Upload
rexradloff
View
2.191
Download
3
Tags:
Embed Size (px)
Citation preview
Modeling of the Active Wedge behind a Gravity Retaining Wall
By: Rex Radloff
14.531 Advanced Soil MechanicsUniversity of Massachusetts LowellDepartment of Civil Engineering
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
2
Rankine Active Wedge
• Wall friction is neglected• Active force acts 1/3H from the
base• Front face angle (θ) is 90⁰• Overburden slope (β) is 0⁰• Back face angle (α) is 45 + φ/2
What if the above was not held constant? • How would the magnitude
and location of Pa change?• What are its effects on the
failure criteria?• To what degree?
Rankine wedge behind a gravity retaining wall.
Rankine stress distribution behind a gravity retaining wall.
φw
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
3
Active Wedge with Reactions
Active wedge with consideration towards θ, β, and φw
Note:The weight and centroid of the active wedge is a function of H, Wh, WBs, γ, α , θ, and β
The weight and centroid of the backfill is a function of H, WBs, Wh, γ, θ, and β
W
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
4
Modeling the Active WedgeKnown
Wall dimensions and materials. Soil and wall shear strength
parameters. Grade of overburden soil.
Solve for Angle θ of the front face of the active wedge. Angle α of the back face of the active wedge. Active force, Pa, that is produced by the active wedge. The location of the Pa resultant. FS against overturning and sliding. Eccentricity Maximum stress, qMAX underneath the foundation.
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
5
Example Retaining Wall: Case 1
γ1 = 117 pcfγconc = 150 pcfφ1’ = 34⁰δ’ = 18⁰Ca = 800 psf
Assume firm underlying soil
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
6
θ = 90⁰, φw = 34⁰ Pa = 6.51 Kips, α = 56.8⁰ θ = 90⁰, φw = 0⁰ Pa = 7.12 Kips, α = 62.0⁰ (45 + φw/2)θ = 71⁰, φw = 34⁰ Pa = 11.24 Kips, α =61.2⁰ θ = 71⁰, φw = 0⁰ Pa = 10.84 Kips, α = 71.2⁰
γ = 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft., Wh = 6.00 ft., φ = 34⁰, β = 0⁰
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
7
Failure Analysis: H/Wh = 3.4
Case FS Over Turning
FS Sliding
Eccentricity, ft.
qMAX, ksf
θ = 90⁰, φw = 34⁰,Pa = 6.51 Kips, α = 56.8⁰
3.53 3.47 -1.45 31
θ = 90⁰, φw = 0⁰ , Pa = 7.12 Kips, α = 62.0⁰
3.69 2.46 0.53 18
θ = 71⁰, φw = 34⁰, Pa = 11.24 Kips, α =61.2⁰
0.53 2.68 -0.04 17
θ = 71⁰, φw = 0⁰, Pa = 10.84 Kips, α = 71.2⁰
0.85 1.61 3.48 32The increase in Pa lead to a below acceptable FS for overturning.
The greater distance between the Pa vector and pt.O led to a below acceptable eccentricity.
|e| 2.08 ft
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
8
Example Retaining Wall: Case 2
γ1 = 117 pcfγconc = 150 pcfφ1’ = 34⁰δ’ = 18⁰Ca = 800 psf
Assume firm underlying soil
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
9
θ = 90⁰, φw = 34⁰ Pa = 6.51 Kips, α = 56.8⁰ θ = 90⁰, φw = 0⁰ Pa = 7.12 Kips, α = 62.0⁰ (45 + φw/2)θ = 45⁰, φw = 34⁰ Pa = 29.03 Kips, α =57.2⁰ θ = 45⁰, φw = 0⁰ Pa = 21.41 Kips, α = 84.50⁰
γ = 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft., Wh = 20.75 ft., φ = 34⁰, β = 0⁰
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
10
Failure Analysis: H/Wh = 1.0
Case FS Over Turning
FS Sliding
Eccentricity, ft.
qMAX, ksf
θ = 90⁰, φw = 34⁰,Pa = 6.51 Kips, α = 56.8⁰
7.11 7.07 -1.93 50
θ = 90⁰, φw = 0⁰ , Pa = 7.12 Kips, α = 62.0⁰
15.04 5.19 -0.70 35
θ = 71⁰, φw = 34⁰, Pa = 11.24 Kips, α =61.2⁰
∞ 7.23 -1.90 55
θ = 71⁰, φw = 0⁰, Pa = 10.84 Kips, α = 71.2⁰
∞ 2.36 2.66 42
|e| 4.46 ft
+58%
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
11
Pa vs. α vs. θ
φw = 34⁰
φw = 0⁰ Li
ne o
f in
ters
ectio
n
Produced using Mathematica
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
12
Conclusion The active force (Pa) depends on the angles θ, α, and
φw found among the active wedge.
Any deviation between the calculated active force behind the same retaining wall depends on the combining effects of θ, α, and φw found among the active wedge.
As the angle θ decreases, the Pa will increase as well as the variance between the Pa calculated with and without wall friction.
Hard to openly predict the influence on the failure criteria
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
13
Is it worth consideration?Yes
University of Massachusetts Lowell 14.531 Advanced Soil Mechanics
14
References Das, B.M., (2006). “Principles of Geotechnical Engineering – Sixth Edition”
Lambe, T.W., and R.V. Whitman, (1969). “Series in Soil Engineering - Soil Mechanics”
Mangano, Sal, (2010). “Mathematica Cookbook”
AutoCAD 2011®
Wolfram Mathematica®