14.330 Soil Stresses - · PDF fileRevised 02/2015 Slide 2 of 24 14.330 SOIL MECHANICS Soil...
Transcript of 14.330 Soil Stresses - · PDF fileRevised 02/2015 Slide 2 of 24 14.330 SOIL MECHANICS Soil...
Slide 1 of 24Revised 02/2015
14.330 SOIL MECHANICS
Soil Stresses
VERTICAL STRESS INCREASES IN SOIL
TYPES 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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14.330 SOIL MECHANICS
Soil Stresses
Examples:- Exterior Wall Foundations
Examples:- Column Footings
Area Loads (q)Strip Loads (q)
VERTICAL STRESS INCREASES IN SOIL
TYPES OF LOADING
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14.330 SOIL MECHANICS
Soil Stresses
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.
VERTICAL STRESS INCREASES IN SOIL
ANALYSIS METHODS: BOUSSINESQ (1993)
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14.330 SOIL MECHANICS
Soil Stresses
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 stresses
substantially 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 with
the Boussinesq method for depths between B and 4B below the
foundation.
VERTICAL STRESS INCREASES IN SOIL
ANALYSIS METHODS: WESTERGAARD
VERTICAL STRESS INCREASES IN SOIL
ANALYSIS METHODS: 2V:1H METHOD
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14.330 SOIL MECHANICS
Soil Stresses
2/522
3
5
3
)(2
3
2
3
zr
zP
L
zPz
+==∆
ππσ
( )[ ] 122/5221/
1
2
3I
z
P
zrz
Pz =
+=∆
πσ
Stresses in an Elastic Medium Caused by Point LoadingFigure 6.11. Das FGE (2005).
Where:
∆σz = Change in Vertical Stress
P = Point Load
I1 =3
2π
1
r / z( )2
+1
5/2
*Based on homogeneous, elastic, isotropic infinitely large half-space
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
POINT LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Table 6.1 Variation of I1 (Das, FGE 2006).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
POINT LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
22
222
3
1
2
)/(
)(
2
+
=∆
+=∆
z
xzq
zx
qz
π
σ
πσ
Where:∆σ = 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 (∆σ∆σ∆σ∆σZ) IN SOIL
LINE LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Table 6.3 Variation of ∆σ/(q/z) with x/z (Das, FGE 2006).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
LINE LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
[ ])2cos(sin δβββπ
σ ++=∆q
Flexible Strip Load over the Surface of
a Semi-infinite Soil MassFigure 6.13. Das FGE (2005).
Where:
∆σ = 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 (∆σ∆σ∆σ∆σZ) IN SOIL
STRIP LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Table 6.4 Variation of ∆σ/q with 2z/B and 2x/B (Das, FGE 2006).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
STRIP LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
∆σ = q 1−1
(R / z)2 +1
3/2
Vertical Stress Below Center of Uniformly Loaded
Flexible Circular AreaFigure 6.15. Das FGE (2005).
Where:∆σ = Change in Vertical Stressq = Load per Unit Areaz = DepthR = Radius
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
CIRCULAR LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Table 6.5 Variation of ∆σ/q with z/R (Das, FGE 2006).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
CIRCULAR LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
+−+
+++
++
++
+++
++
=
−
1
12tan
1
2
1
12
4
1
2222
221
22
22
2222
22
2
nmnm
nmmn
nm
nm
nmnm
nmmn
Iπ
∫ ∫ ∫= =
=++
==∆
B
y
L
x
qIzyx
dxdyqzd
0 0
22/5222
3
)(2
)(3
πσσ
Vertical Stress Below Corner of Uniformly
Loaded Flexible Rectangular Area
Figure 6.16. Das FGE (2005).
Where:∆σ = Change in Vertical Stressq = Load per Unit Areaz = Depth
z
Ln
z
Bm == ;
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
RECTANGULAR LOADING (BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Variation of I2 with m and n.Figure 6.17. Das FGE (2005).
VERTICAL STRESS
INCREASE (∆σ∆σ∆σ∆σZ) INSOIL
RECTANGULAR
LOADING
(BOUSSINESQ 1883)
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14.330 SOIL MECHANICS
Soil Stresses
Figure 12. NAVFAC DM7.01.
VERTICAL STRESS
INCREASE (∆σ∆σ∆σ∆σZ) INSOIL
RECTANGULAR
LOADING
(WESTERGAARD)
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14.330 SOIL MECHANICS
Soil Stresses
∆σ = q I2(1) + I2(2) + I2(3) + I2(4)
∆σ c = qIc
Ic = f (m1, n1)
m1 =L
B;n1 =
z
B
2
Within a Rectangular Loaded Area:
Under Center of Footing:
Figure 6.18. Das FGE (2005).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
RECTANGULAR LOADED AREA
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14.330 SOIL MECHANICS
Soil Stresses
Table 6.6 Variation of Ic with m1 and n1 (Das, FGE 2006).
VERTICAL STRESS INCREASE (∆σ∆σ∆σ∆σZ) IN SOIL
CENTER OF RECTANGULAR LOADED AREA
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14.330 SOIL MECHANICS
Soil Stresses
BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 TABLE C-1)
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14.330 SOIL MECHANICS
Soil Stresses
BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 TABLE C-1)
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14.330 SOIL MECHANICS
Soil Stresses
BOUSSINESQ SOLUTIONS SUMMARY(EM 1110-1-1904 TABLE C-1)
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14.330 SOIL MECHANICS
Soil Stresses
STRIPFOOTING
SQUAREFOOTING
BOUSSINESQ
GRAPHICAL
SOLUTION(EM 1110-1-1904
FIGURE 1-2)
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14.330 SOIL MECHANICS
Soil Stresses
WESTERGAARD
GRAPHICAL
SOLUTION(NAVFAC DM7.01 FIGURE 11)
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14.330 SOIL MECHANICS
Soil Stresses
WESTERGAARD
GRAPHICAL
SOLUTION(NAVFAC DM7.01 FIGURE 11)
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14.330 SOIL MECHANICS
Soil Stresses
))(( zLzB
Qz
++=∆σ
Figure C-1. USACE EM1110-1-1904.
Where:∆σz = Change in Total
Vertical StressQ = Applied Foundation
LoadB = Foundation WidthL = Foundation Length
VERTICAL STRESS INCREASES IN SOIL
ANALYSIS METHODS: 2V:1H METHOD