本文以三維漸近展開的解析模式來分析複合層樑的力學行為。基本作法是視疊層樑沿厚度方向為異質性材料所組成的構件，先從Hellinger- Reissner ( H-R ) 能量泛函式出發，不對位移場及應力場先預作假設， 將平面應力場表為位移場及橫向應力場，再代入泛函能量式中，對泛函式中的位移場及橫向應力場做變分，根據泛函能量極值條件，可以求得二維彈性力學基本方程式。接著再將各場量施以適當的無因次化，設立輔助變數使得剪力變形的影響于首階即出現。並將位移與應力分量對一個與板厚相關之微小參數作漸進展開，可得漸進展開模式中的各階層控制方程式。接著再搭配「微分再生核近似法」原理，處理高階微分及複雜迭代的問題。和一般數值方法不同的是，微分再生核近似法在求解形狀函數之高階導數時具有高度的效率及精確性，很適合分析漸進展開後具有高階微分狀態的各階層控制方程式。
本文最後選取了一複合層樑來分別施加三種不同形式的載重及邊界支承。而所得之結果和聯立O.D.E解析解相比較後，都證明我們可以得到合理準確的數據。故漸進展開解析理論搭配再生核近似法作數值分析，實為一良好的選擇。

In this project, we analyze the mechanics of laminated beams with 3D asymptotic analysis method. The basic elements are laminated beams which were composed with anisotropic materials in the direction of the thickness. The analysis is from the Hellinger – Reissner (H-R) general energy equation and not assuming the displacement and stress field in advance. Then, displacement and lateral stress fields were represented instead of plane stress field in the H-R equation. In the next step, calculus of variations is adopted for displacement and lateral stress fields. The 2D elastic basic equations were obtained according to the extreme value of the general energy equation after. All fields were nondimensionlized appropriately and one assistant variable was adopted for highlighting the influence of shear deformation in the first-order equations. And the asymptotic analysis about one slight parameter related to the thickness of the beam was performed to the displacement and components of stress, therefore, the governing equations of each order were obtained. Finally, the differential reproducing kernel approximation method was collocated to handle high-order differentiation and complicated iteration problems. This numerical analysis was provided with efficiency and accuracy.
In the end of this paper, one laminated beam with three different kinds of loading and boundary support individually was analyzed. After the numerical result was compared with the exact solution of terraced O.D.E. (ordinary differential equation), it proved that the data one gained finally was reasonable and accurate. Therefore, to perform a numerical analysis with asymptotic development and the differential reproducing kernel method is the suitable performance.

參考文獻

[1].Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P., ” Meshless
method: An overview and recent developments.” Comput. Methods Appl. Mech. Engrg. Vol. 139, pp.3-47 (1996)
[2]. Hoff , N. J. , ” Strength of Laminates and Sandwich Structural Elements ” Chapter 1, Engineering Laminates, edited by Dietz, A. G. H., John Wiley & Sons, New York (1949)
[3]. Khdeir, A. A., Reddy, J. N. and Frederick, D. M.,” A Study of Bending, Vibreation and BuckLing of Cross-ply circular Cylindrical shells with Various Shell theories ”, Int. J. Engeg. Sci. 27, pp.1337-1351 (1989)
[4]. Librescu, L. ,Khdeir, A. A. and Frederick, D. M. ” A Shear Deformable Theory of Laminated Composite Shallow Shell-Type Panels and their Response Analysis I ” Free Vibration and Buckling. Acta Mech. 76, pp. 1-33 (1989)
[5]. Lo. K. H., Christensen. R. M., and Wu. E. M., ” A high order theory of plate deformation. Part2: Laminated plates, ” J. Appl. Mech., ASME, Vol. 44 (1977)
[6]. Onate, E., S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor and C. Sacco, ” A Stabilized Finite Point Method for Analysis of Fluid Mechanics Problem ”, Computer Methods in Applied Mechanics and Engineering Vol. 139, pp. 315-346 (1996)
[7]. Pagono, N. J.,” Analysis of the Flexture Test of Bidirectional Composites, ” J.Compos. Mater., Vol. 1 (1967)
[8]. Paris R.B. & Wood A.D..,” Asymptotics of high order differential equations ”, Harlow, Essex, England, Longman Scientific & Technical , New York (1986)
[9]. R. Rwo Valisetty, Lawrence W. Rehfield.,” A theory for stress analysis of composite laminates ”, Washington, D. C. (1983)
[10].Reddy, J. N., ” Exact Solution of Moderatively Thick Laminated Shells ”, J. Engrg. Mech. ASCE 110, pp.794-808 (1984)
[11].Reddy, J. N. and Liu, C. F. ” A Higher-Order Shear Deformation Theory of Laminated Elastic Shells ”, Int. J. Engrg. Sci. 23, pp.319-330 (1985)
[12].Reddy, J. N. and N.D. Phan., ” Dynamic analysis of laminated plates using a higher-order theory ” , Washington, D. C. (1984)
[13].Tarn, J. Q. and Wang, Y. M., ” A Asympotic Theory for Dynamic Response of Anisotropic Inhomogeneous and Laminated Plates, ” Int. J. Solids Struct., Vol.31 (1994)
[14].Timoshenko, S. P. and Goodier, J. N.,” Theory of Elasticity ”, 3rded., Mcgraw Hill Book Co. New York (1970)
[15].Timoshenko,S. Woinowsky-Krieger.,” Theory of Plates and Shells ”, McGraw-Hill , New York (1959)
[16].Verhulst. F. ” Asymptotic analysis from theory to application ”, Berlin , Springer-Verlag, New York (1979)
[17].Wang, Y. M. and Tarn, J. Q., ” A Three-Dimensional Analysis for Anisotropic Inhomogeneous and Laminated Plates ”, Int. J. Solids Struct. Vol. 31 (1994)
[18].Whitney, J. M., ”Structure Analysis of Laminates Anisotropic Plates,” Technomic , Publishing Co. Lancaster , Pa (1987)
[19].Wing - Kam Liu , Shaofan Li , Ted Belytschko , ” Moving least-square
reproducing kernel methods ” (I) Methodology and convergence. Comput. Methods Appl. Mech. Engrg. 143 , pp.113-154 (1997)
[20].錢偉長 , ’’彈性力學’’ , 亞東書局 (1991)