Boussinesqs equation boussinesq published in 1885 a solution for the stresses beneath a point load on the surface of a material which had the following properties. Closedform solutions based on the boussinesq equations are presented in most introductory soil mechanics textbooks for this case. Vertical stress in soil vertical stress caused by a strip load the fundamental equation for the vertical stress increase at a point in a soil mass as the result of a line load can be used to determine the vertical stress at a point caused by a flexible strip load of width b. In the boussinesq approximation, variations in fluid properties other than density. Use of the boussinesq equation for determining the.
Pdf solution of the boussinesq problem of half space. For soil, the soil mass is elastic, isotropic, homogeneous and semiinfinite. Otherwise stated, this means that the influence of the stratification on the inertia terms is ignored in comparison to the influence related to the gravity. Boussinesq equations, stress isobar and pressure bulb concept, pressure. Boussinesq equation an overview sciencedirect topics. However, as the soil is not fully elastic the stresses determined are necessarily approximate. Pdf use of the boussinesq solution in geotechnical and road. Boussinesq s equation considers a point load on the surface of a. Uses elastic theory to get change in stress with depth. The classical boussinesq equation includes the weak nonlinear and dispersive effects. The highest bearing capacity of the soil which was at 7. The term strip loading will be used to indicate a loading that has. Fhwa nhi06088 2 stress and strain in soils soils and foundations volume i 2 4 december 2006 x100 e e e e d max min max r.
Stress distribution of the soil linkedin slideshare. The boussinesq solution for the distribution of stresses in a halfspace resulting from surface loads is largely used in geotechnical and road engineering. The equation of motion corresponding to lagrangian 1 is the. Experimental determination and numerical analysis of vertical. Ye yincan et al, in marine geohazards in china, 2017.
The boussinesq equation describes propagation of waves in weakly nonlinear and weakly dispersive media 1. Subsurface stress increment due to a point load 1 for homogeneous properties are the same everywhere, and isotropic properties are the same in all direction soils boussinesq 1885 french mathematician developed the following equation to calculate vertical stress increment. The most common type of soils that are met in nature are the water deposited sedimentary soils. Vertical stress in a soil mass university of waterloo. Determine vertical soil pressure under footing ceref. It provides examples of distinct rational solutions of the boussinesq equation which have the same function as limit when t goes to zero. However, the soil is neither isotropic nor homogeneous. Lecture 7 stress distribution in soil linkedin slideshare. Pdf use of the boussinesq solution in geotechnical and. In fluid dynamics, the boussinesq approximation for water waves is an approximation valid for weakly nonlinear and fairly long waves. The maximum tensile stress at failure for a dry specimen, as determined by the boussinesq equation for the diametrical point load test, was found to be in very good agreement with the diametrical point load tensile strength is as defined by isrm 1985. If the surface layer is the more rigid, it acts as a distributing mat and the vertical stresses in the underlying soil layer are less than boussinesq values. The force at failure for specimens of different geometry was used to determine the stress distribution along the line of loading.
Lateral pressures on retaining walls due to backfill. Kerr summary in railway engineering, in order to reduce the rate of subgrade deterioration, there is a need to know the vertical normal stress distribution. Equations have been developed to compute stresses at any point in a soil mass on. Missouri university of science and technology scholars mine.
Selvadurai department of civil engineering and applied mechanics, mcgill university, 817 sherbrooke street west, montreal, quebec, canada h3a 2k6 received 14 may 1999. To see more videos on different topic go to playlist for more updates subscribe the channel. Theoretical stress distribution in an elastic multilayered medium malati kesaree charyulu iowa state university. One of the most famous methods is boussinesq strip method. If the stress area is a square then use the square contour chart.
Higherorder partial differential equations boussinesq equation 1. Boussinesq equations deformation engineering civil. An improved analysis for the determination of required. The boussinesq approximation is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and heat transfer. The derived allowable bearing capacity of the soil in relationship to the actual load of the telecommunication mast would be used in the simulation using java programme to determine the cause of failure. Semiinfinite this means infinite below the surface therefore providing no boundaries of the material apart from the surface homogeneous the same properties at. Uses elastic theory to get change in stress with depth below point load. Boussinesq 1885 point load solution uses elastic theory to get change in stress with depth below point load solution based on. However, in the oberbeckboussinesq equations the constraint a2 holds to within the order of approximation.
Boussinesq assumed that the soil is elastic, isotropic and homogeneous for the development of a point load formula. Boussinesq stress contour chart is used to find the stress increase at a depth say y with a given or calculated uniform stress psf or knm2. For example if the vertical stress decrease due to an excavation is required, the negative load is equal to the weight of the soil removed. A practical guide to analyzing soil and structural systems using excel spreadsheets and vba macro programs in opensource code that are provided on the accompanying cd. Equations have been developed to compute stresses at any point in a soil mass. Solutions of the boussinesq equation sciencedirect. The approximation is named after joseph boussinesq, who first derived them in response to the observation by john scott russell of the wave of translation also known as solitary wave or soliton. Its basic assumption is similar to that of the equation of shallow water long wave. Boussinesq equations, stress isobar and pressure bulb concept, pressure distribution on horizontal and vertical planes, stresses due to point load, line load, strip load, uniformly loaded circular and rectangular areas. The stress on the soil element from the contact pressure qo on the surface area da of fig. Application of bousinesqs and westergaards formulae in.
The fluid satisfies conservation of mass, conservation of momentum and conservation of energy. To derive the boussinesq equation for some physical model, one should start from a lagrangian l dx 3 4. The exact equations of the boussinesq approximation the basis of this approximation is that there are flows in which the temperature varies little, and therefore the density varies little, yet in which the buoyancy drives the motion. Pdf the boussinesq solution for the distribution of stresses in a. An approximate stress distribution assumes that the total applied load on the surface of the soil. Suppose wx,t is a solution of the boussinesq equation in question. This yields this yields where the temperature and pressuredependent density, have been replaced by a constant density, except in the body force term representing the buoyancy force. The effect of the westergaard assumption is to reduce the stresses substantially below those obtained by. Boussinesq 1842 1929 evolved equations that can be used to determine stresses at any point p at a depth z as a result of a surface point load. The boussinesq approximation states that the density variation is only important in the buoyancy term, and can be neglected in the rest of the equation.
In the boussinesq approximation, which is appropriate for an almost incompressible. Oberbeckboussinesq equations encyclopedia of mathematics. Stress distribution in soils under pile cap of tapered. This equation arises in hydrodynamics and some physical applications. Stress distribution in soils gate 2019 civil boussinesq. The aim of this study is to use the green and zerna displacement potential function method to solve the boussinesq problem in the theory of elasticity of soil mechanics. Boussinesq 1985 evolved equations that can be used to determine stresses at any point p at a depth z as a result of a surface point load. Stresses beneath dynamically applied vertical point. In newmarks chart, if the length of one unit equals to d, then the radius of the ring is equal to r. Theoretical stress distribution in an elastic multi. An approximate stress distribution assumes that the total applied load on the surface of the soil is distributed over an area of the same shape as the loaded area on the surface, but with dimensions that increase by an amount equal to the depth below the surface.
For most practical analyses of the settlement behavior of soils, it is assumed that the volume of the soil is controlled exclusively by the vertical stress. Boussinesq equations free download as powerpoint presentation. Estimation of vertical stresses at any point in a soil mass due to external. Stress distribution in soils gate boussinesq equation soil mechanics boussinesq theory in soil mechanics westergaard theory soil mechanics westergaard theory. An improved analysis for the determination of required ballast depth by lucas a. Linear elastic, homogeneous, isotropic medium with semiinfinite depth. Boussinesqs solved the problem of stresses produced at any point a due to point load q. Hence, at first, we will discuss the general principles for. The estimation of vertical stresses at any point in a soil mass due to external loading is essential to the. This book gives readers the tools to understand the methods such as finite element analysis used to analyze common problems in structural engineering, foundation engineering and soilstructure interaction.
Boussinesqs equation geotechnical engineering civil. At e emin the soil is as dense as it can get and the relative density equals 100%. An important function in the study of soil mechanics is. The boussinesq equation can also be used to determine the vertical stress below the centre of a flexible circularly loaded area. Let denote the acceleration due to gravity, let be a typical length scale usually the thickness of the layer of the fluid, a representative density, the. The 1872 paper of boussinesq introduces the equations now known as. The boussinesq method uses the theory of elasticity to calculate the stress under a point load in a homogeneous, semiinfinite half space see, for example, bowles, 1996. Useful solutions for stresses under different footing shapes can be obtained by integrating over the area of the footing. These equations are written in terms of total stresses.