Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method
The typically-used element torsional stiffness <i>GJ</i>/<i>L</i> (where <i>G</i> is the shear modulus, <i>J</i> the St. Venant torsion constant, and <i>L</i> the element length) may severely underestimate the torsional stiffness of thin-wa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Nanomaterials |
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| Online Access: | https://www.mdpi.com/2079-4991/12/3/538 |
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| author | Wen-Hao Pan Chuan-Hao Zhao Yuan Tian Kai-Qi Lin |
| author_facet | Wen-Hao Pan Chuan-Hao Zhao Yuan Tian Kai-Qi Lin |
| author_sort | Wen-Hao Pan |
| collection | DOAJ |
| description | The typically-used element torsional stiffness <i>GJ</i>/<i>L</i> (where <i>G</i> is the shear modulus, <i>J</i> the St. Venant torsion constant, and <i>L</i> the element length) may severely underestimate the torsional stiffness of thin-walled nanostructural members, due to neglecting element warping deformations. In order to investigate the exact element torsional stiffness considering warping deformations, this paper presents a matrix stiffness method for the torsion and warping analysis of beam-columns. The equilibrium analysis of an axial-loaded torsion member is conducted, and the torsion-warping problem is solved based on a general solution of the established governing differential equation for the angle of twist. A dimensionless factor is defined to consider the effect of axial force and St. Venant torsion. The exact element stiffness matrix governing the relationship between the element-end torsion/warping deformations (angle and rate of twist) and the corresponding stress resultants (torque and bimoment) is derived based on a matrix formulation. Based on the matrix stiffness method, the exact element torsional stiffness considering the interaction of torsion and warping is derived for three typical element-end warping conditions. Then, the exact element second-order stiffness matrix of three-dimensional beam-columns is further assembled. Some classical torsion-warping problems are analyzed to demonstrate the established matrix stiffness method. |
| first_indexed | 2024-03-09T23:23:55Z |
| format | Article |
| id | doaj.art-e6a1a353216d4a37abd3e09232d4aa0f |
| institution | Directory Open Access Journal |
| issn | 2079-4991 |
| language | English |
| last_indexed | 2024-03-09T23:23:55Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Nanomaterials |
| spelling | doaj.art-e6a1a353216d4a37abd3e09232d4aa0f2023-11-23T17:22:26ZengMDPI AGNanomaterials2079-49912022-02-0112353810.3390/nano12030538Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness MethodWen-Hao Pan0Chuan-Hao Zhao1Yuan Tian2Kai-Qi Lin3College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, ChinaCollege of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, ChinaDepartment of Civil Engineering, Tsinghua University, Beijing 100084, ChinaCollege of Civil Engineering, Fuzhou University, Fuzhou 350000, ChinaThe typically-used element torsional stiffness <i>GJ</i>/<i>L</i> (where <i>G</i> is the shear modulus, <i>J</i> the St. Venant torsion constant, and <i>L</i> the element length) may severely underestimate the torsional stiffness of thin-walled nanostructural members, due to neglecting element warping deformations. In order to investigate the exact element torsional stiffness considering warping deformations, this paper presents a matrix stiffness method for the torsion and warping analysis of beam-columns. The equilibrium analysis of an axial-loaded torsion member is conducted, and the torsion-warping problem is solved based on a general solution of the established governing differential equation for the angle of twist. A dimensionless factor is defined to consider the effect of axial force and St. Venant torsion. The exact element stiffness matrix governing the relationship between the element-end torsion/warping deformations (angle and rate of twist) and the corresponding stress resultants (torque and bimoment) is derived based on a matrix formulation. Based on the matrix stiffness method, the exact element torsional stiffness considering the interaction of torsion and warping is derived for three typical element-end warping conditions. Then, the exact element second-order stiffness matrix of three-dimensional beam-columns is further assembled. Some classical torsion-warping problems are analyzed to demonstrate the established matrix stiffness method.https://www.mdpi.com/2079-4991/12/3/538matrix stiffness methodelement stiffness matrixtorsionwarpingequilibrium analysiselastic buckling analysis |
| spellingShingle | Wen-Hao Pan Chuan-Hao Zhao Yuan Tian Kai-Qi Lin Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method Nanomaterials matrix stiffness method element stiffness matrix torsion warping equilibrium analysis elastic buckling analysis |
| title | Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method |
| title_full | Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method |
| title_fullStr | Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method |
| title_full_unstemmed | Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method |
| title_short | Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method |
| title_sort | exact solutions for torsion and warping of axial loaded beam columns based on matrix stiffness method |
| topic | matrix stiffness method element stiffness matrix torsion warping equilibrium analysis elastic buckling analysis |
| url | https://www.mdpi.com/2079-4991/12/3/538 |
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