Strong and Weak Formulations of a Mixed Higher-Order Shear Deformation Theory for the Static Analysis of Functionally Graded Beams under Thermo-Mechanical Loads
Abstract
:1. Introduction
2. Effective Material Properties
3. The Strong Formulation and Its Application
4. The Weak Formulation and Its Application
5. Boundary Conditions
6. Numerical Examples
6.1. Mechanical Loads
6.2. Thermal loads
7. Concluding Remarks
- In the LW HSDT, the transverse shear and normal stress components induced in the thermo-mechanically loaded FG beams can be obtained using three different approaches. In approach 1, they are obtained using constitutive equations when the displacement components are determined; in approach 2, they are obtained using stress equilibrium equations when the displacement components are determined; in approach 3, they are directly obtained using the Lagrange multipliers. Implementation of the three approaches shows that the accuracy of these approaches is in the following order: approach 2 > approach 3 > approach 1, where the symbol “>” indicates a greater degree of accuracy.
- The accuracy and convergence rates of the LW1, LW2, and LW3 theories are in the following order: LW3 > LW2 > LW1, where the symbol “>” indicates a faster convergence rate and a more accurate solution.
- Variations in the through-thickness distributions of the transverse shear and normal stresses with the length–thickness ratio and the material-property gradient index for the thermal load cases are more drastic than the variations for the mechanical load cases.
- Based on the convergent solutions of the LW HSDT, the accuracy of the various advanced and refined beam theories is in the following order: ESDT > SSDT > (RHSDT, TSDT) > HSDT, where the symbol “>” represents a greater degree of accuracy.
Author Contributions
Funding
Conflicts of Interest
Appendix A. Relationships between the Resultant Forces/Moments and the Displacement Components
Appendix B. The Solution Process of the Fourier Series Expansion Method
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L/h | Theories | (L/2, 0) | (0, −h/2) | (L/2, h/2) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Approach 1 | Approach 2 | Approach 3 | Approach 1 | Approach 2 | Approach 3 | ||||||
5 | 5 | Current L1 (nl = 4) | 9.8459 | 3.6559 | 8.1050 | 0.5432 | 0.6346 | 0.6392 | −0.0550 | -0.0842 | -0.0818 |
Current L1 (nl = 8) | 9.8461 | 3.6768 | 8.1471 | 0.6070 | 0.6380 | 0.6392 | −0.0693 | −0.0824 | −0.0818 | ||
Current L1 (nl = 16) | 9.8462 | 3.6826 | 8.1551 | 0.6273 | 0.6389 | 0.6392 | −0.0759 | −0.0819 | −0.0818 | ||
Current L2 (nl = 2) | 9.8444 | 3.6618 | 8.1498 | 0.7386 | 0.6427 | 0.6428 | −0.0720 | −0.0915 | −0.0818 | ||
Current L2 (nl = 4) | 9.8460 | 3.6822 | 8.1635 | 0.6885 | 0.6392 | 0.6392 | −0.0832 | −0.0844 | −0.0818 | ||
Current L2 (nl = 8) | 9.8463 | 3.6844 | 8.1588 | 0.6546 | 0.6392 | 0.6392 | −0.0829 | −0.0824 | −0.0818 | ||
Current L3 (nl = 2) | 9.8456 | 3.6845 | 8.1581 | 0.6710 | 0.6428 | 0.6428 | −0.0858 | −0.0818 | −0.0818 | ||
Current L3 (nl = 4) | 9.8463 | 3.6847 | 8.1571 | 0.6436 | 0.6392 | 0.6392 | −0.0826 | −0.0818 | −0.0818 | ||
Current L3 (nl = 8) | 9.8463 | 3.6847 | 8.1565 | 0.6395 | 0.6392 | 0.6392 | −0.0819 | −0.0818 | −0.0818 | ||
HSDT [33] | 9.7802 | 3.7089 | 8.1030 | 0.5790 | NA | NA | NA | NA | NA | ||
RSDT [34] | 9.8281 | 3.7100 | 8.1106 | 0.5905 | NA | NA | NA | NA | NA | ||
SSDT [35] | 9.8367 | 3.7140 | 8.1222 | 0.6155 | NA | NA | NA | NA | NA | ||
TSDT [36] | 9.8271 | 3.7097 | 8.1095 | 0.5883 | NA | NA | NA | NA | NA | ||
ESDT [37] | 9.8414 | 3.7177 | 8.1329 | 0.6404 | NA | NA | NA | NA | NA | ||
CBT [16] | 8.7508 | 3.6496 | 7.9428 | NA | NA | NA | NA | NA | NA | ||
20 | 5 | Current L1 (nl = 4) | 8.8197 | 0.9126 | 31.8104 | 0.5725 | 0.6704 | 0.6715 | −0.0137 | −0.0210 | −0.0204 |
Current L1 (nl = 8) | 8.8197 | 0.9130 | 31.8209 | 0.6383 | 0.6712 | 0.6715 | −0.0173 | −0.0205 | −0.0204 | ||
Current L1 (nl = 16) | 8.8197 | 0.9131 | 31.8233 | 0.6592 | 0.6714 | 0.6715 | −0.0189 | −0.0204 | −0.0204 | ||
Current L2 (nl = 2) | 8.8197 | 0.9127 | 31.8219 | 0.7500 | 0.6715 | 0.6715 | −0.0179 | −0.0228 | −0.0203 | ||
Current L2 (nl = 4) | 8.8197 | 0.9131 | 31.8250 | 0.7228 | 0.6715 | 0.6715 | −0.0207 | −0.0210 | −0.0204 | ||
Current L2 (nl = 8) | 8.8197 | 0.9131 | 31.8242 | 0.6878 | 0.6715 | 0.6715 | −0.0206 | −0.0205 | −0.0204 | ||
Current L3 (nl = 2) | 8.8197 | 0.9131 | 31.8240 | 0.7172 | 0.6715 | 0.6715 | −0.0214 | −0.0203 | −0.0203 | ||
Current L3 (nl = 4) | 8.8197 | 0.9131 | 31.8241 | 0.6766 | 0.6715 | 0.6715 | −0.0206 | −0.0204 | −0.0204 | ||
Current L3 (nl = 8) | 8.8197 | 0.9131 | 31.8241 | 0.6718 | 0.6715 | 0.6715 | −0.0204 | −0.0204 | −0.0204 | ||
HSDT [33] | 8.8151 | 0.9133 | 31.8112 | 0.5790 | NA | NA | NA | NA | NA | ||
RSDT [34] | 8.8182 | 0.9134 | 31.8130 | 0.6023 | NA | NA | NA | NA | NA | ||
SSDT [35] | 8.8188 | 0.9134 | 31.8159 | 0.6292 | NA | NA | NA | NA | NA | ||
TSDT [36] | 8.8181 | 0.9134 | 31.8127 | 0.5998 | NA | NA | NA | NA | NA | ||
ESDT [37] | 8.8191 | 0.9135 | 31.8185 | 0.6562 | NA | NA | NA | NA | NA | ||
CBT [16] | 8.7508 | 0.9124 | 31.7711 | NA | NA | NA | NA | NA | NA |
L/h | Boundary Conditions | Current LW3 FEM (ne × nl) and Analytical Solutions | (L/2, 0) | (L/4, −h/2) | (L/2, h/2) | (L/4, 0) | (L/4, 0) | |
---|---|---|---|---|---|---|---|---|
5 | 5 | S–S | Current LW3 FEM (2 × 16) | 9.3110 | 2.3283 | 8.0867 | 0.1956 | −0.1040 |
Current LW3 FEM (4 × 16) | 9.8064 | 2.5412 | 8.3122 | 0.4350 | −0.0762 | |||
Current LW3 FEM (8 × 16) | 9.8435 | 2.5341 | 8.2195 | 0.3667 | −0.0822 | |||
Current LW3 FEM (16 × 16) | 9.8461 | 2.5329 | 8.1780 | 0.3479 | −0.0825 | |||
Current LW3 FEM (32 × 16) | 9.8463 | 2.5327 | 8.1650 | 0.3434 | −0.0825 | |||
Current LW3 analytical solutions | 9.8463 | 2.5326 | 8.1565 | 0.3419 | −0.0825 | |||
10 | 5 | C–C | Current LW3 FEM (2 × 16) | 0.8045 | 0.1171 | 2.7839 | 0.1436 | −0.0322 |
Current LW3 FEM (4 × 16) | 1.8644 | 0.3471 | 5.3558 | 0.5644 | −0.0459 | |||
Current LW3 FEM (8 × 16) | 2.0049 | 0.3460 | 5.4767 | 0.4296 | −0.0379 | |||
Current LW3 FEM (16 × 16) | 2.0190 | 0.3453 | 5.4335 | 0.3657 | −0.0409 | |||
Current LW3 FEM (32 × 16) | 2.0206 | 0.3451 | 5.4128 | 0.3478 | −0.0409 | |||
10 | 5 | S–S | Current LW3 FEM (2 × 16) | 7.9337 | 1.0465 | 13.4887 | 0.0036 | −0.0663 |
Current LW3 FEM (4 × 16) | 8.9089 | 1.2609 | 15.9385 | 0.6858 | −0.0119 | |||
Current LW3 FEM (8 × 16) | 9.0168 | 1.2585 | 16.0672 | 0.4423 | −0.0386 | |||
Current LW3 FEM (16 × 16) | 9.0255 | 1.2577 | 16.0239 | 0.3662 | −0.0409 | |||
Current LW3 FEM (32 × 16) | 9.0261 | 1.2575 | 16.0032 | 0.3478 | −0.0409 | |||
Current LW3 analytical solutions | 9.0261 | 1.2575 | 15.9921 | 0.3417 | −0.0409 | |||
10 | 5 | C–S | Current LW3 FEM (2 × 16) | 2.4077 | 0.3325 | 7.9771 | 0.1791 | −0.0262 |
Current LW3 FEM (4 × 16) | 3.6503 | 0.6304 | 8.8481 | 0.8795 | −0.0550 | |||
Current LW3 FEM (8 × 16) | 3.8062 | 0.6224 | 8.4294 | 0.6455 | −0.0377 | |||
Current LW3 FEM (16 × 16) | 3.8211 | 0.6206 | 8.2202 | 0.5470 | −0.0409 | |||
Current LW3 FEM (32 × 16) | 3.8226 | 0.6203 | 8.1529 | 0.5200 | −0.0409 | |||
10 | 5 | C–F | Current LW3 FEM (2 × 16) | 27.1018 | 3.5694 | 7.7092 | 0.2246 | 0.0102 |
Current LW3 FEM (4 × 16) | 30.1461 | 4.2363 | 12.5271 | 1.8370 | −0.0830 | |||
Current LW3 FEM (8 × 16) | 30.5292 | 4.2316 | 14.7599 | 1.3013 | −0.0371 | |||
Current LW3 FEM (16 × 16) | 30.5660 | 4.2293 | 15.4893 | 1.0976 | −0.0409 | |||
Current LW3 FEM (32 × 16) | 30.5698 | 4.2288 | 15.7002 | 1.0433 | −0.0409 |
h/L | Theories | 103 (L/2, −h/2) | (L/2, −h/2) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Approach 1 | Approach 2 | Approach 3 | Approach 1 | Approach 2 | Approach 3 | |||||
0.1 | 230 | Current LW1 (nl = 16) | 9.7662 | −1.5544 | 1.6977 | 6.5887 | 6.5590 | 0.5622 | 0.0114 | 0.0113 |
Current LW1 (nl = 32) | 9.7662 | −1.5564 | 4.1835 | 6.5664 | 6.5590 | 0.2802 | 0.0113 | 0.0113 | ||
Current LW1 (nl = 64) | 9.7662 | −1.5573 | 5.3849 | 6.5609 | 6.5590 | 0.1441 | 0.0113 | 0.0113 | ||
Current LW2 (nl = 8) | 9.7662 | −1.5610 | 6.9982 | 6.6113 | 6.5590 | −0.1047 | 0.0116 | 0.0113 | ||
Current LW2 (nl = 16) | 9.7662 | −1.5588 | 6.6732 | 6.5721 | 6.5590 | −0.0159 | 0.0113 | 0.0113 | ||
Current LW2 (nl = 32) | 9.7662 | −1.5593 | 6.5882 | 6.5623 | 6.5590 | 0.0047 | 0.0113 | 0.0113 | ||
Current LW3 (nl = 4) | 9.7662 | −1.5576 | 6.6304 | 6.5590 | 6.5590 | 0.0344 | 0.0113 | 0.0113 | ||
Current LW3 (nl = 8) | 9.7662 | −1.5580 | 6.5693 | 6.5590 | 6.5590 | 0.0137 | 0.0113 | 0.0113 | ||
Current LW3 (nl = 16) | 9.7662 | −1.5581 | 6.5604 | 6.5590 | 6.5590 | 0.0115 | 0.0113 | 0.0113 | ||
SSDQM [31] | 9.9772 | −1.5648 | 6.6004 | NA | NA | NA | NA | NA | ||
FEM [31] | 9.7944 | −1.5573 | 6.0427 | NA | NA | NA | NA | NA | ||
0.2 | 230 | Current LW1 (nl = 16) | 33.316 | −41.037 | 1.1403 × 103 | 1.4261 × 103 | 1.4255 × 103 | 9.5825 | 2.4869 | 2.4720 |
Current LW1 (nl = 32) | 33.315 | −40.968 | 1.2850 × 103 | 1.4256 × 103 | 1.4255 × 103 | 5.7645 | 2.4757 | 2.4720 | ||
Current LW1 (nl = 64) | 33.315 | −40.930 | 1.3558 × 103 | 1.4255 × 103 | 1.4255 × 103 | 4.0550 | 2.4729 | 2.4720 | ||
Current LW2 (nl = 8) | 33.315 | −40.901 | 1.4451 × 103 | 1.4277 × 103 | 1.4255 × 103 | −2.7201 | 2.5045 | 2.4720 | ||
Current LW2 (nl = 16) | 33.315 | −40.893 | 1.4308 × 103 | 1.4260 × 103 | 1.4255 × 103 | 1.3386 | 2.4800 | 2.4720 | ||
Current LW2 (nl = 32) | 33.315 | −40.890 | 1.4269 × 103 | 1.4256 × 103 | 1.4255 × 103 | 2.2068 | 2.4740 | 2.4720 | ||
Current LW3 (nl = 4) | 33.315 | −40.887 | 1.4312 × 103 | 1.4255 × 103 | 1.4255 × 103 | 5.2356 | 2.4720 | 2.4720 | ||
Current LW3 (nl = 8) | 33.315 | −40.889 | 1.4265 × 103 | 1.4255 × 103 | 1.4255 × 103 | 2.7039 | 2.4720 | 2.4720 | ||
Current LW3 (nl = 16) | 33.315 | −40.889 | 1.4256 × 103 | 1.4255 × 103 | 1.4255 × 103 | 2.4959 | 2.4720 | 2.4720 | ||
0.1 | 100 | Current LW1 (nl = 16) | 1.6955 | −0.1008 | −0.7732 | 0.1256 | 0.1226 | 0.0688 | 0.0005 | 0.0005 |
Current LW1 (nl = 32) | 1.6956 | −0.1035 | −0.3189 | 0.1233 | 0.1226 | 0.0343 | 0.0005 | 0.0005 | ||
Current LW1 (nl = 64) | 1.6956 | −0.1048 | −0.0966 | 0.1228 | 0.1226 | 0.0173 | 0.0005 | 0.0005 | ||
Current LW2 (nl = 8) | 1.6956 | −0.1066 | 0.1724 | 0.1269 | 0.1226 | −0.0050 | 0.0005 | 0.0005 | ||
Current LW2 (nl = 16) | 1.6956 | −0.1062 | 0.1352 | 0.1236 | 0.1226 | −0.0008 | 0.0005 | 0.0005 | ||
Current LW2 (nl = 32) | 1.6956 | −0.1062 | 0.1257 | 0.1228 | 0.1226 | 0.0002 | 0.0005 | 0.0005 | ||
Current LW3 (nl = 4) | 1.6956 | −0.1061 | 0.1257 | 0.1226 | 0.1226 | 0.0009 | 0.0005 | 0.0005 | ||
Current LW3 (nl = 8) | 1.6956 | −0.1061 | 0.1230 | 0.1226 | 0.1226 | 0.0005 | 0.0005 | 0.0005 | ||
Current LW3 (nl = 16) | 1.6956 | −0.1061 | 0.1226 | 0.1226 | 0.1226 | 0.0005 | 0.0005 | 0.0005 |
Boundary Conditions | Current LW3 FEM (nx × nz) and Analytical Solutions | 103 (L/2, −h/2) | |||
---|---|---|---|---|---|
230 | S–S | Current LW3 FEM (16 × 4) | 9.7651 | −1.5573 | 11.1888 |
Current LW3 FEM (32 × 8) | 9.7661 | −1.5584 | 6.2978 | ||
Current LW3 FEM (64 × 8) | 9.7662 | −1.5582 | 5.0544 | ||
Current LW3 FEM (128 × 8) | 9.7662 | −1.5581 | 4.7425 | ||
Current LW3 FEM (256 × 8) | 9.7662 | −1.5581 | 4.6729 | ||
Current LW3 analytical solutions | 9.7662 | −1.5581 | 4.6379 | ||
100 | C−C | Current LW3 FEM (16 × 4) | 0.2911 | −0.5075 | 0.7075 |
Current LW3 FEM (32 × 8) | 0.2920 | −0.5096 | 0.2412 | ||
Current LW3 FEM (64 × 8) | 0.2923 | −0.5100 | 0.1254 | ||
Current LW3 FEM (128 × 8) | 0.2924 | −0.5101 | 0.0963 | ||
Current LW3 FEM (256 × 8) | 0.2924 | −0.5101 | 0.0890 | ||
100 | S−S | Current LW3 FEM (16 × 4) | 1.6954 | −0.1036 | 0.6970 |
Current LW3 FEM (32 × 8) | 1.6956 | −0.1055 | 0.2413 | ||
Current LW3 FEM (64 × 8) | 1.6956 | −0.1060 | 0.1255 | ||
Current LW3 FEM (128 × 8) | 1.6956 | −0.1061 | 0.0964 | ||
Current LW3 FEM (256 × 8) | 1.6956 | −0.1061 | 0.0884 | ||
Current LW3 analytical solutions | 1.6956 | −0.1061 | 0.0867 | ||
100 | C−S | Current LW3 FEM (16 × 4) | 0.6607 | 0.2565 | 3.4814 |
Current LW3 FEM (32 × 8) | 0.6614 | 0.2559 | 2.8198 | ||
Current LW3 FEM (64 × 8) | 0.6616 | 0.2558 | 2.6534 | ||
Current LW3 FEM (128 × 8) | 0.6617 | 0.2558 | 2.6117 | ||
Current LW3 FEM (256 × 8) | 0.6617 | 0.2558 | 2.6009 | ||
100 | C−F | Current LW3 FEM (16 × 4) | −1.0853 | −0.1036 | 0.6973 |
Current LW3 FEM (32 × 8) | −1.0850 | −0.1055 | 0.2413 | ||
Current LW3 FEM (64 × 8) | −1.0850 | −0.1060 | 0.1255 | ||
Current LW3 FEM (128 × 8) | −1.0850 | −0.1061 | 0.0964 | ||
Current LW3 FEM (256 × 8) | −1.0850 | −0.1061 | 0.0892 |
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Wu, C.-P.; Xu, Z.-R. Strong and Weak Formulations of a Mixed Higher-Order Shear Deformation Theory for the Static Analysis of Functionally Graded Beams under Thermo-Mechanical Loads. J. Compos. Sci. 2020, 4, 158. https://doi.org/10.3390/jcs4040158
Wu C-P, Xu Z-R. Strong and Weak Formulations of a Mixed Higher-Order Shear Deformation Theory for the Static Analysis of Functionally Graded Beams under Thermo-Mechanical Loads. Journal of Composites Science. 2020; 4(4):158. https://doi.org/10.3390/jcs4040158
Chicago/Turabian StyleWu, Chih-Ping, and Zhan-Rong Xu. 2020. "Strong and Weak Formulations of a Mixed Higher-Order Shear Deformation Theory for the Static Analysis of Functionally Graded Beams under Thermo-Mechanical Loads" Journal of Composites Science 4, no. 4: 158. https://doi.org/10.3390/jcs4040158