EJGE Paper

Shear Ring Method for Soil-Structure Interaction Analysis in Floodwalls

Mete Oner, William P. Dawkins

Professors, School of Civil Engineering, Oklahoma State University, Stillwater, OK

and

Reed Mosher

Research Civil Engineer, Information Technology Laboratory, U.S. Army Engineer, Waterways Experiment Station, Vicksburg, MS.

ABSTRACT

A new method is introduced for engineering design/analysis of floodwalls. The method is based on the earlier findings on the behavior of floodwalls: (1) the principal mode of motion is horizontal, but with a rotation component, (2) the soil, as opposed to the pile, is the major component of the system which determines the overall deformations. The development of the method and its verification through comparisons with comprehensive finite element analyses are described.

Keywords: Analysis, Soft Clays, Sheetpiles, Deformation

INTRODUCTION

The companion paper (Ref.5) presented the results of the first phase of a research effort towards the development of a design oriented analysis technique for floodwalls. The first phase involved parametric analyses of typical floodwall systems utilizing a comprehensive analysis technique based on a plane strain finite element method. The main purpose of these analyses was to understand the mechanisms involved in typical floodwalls and, in the light of the results obtained, to develop a simpler, yet reasonably accurate, soil-structure-interaction (SSI) technique suitable for design purposes. Initially, the possibility of deriving a set of "nonlinear soil response curves" was explored. Such curves, representing the relationship between the displacement and pressure at the wall-soil interface, could be incorporated in the conventional, one dimensional SSI procedure (e.g., Ref. 2). This would produce an economical method for engineering analysis. However, the finite element analyses revealed that the nature of the floodwall problem is more complex than ordinary sheetpile walls and the need to develop a new techniqe became apparent. Therefore the second phase of this study was devoted to the development of the a technique, tentatively called the "shear ring" method.

The method is based on the findings that (1) the principal mode of motion is is essentially a rigid body rotation of the entire wall/soil system; and (2) the soil (as opposed to the pile) is the major component of the system which determines the overall deformations.

SOIL ONLY ANALYSIS

The first step in the development of the shear ring method was the construction of a model that only involved the soil (i.e., the stiffness of the sheet pile wall was not included in the model). The original purpose of this was to test the basic idea of a rotational model for predicting levee deformation approximately. However, it has proven to be more useful than that, and has become an essential part of the final method. Therefore a description of the soil-only method is given in this section.

Assumptions

The main assumption is that the levee and its foundation soil rotate about a "pivot" point at some height above the levee. In this first version of the method the soils involved are assumed to be perfectly undrained, saturated clays. The problem is analyzed numerically, to accomodate any soil profile, by dividing the section into a number of rings (Figure 1).

Figure 1. Shear ring model of a floodwall

Each ring boundary is assumed to undergo a rigid body displacement, rotating around the pivot. A relative displacement between the two boundaries of a ring will produce a uniform shear strain along that ring. The corresponding shear stress, however, is non-uniform because of the inhomogeneity of the soil rigidity. Although the stiffness of the pile is ignored the hydrostatic force acting on it is included in the loading. The system idealized in this manner is statically determinate. Basically, a floodwall section does not have to be perfectly symmetrical about the centerline, but an extreme deviation from symmetry may render the basic rotational deformation assumption invalid.

Spring Constant and Shear Modulus

The spring constant representing the stiffness of a ring in the system is a rotational shear spring, i.e., the ratio of the total shear force in the ring to the relative displacement between the two circular boundaries of the ring (The term shear force is used for convenience; more precisely, it is the integral of the shear stress along the ring, or equivalently, the total moment of the shear stresses around the pivot divided by the radius). Referring to Figure 2, it is a simple matter to show that the spring constant, k, for a given ring can be expressed as:

k = G l_i / d (1)

Figure 2. Derivation of a soil-only ring's stiffness (shear spring constant) k

where G is the average shear modulus, l_i is the arc length, and d is the radial thickness of the ring. The geometric constants are known from the levee cross section and the ring subdivision utilized. The determination of the average shear modulus is explained below.

To be able to compare the results with the finite element solutions performed earlier (Ref. 5) the same f model is used which gives the tangent modulus as

(2)

or the equivalent secant modulus (for primary loading) as

(3)

where f at a point is calculated as the maximum shear stress at that point divided by the undrained shear strength at the same point. The required model parameter, Gi (shear modulus at f = 0) is found, as in the finite element analyses, from a prescribed E50/cu ratio (Ref. 5),

(4)

The average shear strength of a given ring is evaluated by dividing the ring into a number of equal angular segments, and finding the value of cu at the mid-point of each segment from the prescribed shear strength profile.

Either the tangent or the secant modulus can be used in this method; in general, a step-by-step loading needs to be followed in the former and an iterative method should be used in the latter case. These methods are established procedures in geotechnical analysis (e.g., see Refs. 3 and 4). Both methods have been employed here and procuced essentially the same results. The secant modulus method, however, was found to be more practical in the soil-only analysis, because it can be formulated without requiring any iterations in this case, as shown below.

The loading considered on the system is the hydrostatic pressure acting on the exposed surfaces on the flood side. The external loads are calculated for each ring edge, and accumulated, proceeding from the top ring towards the bottom ring to find the shear load carried by each ring. The incremental force for a given ring is found as follows. First, the water pressure is evaluated at the points of intersection of the ring boundaries with the surface line. To develop the working equations, consider one segment of the surface line, the edge of one ring (Figure 3a). Calling the pressures at the two ends of the segment p_u and p_l, the resultants for the two halves of the segment, P_u and P_l , are found as

P_u = l (3 p_u + p_l)/8 (5)

Figure 3(a). Hydrostatic pressures (p) acting on the upper and lower boundaries of a ring ring boundary
Note that the pressure is normal to the ground surface, but not necessarily in the tangential direction

Figure 3(b). The resolution of a resultant force, P_u in the directions tangential and normal to the ring boundaries

P_l = l (p_u + 3 p_l)/8 (6)

The rotational components of these forces (i.e., in the direction normal to the local radius), indicated as P_uq components in Figure 3b, are added to the accumulated forces. Using the geometric construction in Figure 3b, these components are found to be

P_nu = p_u d/2 (7)

P_nl = p_l d/2 (8)

The two halves need to be evaluated separately because the shear force acting on the central arc is required in the deformation calculation. The total shear force, S, acting in a ring is the sum of all the partial forces acting above the center of that ring. The average shear stress acting in any ring is found by dividing the total shear force, S, in the ring by the central arc length of the ring, li:

t_ave = S / l_i (9)

Next the average f is evaluated, which in turn yields the average shear modulus for that ring (through the model, Eq. 2 or 3), and finally this shear modulus is used to calculate the shear spring constant for the ring by Eq. 1. It should be noted that this procedure does not require any iteration for nonlinearity when the secant modulus method is used.

Deformation Calculation

Once the shear forces and spring constants are determined for each ring, the relative displacements can be evaluated as

Dd = S / k (10)

These relative displacements are "integrated" numerically to obtain the displacement profile along the centerline, assuming that the absolute displacement of the bottom of the model is stationary, and working upwards.

It should be noted that the relative displacements cannot be summed directly in the shear ring model because of the basic assumption of rotation about a pivot point. A correction must be applied for this rotation as follows. Referring to Figure 4, suppose that the total displacement of the arc numbered i+1 has been computed earlier to be d_i+1 and the displacement of the arc numbered i, d_i, is sought. The average shear strain in the ring between these two arcs, denoted by g_i, is equal to the relative displacement computed for that ring divided by the ring thickness. This relationship gives the relative displacement Dd_i. With the pivotal angles denoted by thetas, if q_i+1 is known, the increment q_i is first found from d_i/R_i, and these two are added to obtain q_i from which the required absolute displacement is found (Figure 4),

Figure 4. Relationships among angles and displacements needed for integratiion of displacements

(11)

The working formula thus obtained amounts to modifying d_i+1 by the factor (R_i/R_i+1) before adding the relative displacement:

(12)

Numerical Accuracy

As in any other numerical method, sensitivity of the results to the various approximations made must be studied. In this particular method, there are two arbitrary selections that must be made: (1) radius increment, or "depth" of a ring, d, and (2) the number of subdivisions used for computing an average shear strength for a ring. Table 1 illustrates the sensitivity of the results to these two factors. The results listed in the table are the calculated crest displacements for the "high strength profile" case at 16 feet water head (Ref. 5). The d values are (as the label indicates) average values. This is because d is not constant since the subdivision is done in such a way that the break points in the soil surface profile are automatically selected as ring boundaries, and the coarse R-intervals thus obtained are subdivided equally. The " nq " values are the number of points taken for cu averaging along a ring. Because of symmetry, only one half of a ring is used in this averaging. The pivot elevation used in these cases is 110 ft (100 ft from levee crest). Table 1 shows that the results are not very sensitive to these factors, and the R-subdivision seems to be relatively less important than the angular subdivision.

Table 1. Sensitivity of calculated crest displacement to subdivisioning
(in inches, 1 inch = 25.4 mm, 1 ft = 0.305 m)

nq = 2 nq = 2 nq = 4 nq = 8 nq = 16

5 2.0495 2.0730 2.0450 2.0270 2.026

2 2.0603 2.0882 2.0598 2.0413 2.036

1 2.0728 2.1003 2.0712 2.0525 2.048

Effects of Pivot Elevation

The selection of a pivot elevation is not immediately clear. First, the effect of the pivot point elevation, hp, will be studied to understand the sensitivity of the results. Next, a rational method will be shown for determination of the correct pivot elevation.

Taking the same case used in the previous paragraph, with d_ave = 1 ft and nq = 8, the variation shown in Figure 5 is obtained for the crest displacement. The maximum displacement, also plotted, is not necessarily the same as the crest displacement due to rotation. Note that a pivot elevation of 10 ft corresponds to placing the pivot point at the levee crest (i.e., assuming that the whole system is rotating about that point), and this naturally results in a zero displacement. It is observed in Figure 5 that the dependence of the solution to pivot elevation is significant, but over a wide range, from approximately 100 ft to 200 ft in this particular case, the calculated crest displacement value is quite stable. In this range the calculated crest displacement agrees amazingly closely with the finite element results (the corresponding finite element results for 10 to 30 ft pile penetration are in the range of 1.9 to 2.1 inches). The calculated crest displacement decreases gradually after the peak that occurs at about a pivot elevation of 120 ft.

Figure 5. Variation of calculated displacements with pivot elevation
High strength profile, 16 ft head (1 ft = 0.305 m, 1 inch = 25.4 mm)

Variation of calculated displacements with pivot elevation

To illustrate the dependence on pivot elevation more closely, the calculated lateral displacement profiles are plotted in Figure 6 for a wide range of pivot elevation (50 to 150 ft). It is observed that lower pivot elevations introduce very large "backwards" distortions in the shallow regions, and this effect decreases as the pivot elevation is increased.

Figure 6. Effect of pivot elevation, hp, on displacement profile
High strength profile, 16 ft head. hp=100 provides the perfect fit
(1 ft = 0.305 m, 1 inch = 25.4 mm)

The question at this point is how to choose the pivot elevation in analyzing a new problem.

Determination of the Pivot Elevation

The basis of the method used for determination of the pivot elevation is a generalization of the principle of minimum potential energy as used in variational finite element formulations (e.g., Ref. 3), which may be stated as "of all statically admissible states of stress and kinematically admissible states of deformation, the one that minimizes the total potential energy functional is the correct solution." Of course this would allow us find the correct shear ring solutions, but not necessarily the "exact" solution of a given problem. The total potential energy functional can be evaluated in a shear ring model, after the solution, as

TPE = SE - PE (13a)

where SE is the total strain energy stored in the system that can be obtained from the shear stresses and strains that are already computed,

(13b)

and PE is the potential energy loss of external loads as the system deforms,

which can be found by lumping the Pnu and Pnl forces at the nodes, Pi, and pairing these with the corresponding displacements, ui,

(13c)

The results of this calculation, for the same case used in the previous example, are plotted in Figure 7. Indeed, the potential energy functional is a minimum at hp = 120 ft which also gives, roughly, the maximum displacement.

Figure 7. High strength profile, 16 ft head (1 ft = 0.305 m, 1 inch = 25.4 mm, 1 lb = 4.5 kN)/p>

The results for different water head values are summarized in Figure 8. In this figure it is observed that the "critical" pivot elevation decreases with water head: for 2 ft water head, the maximum displacement is obtained with a pivot at 180 ft, and for a 16 ft water head the critical pivot elevation is about 120 ft. This indicates that at higher heads the rotational character of the system deformation becomes more pronounced.

Figure 8. Variation of critical pivot elevation with water head
High strength profile (1 ft = 0.305 m, 1 inch = 25.4 mm)

These results indicate that the method is capable of selecting the "right" pivot elevation on its own, but a numerical minimization procedure is required, just as in a slope stability analysis.

Parametric Results and Discussion

Calculations have been performed for the "high strength" and "medium strength" profiles discussed in Ref. 5, and for 2 through 16 ft of water head. The levee crest displacements computed using the shear ring method described above are compared with the finite element results in Figure 9. The agreement of the two is excellent in general. Both the trends and the absolute magnitudes agree quite well.

Figure 9. Comparison of finite element and shear ring solutions -
Crest displacements (1 ft = 0.305 m, 1 inch = 25.4 mm)

The next logical step in developing the shear ring method is splitting the rings in the middle and inserting a pile (bending elements). This will result in a system of linear equations to be solved and some iterative (or step by step) nonlinear scheme will have to be developed. The effect of the pile on the overall displacements may be small, but the representation of the pile in the model is necessary to calculate the pile bending moments for design purposes. Parallel to this extension, it will be necessary to generalize the soil modulus variation within a ring, especially in the hoop direction, such that the local effects such as the passive zone in front of the pile, and stress concentrations around the tip are represented properly in the model.

COMPLETE MODEL WITH PILE

General

The "soil-only" method described in the previous section demonstrated the potential of the shear ring approach. It was found that the deformation of the system could be calculated reasonably accurately with this idealization for both medium and high strength soil profiles and for a range of water heads. However, the bending moments in the pile cannot be computed from the soil-only model, nor can it account for the effect of pile penetration on soil stresses and deformation.

In the following, (a) the sheet pile is added to the model, (b) the shear ring stiffness is generalized to take into account the local stress conditions in the soil, especially in the vicinity of the pile, and (c) the validity and accuracy of the resulting technique is studied for the six "typical floodwall" cases that were analyzed earlier in plane strain finite element study (Ref. 5).

During the development of the method the policy has been to increase the complexity of the procedure only one small step at a time, in order to obtain the accuracy desired with the simplest possible technique. The geometry is represented by dividing the levee section into a number of concentric rings (as before, Figure 1) and the sheet pile is represented by linear-elastic flexural elements.

Upon loading, each ring boundary (a circular arc) is assumed to undergo a displacement tangent to the circle, producing the rotation around the pivot. To be able to take into consideration the local effects, this tangential displacement is now allowed to vary along the arc. To represent the local (active/passive) stress conditions in the vicinity of the pile the shear rings are split in the middle, at the location of the pile, producing a "right ring" and a "left ring" which are connected to the pile elements at discrete nodes.

A linear equation system is set up to represent this model, following the well known "displacement method" of mechanics. The unknowns of the equation system are the nodal point displacements; the "right-hand side" vector contains the corresponding loads; and, the coefficient matrix is composed of an orderly assemblage of the element stiffness matrices. Due to soil nonlinearity this system is solved several times with updated soil parameters for each loading increment.

Soil Nonlinearity

Shear modulus is evaluated for each ring element (separately for right and left halves) at three points: one at each end of the median arc, and one at the center (Figure 10a). This three-point scheme allows a quadratic interpolation of G along a ring:

(14)

where the subscripts L, M, and R refer to the left end, middle, and right end, and the Ni are quadratic interpolation functions in terms of a dimensionless distance parameter, t, that varies from 0 to 1 along the median arc of a ring (Figure 10b); viz:

Figure 10. Elements of three point (quadratic) interpolation
(a) Points where shear modulus is evaluated
(b) Quadratic interpolation functions

(15)

In the initial step, the soil-only solution, the ring spring constant is calculated using the integral average,

(16)

but in the total solution the modulus variation is combined with the displacement interpolation functions as shown below.

The loading considered on the system is the hydrostatic pressure acting on the exposed surfaces on the flood side. The load vector for the equation system is set up using the nodal loads calculated as in the soil-only case (Eqs. 5 - 8).

Beam Element Stiffness

The elements used to model the sheet pile are the commonplace flexural members. For the sake of completeness the beam element stiffness matrix used in this work is given in Appendix I.

Linear Ring Element

At an earlier step during the development of the shear ring method a four node shear element was considered. This type of element is the simplest possible idealization when the free surfaces and the soil-pile interface plane are to be represented. It will be described first for clarity of discussion. A four-node geometry allows a linear interpolation of diplacement in both radial and hoop directions, resulting in strain variations as follows:

Hoop strain,

(17)

Shear strain,

(18)

in which ui are the nodal displacements in the tangential direction, r and t are dimensionless distances in R direction and q direction,

(19a)

(19b)

and r is the radius ratio,

(20)

where R₁ and R₂ are the radii of the top and bottom boundaries of the ring, and q₁ and q₂ are the right and left end angles for a ring. The innocent looking r factor has a very significant role here; it is what represents the rotational character of deformation in the mathematical model (through Eq. 18).

Notice that the hoop strain is constant in the hoop (t) direction, but varies linearly in the radial (R) direction. Shear strain is constant in the radial (R) direction, and varies linearly in hoop (t) direction. This is the highest degree of interpolation possible with a four-node element. To determine whether this degree of approximation is adequate, it was implemented and tested, using the plane strain finite element analysis results (Ref. 5) as a yardstick. The main steps of the derivation will be described now as these are shared by the more refined ring elements presented subsequently.

To derive the element stiffness matrix, the strain-displacement relationships are first expressed in matrix form as

(21)

where

(22)

(23)

(24)

which is then combined with the stress-strain relationship to give stresses,

(25)

(26)

where G is the shear modulus, and H is called "hoop modulus," explained in Appendix I. Finally, the element stiffness matrix, [k^R], is obtained as (e.g., see Ref. 3),

(27)

The bandwidth of the global linear equation system to be solved is 8. It is indeed an economical system. However, the main limitation is that the local (active/passive-like) effects cannot be accounted for accurately, since a linear interpolation between the two ends of a long arc is a crude approximation. Notice again that the hoop stress is constant in that direction in a four-node element. Comparisons with results of plane strain finite element analysis also showed that the bending moments could not be predicted sufficiently accurately. Therefore, it was necessary to go to the next level of approximation, a quadratic model.

Quadratic Ring Element

It is clear that the local (active/passive) effects would be more closely represented by allowing higher order interpolation functions for displacements. The price of this improvement is increased computational effort due to the addition of a central node along the arcs. This addition increases the number of unknowns by two per ring level used, and the bandwidth of the equation system increases from 8 to 10.

The derivation of the element stiffness matrix follows the same basic steps as in the linear element. Since the shear strain in an element can now be calculated at three points in the hoop direction (at the two ends and the middle), a quadratic interpolation is achieved:

(28a)

(28b)

(28c)

(29)

where N's are the same quadratic interpolation poynomials (shape functions) used for shear modulus (Eq. 15).

The normal strain in the hoop direction, being the first derivative of the quadratic displacement function in that direction, varies linearly with t. It also varies linearly in the radial direction as it can be interpolated between the top and the bottom of the element, thus,

(30)

where l1 and l2 are the top and bottom boundary lengths of the ring, and N-primes are the derivatives of shape functions with respect to t;

(31)

The strain-displacement relationships formulated in this manner are then put in matrix form, giving the [B] matrix, and this leads to the element stiffness matrix by the same integration as before. The element stiffness matrix obtained in this manner is given in Appendix II in an abbreviated form, in which Iij and Jij are the integrals (of up to sixth order polynomials)

(32)

(33)

For a fast evaluation in the computer program these integrals were taken in closed form, yielding the formulas given in Appendix II.

Quasi-Cubic Extension

Although the accuracy obtained with the quadratic modael was reasonable, another improvement step was undertaken. With the anticipation of a larger effect of localized stresses near the sheet pile in a drained case, it was felt that a higher order interpolation function would be desirable. The obvious way of achieving this is by adding more degrees of freedom to the ring element. However, this course was not taken as it would increase the complexity (and cost) of computations. Instead, it was assumed that the localization effects can be represented by a function that has a higher rate of decay in the vicinity of the pile. The following modified functions were derived to provide the desired effect:

(34a)

(34b)

(34c)

Replacing the N functions by these M functions the derivations described in the preceeding section were repeated. The modified stiffness matrix has the same closed form representation as in the quadratic case, with the I and J integrals changed to alternate forms as presented in Appendix II.

Initial stresses

Since the stress-strain model for the soil is expressed in terms of stresses, both the initial stresses and the changes due to loading are needed. Stress changes due to loading may be obtained after solving for displacements, but the shear ring method, being essentially a one dimensional analysis, is not capable of performing a "gravity turn-on" analysis. Therefore, a suitable approximation must be used to estimate the initial stresses.

The approximations adopted at this stage are as follows. The initial vertical stress is taken as the soil overburden pressure, and the horizontal stress as some K0 times that. From the lessons learned by the static stress distribution in earth dams, and the results of previous floodwall analyses (Ref. 5), it can be concluded that the main limitation of this simple procedure is the potential difficulty in choosing an appropriate K0 value, especially within and immediately below the levee fill. The lateral stress coefficient to be used in that area is known to be less than the "normal" K0 due to the lateral spreading effect. Therefore, as a first approximation, one may use 1/2 K0 in that area, and let the coefficient increase to its full value at some depth. In the current implementation, the depth where full K0 is reached has been chosen as the height of the levee.

Determination of the Pivot Elevation

The selection of the correct pivot elevation is based on energy considerations as in the soil-only case. Of course, the entire calculation can be repeated searching for the minimum energy, but this was found to be unnecessary.

The six "typical" levee problems analyzed by plane strain finite elements (Ref. 5) were analyzed with the new method. In these analyses the soil-only solutions discussed earlier in this paper were used as a guide in choosing the trial pivot locations. It was observed consistently that the strain energy stored in the sheet pile is a negligible fraction of the total energy (typically on the order of 0.1% to 1% of that stored in the soil elements). Of course this explains why it was possible to calculate displacements accurately by the soil-only method; it is the soil that determines the major aspects of the deformation of the system. Therefore, the pile is not considered in the analyses for the determination of the pivot elevation. This approximation enormously simplifies the procedure. The entire iterative equation system solution does not have to be repeated using different pivots. The current computer program internally repeats the soil-only solution to determine the appropriate pivot elevation, and then starts the main solution procedure using only that pivot.

RESULTS OF THE ANALYSES

Ref. 5 presents the results for plane strain finite element analyses for three penetration depths (10 ft, 20 ft, and 30 ft) and two idealized shear strength profiles making up altogether six cases. The results of the analyses using the shear ring method are compared with those of the plane strain finite element method in Figures 11, 12, and 14 through 16.

Figure 11. Comparison of finite element (FEM) and shear ring (SRM) results
High strength profile, all three piles, at 16 ft head (1 ft = 0.305 m, 1 k = 4.5 kN)

Figure 11 shows the moments, for all three penetrations, for the high strength soil profile. Clearly, the trends are predicted correctly. The 10ft pile behaves as assumed in the classical design procedures where moment reversal does not occur. The deeper piles exhibit a reversal at the same points as the plane strain finite element analyses show. As compared to the finite element results, the maximum moments for 10-ft and 20-ft piles are larger in magnitude, and the moments in the 30-ft pile case are lower close to the tip. Overall, the agreement seems satisfactory.

Figure 12. Comparison of finite element (FEM) and shear ring (SRM) results
Medium strength profile, 16 ft head (1 ft = 0.305 m, 1 k = 4.5 kN)

Figure 13. Comparison of FEM and conventional SSI results
High strength profile, 16 ft head (1 ft = 0.305 m, 1 k = 4.5 kN)

Figure 12 shows the results for the medium strength soil profile. Again, the agreement with the finite element results are quite favorable. It should be noted, in evaluating the new method, that the "conventional SSI" (Ref. 1) gives results that are only reasonable in the 10-ft case (Figure 13).

Figure 14. Variation of moment as water head increases
Medium strength profile, 10 ft pile (1 ft = 0.305 m, 1 k = 4.5 kN)

Figure 15. Variation of moment as water head increases
High strength profile, 30 ft pile (1 ft = 0.305 m, 1 k = 4.5 kN)

The water head applied in the previous results was the maximum, 16 ft (measured from the base of the levee). To see if the new method is capable of reproducing the effects of the rising water head, a few lower heads were also computed. Figure 14 shows the results for the 10-ft pile in the medium strength soil, and Figure 15 shows those for the 30-ft pile in the high strength soil. The nature of the error (i.e., the difference between shear ring and finite elements) is the same for all three heads. Aside from that, both the medium and high strength cases show remarkable agreement.

The net lateral earth pressure, the difference between the contact pressures on the two faces of the sheet pile, are plotted in Figure 16 for two cases. The agreement is quite good.

Figure 16. Net earth pressure on the pile at water head of 16 ft
High strength case, 10 and 20 ft piles (1 ft = 0.305 m)

These and other discrepancies were observed closely in all runs. It was found that the errors in the earlier, more approximate, linear ring model were of the same character as the quadratic ring case discussed above, only larger in magnitude. This indicates that the resistance of the pile to comply with the soil deformations creates stress conditions that are highly localized in the soil-pile contact area, and that the quadratic variation of displacement is not sufficiently flexible to represent this localization. Repetition of the six typical levee cases with the modified model produced results that were almost identical to 2D finite element results (Ref. 6).

CONCLUSIONS

As the results of comprehensive finite element analyses have revealed that a typical floodwall will not behave as it is usually assumed in a conventional design-analysis method. This observation has lead to the development of a new method of analysis, the "shear ring" method. Based on the results of the studies presented in this report it can be concluded that

(1) The new method reproduces the finite element results reasonably accurately at a small fraction of the cost,

(2) It is only slightly more complicated than a conventional SSI procedure, thus capable of serving as a design method,

(3) The two dimensional soil profile is represented in the model allowing the influence of levee slopes be taken into account without any additional effort, and

(4) The conventional soil properties, such as the shear strength parameters, are used as input to the shear ring method as opposed to one dimensional soil response curves required in a conventional SSI analysis.

Preliminary results for the drained case and other sheetpile problems have indicated that the new method can be extended to perform a drained analysis. Thus the shear ring method not only solves the floodwall problem but also shows a good promise for further generalization to cover other soil structure interaction problems.

ACKNOWLEDGMENT

The results presented in this paper are part of the research and development work conducted under the Soil-Structure Interaction Project sponsored by the U.S. Army Corps of Engineers, Civil Works Research and Development, Structural Engineering Program. Permission was granted by the Chief of Engineers to publish this information.

Appendix I. Derivation of Hoop Modulus

Recalling the generalized Hooke's law, in r, s, t coordinates,

Under plane strain conditions, working in the r-t plane, where t direction is along the arc, and r denotes the radial direction, and s being the third direction perpendicular to the analysis plane, Hooke's law can be specialized by setting e_s = 0, as

In the formulation of shear ring stiffness matrices, a "hoop modulus," H, is required that relates the hoop stress to hoop strain,

It can easily be derived from the above expressions that (a) if s_r = 0, i.e., the ring offers no resistance to radial spreading under compression, or is "free" laterally, then

and (b) if, in the other extreme, e_r = 0, i.e., the ring has no freedom to expand laterally under compression, or is "fixed," then

A judgment has to be made as to which H better represents the conditions of the problem being analyzed. In a floodwall problem it may be argued that the "free" condition is more appropriate for the regions close to the ground surface while it seems more appropriate to adopt the "fixed" condition at larger depths. In the current version of the shear ring computer program H is linearly varied between H_free and H_fixed between the top and the bottom of the cross section being analyzed.

Beam Element Stiffness Matrix

The elements used to model the sheet pile are the commonplace flexural members. For the sake of completeness the beam element stiffness matrix used in this work is given below.

where L is the length of the beam element, E is the Young's modulus, I is the moment of inertia per unit width. The generalized coordinates are ordered as top displacement, top rotation, bottom displacement, and bottom rotation.

Appendix II. Stiffness matrix of the quadratic shear ring element

The quadratic ring element stiffness matrix is obtained by evaluating the integrals in Eq. 27 using Eqs. 24 and 26, as follows.

[k^R] = d (l₁ + l₂)/2 times the following matrix

with l₁ and l₂ being the top and bottom arc lengths, d being the ring thickness, subscripts L, M, R to indicating the Left, Middle, and Right points of the ring, the abbreviations I_ij and J_ij stand for:

I₁₁ = (37 H_L + 36 H_M - 3 H_R) / 30

I₁₂ = -(22 H_L + 16 H_M + 2 H_R) / 15

I₁₃ = ( 7 H_L - 4 H_M + 7 H_R) / 30

I₂₂ = (24 H_L + 32 H_M + 24 H_R) / 15

I₂₃ = -( 2 H_L + 16 H_M + 22 H_R) / 15

I₃₃ = (-3 H_L + 36 H_M + 37 H_R) / 30

J₁₁ = (39 G_L + 20 G_M - 3 G_R) / 420

J₁₂ = ( 5 G_L + 4 G_M - 2 G_R) / 105

J₁₃ = -( 3 G_L + 8 G_M + 3 G_R) / 420

J₂₂ = ( 4 G_L + 48 G_M + 4 G_R) / 105

J₂₃ = (-2 G_L + 4 G_M + 5 G_R) / 105

J₃₃ = (-3 G_L + 20 G_M +39 G_R) / 420

(b) With Quasi-Cubic Interpolation Functions:

I₁₁ = (161 H_L + 108 H_M - 3 H_R) / 105

I₁₂ = -(166 H_L + 112 H_M + 2 H_R) / 105

I₁₃ = ( 5 H_L + 4 H_M + 5 H_R) / 105

I₂₂ = ( 24 H_L + 32 H_M + 24 H_R) / 15

I₂₃ = -( 2 H_L + 112 H_M + 166 H_R) / 105

I₃₃ = ( -3 H_L + 108 H_M + 161 H_R) / 105

J₁₁ = (93 G_L + 40 G_M - G_R) / 1260

J₁₂ = (10 G_L + 12 G_M - G_R) / 315

J₁₃ = -( G_L + 4 G_M + G_R) / 1260

J₂₂ = (4 G_L + 48 G_M + 4 G_R) / 105

J₂₃ = (-G_L + 12 G_M + 10 G_R) / 315

J₃₃ = (-G_L + 40 G_M + 93 G_R) / 1260

The subscripts L, M, and R denote the left-end, middle, and right-end values of the parameter.

REFERENCES

1. Dawkins, W. P., User's Guide: Computer Program for Soil-Structure Interaction Analysis of Sheet Pile Retaining Walls (CSHTSSI). Report to ADP Center, U.S.A.E. Waterways Experiment Station, Vicksburg, MS, 1982.

2. Dawkins, W. P., User's Guide: Computer Program for Analysis of Beam-Column Structures for Nonlinear Supports (CBEAMC). Instruction Report K-82-6, U.S.A. E. Waterways Experiment Station, Vicksburg, MS, 1982.

3. Desai, C.S., and J.F. Abel, Introduction to the Finite Element Method, Van Nostrand Reinhold Co., New York, N.Y., 1972

4. Desai, C.S., and Christian, eds., Numerical Methods in Geotechnical Engineering

5. Oner, M., W. P. Dawkins, and I. Hallal, "Soil-Structure Interaction Effects in Floodwalls", The Electronic Journal of Geotechnical Engineering, ejge, Vol. 2, 1997.

6. Oner, M., W. P. Dawkins, I. Hallal, and C.C. Lai, Development of a New Method for Soil Structure Interaction Analysis of Floodwalls, Report presented to U.S.A.E. Waterways Experiment Station, Vicksburg, MS, December, 1988.


	nq = 2	nq = 2	nq = 4	nq = 8	nq = 16

5	2.0495	2.0730	2.0450	2.0270	2.026
2	2.0603	2.0882	2.0598	2.0413	2.036
1	2.0728	2.1003	2.0712	2.0525	2.048