Stress due to surface load

Transcription

1 Stress due to surface load To analyze problems such as compressibility of soils, bearing capacity of foundations, stability of embankments, and lateral pressure on earth retaining structures, we need to know the nature of the distribution of stress along a given cross section of the soil profile. When a load is applied to the soil surface, it increases the vertical stresses within the soil mass. The increased stresses are greatest directly under the loaded area, but extend indefinitely in all directions. Allowable settlement, usually set by building codes, may control the allowable bearing capacity. The vertical stress increase with depth must be determined to calculate the amount of settlement that a foundation may undergo. Foundations and structures placed on the surface of the earth will produce stresses in the soil. These stresses will decrease with the distance from the load. How these stresses decrease depends upon the nature of the soil bearing the load. Stress Due to a Concentrated Load Individual column footings or wheel loads may be replaced by equivalent point loads provided that the stresses are to be calculated at points sufficiently far from the point of application of the point load. Vertical stress due to a concentrated load: Boussinesq s Formula Wastergaard Formula Boussinesq s Formula for Point Loads: In 1885, Boussinesq developed the mathematical relationships for determining the normal and shear stresses at any point inside homogenous, elastic and isotropic mediums due to a concentrated point loads located at the surface. Assumption: The soil mass is elastic, isotropic (having identical properties in all direction throughout), homogeneous (identical elastic properties) and semi-infinite depth. The soil is weightless. The distribution of σz in the elastic medium is apparently radially symmetrical. The stress is infinite at the surface directly beneath the point load and decreases with the square of the depth. At any given non-zero radius, r, from the point of load application, the vertical stress is zero at the surface, increases to a maximum value at a depth where Ɵ= 39.25, approximately, and then decreases with depth.

2 According to Boussinesq s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by According to Boussinesq s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by: Equation shows that the vertical stress is Directly proportional to the load Inversely proportional to the depth squared, and Proportional to some function of the ratio (r/z). It should be noted that the expression for z is independent of elastic modulus (E) and Poisson s ratio (μ), i.e. stress increase with depth is a function of geometry only. Pressure Distribution Diagram Equation may be used to draw three types of pressure distribution diagram. They are: The vertical stress distribution on a horizontal plane at depth of z below the ground surface. The vertical stress distribution on a vertical plane at a distance of r from the load point, and The stress isobar. Distribution on a horizontal plane The vertical stress distribution on a horizontal plane at depth of z below the ground surface

3 Distribution on a vertical plane The vertical stress distribution on a vertical plane at a distance of r from the point load Stress isobars An isobar is a line which connects all points of equal stress below the ground surface. In other words, an isobar is a stress contour. Stress below a Line Load The vertical stress increase due to line load, (σz), inside the soil mass can be determined by using the principles of the theory of elasticity, or

4 This equation can be rewritten as: Vertical Stress caused by a horizontal line load The vertical stress increase (σσ) at point A in the soil mass caused by a horizontal line load can be given as: Vertical Stress caused by a strip load The term strip loading will be used to indicate a loading that has a finite width along the x axis but an infinite length along the y axis. 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. Vertical stress at point A can be determined by equation:

5 Vertical Stress Due to Embankment Loading The vertical stress increase in the soil mass due to an embankment of height H may be expressed as: Where: Vertical Stress due to a uniformly loaded circular area 1- Under the center: The increase in the vertical stress (σz) at depth z (point A) under the center of a circular area of diameter D = 2R carrying a uniform pressure q is given by: 2- At any point: The increase in the vertical stress (σz) at any point located at a depth z at any distance r from the center of the loaded area can be given

6 Vertical Stress Caused by a Rectangular loaded area The increase in the vertical stress (σz)) at depth z under a corner of a rectangular area of dimensions B = m z and, L = n z carrying a uniform pressure q is given by Where: The increase in the stress at any point below a rectangular loaded area can be found by dividing the area into four rectangles. The point A is the corner common to all four rectangles.

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