Innovative Insights on the Thin Square Plate Large Deflection Problem
Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, a...
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| Format: | Article |
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MDPI AG
2023-10-01
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| Series: | Materials |
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| Online Access: | https://www.mdpi.com/1996-1944/16/21/6967 |
| _version_ | 1797631633002594304 |
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| author | Gilad Hakim Haim Abramovich |
| author_facet | Gilad Hakim Haim Abramovich |
| author_sort | Gilad Hakim |
| collection | DOAJ |
| description | Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, as displayed in the well-known von Kármán equations set. The present study aims at providing explicit expressions for the membrane stresses, the deflections, and the Airy stress function for a general square plate area vertically uniformly loaded to reach large deflection state. This was obtained by using the results of a high-fidelity finite element analysis applied on a lateral loaded simply supported thin square plate, which are then casted to yield approximate Fourier series expressions for the membrane stresses, deflections, and the Airy stress function. The stress map figures provide a good understanding of the critical points on the plate, while the explicit mathematical expressions enabled the calculation of deflections and stresses for the entire plate area. Among other interesting findings, the presence of relatively high tensile and compressive membrane stresses existing near the plate edges was revealed, which might lead to potential failure hazards. The derivatives of the deflections and the Airy stress function enabled the validation of the large deflections von Kármán equations set for the investigated case, and it turned out that the generated expressions for the stresses and the lateral deflection based on a high-fidelity finite element model satisfy the second equation with a good accuracy, while the first one remains to further be investigated. Moreover, using the generated explicit equations, the load influence on the deflections and stresses was also analyzed to yield general novel expressions for the medium and very large deflections states. To generalize the investigated case, the stresses and the deflections were non-dimensionalized so they can be used for any material and plate dimensions. |
| first_indexed | 2024-03-11T11:26:10Z |
| format | Article |
| id | doaj.art-485f47793c3b49f8bf931d8b3e1312cb |
| institution | Directory Open Access Journal |
| issn | 1996-1944 |
| language | English |
| last_indexed | 2024-03-11T11:26:10Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Materials |
| spelling | doaj.art-485f47793c3b49f8bf931d8b3e1312cb2023-11-10T15:07:29ZengMDPI AGMaterials1996-19442023-10-011621696710.3390/ma16216967Innovative Insights on the Thin Square Plate Large Deflection ProblemGilad Hakim0Haim Abramovich1Technion Faculty of Aerospace Engineering, Israel Institute of Technology, I.I.T., Haifa 32000, IsraelTechnion Faculty of Aerospace Engineering, Israel Institute of Technology, I.I.T., Haifa 32000, IsraelThin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, as displayed in the well-known von Kármán equations set. The present study aims at providing explicit expressions for the membrane stresses, the deflections, and the Airy stress function for a general square plate area vertically uniformly loaded to reach large deflection state. This was obtained by using the results of a high-fidelity finite element analysis applied on a lateral loaded simply supported thin square plate, which are then casted to yield approximate Fourier series expressions for the membrane stresses, deflections, and the Airy stress function. The stress map figures provide a good understanding of the critical points on the plate, while the explicit mathematical expressions enabled the calculation of deflections and stresses for the entire plate area. Among other interesting findings, the presence of relatively high tensile and compressive membrane stresses existing near the plate edges was revealed, which might lead to potential failure hazards. The derivatives of the deflections and the Airy stress function enabled the validation of the large deflections von Kármán equations set for the investigated case, and it turned out that the generated expressions for the stresses and the lateral deflection based on a high-fidelity finite element model satisfy the second equation with a good accuracy, while the first one remains to further be investigated. Moreover, using the generated explicit equations, the load influence on the deflections and stresses was also analyzed to yield general novel expressions for the medium and very large deflections states. To generalize the investigated case, the stresses and the deflections were non-dimensionalized so they can be used for any material and plate dimensions.https://www.mdpi.com/1996-1944/16/21/6967square thin platelarge deflectionvon Kármán equationsmembrane stressnon-linear load–deflection curvefinite element analysis |
| spellingShingle | Gilad Hakim Haim Abramovich Innovative Insights on the Thin Square Plate Large Deflection Problem Materials square thin plate large deflection von Kármán equations membrane stress non-linear load–deflection curve finite element analysis |
| title | Innovative Insights on the Thin Square Plate Large Deflection Problem |
| title_full | Innovative Insights on the Thin Square Plate Large Deflection Problem |
| title_fullStr | Innovative Insights on the Thin Square Plate Large Deflection Problem |
| title_full_unstemmed | Innovative Insights on the Thin Square Plate Large Deflection Problem |
| title_short | Innovative Insights on the Thin Square Plate Large Deflection Problem |
| title_sort | innovative insights on the thin square plate large deflection problem |
| topic | square thin plate large deflection von Kármán equations membrane stress non-linear load–deflection curve finite element analysis |
| url | https://www.mdpi.com/1996-1944/16/21/6967 |
| work_keys_str_mv | AT giladhakim innovativeinsightsonthethinsquareplatelargedeflectionproblem AT haimabramovich innovativeinsightsonthethinsquareplatelargedeflectionproblem |