The Behavior of Long Thin Rectangular Plates under Normal Pressure—A Thorough Investigation
Abstract
:1. Introduction
2. Materials, Methods, and Results
3. Discussion
4. Conclusions
- High-AR rectangular plates (above a certain value) with movable edges cannot be considered infinitely long plates, but rather, the end influences are preserved near the ends.
- The midpoint deflections of the SSSS-M case with 10 < AR < 20 and of the CCCC-M case with 6 < AR < 15 are higher than that of an infinitely long plate. A plausible explanation for this phenomenon might be the unusual shape of the deflections for these unique boundary conditions.
- This behavior has also been detected for small-deflection theory based on the classical Navier solution and all-around clamped plates in the small-deflection regime.
- The influence of the plate’s ends on the deflection and on the in-plane stresses penetrates from the far ends into the plate’s midpoint in different ways.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. (Based on [21])
- Finite element analysis (FEA) process description
- Model preparation:
- Geometry: create surfaces with the shape and dimensions of the requested plates—squares and rectangles.
- Define the material for the plate: isotropic polycarbonate with an E modulus of 2.4 GPa and Poisson’s ratio of 0.38. These two constants are enough here.
- Define the property for the plate element using the material from step 2 with a thickness of 0.005 m (5 mm); give all other inputs the default values.
- Mesh the surface with 50 × 50 mm quad plate elements using the property from step 3.
- Set the distributed load on the surface; use force per area on surface for a directional vector load and pressure on surface for a follower load. Set the load value to 1000 Pa in the Z direction.
- Define the boundary conditions, which are called constraints in the software. TX-1, TY-2, and TZ-3 relate to translation in the x,y,z directions. RX-4, RY-5, and RZ-6 relate to rotations around the x,y,z axes. The four types of BCs (simply supported, clamped, movable, immovable) are alternatively set on the surface perimeter:
- SSSS-M: TZ, designated 3 (in-plane movable BC);
- CCCC-M: TZ, RX, RY, designated 345 (in-plane movable BC);
- SSSS-I: TX, TY, TZ, designated T (in-plane immovable BC);
- CCCC-I: TX, TY, TZ, RX, RY, RZ, designated F (in-plane immovable BC).
Additionally, the FEA process requires that all free-body degrees of freedom (DOF) will be eliminated. For that, virtual (not participating) BCs are added where necessary: the plate midpoint may have a TX and TY designated 12, and one edge may have an additional TX or TY. - The last step is to define the type of analysis to run. Since large deflections with nonlinear effects are expected, we chose the analysis type “10..Nonlinear Static”. This analysis type has many input parameters that control its operation. From these, we set “Increments or Time Steps” to 20, and in “Output Control”, we collected “All” intermediate results. These settings enabled us to later see how various plate features developed as the load increased.
- Analyzing the model:
- Post-processing:
- Total translation (deflection);
- Plate X membrane force;
- Plate Y membrane force;
- Plate XY membrane force.
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Young’s modulus E | 2.4 GPa |
Poisson’s ratio ν | 0.38 |
Thickness h | 0.005 m |
Length a | varies |
Width b | 1 m |
Distributed load q | 100 Pa |
The calculated bending rigidity D D = Eh3/12(1 − ν2) | 29.219 Nm |
Summation indexes m and n | 1, 3, 5, …, 31 |
Equivalent Eyt modulus | 228.56 | MPa |
Equivalent Ext modulus | 318.77 | MPa |
Equivalent Gxyt modulus | 83.417 | MPa |
Poisson’s ratio νxyt | 0.37 | |
Equivalent Exb modulus | 647.69 | MPa |
Equivalent Eyb modulus | 370.51 | MPa |
Equivalent Gxyb modulus | 13.55 | MPa |
Poisson’s ratio νxyb | 0.37 | |
Shear rigidity Sx | 136.27 | N/mm |
Shear rigidity Sy | 3.8697 | N/mm |
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Hakim, G.; Abramovich, H. The Behavior of Long Thin Rectangular Plates under Normal Pressure—A Thorough Investigation. Materials 2024, 17, 2902. https://doi.org/10.3390/ma17122902
Hakim G, Abramovich H. The Behavior of Long Thin Rectangular Plates under Normal Pressure—A Thorough Investigation. Materials. 2024; 17(12):2902. https://doi.org/10.3390/ma17122902
Chicago/Turabian StyleHakim, Gilad, and Haim Abramovich. 2024. "The Behavior of Long Thin Rectangular Plates under Normal Pressure—A Thorough Investigation" Materials 17, no. 12: 2902. https://doi.org/10.3390/ma17122902
APA StyleHakim, G., & Abramovich, H. (2024). The Behavior of Long Thin Rectangular Plates under Normal Pressure—A Thorough Investigation. Materials, 17(12), 2902. https://doi.org/10.3390/ma17122902