1. Modeling of the Active Wedge behind a Gravity Retaining Wall By: Rex Radloff 14.531 Advanced Soil Mechanics University of Massachusetts Lowell Department of Civil Engineering

3. Active force acts 1/3H from the base

4. Front face angle (θ) is 90⁰

5. Overburden slope (β) is 0⁰

7. What are its effects on the failure criteria?

8. To what degree?Rankine stress distribution behind a gravity retaining wall. University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 2

9. Active Wedge with Reactions W 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 Active wedge with consideration towards θ, β, and φw University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 3

11. Angle α of the back face of the active wedge.

12. Active force, Pa, that is produced by the active wedge.

13. The location of the Pa resultant.

14. FS against overturning and sliding.

15. Eccentricity

16. Maximum stress, qMAX underneath the foundation.University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 4

17. Example Retaining Wall: Case 1 University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 5 γ1 = 117 pcfγconc = 150 pcf φ1’ = 34⁰ δ’ = 18⁰ Ca = 800 psf Assume firm underlying soil

18. γ= 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft., Wh = 6.00 ft., φ = 34⁰, β = 0⁰ 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 University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 6

19. Failure Analysis: H/Wh = 3.4 |e| ≤𝑏6 2.08 ft The 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. University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 7

20. Example Retaining Wall: Case 2 University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 8 γ1 = 117 pcfγconc = 150 pcf φ1’ = 34⁰ δ’ = 18⁰ Ca = 800 psf Assume firm underlying soil

21. γ= 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft., Wh = 20.75 ft., φ = 34⁰, β = 0⁰ 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 University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 9

22. Failure Analysis: H/Wh = 1.0 |e| ≤𝑏6 4.46 ft University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 10 +58%

23. Pa vs. α vs. θ φw = 34⁰ Line of intersection φw = 0⁰ Produced using Mathematica University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 11

24. 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 12

25. Is it worth consideration? University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 13 Yes

26. References University of Massachusetts Lowell 14.531 Advanced Soil Mechanics 14 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®