應用SCM法於厚板分析

Abstract

應用SCM法在許多的工程問題上，皆被證實有相當好的準確性和效率。在處理薄板板時，SCM也被證實有極佳的表現。然而，在面對板的厚長比大於0.1的板的分析，不再適用於由Kirchhoff和Love所發展的傳統板理論，該理論假設板垂直中間面的斷面在板受力變形之後仍然垂直中間面，因此就忽略了剪應變的效應。因為這個先天上的限制，傳統板理論必須被修正，考慮剪應變效應才可以解決厚板的工程問題。因此學者致力於發展出一套二維的板理論，能夠有比三維板理論更簡單的條件，卻能有效修正傳統板理論的誤差。REISSNER和MINDLIN分別各自發展出他們的厚板理論，其共通點為使用三個變數分別是w(橫向位移)、ψx(x向旋轉角)、ψy(y向旋轉角)建構由三條偏微分方程式組成的厚板控制方程式。本文應用一種重疊式結點佈置法，將三個控制方程式的問題成功使用SCM法進行分析得到數值解。使用商用軟體MATLAB撰寫程式，針對不同邊界條件與不同厚長比的板進行分析，探討其收斂性和準確性。成功的將SCM法應用在厚板的分析上，並且維持SCM法一貫的快速和準確。

The interest of engineers in numerical methods has grown exponentially in the last decades. High-capacity personal computers are now mass produced and are available for a very reasonable price. SCM is one of numerical methods which had been applied in several engineering problem ,and be proved that SCM yields sufficiently accurate result and SCM is fast. In thin plate problem, SCM also provides reliable analyses. However, for the plate with thickness-to-span ratio h/L bigger than 0.1, Kirchhoff and Love’s classical plate theory’s accuracy decrease with growing thickness of the plate. There is an inherent limitation of classical plate theory due to the neglect of the effect of transverse shear strain, since it is assumed the normal to the middle plane remain normal to the deflected middle plane. It’s necessary to refined theories by introducing the effect of transverse shear strains in order to obtain reliable result for the behavior of the thick plate. Developing a refined two-dimensional equation which has more reliable result than classical theory and not so complex like three-dimensional equation interesting engineers. REISSNER and MINDLIN all formulate their thick plate equations by using three variables w(transverse deflection),ψx(rotation of the normals to the middle surface along X axis)、ψy(rotation of the normals to the middle surface along X axis.) In this paper, we apply SCM in thick plate with three governing equation analysis by using redefined overlap collocation. By using MATLAB, we have several case with different thickness-to-span ratio, and proved that SCM has economical and reliable result of thick plate analyses.