =\frac{3\cdot \left( {1+\nu } \right)^{2}\cdot \left( {a^{2}+3\cdot b^{2}} \right)^{2}\cdot b^{2}}{\left( {1+\nu } \right)^{2}\cdot \left( {22\cdot a^{2}+30\cdot b^{2}} \right)\cdot b^{4}+\left( {2\cdot \nu ^{2}\cdot a^{2}+\left( {4+8\cdot \nu +10\cdot \nu^{2}} \right)\cdot b^{2}} \right)\cdot a^{4}}\cdot \pi \cdot a\cdot b;$ $A_{v,z} =\frac{3\cdot \left( {1+\nu } \right)^{2}\cdot \left( {3\cdot a^{2}+b^{2}} \right)^{2}\cdot a^{2}}{\left( {1+\nu } \right)^{2}\cdot \left( {30\cdot a^{2}+22\cdot b^{2}} \right)\cdot a^{4}+\left( {\left( {4+8\cdot \nu +10\cdot \nu^{2}} \right)\cdot a^{2}+2\cdot \nu^{2}\cdot b^{2}} \right)\cdot b^{4}}\cdot \pi \cdot a\cdot b;$ $\alpha =0; \quad I_{y}=I_{v} =I_{1} =\frac{\pi \cdot a\cdot b^{3}}{4}; \quad I_{z} =I_{u} =I_{2} =\frac{\pi \cdot a^{3}\cdot b}{4};$ $I_{t} =\frac{\pi \cdot a^{3}\cdot b^{3}}{a^{2}+b^{2}}; \quad I_{w} =\frac{\pi \cdot a^{3}\cdot b^{3}}{24}\cdot \left( {\frac{a^{2}-b^{2}}{a^{2}+b^{2}}} \right)^{2};$ $i_{y} =i_{v} =\frac{b}{2}; \quad i_{z} =i_{u} =\frac{a}{2};$ $W_{u+} =W_{u-} =\frac{\pi \cdot a^{2}\cdot b}{4}; \quad W_{v+} =W_{v-} =\frac{\pi \cdot a\cdot b^{2}}{4};$ $W_{pl,u} =\frac{4\cdot a^{2}\cdot b}{3}; \quad W_{pl,v} =\frac{4\cdot a\cdot b^{2}}{3};$ $a_{u+} =a_{u-} =\frac{b}{4}; \quad a_{v+} =a_{v-} =\frac{a}{4};$ $y_{m} =y_{b} =z_{m} =z_{b} =0;$ $P=P_{e} =4\cdot a\cdot E\left( {\frac{\sqrt {a^{2}-b^{2}} }{a}} \right), \quad where:\quad E\left( x \right) \text {Legendre complete elliptic integral of the second kind;}$ $P\approx 4\cdot \left( {a+b} \right)-\frac{2\cdot \left( {4-\pi } \right)\cdot a\cdot b}{\sqrt[{\frac{3\cdot \pi -8}{8-2\cdot \pi }}]{\frac{a^{\frac{3\cdot \pi -8}{8-2\cdot \pi }}+b^{\frac{3\cdot \pi -8}{8-2\cdot \pi }}}{2}}}; \quad P\approx \pi \cdot \left( {a+b} \right); \quad P_{i} =0;$ $I_{12} =0; \quad I_{p} =\frac{\pi \cdot a\cdot b\cdot \left( {a^{2}+b^{2}} \right)}{4}; \quad i_{p} =\frac{\sqrt {a^{2}+b^{2}} }{2}; \quad W_{p} =\frac{\pi \cdot b\cdot \left( {a^{2}+b^{2}} \right)}{4}.$