Effective Stress and Equilibrium Equation for Soil Mechanics

Longtan Shao, Xiaoxia Guo, Shiyi Liu, Guofeng Zheng

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9781138092310
• eBook ISBN: 9781351607414
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

The concept of effective stress and the effective stress equation is fundamental for establishing the theory of strength and the relationship of stress and strain in soil mechanics and poromechanics. However, up till now, the physical meaning of effective stress has not been explained clearly, and the theoretical basis of the effective stress equation has not been proposed. Researchers have not yet reached a common understanding of the feasibility of the concept of effective stress and effective stress equation for unsaturated soils.
Effective Stress and Equilibrium Equation for Soil Mechanics discusses the definition of the soil skeleton at first and clarifies that the soil skeleton should include a fraction of pore water. When a free body of soil skeleton is taken to conduct internal force analysis, the stress on the surface of the free body has two parts: one is induced by pore fluid pressure that only includes normal stress; the other is produced by all the other external forces excluding pore fluid pressure. If the effective stress is defined as the soil skeleton stress due to all the external forces excluding pore fluid pressure, the effective stress equation can be easily obtained by the internal force equilibrium analysis. This equation reflects the relationship between the effective stress, total stress and pore fluid pressure, which does not change with the soil property. The effective stress equation of saturated soils and unsaturated soils is unified, i.e., o˜=o˜t –Seuw–(1–Se)ua. For multiphase porous medium, o˜=o˜t –u*,u*=Seiui(i=1,2,...,M). In this book, a theoretical formula of the coefficient of permeability for unsaturated soils is derived. The formula of the seepage force is modified based on the equilibrium differential equation of the pore water. The relationship between the effective stress and the shear strength and deformation of unsaturated soils is preliminarily verified. Finally, some possibly controversial problems are discussed to provide a better understanding of the role of the equilibrium equation and the concept of effective stress.

Chapter 1

Introduction

While soil has played a vital role since the early stage of the development of human society, the science of soil mechanics has a rather short history. The earliest documented theoretical research on soil mechanics can be traced back to the 1770s. In 1773, Coulomb researched material strength and developed the Coulomb’s law of shear strength, based upon which Mohr established the Mohr-Coulomb strength theory of soil. This theory has since become the groundwork for the strength and failure study of soil and provided theoretical framework for computation of soil compression, bearing capacity of foundation soil and stability analysis on soil slopes. In a report published in 1776, Coulomb discussed soil pressure theory based on the equilibrium analysis of soil wedge and developed the computation methods for soil pressure of retaining walls. Rankine investigated soil pressure theory using plastic equilibrium analysis of the soil body in 1857. Darcy proposed Darcy’s seepage law through extensive laboratory experiments in 1856 and laid a solid foundation for the percolation theory and seepage flow mechanics. Boussinesq and Flamant proposed the displacement and stress distribution theory of isotropic and homogeneous semi-infinite surfaces under vertical concentrated loading and linear loading in 1885 and 1892, respectively. In the early 20th century, according to the plastic equilibrium principle, Prandtl investigated the process of hard objects pressed into relatively soft, isotropic and homogeneous materials and derived the ultimate bearing capacity equation for soil. Later on, Terzaghi, Meyerhof, Vesic and Hansen further improved this equation and proposed their own versions of ultimate bearing capacity equations for foundations.

Despite the numerous outcomes and achievements on the study of soil mechanics, those studies were scattered and not well-organized in a systematic way before Terzaghi, who is considered a paramount figure in establishing the subject of soil mechanics, mainly in two respects. First, soil was classified into clay and sand according to physical properties, and the various laws, principles and theories on soil mechanics were summarized to form a basic framework for the study of soil mechanics. Second, effective stress principle for saturated soils and the one-dimensional consolidation theory were established. Soil Mechanics, published by Terzaghi in 1925, is the earliest work systematically discussing the knowledge system of soil mechanics. He published Theoretical Soil Mechanics in 1943 and Soil Mechanics in Engineering Practices with Peck in 1948 and primarily built a relatively complete knowledge system of soil mechanics and geotechnical engineering. Therefore, Terzaghi is considered the founder for the subject of soil mechanics. Based on Terzaghi’s effective stress principle and consolidation theory, Biot developed the basic equilibrium equation in soil mechanics, i.e., Biot’s consolidation equation. The effective stress principle along with Biot’s consolidation equation has become the cornerstone for soil mechanics.

1.1 Effective stresses

1.1.1 Effective stresses of saturated soils

Terzaghi first proposed the concept of effective stress in 1936, with the original quote:

The stresses in any point of a section through a mass of earth can be computed from the total principal stresses n′_I, n′_II and n′_III which act in this point. If the voids of the earth are filled with water under a stress n_w, the total principal stresses consist of two parts. One part, n_w, acts in the water and in the solid in every direction with equal intensity. It is called the neutral stress. The balance, n_I = n′_I – n_w, n_II – n′_II – n_w and n_III = n′_III – n_w, represents an excess over the neutral stress n_w and it has its seat exclusively in the solid phase of the earth.

This fraction of the total principal stresses is called the effective principal stresses or equal values of the total principal stresses, with the effective stresses dependent on the value of n_w. In order to determine the effect of a change of n_w at a constant value of the effective stresses, numerous tests were made on sand, clay and concrete, in which n_w was varied between zero and several hundred atmosphere pressure. All these tests led to the following conclusions, valid for the materials mentioned.

A change of the neutral stress n_w produces practically no volumetric change and has practically no influence on the stress conditions for failure. Each of the porous materials mentioned was found to react a change of n_w as if it were incompressible and as if its internal friction were equal to zero. All the measurable effects of a change of the stress, such as compression, distortion and a change of the bearing resistance, are exclusively due to changes in the effective stresses, n_I, n_II and n_III. Hence the investigation of the stability of a saturated body of earth requires the knowledge of both the total and the neutral stresses.

This statement by Terzaghi revealed the following facts: (1) traditionally, the stress of soils people calculated is the total stress; (2) the total stress of soils is composed of effective stress and pore water pressure; and (3) the volume and strength change induced by pore water pressure are too small to be observed, whereas the effective stress is the cause for the deformation and strength change of soils.

The validity of the effective stress pr...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• Preface
• 1 Introduction
• 2 Equilibrium differential equations of a soil
• 3 Effective stress
• 4 Seepage equation of unsaturated soils
• 5 Discussion on some issues related to effective stress
• Units and symbols
• References
• Subject Index