用交叉梁系比拟求解正交各向异性薄板弯曲问题

袁全; 袁驷

doi:10.21656/1000-0887.440232

用交叉梁系比拟求解正交各向异性薄板弯曲问题

doi: 10.21656/1000-0887.440232

袁全^,,
袁驷

清华大学土木工程系，北京 100084

基金项目:

国家自然科学基金 51878383

国家自然科学基金 51378293

详细信息

通讯作者:
袁全(1993—)，男，博士(通讯作者. E-mail: quany@tsinghua.edu.cn)

中图分类号: O342
计量
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- PDF下载量: 82
- 被引次数: 0
出版历程
- 收稿日期: 2023-07-29
- 修回日期: 2023-11-29
- 刊出日期: 2024-05-01

Analogic Analysis of Orthotropic Plate Bending Problems With Gridwork Systems

YUAN Quan^,,
YUAN Si

Department of Civil Engineering, Tsinghua University, Beijing 100084, P.R.China

摘要

摘要: 采用交叉梁系结构比拟求解正交各向异性薄板结构，给出了两种结构在静力分析和自由振动分析中的相容性条件；对于满足相容性条件且仅含有简支和固支组合边界的相容问题，论证了其解答随着交叉梁系网格加密可收敛到正交各向异性板的理论解. 进一步建立了所有类型内力的计算公式，并给出了采用3D结构力学求解器求解的算法实施和数值算例(包括矩形板和圆形板问题)，用以验证理论分析的正确性.
- 交叉梁系 /
- 正交各向异性薄板 /
- 相容性条件 /
- 3D结构力学求解器 /
- 数值计算
Abstract: Orthotropic plates were analogized to gridwork systems, and the compatibility and incompatibility between these 2 structure types in static and dynamic analyses were discussed. The results show that, the gridwork results converge to the theoretical solution of plates through the grid refinement for compatible problems with simply supported and/or clamped boundaries. All the formulas for calculation of various internal forces were derived, and the algorithms required in solving the gridwork systems were proposed by means of the 3D structural mechanics solver. A wide range of numerical examples, including rectangular and circular plates, were given to verify the validity of the proposed theory and algorithms.
- gridwork /
- orthotropic thin plate /
- compatibility /
- 3D structural mechanics solver /
- numerical computation

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图 1 交叉梁系结构

Figure 1. The gridwork system

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图 2 圆形板的8×8网格

Figure 2. The 8×8 grid of a circular plate

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图 3 不同网格前70阶频率系数误差分布

Figure 3. Error distributions of the 1st 70 frequency coefficients on different grids

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图 4 第5阶和第10阶振型(16×16网格)

Figure 4. The 5th and 10th vibration modes (16×16 grid)

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表 1 内力计算公式汇总

Table 1. Formulas for calculation of internal forces

	node average	formulas
bending moments	$\tilde{M}_x=\frac{\tilde{M}_{x-}+\tilde{M}_{x+}}{2 b}, \tilde{M}_y=\frac{\tilde{M}_{y-}+\tilde{M}_{y+}}{2 b}$	$M_x=\tilde{M}_x+\frac{D_1}{D_y} \tilde{M}_y, M_y=\tilde{M}_y+\frac{D_1}{D_x} \tilde{M}_x$
torques	$\widetilde{T}_{x y}=\frac{\widetilde{T}_{x-}+\widetilde{T}_{x+}+\widetilde{T}_{y+}+\widetilde{T}_{y-}}{4 b}$	$M_{x y}=-\frac{2 D_{x y}}{H} \widetilde{T}_{x y}$
shear forces	$\widetilde{Q}_x=\frac{\widetilde{Q}_{x-}+\widetilde{Q}_{x+}}{2 b}, \widetilde{Q}_y=\frac{\widetilde{Q}_{y-}+\widetilde{Q}_{y+}}{2 b}$	$Q_x=\widetilde{Q}_x, Q_y=\widetilde{Q}_y$

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表 2 ε=1.2时，工况Ⅰ的四边简支板的结果(a/b=1.2, $\sqrt[4]{D_y / D_x}$=1)

Table 2. Results of simply supported square plates under uniform loads for case Ⅰ with ε=1.2 (a/b=1.2, $\sqrt[4]{D_y / D_x}$=1)

N_x×N_y	α	error δ/%	β₁	error δ/%	β₂	error δ/%
12×10	0.532 5	5.75	3.316	3.61	5.048	3.66
24×20	0.546 8	3.22	3.365	2.19	5.115	2.49
48×40	0.555 4	1.70	3.398	1.23	5.169	1.36
96×80	0.560 1	0.87	3.417	0.68	5.201	0.75
analytical solution^[1]	0.565		3.44		5.24

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表 3 ε=1.2时，工况Ⅱ的四边简支板的结果(a/b=1, $\sqrt[4]{D_y / D_x}$=1.2)

Table 3. Results of simply supported square plates under uniform loads for case Ⅱ with ε=1.2 (a/b=1, $\sqrt[4]{D_y / D_x}$=1.2)

N×N	α	error δ/%	β₁	error δ/%	β₂	error δ/%
4×4	0.507 3	10.2	3.331	3.15	5.137	1.97
8×8	0.525 4	7.00	3.325	3.34	5.016	4.28
16×16	0.541 7	4.13	3.359	2.36	5.077	3.12
32×32	0.552 4	2.24	3.392	1.39	5.144	1.83
64×64	0.558 4	1.16	3.413	0.79	5.187	1.01
analytical solution^[1]	0.565		3.44		5.24

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表 4 ε=2时，工况Ⅰ的四边简支板的结果(a/b=2, $\sqrt[4]{D_y / D_x}$=1)

Table 4. Results of simply supported square plates under uniform loads for case Ⅰ with ε=2 (a/b=2, $\sqrt[4]{D_y / D_x}$=1)

N_x×N_y	α	error δ/%	β₁	error δ/%	β₂	error δ/%
8×4	0.933 9	7.81	1.495	14.1	9.381	2.69
16×8	0.970 1	4.24	1.684	3.22	9.386	2.63
32×16	0.990 4	2.23	1.728	0.69	9.483	1.63
64×32	1.001 3	1.15	1.739	0.08	9.556	0.87
analytical solution^[1]	1.013		1.74		9.64

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表 5 ε=2时，工况Ⅱ的四边简支板的结果(a/b=1, $\sqrt[4]{D_y / D_x}$=2)

Table 5. Results of simply supported square plates under uniform loads for case Ⅱ with ε=2 (a/b=1, $\sqrt[4]{D_y / D_x}$=2)

N×N	α	error δ/%	β₁	error δ/%	β₂	error δ/%
4×4	0.952 0	6.02	0.986	43.4	9.580	0.62
8×8	0.978 6	3.40	1.637	5.89	9.456	1.91
16×16	0.992 5	2.03	1.739	0.04	9.490	1.56
32×32	1.001 5	1.13	1.752	0.68	9.550	0.93
64×64	1.006 9	0.61	1.749	0.54	9.593	0.48
analytical solution^[1]	1.013		1.74		9.64

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表 6 固支圆板中心挠度和弯矩的结果(ε=1.2, H=$\sqrt{D_x D_y}$)

Table 6. Results of central deflections and bending moments of clamped circular plates (ε=1.2, H=$\sqrt{D_x D_y}$)

N×N	w_c	error δ/%	M_xc	error δ/%	M_yc	error δ/%
4×4	2.267	5.84	4.207	1.82	9.337	8.98
8×8	2.170	1.31	4.145	0.32	8.766	2.31
16×16	2.149	0.31	4.134	0.04	8.618	0.58
32×32	2.144	0.07	4.132	0.01	8.580	0.14
64×64	2.142	0.00	4.132	0.00	8.571	0.03
analytical solution^[1]	2.142		4.132		8.568
multiplier	qa⁴/(100D_y)		qa²/100		qa²/100

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表 7 固支圆板内点剪力和扭矩的结果(ε=1.2, H=$\sqrt{D_x D_y}$)

Table 7. Results of interior shear forces and tortional moments of clamped circular plates (ε=1.2, H=$\sqrt{D_x D_y}$)

N×N	Q_x(a/2, a/2)	error δ/%	Q_y(a/2, a/2)	error δ/%	M_xy(a/2, a/2)	error δ/%
4×4	18.462 3	0.64	32.775 4	3.54	2.714 7	8.75
8×8	18.036 7	1.68	31.943 9	0.92	2.962 6	0.42
16×16	18.287 5	0.31	31.662 9	0.03	2.976 1	0.04
32×32	18.329 2	0.09	31.647 6	0.02	2.975 4	0.01
64×64	18.340 8	0.02	31.652 0	0.01	2.975 2	0.00
analytical solution^[1]	18.345	31.654	2.975
multiplier	qa/100		qa/100		qa²/100

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表 8 固支圆板边界点剪力和扭矩的结果(ε=1.2, H=$\sqrt{D_x D_y}$)

Table 8. Results of boundary shear forces and tortional moments of clamped circular plates (ε=1.2, H=$\sqrt{D_x D_y}$)

N×N	Q_x(a, 0)	error δ/%	Q_y(0, a)	error δ/%	M_xy($\sqrt{3}$a/2, a/2)	error δ/%
4×4	37.703	2.76	59.821	5.51	4.136	19.73
8×8	36.950	0.70	61.331	3.12	4.817	6.53
16×16	36.620	0.20	62.141	1.84	4.816	6.54
32×32	36.528	0.45	62.653	1.04	5.000	2.96
64×64	36.553	0.37	62.953	0.54	5.091	1.15
analytical solution^[1]	36.69		63.31		5.15
multiplier	qa/100		qa/100		qa²/100

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表 9 固支圆板中心挠度和弯矩的结果(ε=2, H=$\sqrt{D_x D_y}$)

Table 9. Results of central deflections and bending moments of clamped circular plates (ε=2, H=$\sqrt{D_x D_y}$)

N×N	w_c	error δ/%	M_xc	error δ/%	M_yc	error δ/%
4×4	3.511	3.57	0.205 1	75.8	14.79	9.06
8×8	3.417	0.79	0.683 0	19.4	13.86	2.23
16×16	3.395	0.17	0.805 4	4.96	13.63	0.54
32×32	3.391	0.04	0.836 8	1.26	13.58	0.14
64×64	3.390	0.00	0.844 7	0.33	13.56	0.02
analytical solution^[1]	3.390		0.847 5		13.56
multiplier	qa⁴/(100D_y)		qa²/100		qa²/100

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表 10 固支圆板内点剪力和扭矩的结果(ε=2, H=$\sqrt{D_x D_y}$)

Table 10. Results of interior shear forces and tortional moments of clamped circular plates (ε=2, H=$\sqrt{D_x D_y}$)

N×N	Q_x(a/2, a/2)	error δ/%	Q_y(a/2, a/2)	error δ/%	M_xy(a/2, a/2)	error δ/%
4×4	9.104 9	53.48	45.098	2.34	1.437 4	15.19
8×8	5.551 9	6.41	44.721	1.48	1.652 2	2.52
16×16	5.876 9	0.93	44.107	0.09	1.689 4	0.33
32×32	5.923 2	0.15	44.076	0.02	1.693 6	0.08
64×64	5.929 9	0.04	44.071	0.00	1.694 6	0.01
analytical solution^[1]	5.932 2		44.068		1.694 9
multiplier	qa/100		qa/100		qa²/100

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表 11 固支圆板边界点剪力和扭矩的结果(ε=2, H=$\sqrt{D_x D_y}$)

Table 11. Results of boundary shear forces and tortional moments of clamped circular plates (ε=2, H=$\sqrt{D_x D_y}$)

N×N	Q_x(a, 0)	error δ/%	Q_y(0, a)	error δ/%	M_xy($\sqrt{3}$a/2, a/2)	error δ/%
4×4	18.482	55.77	79.800	9.46	2.370 8	19.24
8×8	14.532	22.49	87.166	1.10	2.793 0	4.86
16×16	12.924	8.93	85.884	2.56	2.772 4	5.56
32×32	12.253	3.28	86.990	1.30	2.866 1	2.37
64×64	11.971	0.90	87.556	0.66	2.911 3	0.83
analytical solution^[1]	11.864		88.136		2.935 7
multiplier	qa/100		qa/100		qa²/100

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表 12 均布荷载简支方板的结果(例3，a/b=1, $\sqrt[4]{D_y / D_x}$=$\sqrt{1/5}$, ν=0.25, H=1.25D_y)

Table 12. Result of simply supported square plates under uniform loads (example 3, a/b=1, $\sqrt[4]{D_y / D_x}$=$\sqrt{1/5}$, ν=0.25, H=1.25D_y)

N×N	w_c	error δ/%	M_xc	error δ/%	M_yc	error δ/%
4×4	0.613	3.7	0.1276 8	2.61	0.3871 2	4.81
8×8	0.637	2.0	0.1293 0	1.38	0.4116 0	1.21
16×16	0.645	0.7	0.1301 8	0.70	0.4133 7	1.65
32×32	0.648	0.3	0.1306 3	0.36	0.4111 4	1.10
64×64	0.650	0.0	0.1308 6	0.18	0.4094 9	0.70
analytical solution^[15]	0.65		0.131 1		0.406 7
multiplier	qa⁴/(1 200D_y)		qa²		qa²/100

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表 13 四边简支方板前10阶频率系数的收敛比较(a/b=1, $\sqrt[4]{D_y / D_x}$=1.2)

Table 13. Convergence comparison of the 1st 10 frequency coefficients of a simply supported plate (a/b=1, $\sqrt[4]{D_y / D_x}$=1.2)

order k	8×8	error δ/%	16×16	error δ/%	32×32	error δ/%	64×64	error δ/%	exact solution
1	17.192	2.80	17.052	1.96	16.911	1.12	16.823	0.597	16.72
2	37.713	1.15	37.771	1.30	37.597	0.84	37.458	0.464	37.29
3	46.090	-0.52	46.575	0.52	46.547	0.46	46.463	0.281	46.33
4	66.522	-0.56	67.601	1.06	67.494	0.90	67.256	0.542	66.89
5	71.057	-0.70	71.870	0.44	71.858	0.42	71.742	0.262	71.55
6	92.493	-3.33	95.215	0.49	95.735	0.06	95.778	0.102	95.68
7	99.007	-2.13	101.749	0.58	101.938	0.77	101.673	0.504	101.16
8	112.074	-3.59	116.096	0.13	116.747	0.44	116.643	0.345	116.24
9	117.058	-2.07	119.392	0.12	119.737	0.17	119.703	0.143	119.53
10	143.185	-3.99	148.933	0.14	149.838	0.47	149.697	0.373	149.14

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表 14 前70阶频率系数的最大误差(=第70阶频率系数的误差)

Table 14. Max errors of the 1st 70 frequency coefficients (=the error of the 70th frequency coefficient)

mesh	8×8	16×16	32×32	64×64
max error of the first 70 frequency coefficients δ/%	24.8	8.0	1.9	0.42

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参考文献(15)

[1]	TIMOSHENKO S, WOINOWSKY-KRIEGER S. Theory of Plates and Shells[M]. 2nd ed. New Jersey: McGraw-Hill, 1987.
[2]	古国纪. 交叉梁系的计算[J]. 土木工程学报, 1958, 5(3): 165-178. https://www.cnki.com.cn/Article/CJFDTOTAL-TMGC195803000.htm GU Guoji. The analysis of caisson-beam[J]. China Civil Engineering Journal, 1958, 5(3): 165-178. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-TMGC195803000.htm
[3]	张玉敏, 鲍鹏, 崔奕. 板与交叉梁系的相似性及其简化计算[J]. 河南大学学报(自然科学版), 2002, 32(1): 83-86. doi: 10.3969/j.issn.1003-4978.2002.01.024 ZHANG Yumin, BAO Peng, CUI Yi. The theoretical similarity between a plate and a grillage beam[J]. Journal of Henan University (Natural Science), 2002, 32(1): 83-86. (in Chinese) doi: 10.3969/j.issn.1003-4978.2002.01.024
[4]	袁驷, 袁全. 交叉梁系比拟求解薄板弯曲问题的相容性及其计算[J]. 土木工程学报, 2023, 56(3): 1-8. https://www.cnki.com.cn/Article/CJFDTOTAL-TMGC202303001.htm YUAN Si, YUAN Quan. Compatibility in analogy of gridworks to thin plate bending and its numerical solution[J]. China Civil Engineering Journal, 2023, 56(3): 1-8. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-TMGC202303001.htm
[5]	王森林, 李进宝, 马红艳, 等. 基于辛叠加方法的正交各向异性矩形悬臂薄板受迫振动解析解[J/OL]. 应用数学和力学, 2023[2023-11-29]. https://link.cnki.net/urlid/50.1060.o3.20231110.1245.002. WANG Senlin, LI Jinbao, MA Hongyan, et al. Analytical forced vibration solutions of orthotropic cantilever rectangular thin plates by the symplectic superposition method[J]. Applied Mathematics and Mechanics, 2023[2023-11-29]. https://link.cnki.net/urlid/50.1060.o3.20231110.1245.002. (in Chinese)
[6]	黄炎, 雷勇军, 申慧君. 各向异性矩形板自由振动的一般解析解法[J]. 应用数学和力学, 2006, 27(4): 411-416. doi: 10.3321/j.issn:1000-0887.2006.04.005 HUANG Yan, LEI Yongjun, SHEN Huijun. Free vibration of anisotropic rectangular plates by general analytical method[J]. Applied Mathematics and Mechanics, 2006, 27(4): 411-416. (in Chinese) doi: 10.3321/j.issn:1000-0887.2006.04.005
[7]	李晶晶, 程昌钧. 考虑高阶横向剪切正交各向异性板非线性弯曲的微分求积分析[J]. 应用数学和力学, 2004, 25(8): 801-808. doi: 10.3321/j.issn:1000-0887.2004.08.005 LI Jingjing, CHENG Changjun. Differential quadrature method for bending of orthotropic plates with finite deformation and transverse shear effects[J]. Applied Mathematics and Mechanics, 2004, 25(8): 801-808. (in Chinese) doi: 10.3321/j.issn:1000-0887.2004.08.005
[8]	黄家寅. 变厚度正交各向异性矩形板非线性非对称弯曲问题的基本方程[J]. 应用数学和力学, 2004, 25(7): 741-744. doi: 10.3321/j.issn:1000-0887.2004.07.012 HUANG Jiayin. Basic equations of the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with variable thickness[J]. Applied Mathematics and Mechanics, 2004, 25(7): 741-744. (in Chinese) doi: 10.3321/j.issn:1000-0887.2004.07.012
[9]	黄家寅. 四边固定变厚度正交各向异性矩形板的非线性非对称弯曲问题的一致有效渐近解[J]. 应用数学和力学, 2004, 25(7): 745-754. doi: 10.3321/j.issn:1000-0887.2004.07.013 HUANG Jiayin. Uniformly valid asymptotic solutions of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate of four clamped edges with variable thickness[J]. Applied Mathematics and Mechanics, 2004, 25(7): 745-754. (in Chinese) doi: 10.3321/j.issn:1000-0887.2004.07.013
[10]	额布日力吐, 冯璐, 阿拉坦仓. 四边固支正交各向异性矩形薄板弯曲问题的辛叠加方法[J]. 应用数学和力学, 2018, 39(3): 311-323. doi: 10.21656/1000-0887.380092 EBURILITU, FENG Lu, ALATANCANG. Analytical bending solutions of clamped orthotropic rectangular thin plates with the symplectic superposition method[J]. Applied Mathematics and Mechanics, 2018, 39(3): 311-323. (in Chinese) doi: 10.21656/1000-0887.380092
[11]	王新志, 赵永刚, 叶开沅, 等. 正交各向异性板的非对称大变形问题[J]. 应用数学和力学, 2002, 23(9): 881-888. doi: 10.3321/j.issn:1000-0887.2002.09.001 WANG Xinzhi, ZHAO Yonggang, YEH Kaiyuan, et al. Unsymmetrical large deformation problem of orthotropic plates[J]. 应用数学和力学, 2002, 23(9): 881-888. (in Chinese) doi: 10.3321/j.issn:1000-0887.2002.09.001
[12]	袁驷. 结构力学Ⅰ: 基础教程(第4版) 求解器[EB/OL]. V3.0. 北京: 高等教育出版社. (2018-07-13)[2023-07-29]. https://abook.hep.com.cn/12202428. YUAN Si. Structural mechanical: fundamental course (4th ed) solver[EB/OL]. V3.0. Beijing: Higer Education Press. (2018-07-13) [2023-07-29]. (in Chineses)
[13]	龙驭球, 包世华, 袁驷. 结构力学Ⅰ[M]. 4版. 北京: 高等教育出版社, 2018. LONG Yuqiu, BAO Shihua, YUAN Si. Structure Mechanics [M]. 4th ed. Beijing: Higher Education Press, 2018. (in Chinese)
[14]	袁驷. 程序结构力学[M]. 2版. 北京: 高等教育出版社, 2008. YUAN Si. Programming Structural Mechanics[M]. 2nd ed. Beijing: Higher Education Press, 2008. (in Chinese)
[15]	REDDY J N. Theory and Analysis of Elastic Plates and Shells[M]. 2nd ed. Boca Raton: CRC Press, 2007.