The Torsional Rigidity of a Rectangular Prism
Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced b...
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/13/2194 |
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| author | Cho-Liang Tsai Chih-Hsing Wang Sun-Fa Hwang Wei-Tong Chen Chin-Yi Cheng |
| author_facet | Cho-Liang Tsai Chih-Hsing Wang Sun-Fa Hwang Wei-Tong Chen Chin-Yi Cheng |
| author_sort | Cho-Liang Tsai |
| collection | DOAJ |
| description | Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko’s process. To confirm the correctness of Timoshenko’s solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko’s process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko’s or other researchers’ solutions, either for torsional rigidity or for the induced shear stresses and displacements. |
| first_indexed | 2024-03-09T04:02:48Z |
| format | Article |
| id | doaj.art-b38bdf59aad3411390f6d7f194584979 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:02:48Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b38bdf59aad3411390f6d7f1945849792023-12-03T14:11:42ZengMDPI AGMathematics2227-73902022-06-011013219410.3390/math10132194The Torsional Rigidity of a Rectangular PrismCho-Liang Tsai0Chih-Hsing Wang1Sun-Fa Hwang2Wei-Tong Chen3Chin-Yi Cheng4Department of Civil and Construction Engineering, National Yunlin University of Science and Technology, Yunlin 64002, TaiwanCollege of Future, Bachelor Program in Industrial Projects, National Yunlin University of Science and Technology, Yunlin 64002, TaiwanDepartment of Mechanical Engineering, National Yunlin University of Science and Technology, Yunlin 64002, TaiwanDepartment of Civil and Construction Engineering, National Yunlin University of Science and Technology, Yunlin 64002, TaiwanDepartment of Mechanical Engineering, National Yunlin University of Science and Technology, Yunlin 64002, TaiwanUsing the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko’s process. To confirm the correctness of Timoshenko’s solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko’s process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko’s or other researchers’ solutions, either for torsional rigidity or for the induced shear stresses and displacements.https://www.mdpi.com/2227-7390/10/13/2194energy conservationrectangular prismtorsional rigidityTSAI technique |
| spellingShingle | Cho-Liang Tsai Chih-Hsing Wang Sun-Fa Hwang Wei-Tong Chen Chin-Yi Cheng The Torsional Rigidity of a Rectangular Prism Mathematics energy conservation rectangular prism torsional rigidity TSAI technique |
| title | The Torsional Rigidity of a Rectangular Prism |
| title_full | The Torsional Rigidity of a Rectangular Prism |
| title_fullStr | The Torsional Rigidity of a Rectangular Prism |
| title_full_unstemmed | The Torsional Rigidity of a Rectangular Prism |
| title_short | The Torsional Rigidity of a Rectangular Prism |
| title_sort | torsional rigidity of a rectangular prism |
| topic | energy conservation rectangular prism torsional rigidity TSAI technique |
| url | https://www.mdpi.com/2227-7390/10/13/2194 |
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