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14.532 THEORETICAL SOIL MECHANICSStresses in a Soil Mass
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VERTICAL STRESS INCREASES IN SOILSTYPES OF LOADING
Point Loads (P)
Figure 6.11. Das FGE (2005).
Examples:- Railroad track
Examples:- Posts
Line Loads (q/unit length)
Figure 6.12. Das FGE (2005).
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Examples:- Exterior Wall Foundations
Examples:- Column Footings
VERTICAL STRESS INCREASES IN SOILSTYPES OF LOADING
Area Loads (q)Strip Loads (q)
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VERTICAL STRESS INCREASE IN SOILSANALYSIS METHODS - BOUSSINESQ (1883)Based on homogeneous, weightless, elastic, isotropic infinitely large half-space free of initial stress and deformation. The modulus of elasticity is assumed constant and the principle of linear superposition is assumed valid (EM1110-1-1904, 1990). Not accurate for layered soil stratigraphy with substantial thickness (NAVFAC DM7.01, 1986).
Rigid Surface Layer Over Weaker Underlying Layer: 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.
Weaker Surface Layer Over Stronger Underlying Layers: If the surface layer is less rigid than the underlying layer, then vertical stresses in both layers exceed the Boussinesq values.
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VERTICAL STRESS INCREASE IN SOILSANALYSIS METHODS - WESTERGAARD
VERTICAL STRESS INCREASE IN SOILSANALYSIS METHODS – 2V:1H METHOD
Based on the assumption that the soil on which load is applied is reinforcedby closely spaced horizontal layers which prevent horizontal displacement.The effect of the Westergaard assumption is to reduce the stressessubstantially below those obtained by the Boussinesq equations.
An approximate stress distribution assumes that the total applied load onthe surface of the soil is distributed over an area of the same shape as theloaded area on the surface, but with dimensions that increase by anamount equal to the depth below the surface.Vertical stresses calculated 2V:1H method agree reasonably well withthe Boussinesq method for depths between B and 4B below thefoundation.
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14.532 THEORETICAL SOIL MECHANICSStresses in a Soil Mass
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2/522
3
5
3
)(23
23
zrzP
LzP
z
122/5221/
123 I
zP
zrzP
z
VERTICAL STRESS INCREASE (z) IN SOILSPOINT LOADING (Boussinesq, 1883)
Stresses in an Elastic Medium Caused by Point LoadingFigure 6.11. Das FGE (2005).
Where:z = Change in Vertical StressP = Point Load
I1 3
21
r / z 2 1
5/2
*Based on homogeneous, elastic, isotropic infinitely large half-space
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Table 6.1 Variation of I1 (Das, FGE 2006).
VERTICAL STRESS INCREASE (z) IN SOILSPOINT LOADING (Boussinesq, 1883)
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22
222
3
1
2)/(
)(2
zxzq
zxqz
Where: = Change in Vertical Stressq = Load per Unit Lengthz = Depthx = Distance from Line Load *Based on flexible line load of infinite length on a
homogeneous, elastic, isotropic semi-infinite half-space
or
DimensionlessForm
Line Load over the Surface ofa Semi-infinite Soil MassFigure 6.12. Das FGE (2005).
VERTICAL STRESS INCREASE (z) IN SOILSLINE LOADING (Boussinesq, 1883)
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Table 6.3 Variation of /(q/z) with x/z (Das, FGE 2006).
VERTICAL STRESS INCREASE (z) IN SOILSLINE LOADING (Boussinesq, 1883)
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)2cos(sin
q
Flexible Strip Load over the Surface ofa Semi-infinite Soil MassFigure 6.13. Das FGE (2005).
Where:
= Change in Vertical Stressq = Load per Unit Areaz = Depthx = Distance from Line Load
Angles measured in counter-clockwise direction are taken as
positive
VERTICAL STRESS INCREASE (z) IN SOILSSTRIP LOADING (Boussinesq, 1883)
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Table 6.4 Variation of /q with 2z/B and 2x/B (Das, FGE 2006).
VERTICAL STRESS INCREASE (z) IN SOILSSTRIP LOADING (Boussinesq, 1883)
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q 1 1
(R / z)2 1 3/2
Vertical Stress Below Center of Uniformly LoadedFlexible Circular Area
Figure 6.15. Das FGE (2005).
Where:
= Change in Vertical Stressq = Load per Unit Areaz = DepthR = Radius
VERTICAL STRESS INCREASE (z) IN SOILSCIRCULAR LOADING (Boussinesq, 1883)
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Table 6.5 Variation of /q with z/R (Das, FGE 2006).
VERTICAL STRESS INCREASE (z) IN SOILSCIRCULAR LOADING (Boussinesq, 1883)
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112tan
12
112
41
2222
221
22
22
2222
22
2
nmnmnmmn
nmnm
nmnmnmmn
I
B
y
L
x
qIzyx
dxdyqzd0 0
22/5222
3
)(2)(3
Vertical Stress Below Corner of UniformlyLoaded Flexible Rectangular Area
Figure 6.16. Das FGE (2005).
Where:
= Change in Vertical Stressq = Load per Unit Areaz = Depth
zLn
zBm ;
VERTICAL STRESS INCREASE (z) IN SOILSRECATNGULAR LOADING (Boussinesq, 1883)
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Variation of I2 with m and n.Figure 6.17. Das FGE (2005).
VERTICAL STRESS INCREASE (z) IN
SOILSRECTANGULAR
LOADING (Boussinesq, 1883)
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Slide 15 of 23Figure 12. NAVFAC DM7.01.
VERTICAL STRESS INCREASE (z) IN
SOILSRECTANGULAR
LOADING (Westergaard)
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VERTICAL STRESS INCREASE (z) IN SOILSRECTANGULARLY LOADED AREA
q I2(1) I2(2) I2(3) I2(4)
c qIc
Ic f (m1,n1)
m1 LB
;n1 zB2
Within a Rectangularly Loaded Area:
Under Center of Footing:
Figure 6.18. Das FGE (2005).
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VERTICAL STRESS INCREASE (z) IN SOILSCENTER OF RECTANGULARLY LOADED AREA
(Boussinesq)Table 6.6 Variation of Ic with m1 and n1 (Das, FGE 2006).
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BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 Table C-1)
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BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 Table C-1)
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BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 Table C-1)
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STRIPFOOTING
SQUAREFOOTINGBOUSSINESQ
GRAPHICAL SOLUTION
(EM 1110-1-1904Figure 1-2)
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WESTERGAARD GRAPHICAL SOLUTION
(NAVFAC DM7.01 Figure 11)
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WESTERGAARD GRAPHICAL SOLUTION
(NAVFAC DM7.01 Figure 11)
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2V:1H DISTRIBUTION METHOD
))(( zLzBQ
z
Figure C-1. USACE EM1110-1-1904.
Where:z = Change in Total Vertical
StressQ = Applied Foundation LoadB = Foundation WidthL = Foundation Length
