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# Modeling of the Active Wedge behind a Gravity Retaining Wall

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The Rankine horizontal stress method is a very common and simple approach in calculating the active force behind a gravity retaining wall. However, because numerous assumptions have to be made a deviation among results will arise, although the degree of this discrepancy have been previously defined as negligible and is typically ignored.
A full wedge analysis was performed using the program Wolfram Mathematica<sup>®</sup> to specifically outline the degree of this discrepancy. Results showed that the deviation among the calculated active force was relative to the conditions of the retaining system, and can be substantial at times. Additionally, it was revealed that the difference among results could not be defined as negligible or substantial unless a full wedge analysis was performed.

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### Modeling of the Active Wedge behind a Gravity Retaining Wall

1. 1. 14.531 Advanced Soil Mechanics Modeling of the Active Wedge behind a Gravity Retaining Wall By: Rex RadloffThe active force (Pa) as a function of the front face angle (θ) and the back face angle (α) found on the activewedge. Produced using Wolfram Mathematica®Abstract The Rankine horizontal stress method is a very common and simple approach incalculating the active force behind a gravity retaining wall. However, because numerousassumptions have to be made a deviation among results will arise, although the degree of thisdiscrepancy have been previously defined as negligible and is typically ignored. A full wedge analysis was performed using the program Wolfram Mathematica® tospecifically outline the degree of this discrepancy. Results showed that the deviation among thecalculated active force was relative to the conditions of the retaining system, and can besubstantial at times. Additionally, it was revealed that the difference among results could notbe defined as negligible or substantial unless a full wedge analysis was performed. 1
2. 2. 14.531 Advanced Soil Mechanics Table of ContentsIntroduction……………………………………………….……………………………………………………………..…………3Rankine Active Wedge………………………………………………………………………………………………………….3Wedge Analysis……………………………………………………………………………………………………………………4Case 1 (θ = 72:  90:, φw = 34:)……………………………………………………………………………………..….5Results: Case 1 (θ = 72:  90:, φw = 34:)……………………………………………………………………………5Failure Criteria: Case 1 (θ = 72:  90:, φw = 34:)…………………………………………………………….….6Case 2 (θ = 45:  90:, φw = 34:)……………………………………………………………………………………..….7Results: Case 2 (θ = 45:  90:, φw = 34:)……………………………………………………………………………8Failure Criteria: Case 2 (θ = 4:  90:, φw = 34:)………………………………………………………………….8Conclusion……………………………………………………………………………………..………………………………….…9References.…………………………………………………………………………………………………………………..…...12Appendices.………………..………………………………………………………………………………………………..…...13 2
3. 3. 14.531 Advanced Soil MechanicsIntroduction As an engineer it is important to accurately produce results in a short amount of time.Producing incorrect results quickly is unacceptable as the same goes for producing the correctresult over an extended period of time. It is common practice for an engineer to minimize timeby making assumptions and producing results that are within the range of error. However, ifthis practice is fully adopted, then there may be times when an assumption can be at theexpense of an accurate result. The following will attempt to place the degree of significance ofputting more time into improving the results calculated from the Rankine method whenanalyzing a retaining wall, and determining frankly, “is it worth it?”Rankine Active Wedge The Rankine horizontal stress method is a very common and simple approach incalculating the active force behind a gravity retaining wall. The assumptions associated with thismethod is that the wall friction is φw = 0:, front face of the wedge is θ = 90:, back face of thewedge is α = 45 + φ/2, overburden grade is β = 0:, and the resultant of the active force acts at adistance of H/3 from the base. Figures 1 and 2 demonstrate these constraints by showing thegeometry of the driving wedge and stress distribution behind the retaining wall. However, again, this analysis represents the case with all of the above assumptions andthere is no way of knowing if these conditions are those that produce the highest active force.Therefore, a full wedge analysis will need to be performed while these variables are free todeviate, and the conditions which produce the largest active force should be compared withthose from the above method. It is important to stress that it is not the difference betweeneach calculated active force that is of concern, rather the difference of produced moment onthe foundation, which is a function of the resultants location. Figure 1 Geometry of the driving wedge under the Rankine active wedge analysis. 3
4. 4. 14.531 Advanced Soil MechanicsFigure 2 Stress distribution of the active pressure behind a gravity retaining wall using the Rankine horizontalactive stress parameter Ka.Wedge Analysis To perform an accurate wedge analysis behind a gravity retaining wall, Figure 3 must beconsidered in its entirety. The known variables are as follows: wall dimensions and materials,soil and wall shear strength parameters, and the grade of the overburden soil. The variables tobe solved for, at the maximum active force, is the angle of the front face (θ), the angle of theback face (α), and location of the active force. To model such a problem, it is preferred to use amodeling program such as Wolfram Mathematica® to plot and interpret data. Figure 3 Active wedge with consideration towards θ, β, and φw 4
5. 5. 14.531 Advanced Soil MechanicsCase 1 (θ = 72⁰  90⁰, φw = 34⁰) The example which will be interpreted for comparison purpose can be seen in Figure 4.The analysis will consider the blue and red wedge with and without wall friction (φw). Theexample did limit the overburden grade (β) to 0: in hopes to recognize any deviation specificallyrelated to the Rankine method. Also the maximum angle (θ) of the front face of the wedge inred is controlled by the wall dimensions. Figure 4 Example 1 for an active wedge analysisResults: Case 1 (θ = 72⁰  90⁰, φw = 34⁰) The data from case 1 was analyzed using Wolfram Mathematica and can been seen inFigure 5. The active force (Pa) and angle of the back face (α) of the blue wedge (θ = 90:) are6.51 Kips and 56.8: for the analysis with wall friction, and 7.12 Kips and 62.0: for the analysiswithout wall friction. It should be noted that α = 62.0: is also equal to 45 + φw/2. The resultsyielded are fairly close to one another; however the differences in moments still need to beinterpreted to actually understand its influence on the design. The active force (Pa) and angle of the back face (α) of the red wedge (θ = 71:) are 11.24Kips and 61.2: for the analysis with wall friction, and 10.84 Kips and 71.2: for the analysiswithout wall friction. The active forces calculated are fairly close to one another; however theangles of the back face (α) deviate by 10:. This is especially important, as an engineer uses thelocation of the failure in certain types of retaining wall design, i.e. using tiebacks, soil nails, ordead-man anchors. 5
6. 6. 14.531 Advanced Soil MechanicsFigure 5 Wedge analysis using the following: γ = 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft, Wh = 6.00 ft, φ =34⁰, β = 0⁰ Between both sets of values, the calculated active force (Pa) and angle of the back face(α) deviate significantly from one another. These results show an underestimation of the activeforce by a factor of 1.7. With that said, this does not mean the foundation was under designedby the same factor; there still needs to be an analysis done on the retaining walls failurecriteria, which takes into consideration the location of each resultant.Failure Criteria: Case 1 (θ = 72⁰  90⁰, φw = 34⁰) The failure criteria of a retaining wall – shallow foundation system consists ofoverturning, sliding, eccentricity, and bearing capacity, and any deviation in the above resultswill only be considered depending on their effects on this criteria. Table 1 present this failurecriteria for each case. For the analysis of the blue wedge (θ = 90:), the failure criteria deviated by a factor 1.4and 1.7 for the factor of safety against sliding and the maximum bearing stress respectively.This hints that the resultant shifted enough to create a large enough difference in moments andalso verifies the significance of the wall friction, regardless of the similar active forces. For theanalysis of the red wedge (θ = 71:), the failure criteria deviated by 1.6 and 1.9 for the factor ofsafety against sliding and the maximum bearing stress respectively. 6
7. 7. 14.531 Advanced Soil Mechanics Table 1 Values against the failure criteria for each analysis Case FS Over- FS Eccentricity q MAX Turning Sliding (ft) (ksf) θ = 90:, φw = 34:, 3.53 3.47 -1.45 31 Pa = 6.51 Kips, α = 56.8: θ = 90:, φw = 0:, 3.69 2.46 0.53 18 Pa = 7.12 Kips, α = 62.0: θ = 71:, φw = 34:, 0.53 2.68 -0.04 17 Pa = 11.24 Kips, α =61.2: θ = 71:, φw = 0:, 0.85 1.61 3.48 32 Pa = 10.84 Kips, α = 71.2: However, the two wedges of interest are the blue (θ = 90:) and the red (θ = 71:), as theyrepresent the two extremes between making and not making assumptions. The valuespresented in Table 1 show a deviation by a factor of 7.0 for the factor of safety againstoverturning; the remaining values are close enough to ignore any difference. Nevertheless, thedifference in the factor of safety against overturning is enough to consider the two methods.Case 2 (θ = 45⁰  90⁰, φw = 34⁰) The second case that will be interpreted under the wedge analysis can be seen in Figure5. The purpose of this example is to understand if there is any validity to the above results,which can be determined by extending the front face angle (θ) to an extreme case and monitorhow each result varies. The blue wedge will represent the Rankine case being interpreted withand without wall friction (as analyzed before) and the red wedge will represent the extremecase with the front face angle (θ) equal to 0, with and without wall friction. Figure 5 Example 2 for an active wedge analysis 7
8. 8. 14.531 Advanced Soil MechanicsResults: Case 2 (θ = 45⁰  90⁰, φw = 34⁰) The data from case 2 was analyzed using Wolfram Mathematica and can been see inFigure 6. The active force (Pa) and angle of the back face (α) of the red wedge (θ = 45:) are29.03 Kips and 62.0: for the analysis with wall friction, and 21.41 Kips and 84.5: for the analysiswithout wall friction. The active forces calculated now begin to separate much more thanbefore, as does the difference in the angle (α) of the wedges back face. Also, the active forceand angle of the back face found under the blue wedge (θ = 45:) are the same from theprevious example.Figure 6 Wedge analysis using the following: γ = 117 pcf, Cs = 0 psf, Cw = 0 psf, H = 20.75 ft, Wh = 6.00 ft, φ =34⁰, β = 0⁰Failure Criteria: Case 2 (θ = 45⁰  90⁰, φw = 34⁰) The two wedges of interest are the Rankine wedge (blue, θ = 90:) without wall frictionand the red wedge (θ = 45:) with wall friction. Results from the failure criteria show a largerdeviation between each case, but more importantly the maximum bearing stress (q MAX)deviates by a factor of 1.57. This proves that increasing the front face angle (θ) to a certaindegree will lead to a significant deviation between the maximum bearing stress. It should alsobe noted that the overturning values approach infinity as the resultant of the active forcepoints below the point of rotation (pt. O). 8
9. 9. 14.531 Advanced Soil Mechanics Table 2 Values against the failure criteria for each analysis Case FS Over- FS Eccentricity q MAX Turning Sliding (ft) (ksf) θ = 90:, φw = 34:, 7.11 7.07 -1.93 50 Pa = 6.51 Kips, α = 56.8: θ = 90:, φw = 0:, 15.04 5.19 0.70 35 Pa = 7.12 Kips, α = 62.0: θ = 45:, φw = 34:, ∞ 7.23 -1.90 55 Pa = 11.24 Kips, α =61.2: θ = 45:, φw = 0:, ∞ 2.36 2.66 42 Pa = 10.84 Kips, α = 71.2:Conclusion The full wedge analysis, for the active case, proves that there is no significant differencebetween regarding and disregarding wall friction when comparing the active force. However,because of a shift in the location of the active forces resultant, a difference in moments iscreated, making for a significant change in the values produced under the failure criteria.Therefore it is important to consider the wall friction when analyzing a specific wedge. The effects of deviating the front face angle (θ) is relative to the conditions. Figure 7shows the two active force curves, with and without wall friction, and its relationship with thefront and back face angles (θ, α) of the failure wedge. The plot shows two aggressive curveswhich actually intersect at a critical location. Visually it can be understood that these functionsare difficult to predict beforehand; also as there is a decrease with the front face angle (θ),there is a larger deviation between the active forces (Pa) calculated with and without wallfriction. However, the moments created and its influence on the failure criteria are of moreimportance, as this is what dictates design. The above proves that by making assumptions to simplify a problem and to save time,the retaining wall – shallow foundation system can be significantly over or under designed. Andbecause it is not realistic to predict the influence of these assumptions beforehand, a fullanalysis should be performed for assurance. 9
10. 10. 14.531 Advanced Soil Mechanics φw = 34: φw = 0:Figure 7 The active force (Pa) as a function of the front face angle (θ) and the back face angle (α) 10
11. 11. 14.531 Advanced Soil MechanicsReferences  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 11