Exact Bending Solutions of Rectangular Moderately Thick Plates Resting on 2-Parameter Foundations With 4 Edges Free With the Finite Integral Transform Method
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摘要: 弹性地基上的中厚板是一类重要的工程承载结构，其承载后弯曲行为的研究具有重要理论意义和实用价值. 同时考虑反力系数和剪切模量的双参数弹性地基模型可以准确描述板和地基之间的相互作用，利用二维有限积分变换方法推导出了双参数地基上四边自由矩形中厚板弯曲问题的精确解. 由于在求解过程中不需要预先人为地选取位移函数，而是直接从问题的基本方程出发，利用有限积分变换的数学方法求出满足四边自由边界条件的精确解，因此问题的求解更加严格. 通过计算实例验证了有限积分变换得到的精确解的正确性，相应的参数分析可以为工程设计提供理论基础.Abstract: Moderately thick plates resting on elastic foundations are an important type of engineering load-bearing structure. The study of their bending behaviors under loads has significant theoretical significance and practical value. The 2-parameter elastic foundation with both the reaction constant and the shear modulus can accurately model the interaction between the plate and the foundation. With the 2D finite integral transform technique, the exact solutions of the displacements and internal forces of a moderately thick rectangular plate with all 4 edges free and supported by a 2-parameter elastic foundation, were derived. The displacement function was not manually selected in advance during the solution process, and instead, the exact solution satisfying the free boundary conditions on all 4 edges was derived directly from the fundamental equations for the problem with the finite integral transform method. The results show that, the exact solution is more rigorous. The accuracy of the exact solution derived from the finite integral transform was validated through computational examples. The presented parameter analysis can provide a theoretical basis for engineering design.
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Key words:
- 2-parameter foundation /
- 4 edges free /
- moderately thick plate /
- finite integral transform /
- exact solution
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图 1 矩形板坐标系示意图
Figure 1. Schematic diagram of the coordinate system on a rectangular plate
图 2 板中心矩形区域载荷分布示意图
Figure 2. Schematic diagram of the load distribution in the rectangular area at the plate center
图 3 不同地基剪切刚度的板中心挠度随长宽比变化
Figure 3. Central deflections of the plate with different foundation shear moduli vs. the aspect ratio
图 4 不同地基剪切刚度的板中心应力随长宽比变化
Figure 4. Central stresses of the plate with different foundation shear moduli vs. the aspect ratio
表 1 中线挠度结果收敛性分析
Table 1. The convergence analysis of deflection results on the central line
M(=N) W/(10-8 m) 5 10 15 20 25 30 x=0.5a 0.592 0.592 0.592 0.591 0.591 0.591 x=0.3a 0.560 0.560 0.560 0.560 0.560 0.560 x=0.1a 0.491 0.494 0.495 0.494 0.494 0.494 
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表 2 中线应力结果收敛性分析
Table 2. The convergence analysis of stress results on the central line
M(=N) σx/Pa 150 200 250 300 350 400 x=0.5a 0.805 0.805 0.807 0.807 0.806 0.806 x=0.3a 0.545 0.545 0.546 0.546 0.545 0.545 x=0.1a 0.029 0.029 0.029 0.029 0.029 0.029
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表 3 中厚板的挠度及应力值(y=0.5)
Table 3. The deflections and stresses of the moderately thick plate (y=0.5)
x W/(10-8 m) σx/Pa FDM[3] Fourier series[8] superposition method[10] present FDM[3] Fourier series[8] superposition method[10] present 0.5 0.581 0.576 0.591 0.591 0.821 0.928 0.823 0.806 0.4 0.574 0.568 0.583 0.583 0.753 0.866 0.762 0.744 0.3 0.550 0.546 0.559 0.560 0.553 0.668 0.567 0.545 0.2 0.528 0.530 0.523 0.524 0.267 0.352 0.256 0.227 0.1 0.501 0.501 0.494 0.494 0.027 0.154 0.061 0.029 0 0.483 0.477 0.474 0.475 0 0.091 0 0
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