探討楊氏模數在具空間變異性土體的變異數折減因子

Abstract

在大地工程設計中，時常假設土壤為均質土壤以方便分析與設計，然而土壤本身在沉澱堆積的過程中產生空間變異性，為了有效模擬空間變異性對結構物的影響，常常使用隨機場分析，透過模擬許多隨機場樣本得到結構物反應(如:基礎位移及基礎反力)的變異程度，然而隨機場分析對工程師而言需耗費較長時間與成本，為方便工程設計前人提出一些空間平均模型取代隨機場分析，並將具變異性土壤以均一值表示，此過程稱為均值化。 Ching et al. (2018) 提出 pseudo incremental energy model 的概念，利用穩態對數隨機場模擬具有空間變異性的土壤，執行一次均質有限元素法得到模型中各元素的權重來計算均質化的楊氏模數(E_wg)，即可得到合理的均質化楊氏模數。由於PIE模型在估計有效楊氏模數有出色表現，因此本研究延續前人研究，將PIE模型得到的變異數折減因子進行回歸，使未來工程師使用PIE模型不需要再執行有限元素分析，而是使用回歸式，較為便利。 本研究透過最小平方法，對大地工程結構物的各種情況做回歸式，並透過比較回歸式結果與隨機場分析所得到的變異數折減因子，驗證回歸式的準確性。

In geotechnical engineering design, it is often assumed that soil is homogeneous for ease of analysis and design. However, the soil exhibits spatial variability due to its sedimentation and accumulation processes. To effectively simulate the impact of spatial variability on structures, random finite element analysis is commonly used. This involves generating numerous random field samples to determine the variability in structural responses (e.g., foundation displacement, foundation reaction). However, random finite element analysis can be time-consuming and costly for engineers. To facilitate engineering design, some spatial averaging models have been proposed to replace random field analysis by representing the soil with variability as a homogeneous value, a process known as homogenization. Ching et al. (2018) proposed the concept of the pseudo incremental energy (PIE) model, which uses a stationary log-normal random field to simulate soil with spatial variability. By performing homogeneous finite element method (FEM) analysis, the weights of each element in the model are calculated to determine the homogenized Young's modulus (E_wg). This provides a reasonable homogenized Young's modulus. Since the PIE model has shown excellent performance in estimating the effective Young's modulus, this study extends previous research by performing a regression on the variance reduction factors obtained from the PIE model. This allows future engineers to use the PIE model without needing to perform FEM analysis, instead utilizing the regression formula for greater convenience. This study uses the least squares method to perform regression analysis for various scenarios of geotechnical structures. By comparing the regression results with the variance reduction factors obtained from random finite element analysis, the accuracy of the regression equations is verified.