Objective
To verify the accuracy of calculations of the geometric properties of a solid elliptical cross‑section.
Reference
A.A. Umansky, Design engineer’s handbook – calculation and theory, 1973, Book 1.
Problem statement
For a solid elliptical cross-section, verify the accuracy of the calculated geometric properties.
Geometry
b = 30 cm; h = 50 cm.
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Analytical model
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Model in the 'Cross-section Design Toolkit' module
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Note
The triangulation mesh size is 10 mm.
Analytical solution
| A = πhb | |
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| φ = 0 |
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| Ry = h/2 | Rz = b/2 |
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| Y = h/4 | Z = b/4 |
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Pint = 0 |
| X0 = b | Y0 = h | yb = zb = 0 |
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Comparison of calculation results
| Parameter | Theory | LIRA-FEM | Error, % |
| Area, A cm2 | 4712,389 | 4711,92 | 0,01 |
| Shear area about the principal Y1-axis, Ay cm2 | 3724,143 | 3792,27 | 1,7965 |
| Shear area about the principal Z1-axis, Az cm2 | 4147,17 | 4149,29 | 0,0511 |
| Rotation of the principal Y1-axis relative to the X-axis of the current coordinate system, φ ° | 0 | 0 | 0 |
| Moment of inertia about the central U-axis, Iu cm4 | 2945243,113 | 2944651,768 | 0,0201 |
| Moment of inertia about the central V-axis, Iv cm4 | 1060287,522 | 1060074,637 | 0,0201 |
| Torsional moment of inertia, It cm4 | 3118492,708 | 3117988,306 | 0,0162 |
| Warping moment of inertia, Iw cm6 | 97835065,337 | 97791264,635 | 0,0448 |
| Radius of gyration about the principal Y1-axis, Ry mm | 250 | 249,99 | 0,004 |
| Radius of gyration about the principal Z1-axis, Rz mm | 150 | 149,99 | 0,0067 |
| Max section modulus about the principal Y1-axis, maxWy cm3 | 58904,862 | 58893,035 | 0,0201 |
| Min section modulus about the principal Y1-axis, minWy cm3 | 58904,862 | 58893,035 | 0,0201 |
| Max section modulus about the principal Z1-axis, maxWz cm3 | 35342,917 | 35335,821 | 0,0201 |
| Min section modulus about the principal Z1-axis, minWz cm3 | 35342,917 | 35335,821 | 0,0201 |
| Plastic section modulus about the Y1pl-axis, Wply cm3 | 100000 | 99984,941 | 0,0151 |
| Plastic section modulus about the Z1pl-axis, Wplz cm3 | 60000 | 59990,965 | 0,0151 |
| Max moment of inertia, maxI cm4 | 2945243,113 | 2944651,768 | 0,0201 |
| Min moment of inertia, minI cm4 | 1060287,521 | 1060074,637 | 0,0201 |
| Max radius of gyration, maxR cm | 250 | 249,99 | 0,004 |
| Min radius of gyration, minR cm | 150 | 149,99 | 0,0067 |
| Core distance in the negative direction of the principal Y1-axis, Y- mm | 75 | 74,99 | 0,0133 |
| Core distance in the positive direction of the principal Y1-axis, Y+ mm | 75 | 74,99 | 0,0133 |
| Core distance in the negative direction of the principal Z1-axis, Z- mm | 125 | 124,99 | 0,008 |
| Core distance in the positive direction of the principal Z1-axis, Z+ mm | 125 | 124,99 | 0,008 |
| X-coordinate of the centroid in the current coordinate system, X0 mm | 300 | 300 | 0 |
| Y-coordinate of the centroid in the current coordinate system, Y0 mm | 500 | 500 | 0 |
| Y-coordinate of the shear centre, yb cm | 0 | 0 | 0 |
| Z-coordinate of the shear centre, zb cm | 0 | 0 | 0 |
| Perimeter, P mm | 2551,8 | 2552,64 | 0,0329 |
| Perimeter of internal contours, Pint mm | 0 | 0 | 0 |
| Perimeter of external contours, Pext mm | 2551,8 | 2552,64 | 0,0329 |
| Polar moment of inertia, Ip cm4 | 4005530,633 | 4004726,405 | 0,02 |
| Polar radius of gyration, ip cm | 29,155 | 29,153 | 0,0069 |
| Polar section modulus, Wp cm3 | 80110,8 | 80094,528 | 0,0203 |
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