以三維基本土體模型探討具空間變異性土體的有效楊氏模數

Abstract

由於土壤是天然材料，其形成受自然環境影響甚巨，材料性質較為複雜，但在大地工程問題中為了簡化分析，經常假設土體為均質。本研究探討具空間變異性土壤之楊氏模數，若此土壤須以一均質土壤來模擬時，該如何決定此均質土壤的楊氏模數?為了回答此問題，本研究以隨機有限元素法模擬具空間變異性土壤的受力變形狀況，對3D簡單土體模型施以軸向位移，以所得應力反推土壤的整體楊氏模數，定義此整體楊氏模數為代表此具空間變異性土壤之有效楊氏模數。 本研究試圖以簡單的數學平均來估計有效楊氏模數，發現在3D簡單土體模型問題中，有效楊氏模數可以適當的空間平均來估計其值以及統計性質(平均值、變異係數)，對於等向性案例，可用幾何平均來估計，對於層狀案例，水平方向可用算術平均，垂直方向可用調和平均來估計，對於柱狀案例，水平方向可用幾何平均，垂直方向可用算術平均來估計；而簡化的彈簧串並聯模式則適用於各種不同方向性的隨機場案例，以及應力應變分布不均的基本土體模型案例。 Fenton and Griffiths (2002, 2005)曾研究過相關問題，他們發現在淺基礎問題中，若其下為等向性空間變異的土壤，則此基礎所感受到土壤的有效楊氏模數的統計性質，可用基礎下方土壤楊氏模數幾何平均的統計性質來估計；本研究在3D基本土體模型問題的主要結論與Fenton and Griffiths一致，且更為強大，我們發現不只是有效楊氏模數的統計性質，它的數值亦可以適當的空間平均來估計，然而，當我們回到淺基礎問題時，卻發現依然僅有統計性質可被估計，對於基本土體模型的結論不適用於淺基礎，我們猜測可能是淺基礎問題應力應變分布不均，再次以應力應變分布不均的基本土體模型問題來驗證，但發現此猜測錯誤，對於兩種問題差異的部分，仍需更進一步的研究。

In order to simplify the geotechnical problems, engineers usually assumed the soil to be homogeneous. However, the soil was formed naturally with complicated processes, it should not be homogeneous. If we have to use a value to represent its property, how to determine this value? This study focus on the Young’s modulus. We use random finite element analysis to simulate a soil mass with spatially variable Young’s modulus subjected to displacement-controlled 1D compression and back-calculate the overall Young’s modulus by the stress responses. Define the overall Young’s modulus as the effective Young’s modulus(Eeff). We investigate whether the effective Young’s modulus can be strongly correlated to any spatial average. For the 3D elementary soil mass problems, we find that the numerical vales and statistics of effective Young’s modulus can be approximated by appropriate spatial averages. For isotropic cases, Eeff can be approximated by geometric mean. For layer cases, Eeff can be approximated by arithmetic mean (Ea) when loading direction is parallel to the layers and can be approximated by harmonic mean (Eg) when loading direction is perpendicular to the layers. For column cases, Eeff can be approximated by harmonic mean(Eh) when loading direction is parallel to the columns and can be approximated by geometric mean when loading direction is perpendicular to the columns. And the unified spatial average model can approximate Eeff in every case above without switch among Ea, Eg and Eh. Fenton and Griffiths (2002, 2005) studied probabilistic foundation settlement, and they found that for foundations on soils with isotropic SOFs, Eeff can be modeled as Eg of the E random field over a prescribed domain under the footing. For elementary soil mass problems, we have the consistent and more stronger results. We find that not only the statistics but also the numerical values of Eeff can be approximated by appropriate apatial averages. However, when we go back to foundation problems, we find only the statistics of Eeff can be approximated by apatial averages. The reason why the results of elementary soil mass can’t be applied on the foundation problems needs more investigation.