Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells
Abstract
:1. Introduction
2. Statement of the Problem
3. Solution Methodology
3.1. Boundary Discontinues Fourier Series Method
3.2. Generalized Differantial Quadrature Method
4. Numerical Results and Discussion
4.1. Validation Study
4.2. Convergence Study
4.3. Parametrice Study
5. Conclusions
- ✓
- It is worth noting that almost no difference resulted from changing from moderately deep (R/a = 10) to (R/a = Plate) between BDM and GDQ for the very well-known SS3 type boundary conditions. The relatively small discrepancies between the present methodologies and the finite element method can be predicated to the variation between strong-form and weak-form formulations. Our proposed solution methodologies for BDM and GDQ methods are based on the strong form, as it satisfies the compulsory and natural boundary conditions. In contrast to strong-form, the finite element method uses the weak-form formulation and does not apply natural or force boundary conditions which are imposed on the secondary variable, such as forces and moments.
- ✓
- A rapid and monotonic convergence was observed with the Fourier series for the terms m, n > 10 for displacements and m, n > 20 for stress in the BDM. The stress convergence was higher than the displacement. This is possibly due to the presence of a discontinuity (complementary boundary constraint) in the derivative of the displacement in expression of the stress. The same convergence characteristics were observed for grid numbers m, n = 13 with the generalized differential quadrature method.
- ✓
- A difference was seen when changing from the range 11.28% (R/a = Plate) to 18.75% (R/a = 10) for moderately thick (a/h = 20) antisymmetric [0°/90°] plates and shell with the roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges were subjected to the simply supported constraint between the BDM and GDQ methods. This is possibly due to the presence of a discontinuity in the derivatives which comes from the bending–stretching matrix (B matrix presence) in antisymmetric lamination. However very close results were obtained for symmetric ([0°/90°/0°] plates and shells using the whole range R/a and a/h ratios. It is also important to state that all three methods (BDM, DQM, and FEM) use different formulations, and among them, BDM provides analytical solutions and satisfies boundary conditions exactly, as is shown in Table 2 compared to FEM and DQM.
- ✓
- The difference between BDM and GDQ for an antisymmetric [0°/90°] shell increases with the increase of (E1/E2). The reason for this may be the dominant discontinuity in the bending–stretching matrix (B matrix presence) in antisymmetric lamination.
- ✓
- The effect of the modulus ratio, E1/E2, was more pronounced in the thin laminates (a/h > 50). Furthermore, this effect was increased by the beam–column effect in the case of an antisymmetric laminate. This is because the bending–stretching coupling is dominant in an antisymmetric laminate and produces a softening effect on the beam–column type, which consequently increases the normalized central deflection.
- ✓
- The effect of the radius-to-length ratio R/a on transverse displacement, w*, also plays a critical role in the thinner shell regime, specifically beginning from the ratio R/a < 40. Bending–stretching coupling is inherited in antisymmetric laminations, and it directly impacts the interaction of membrane action with the beam–column/tie-bar effect. The membrane action due to curvature has a complicated interaction with the stated mechanism. This interaction should be considered during the prior design of composite shells.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
References
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[0°/90°] | [0°/90°/0°] | [0°/90°/90°/0°] | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
R/a | BDM | GDQ | Ref. [18] | BDM | GDQ | Ref. [18] | BDM | GDQ | Ref. [18] | |
10 | 5.5428 | 5.542 | 5.5428 | 3.644 | 3.644 | 3.6445 | 3.720 | 3.720 | 3.720 | |
20 | 11.273 | 11.27 | 11.273 | 5.547 | 5.547 | 5.5473 | 5.661 | 5.661 | 5.661 | |
50 | 15.714 | 15.71 | 15.714 | 6.482 | 6.482 | 6.4827 | 6.614 | 6.614 | 6.614 | |
100 | 16.645 | 16.64 | 16.645 | 6.642 | 6.642 | 6.6421 | 6.777 | 6.777 | 6.777 | |
Plate | 16.979 | 16.97 | 16.980 | 6.697 | 6.696 | 6.6970 | 6.833 | 6.833 | 6.842 | |
R/a | BDM | GDQ | FEM | BDM | GDQ | FEM | BDM | GDQ | FEM | |
10 | 420.17 | 420.2 | 418.98 | 4688.6 | 4688.6 | 4678.5 | 4961.8 | 4961.8 | 4954.4 | |
20 | 868.116 | 868.1 | 866.180 | 6962.9 | 6963 | 6950 | 7238.2 | 7238.2 | 7227 | |
50 | 1196.10 | 1196.1 | 1196 | 7956.6 | 7956.7 | 7985.2 | 8175.8 | 8175.8 | 8209.2 | |
100 | 1256.62 | 1256.6 | 1255.6 | 8081.2 | 8081.3 | 8075 | 8271 | 8271 | 8265.9 | |
Plate | 1268.70 | 1268.7 | 1267.4 | 8073 | 8073 | 8064 | 8229.6 | 8229.5 | 8222.7 |
R/a | BDM | GDQ | FEM | |
---|---|---|---|---|
10 | 14.880 | 18.315 | 20.900 | |
20 | 20.763 | 20.796 | 22.165 | |
Plate | 22.152 | 19.901 | 22.603 | |
10 | 154.778 | 194.918 | 142.100 | |
20 | 172.361 | 176.382 | 129.650 | |
Plate | 136.668 | 118.459 | 105.135 | |
10 | 1850.1 | 2323.4 | 2569.2 | |
20 | 2612.8 | 2614.2 | 2684.3 | |
Plate | 2785.6 | 2466.0 | 2690.4 |
[0°/90°] | [0°/90°/0°] | [0°/90°/90°/0°] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
R/a | a/h | Method | w* | w* | w* | ||||||
10 | 20 | BDM | 14.880 | 154.778 | 1850.1 | 10.950 | 1432.6 | 101.046 | 10.999 | 1489.3 | 110.386 |
GDQ | 18.315 | 194.918 | 2323.4 | 10.811 | 1450.8 | 99.4543 | 10.864 | 1510.8 | 108.887 | ||
50 | BDM | 4.9204 | 206.615 | 1460.8 | 2.485 | 1851.6 | 40.88 | 3.1181 | 2394 | 76.979 | |
GDQ | 8.1924 | 308.405 | 2690 | 2.508 | 1874.7 | 41.746 | 3.1443 | 2419.0 | 78.042 | ||
100 | BDM | 1.3248 | 159.760 | 550.63 | 0.666 | 1778.9 | 9.1941 | 0.9236 | 2499.9 | 36.3349 | |
GDQ | 2.3004 | 201.457 | 1313.9 | 0.671 | 1792.5 | 9.9637 | 0.9304 | 2518.5 | 37.3247 | ||
Plate | 20 | BDM | 22.147 | 136.554 | 2785 | 28.660 | 944.08 | 269.434 | 20.966 | 769.890 | 200.210 |
GDQ | 19.901 | 118.459 | 2466.0 | 27.058 | 1004.4 | 252.13 | 20.190 | 828.9 | 191.707 | ||
50 | BDM | 19.818 | 399.629 | 6406.5 | 19.598 | 3048.2 | 436.80 | 15.786 | 2680.8 | 390.017 | |
GDQ | 17.114 | 338.554 | 5439.2 | 19.469 | 3070.2 | 458.07 | 15.853 | 2681.5 | 389.6595 | ||
100 | BDM | 19.193 | 837.166 | 1244.5 | 16.799 | 6544.3 | 793.02 | 14.137 | 5893.2 | 703.959 | |
GDQ | 16.588 | 695.911 | 1058.2 | 17.976 | 6381.6 | 846.779 | 14.98 | 5622.0 | 741.542 |
R/a | a/h | Method | E1/E2 | ||||||
---|---|---|---|---|---|---|---|---|---|
3 | 10 | 20 | 30 | 40 | 50 | ||||
w* | 10 | 20 | BDM | 33.877 | 23.8006 | 17.047 | 13.171 | 10.655 | 8.902 |
GDQ | 34.559 | 25.9268 | 20.218 | 16.769 | 14.392 | 12.638 | |||
50 | BDM | 16.055 | 9.336 | 5.860 | 4.229 | 3.283 | 2.668 | ||
GDQ | 17.222 | 12.108 | 9.130 | 7.444 | 6.314 | 5.494 | |||
100 | BDM | 5.4233 | 2.7262 | 1.6004 | 1.1292 | 0.8699 | 0.705 | ||
GDQ | 5.8318 | 3.665 | 2.6041 | 2.0677 | 1.7306 | 1.495 | |||
Plate | 20 | BDM | 41.708 | 31.449 | 24.464 | 20.277 | 17.427 | 15.346 | |
GDQ | 39.952 | 28.524 | 21.942 | 18.274 | 15.833 | 14.059 | |||
50 | BDM | 37.953 | 28.482 | 21.977 | 18.077 | 15.429 | 13.500 | ||
GDQ | 36.683 | 25.320 | 19.007 | 15.609 | 13.396 | 11.810 | |||
100 | BDM | 36.995 | 27.699 | 21.311 | 17.487 | 14.895 | 13.010 | ||
GDQ | 36.123 | 24.734 | 18.448 | 15.089 | 12.914 | 11.362 | |||
10 | 20 | BDM | 408.591 | 268.165 | 181.547 | 134.061 | 104.293 | 84.170 | |
GDQ | 420.420 | 294.297 | 218.847 | 175.913 | 147.473 | 127.130 | |||
50 | BDM | 727.107 | 402.740 | 247.767 | 176.540 | 135.605 | 109.158 | ||
GDQ | 755.659 | 480.335 | 346.930 | 278.613 | 234.863 | 203.835 | |||
100 | BDM | 696.610 | 332.097 | 193.171 | 136.115 | 104.819 | 85.020 | ||
GDQ | 706.825 | 363.486 | 233.464 | 178.37 | 146.929 | 126.241 | |||
Plate | 20 | BDM | 344.536 | 231.245 | 159.110 | 118.960 | 93.386 | 75.828 | |
GDQ | 338.443 | 212.727 | 139.872 | 101.943 | 78.762 | 63.292 | |||
50 | BDM | 932.956 | 647.333 | 459.655 | 352.222 | 282.161 | 233.055 | ||
GDQ | 895.587 | 582.226 | 394.657 | 294.155 | 231.333 | 188.603 | |||
100 | BDM | 1905.6 | 1336.1 | 958.7 | 740.8 | 597.8 | 496.9 | ||
GDQ | 1808.3 | 1183.6 | 807.6 | 605.1 | 478 | 391.2 | |||
10 | 20 | BDM | 834.2 | 1380.2 | 1747.4 | 1922.3 | 2009.3 | 2051.6 | |
GDQ | 858 | 1525.6 | 2111.6 | 2499.8 | 2776.9 | 2984.3 | |||
50 | BDM | 1085.2 | 1373.3 | 1453.2 | 1460.3 | 1447.6 | 1428.7 | ||
GDQ | 1184 | 1894 | 2480.7 | 2866 | 3141.9 | 3350 | |||
100 | BDM | 731.916 | 670.952 | 579.850 | 528.011 | 495.387 | 472.958 | ||
GDQ | 811.3 | 1060.7 | 1244.8 | 1374 | 1474.6 | 1557 | |||
Plate | 20 | BDM | 967.9 | 1795.5 | 2521.2 | 3006.7 | 3361.3 | 3634.6 | |
GDQ | 921.1 | 1601.7 | 2225.8 | 2672.7 | 3016.9 | 3293.2 | |||
50 | BDM | 2259.0 | 4162.7 | 5810.8 | 6906.9 | 7708.3 | 8329.4 | ||
GDQ | 2166.5 | 3633 | 4934.7 | 5869.5 | 6600.4 | 7198.3 | |||
100 | BDM | 4428 | 8127 | 11,303 | 13,401 | 14,929 | 16,112 | ||
GDQ | 4285 | 7123 | 9611 | 11,390 | 12,781 | 13,921 |
R/a | a/h | Method | E1/E2 | ||||||
---|---|---|---|---|---|---|---|---|---|
3 | 10 | 20 | 30 | 40 | 50 | ||||
w* | 10 | 20 | BDM | 32.259 | 19.215 | 12.731 | 9.620 | 7.755 | 6.504 |
GDQ | 31.795 | 18.864 | 12.544 | 9.514 | 7.692 | 6.466 | |||
50 | BDM | 13.260 | 5.632 | 3.059 | 2.091 | 1.587 | 1.277 | ||
GDQ | 13.344 | 5.678 | 3.086 | 2.111 | 1.601 | 1.289 | |||
100 | BDM | 4.356 | 1.631 | 0.834 | 0.554 | 0.414 | 0.329 | ||
GDQ | 4.357 | 1.635 | 0.839 | 0.559 | 0.4177 | 0.332 | |||
Plate | 20 | BDM | 43.818 | 34.616 | 30.074 | 27.493 | 25.597 | 24.062 | |
GDQ | 42.631 | 32.968 | 28.419 | 25.954 | 24.192 | 22.785 | |||
50 | BDM | 37.065 | 25.627 | 20.850 | 18.653 | 17.263 | 16.240 | ||
GDQ | 37.146 | 25.559 | 20.728 | 18.520 | 17.132 | 16.116 | |||
100 | BDM | 35.240 | 23.015 | 18.047 | 15.883 | 14.585 | 13.674 | ||
GDQ | 36.173 | 24.169 | 19.235 | 17.042 | 15.701 | 14.742 | |||
10 | 20 | BDM | 875.1 | 1245.6 | 1395.2 | 1459.5 | 1496.1 | 1519.9 | |
GDQ | 886.2 | 1264.1 | 1413.8 | 1477.5 | 1513.6 | 1537.1 | |||
50 | BDM | 1550.2 | 1865.4 | 1867 | 1836.2 | 1809.6 | 1788.7 | ||
GDQ | 1555.8 | 1883.5 | 1889.3 | 1859.8 | 1833.7 | 1813.1 | |||
100 | BDM | 1559 | 1798.8 | 1792.4 | 1765.9 | 1743.5 | 1725.8 | ||
GDQ | 1554.9 | 1802.4 | 1803.4 | 1781.6 | 1762.4 | 1747 | |||
Plate | 20 | BDM | 699.61 | 927.91 | 956.36 | 931.67 | 895.94 | 858.31 | |
GDQ | 720.6 | 972.1 | 1011.4 | 991.6 | 958.5 | 922.4 | |||
50 | BDM | 1998.1 | 2828.2 | 3033.5 | 3041.7 | 2998.1 | 2936.2 | ||
GDQ | 1993.2 | 2839 | 3052.9 | 3065.7 | 3025 | 2965.3 | |||
100 | BDM | 4140.5 | 5968.8 | 6481.1 | 6559.9 | 6519.6 | 6434.3 | ||
GDQ | 4059.6 | 5843.3 | 6328 | 6388.9 | 6334.7 | 6237.9 | |||
10 | 20 | BDM | 361.159 | 190.887 | 119.383 | 87.688 | 69.430 | 57.484 | |
GDQ | 357.635 | 187.504 | 117.347 | 86.424 | 68.601 | 56.918 | |||
50 | BDM | 409.137 | 127.946 | 54.639 | 32.178 | 22.032 | 16.464 | ||
GDQ | 412.912 | 129.886 | 55.701 | 32.898 | 22.574 | 16.899 | |||
100 | BDM | 265.097 | 51.627 | 14.368 | 6.362 | 3.601 | 2.383 | ||
GDQ | 264.017 | 52.361 | 15.190 | 7.074 | 4.208 | 2.906 | |||
Plate | 20 | BDM | 452.400 | 334.082 | 283.944 | 257.786 | 239.216 | 224.367 | |
GDQ | 440.178 | 316.606 | 266.128 | 241.087 | 223.864 | 210.326 | |||
50 | BDM | 993.809 | 632.001 | 496.701 | 439.754 | 405.647 | 381.325 | ||
GDQ | 992.951 | 627.821 | 491.164 | 433.981 | 399.974 | 375.874 | |||
100 | BDM | 1910.4 | 1142 | 859.2 | 746 | 681.9 | 638.5 | ||
GDQ | 1947.4 | 1193.2 | 913.2 | 799.1 | 733.2 | 687.7 |
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Algül, İ.; Oktem, A.S. Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells. Appl. Sci. 2022, 12, 12547. https://doi.org/10.3390/app122412547
Algül İ, Oktem AS. Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells. Applied Sciences. 2022; 12(24):12547. https://doi.org/10.3390/app122412547
Chicago/Turabian StyleAlgül, İlke, and Ahmet Sinan Oktem. 2022. "Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells" Applied Sciences 12, no. 24: 12547. https://doi.org/10.3390/app122412547
APA StyleAlgül, İ., & Oktem, A. S. (2022). Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells. Applied Sciences, 12(24), 12547. https://doi.org/10.3390/app122412547