任意面内荷载作用下圆板的弯曲、振动与屈曲分析
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摘要
本文应用能量法中的Ritz(里茨法)全面、彻底、细致地研究了任意面内力作用下圆板的力学特性，即弹性圆板在任意面内力作用下的横向静力弯曲、屈曲及自由振动特性。首先，导出了用极坐标系描述的一对集中力作用下圆板的面内应力方程，由力的叠加原理可以得到多对面内集中力作用下圆板的面内应力方程，由积分计算得到任意面内分布力作用下圆板的面内应力表达式。然后取幂级数和三角级数的积作为容许函数，应用Ritz(里茨法)分别导出了任意荷载作用下的静力弯曲、屈曲、振动的方程。通过编制相应的Fortran程序，从而求得了任意荷载作用下圆板的弯曲位移、屈曲荷载及固有频率。此方法不但能给出高精度的结果，而且收敛性能够保证、计算量小、分析过程较简单且编程也较容易。
本文主要以简支圆板在多对面内集中力与分布荷载作用下的情况为例进行了数值计算，结果充分说明了本文方法的正确性和可应用性。对于其它的各种情况，也做了一些相应的研究。这些算例清楚地表明，本文方法有很好的收敛性和很高的精度。
This paper uses Ritz theory and presents a method to solve the problem of bend, free vibration and buckling properties of the circular plates subjected to arbitrary forces. Firstly, the expressions of the in-plane stress of circular plates that is subjected to a pair of concentrated forces in polar coordinates are derived and the in-plane stress equations are obtained subjected to many pairs of concentrated forces and arbitrary forces. Secondly, the general matrix equations of bend, free vibration and buckling are obtained by the means of method of Ritz method. Then, this paper use Fortran to obtain solution of bend, free vibration and buckling of a circular plates. Finally, the roots of the bend, buckling, frequency of the equation can be numerically worked out. The method can give the results with high precision. The calculating cost is little, the course of the analysis is rather simple and the programming is easy.
The paper takes simply supported circular as example. The numerical results demonstrate the convergence and accuracy of the present method.
本文主要以简支圆板在多对面内集中力与分布荷载作用下的情况为例进行了数值计算，结果充分说明了本文方法的正确性和可应用性。对于其它的各种情况，也做了一些相应的研究。这些算例清楚地表明，本文方法有很好的收敛性和很高的精度。
This paper uses Ritz theory and presents a method to solve the problem of bend, free vibration and buckling properties of the circular plates subjected to arbitrary forces. Firstly, the expressions of the in-plane stress of circular plates that is subjected to a pair of concentrated forces in polar coordinates are derived and the in-plane stress equations are obtained subjected to many pairs of concentrated forces and arbitrary forces. Secondly, the general matrix equations of bend, free vibration and buckling are obtained by the means of method of Ritz method. Then, this paper use Fortran to obtain solution of bend, free vibration and buckling of a circular plates. Finally, the roots of the bend, buckling, frequency of the equation can be numerically worked out. The method can give the results with high precision. The calculating cost is little, the course of the analysis is rather simple and the programming is easy.
The paper takes simply supported circular as example. The numerical results demonstrate the convergence and accuracy of the present method.
引文
1．曹志远．杨昇田．厚板动力学理论及其应用．北京：科学出版社，1983．
2．吴连元．板壳稳定性理论．武汉：华中理工大学出版社，1996．
3．黄与宏．板结构．北京：人民交通出版社，1992．
4．陈昌平．数学物理方程．南京：南京理工大学出版社，1986．
5．何福保，沈亚朋．板壳理论．西安：西安交通大学出版社，1992．
6．王文亮．结构动力学．上海：复旦大学出版社，1991．
7．铁摩辛柯．板壳理论．北京：科学出版社，1977．
8．俞载道．结构动力学基础．上海：同济大学出版社，1985．
9．汉斯(著)，吴肇之(译)．实用薄板理论．北京：人民交通出版社，1982．
10．黄炎．弹性薄板理论．长沙：国防科技大学出版社，1982．
11．徐芝伦．弹性力学．北京：高等教育出版社，1990．
12．周正威译．工程结构的振动．上海：同济大学出版社，1991．
13．吴福光，蔡承武．振动理论．北京：高等教育出版社，1987．
14．周叮．一类变截面梁横向自由振动的精确解析解．振动与冲击．1996，15(3)：12—15．
15．周叮．点载静力梁函数在求解矩形薄板自振特性中的应用．振动与冲击．1996，15(1)：77—82．
16．周叮．边缘转动受弹性限制矩形板横向振动固有频率的一个近似解法．应用数学和力学．1996，17(5)：433—438．
17．周叮．两对边简支中间有任意多个单向弹性线支矩形板横向振动的一个解析解法．应用数学和力学．1996，17(8)：729—734．
18．周叮．任意形状弹性薄板弯曲的解析解．应用数学和力学．1996，17(12)：1123—1127．
19．周叮．环支圆板横振特性的一个新的解析解法南京理工大学学报．1995，19(3)：214—218．
20．周叮．受任意个同心弹性环支圆板横向自由振动的一个新的解析解法．工程力学．1994，11(3)：38—46．
21．周叮．任意载荷作用下矩形弹性薄板横向弯曲的一个高精度近似解．应用数学和力学．1993，14(3)：225—230．
22．周叮．两对边简支另外两对边任意支承的板内有任意多个点支矩形板横向振动的精确解析解．振动与冲击．1993，12(1)：12—15．
23．周叮．连续矩形板横向振动固有特性的计算．工程力学．1992，9(3)：124—130．
24．周叮．任意跨弹性支承直梁横向自振的一个新解法．工程力学．1991，8(4)：111—125．
25．周叮．板内附着任意个弹性质量的两对边简支矩形板的横向振动．强度与环境．1997，(2)：38—45．
26．周叮．边缘有弹性点支矩形板的横向振动．应用数学和力学．1989，10(10)：929—938．
27．周叮．两对边简支变厚度矩形板横向自振频率的渐近解法．强度与环境．1992，(3)：16—22．
28．李文深．圆形薄板的振动．黑龙江大学自然科学学报．1991，8(1)：47—50．
29．张英世．两边受拉压的矩形薄板的振动．北京航空航天大学学报．1999，25(1)：64—67．
30．余熙莹．圆板承受一般轴对称分布荷载的解析计算方法．力学与实践．2003，25(2)：67—68．
31. Rozsa, Stability Analysis of a Circular Plate Submitted to Two Compressive Forces Acting along a Diameter. Acta Technica Academiae Scientiarum Hungaricae. 1966, 55: 153-172.
32. Y. K. Cheung, D. Zhou Vibrations of tapered Mindlin plates in terms of static Timoshenko beam functions. Journal of Sound and Vibration. 2002
33. Zhou Ding. Vibration of an arbitrarily shaped membrane carrying elastically mounted masses. Computers and Structures. 1992, 42(3): 389-394:
34. Zhou Ding. The application of a type of new admissible functions to the vibration of rectangular plates. Computers and Structures. 1994, 52(2): 199-203.
35. Zhou Ding. Free vibration of arbitrarily shaped plates with elastic or/and rigid concentric ring supports. Computers and Structures. 1994, 50 (5): 685-792.
36. Zhou Ding. The vibrations of a cantilever beam carrying a heavy tip mass with elastic supports. Journal of Sound and Vibration. 1997, 206 (6): 275-279.
37. Y. K. Cheung, D. Zhou. The free vibrations of rectangular composite plates with point-supports using static beam functions. Composite Structures. 1999, 44(12): 145-154.
38. Y. Xiang. Exact vibration solutions for circular Mindlin plates with multiple concentric ring supports. International Journal of Solids and Structures. 2002, (39): 6081—6102.
39. D. Zhou. Vibrations of Mindlinrectangular plates with elastically restrained edges using static Timoshenko beam functions with the Rayleigh—Ritz method. International Journal of Solids and Structures. 2001, (38): 5565-5580.
40. Y. K. Cheung, D. Zhou. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers and Structures. 2000, (78): 757—768.
41. A.W. Leissa, E.F.Ayoud. Vibration and Buckling of a simply supported rectangular plate subjected to a pair of In-plane concentrated forces. Journal of Sound and Vibration. 1988, 127 (1): 155-171.
42. A.W.Leissa, E.F.Ayoud. Tension Buckling of rectangular sheets due to concentrated forces. ASCE Journal of Engineering Mechanics. 1989, 115(2): 2749-2762.
43. A.W. Leissa, Y. Narita. Nature frequencies of simply supported plates. Journal of Sound and Vibration. 1980, 70 (2): 221—229.
44. A.W .Leissa,. E.F. Ayoud. Free vibration and tension buckling of circular plates with diametral point forces. Journal of Applied mechanics. 1990, 57: 995-999.
45. A.W ,Leissa. Three-dimensional vibrations of thick circular and annular plates. Journal of Sound and Vibration. 1998, 209: 15-41.
46. Y. Narita. Vibration and Stability of circular plates under partially distributed or concentrated Inplane loads. ASME Journal of Apllied Mechanics. 1985, 52: 549-552.
47. David Durban. Instability of an elastic circular plate subjected to nonuniform loads. AIAA Journal. 1977, 15 (3): 360-365.
48. N. Yamaki. Buckling of circular plate under locally distributed forces Transactions of the Japanese Society of Mechanical Engeneers. 1956, 22 (115): 119-123.
49. E.F. Ayoud. Vibration and buckling of plates subjected to Inplane concentrated forces. Ph.D. Dissertation, Ohio State University. 1989.
50. N. Yamaki.Buckling of a thin Annular plate under uniform
compression. Journal of Applied Mechanics. 1958, 25: 267-273.
2．吴连元．板壳稳定性理论．武汉：华中理工大学出版社，1996．
3．黄与宏．板结构．北京：人民交通出版社，1992．
4．陈昌平．数学物理方程．南京：南京理工大学出版社，1986．
5．何福保，沈亚朋．板壳理论．西安：西安交通大学出版社，1992．
6．王文亮．结构动力学．上海：复旦大学出版社，1991．
7．铁摩辛柯．板壳理论．北京：科学出版社，1977．
8．俞载道．结构动力学基础．上海：同济大学出版社，1985．
9．汉斯(著)，吴肇之(译)．实用薄板理论．北京：人民交通出版社，1982．
10．黄炎．弹性薄板理论．长沙：国防科技大学出版社，1982．
11．徐芝伦．弹性力学．北京：高等教育出版社，1990．
12．周正威译．工程结构的振动．上海：同济大学出版社，1991．
13．吴福光，蔡承武．振动理论．北京：高等教育出版社，1987．
14．周叮．一类变截面梁横向自由振动的精确解析解．振动与冲击．1996，15(3)：12—15．
15．周叮．点载静力梁函数在求解矩形薄板自振特性中的应用．振动与冲击．1996，15(1)：77—82．
16．周叮．边缘转动受弹性限制矩形板横向振动固有频率的一个近似解法．应用数学和力学．1996，17(5)：433—438．
17．周叮．两对边简支中间有任意多个单向弹性线支矩形板横向振动的一个解析解法．应用数学和力学．1996，17(8)：729—734．
18．周叮．任意形状弹性薄板弯曲的解析解．应用数学和力学．1996，17(12)：1123—1127．
19．周叮．环支圆板横振特性的一个新的解析解法南京理工大学学报．1995，19(3)：214—218．
20．周叮．受任意个同心弹性环支圆板横向自由振动的一个新的解析解法．工程力学．1994，11(3)：38—46．
21．周叮．任意载荷作用下矩形弹性薄板横向弯曲的一个高精度近似解．应用数学和力学．1993，14(3)：225—230．
22．周叮．两对边简支另外两对边任意支承的板内有任意多个点支矩形板横向振动的精确解析解．振动与冲击．1993，12(1)：12—15．
23．周叮．连续矩形板横向振动固有特性的计算．工程力学．1992，9(3)：124—130．
24．周叮．任意跨弹性支承直梁横向自振的一个新解法．工程力学．1991，8(4)：111—125．
25．周叮．板内附着任意个弹性质量的两对边简支矩形板的横向振动．强度与环境．1997，(2)：38—45．
26．周叮．边缘有弹性点支矩形板的横向振动．应用数学和力学．1989，10(10)：929—938．
27．周叮．两对边简支变厚度矩形板横向自振频率的渐近解法．强度与环境．1992，(3)：16—22．
28．李文深．圆形薄板的振动．黑龙江大学自然科学学报．1991，8(1)：47—50．
29．张英世．两边受拉压的矩形薄板的振动．北京航空航天大学学报．1999，25(1)：64—67．
30．余熙莹．圆板承受一般轴对称分布荷载的解析计算方法．力学与实践．2003，25(2)：67—68．
31. Rozsa, Stability Analysis of a Circular Plate Submitted to Two Compressive Forces Acting along a Diameter. Acta Technica Academiae Scientiarum Hungaricae. 1966, 55: 153-172.
32. Y. K. Cheung, D. Zhou Vibrations of tapered Mindlin plates in terms of static Timoshenko beam functions. Journal of Sound and Vibration. 2002
33. Zhou Ding. Vibration of an arbitrarily shaped membrane carrying elastically mounted masses. Computers and Structures. 1992, 42(3): 389-394:
34. Zhou Ding. The application of a type of new admissible functions to the vibration of rectangular plates. Computers and Structures. 1994, 52(2): 199-203.
35. Zhou Ding. Free vibration of arbitrarily shaped plates with elastic or/and rigid concentric ring supports. Computers and Structures. 1994, 50 (5): 685-792.
36. Zhou Ding. The vibrations of a cantilever beam carrying a heavy tip mass with elastic supports. Journal of Sound and Vibration. 1997, 206 (6): 275-279.
37. Y. K. Cheung, D. Zhou. The free vibrations of rectangular composite plates with point-supports using static beam functions. Composite Structures. 1999, 44(12): 145-154.
38. Y. Xiang. Exact vibration solutions for circular Mindlin plates with multiple concentric ring supports. International Journal of Solids and Structures. 2002, (39): 6081—6102.
39. D. Zhou. Vibrations of Mindlinrectangular plates with elastically restrained edges using static Timoshenko beam functions with the Rayleigh—Ritz method. International Journal of Solids and Structures. 2001, (38): 5565-5580.
40. Y. K. Cheung, D. Zhou. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers and Structures. 2000, (78): 757—768.
41. A.W. Leissa, E.F.Ayoud. Vibration and Buckling of a simply supported rectangular plate subjected to a pair of In-plane concentrated forces. Journal of Sound and Vibration. 1988, 127 (1): 155-171.
42. A.W.Leissa, E.F.Ayoud. Tension Buckling of rectangular sheets due to concentrated forces. ASCE Journal of Engineering Mechanics. 1989, 115(2): 2749-2762.
43. A.W. Leissa, Y. Narita. Nature frequencies of simply supported plates. Journal of Sound and Vibration. 1980, 70 (2): 221—229.
44. A.W .Leissa,. E.F. Ayoud. Free vibration and tension buckling of circular plates with diametral point forces. Journal of Applied mechanics. 1990, 57: 995-999.
45. A.W ,Leissa. Three-dimensional vibrations of thick circular and annular plates. Journal of Sound and Vibration. 1998, 209: 15-41.
46. Y. Narita. Vibration and Stability of circular plates under partially distributed or concentrated Inplane loads. ASME Journal of Apllied Mechanics. 1985, 52: 549-552.
47. David Durban. Instability of an elastic circular plate subjected to nonuniform loads. AIAA Journal. 1977, 15 (3): 360-365.
48. N. Yamaki. Buckling of circular plate under locally distributed forces Transactions of the Japanese Society of Mechanical Engeneers. 1956, 22 (115): 119-123.
49. E.F. Ayoud. Vibration and buckling of plates subjected to Inplane concentrated forces. Ph.D. Dissertation, Ohio State University. 1989.
50. N. Yamaki.Buckling of a thin Annular plate under uniform
compression. Journal of Applied Mechanics. 1958, 25: 267-273.
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