垂直橫向等向性材料在半無限空間受三維表面點荷重作用之位移閉合解探討

Abstract

基礎材料受工程結構物作用，常因超額載重及位移變化量而產生破壞，故考慮現地承受大量載重作用下基礎所引產生之「位移和應力」是有必要的。一般岩石或土壤若以明顯之地質構造或彈性對稱方向來看，可分為一般異向性、正交性或橫向等向性材料，這就是「異向性」，其中又以橫向等向性材料最為常見，例如：單組規則節理岩體、層狀岩體或具有葉理的岩石等都可視為橫向等向性材料。由於地質構造作用，岩體內不連續面位態並非全為「水平」，故本文依2009年胡廷秉提出對於半無限空間傾斜橫向等向性材料受載之傳統求解偏微分方程方法，探討半無限空間垂直橫向等向性材料受三維表面點荷重作用的位移閉合解。 本論文主要是先推導在微小變形條件下，根據彈性力學理論，求解由偏微分方程所組成之控制方程，首先將控制方程透過雙傅立葉轉換轉變為常微分方程，並配合邊界條件求出傅立葉定義下之半無限空間垂直橫向等向性材料位移解，然後經由雙傅立葉逆轉換且利用殘數定理積分而得到半無限空間垂直橫向等向性材料受三維點荷重作用的位移閉合解。最後，藉由範例說明，特別針對殘數定理積分部份探討半無限空間垂直橫向等向性材料受三維表面點荷重作用之位移閉合解與受到材料的異向性影響性，並且與2006年Ding 等人和2009年胡廷秉提出之半無限水平橫向等向性空間比較相同，因此，本文採用之方法若能完整解出各殘數積分值，則垂直橫向等向性材料於半無限空間受表面三維點荷重作用之位移閉合解析解應可合理性的被解出。

The failure of a foundation in soil/rock is often caused by over loading or large displacements. This fact is particularly important to analyze stresses and displacements when structures impose very large loads on the underlying soil/rock. However, it is also important for understanding the influence of the "anisotropy" of soil/rock on stresses, strains and displacements. Based on the orientation of geological structures or direction of planes of elastic symmetry, Elastic materials can be divided into general anisotropic, orthogonal or transversely isotropic materials. The nature of anisotropy of soils/rocks is caused by depositing via sedimentation over a long period of time, cutting by regular discontinuities, such as cleavages, foliations, stratifications and joints. Anisotropic soils/rocks are commonly modeled as transversely isotropic materials based on the practical engineering considerations. Nevertheless, the inclination of planes of elastic symmetry is not always horizontal, and hence, this thesis extending the approach proposed by Hu (2009) to study the closed form solutions for displacements in an half space with vertical transversely isotropy subjected to a surface 3D point load. To obtain the closed form solutions, the double Fourier transform was used to reduce the partial differential equations to ordinary differential equations, firstly. Then, the solutions of displacement and stress in Fourier domain can be determinednd from the boundary conditions. Finally, the closed form solutions for stresses and displacements in a vertical transversely isotropic half space material subjected to a 3D point load can be obtained using the double inverse Fourier transform and residue theorem. The present closed-form solutions demonstrate that the material anisotropy could affect the displacements and stresses in a vertical transversely isotropy. The illustrative examples show that the calculated displacements in a horizontal transversely isotropic half space are the same/similar as those presented by Hu (2009) and Ding et, al., (2006). Hence, the closed form solutions for stresses and displacements in a vertical transversely isotropic half space can be reasonably solved if the all of residue of integrals in the inverse Fourier domain can be determined exactly.