What is Boussinesq's hypothesis?
Boussinesq's hypothesis is that the turbulent stresses are related to the mean velocity gradients in almost the same way that the viscous stresses are related to the complete velocity gradients. Boussinesq equations ( Boussinesq, 1871, 1872) are a set of nonlinear partial differential equations.
What is Boussinesq theory in civil engineering?
1. Boussinesq Theory for Vertical Stress due to Concentrated Load: Boussinesq in 1885 gave a solution for stress distribution in a homogeneous and isotropic subgrade subjected to a vertical concentrated load (point load) on the ground surface.
What is a Boussinesq model?
Boussinesq Models are applicable to shallow-water conditions with weak nonlinearity and weak frequency dispersion of the waves. The two-dimensional form of the equations were developed by Peregrine (1967) where η is the water surface elevation and u ¯ is the depth averaged horizontal velocity.
What is the essence of the Boussinesq approximation?
The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations.
What are the three assumptions of Boussinesq theory?
The soil mass is semi-infinite, homogeneous and isotropic. The soil has a linear stress-strain relationship, The soil is weightless. The load is a point load acting on the surface.
What is Boussinesq equation used for?
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.
What is the difference between Westergaard theory and Boussinesq theory?
Boussinesq's theory considers that the medium of soil is isotropic while Westergaard's theory considers that the medium of soil is anisotropic.
What is Westergaard theory?
Explanation: Westergaard's Theory Assumptions: Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain.
What is Boussinesq influence factor?
It may be noted that the Boussinesq influence factor for vertical stress is a function of the ratio r/z and is independent of soil properties. It is maximum when r = 0, that is, below the concentrated load, and is equal to 0.4775. Values of IB are computed for different values of r/z and are presented in Table 8.1.
How do you derive Boussinesq equation?
u is the horizontal flow velocity component: u = ∂φ / ∂x, w is the vertical flow velocity component: w = ∂φ / ∂z, g is the acceleration by gravity. Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions.
What are the assumptions of Westergaard theory?
Westergaard's Theory Assumptions: Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain.
What is Westergaard equation?
For vertical stress computation, he had proposed a formula in 1938. The formula is presented below. Westergaard's Equation for Point Loads. If Q is the point load and σz is the vertical stress due to the point load, σz=Q2πz2×√(1−2μ)/(2−2μ)[(1−2μ)/(2−μ)+(r/z)2)]3/2=Qz2Iw.
Which theory is applicable for layered soil?
Westergaard theory considers infinitely thin elastic layers of soil.
What is the recompression index?
The recompression index ( ) from the oedometer test is used to estimate the consolidation settlement of over-consolidated (OC) clays.
What are isobars in soil?
An isobar or pressure bulb is a stress contour or a line which connects all points below the ground surface at which the vertical pressure is the same.
What is time factor in consolidation?
It is the parameter used to find the consolidation coefficient with time taken for consolidation settlement in soil. It is a non-dimensional number.
What is the purpose of Boussinesq equations?
Boussinesq equations ( Boussinesq, 1871, 1872) are a set of nonlinear partial differential equations. Boussinesq equations incorporate frequency dispersion while the shallow water equations are not frequency-dispersive. Boussinesq equations are widely applied in numerical modeling in the field of coastal engineering to model water waves in shallow ...
What was Boussinesq's contribution to the theory of wave propagation?
In particular, the papers he published between 1868 and 1871 introduced the concepts of eddy viscosity, nonlinear dispersive waves, and density stratification.
Which hypothesis is related to the mean velocity gradient?
Boussinesq's hypothesis is that the turbulent stresses are related to the mean velocity gradients in almost the same way that the viscous stresses are related to the complete velocity gradients.
Is Boussinesq a three dimensional equation?
Although wave simulation in such cases is perfectly described by the Navier-Stokes equations, currently it is extremely difficult to solve the full three-dimensional equations in complicated models.
What is the essence of Boussinesq?
The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids.
What is the Boussinesq approximation?
In fluid dynamics, the Boussinesq approximation ( pronounced [businɛsk], named for Joseph Valentin Boussinesq ) is used in the field of buoyancy-driven flow (also known as natural convection ).
Why is Boussinesq's equation used?
It is simpler than the Westergaard’s equation and since the vertical stress estimates are greater from the Boussinesq’s equation, it can only overestimate the loadings within the soil, and hence be a conservative one.
When did Boussinesq publish his equations?
In 1885, Boussinesq published equations to determine the state of stress in a subgrade material. He investigated the stresses in an elastic, semi-infinite, homogeneous, and isotropic soil solid medium. The assumed solid medium is loaded normally on its upper plane surface by a concentrated point load.
What is Boussinesq's formula?
Boussinesq’s Formula: It was used to find the stress at point P at a depth z due to the load acting on the surface. It was developed in 1885. The assumption made for this formula is: The soil is elastic, isotropic, semi-infinite, and homogenous. The soil is weightless.
When the stress is transferred from grain to grain at the point of contact via a soil mass, it is called?
When the stress is transferred from grain to grain at the point of contact via a soil mass is called effective stress . It is also known as intergranular stress. It is denoted by σ’.
1. Boussinesq Theory for Vertical Stress due to Concentrated Load
Boussinesq in 1885 gave a solution for stress distribution in a homogeneous and isotropic subgrade subjected to a vertical concentrated load (point load) on the ground surface.
3. The 2:1 Distribution Method
For calculation of settlements, vertical stress is often obtained by assuming that the load is dispersed at a slope of 2 Vertical: 1 Horizontal from the base of the footing. Consider a rectangular footing ABCD of width b and length 1 subjected to a uniform pressure q. It is required to determine the vertical stress at depth z below the footing.
Comparison of Different Methods for Determination of Vertical Stress
Figure 8.36 shows the variation in vertical stress with depth as computed by the Boussinesq theory and the 2:1 approximate method. It may be observed from Fig. 8.36 that the vertical stress by the Boussinesq theory gives maximum and minimum values at the center and at the edge of the footing, respectively.
Overview
In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow (also known as natural convection). It ignores density differences except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Sound waves are impossible/neglected whe…
The approximation
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 fluid satisfies conservation of mass, conservation of momentum and conservation of energy. In the Boussinesq approximation, variations in fluid properties other than density ρ are ignored, and density only appears when it is multiplied by g, the gravitational acceleration. If u is the local velocity of a parcel of fluid, the cont…
Advantages
The advantage of the approximation arises because when considering a flow of, say, warm and cold water of density ρ1 and ρ2 one needs only to consider a single density ρ: the difference Δρ = ρ1 − ρ2 is negligible. Dimensional analysis shows that, under these circumstances, the only sensible way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity g′ where
Inversions
One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed. The Boussinesq approximation is inaccurate when the dimensionless density difference Δρ/ρ is of order unity.
For example, consider an open window in a warm room. The warm air inside is less dense than the cold air outside, which flows into the room and down towards the floor. Now imagine the op…
Further reading
• Boussinesq, Joseph (1897). Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section. Vol. 1. Gauthier-Villars. Retrieved 10 October 2015.
• Kleinstreuer, Clement (1997). Engineering Fluid Dynamics An Interdisciplinary Systems Approach. Cambridge University Press. ISBN 978-0-52-101917-0.