slipping strip analysis of reinforced earth

INTERNATlONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANI(S. VOL. 2. 343-366 (1978)

SLIPPING STRIP ANALYSIS OF REINFORCED EARTH

D. J. NAYLOR* AND H. RICHARDS$

Department of Civil Engineering, University of Wales, Swansea, Wales

SUMMARY

Reinforced earth in plane strain is idealized as a homogeneous material with the strips attached to the elastic soil matrix by a conceptual shear zone. A ‘no-slip’ finite element model is derived by assigning a large shear modulus to the shear zone. Relaxation of this modulus using a tangential stiffness algorithm in conjunction with a Mohr-Coulomb strip-slip criterion allows slipping to be simulated.

The finite element formulation is validated and the finite element discretization assessed by comparisons against exact solutions for a simple test problem.

An idealized reinforced earth wall example is used to demonstrate the feasibility of the method and to answer the question: ‘is slipping significant?’ The method is shown to be potentially useful, and slipping is shown to be significant.

INTRODUCI7ON

Numerical models of reinforced earth usually assume that there is no slip between the strips and the ~oil.’’*’~.’~ Whereas this may be a reasonable assumption in reinforced materials with a cohesive matrix such as concrete or fibreglass, it is very likely that at least local slip will occur in reinforced earth.

This paper has two objectives. The first is to present a model of reinforced earth which allows slipping to occur between strips and soil when a yield stress is reached. The model is devised for finite element applications, and a formulation for application by the displacement method is given. The second objective is to assess the importance of slipping by making comparative analyses of a reinforced earth retaining wall in which slipping is first not allowed (‘no-slip’ analysis) and then is allowed (‘slip’ analysis).

The model treats the reinforced earth as a homogeneous material, i.e., it assumes that the strip spacing is small in relation to the overall dimensions. This is a very useful feature for finite element analysis since it allows the element size to be chosen independently of the physical spacing of the strips. It is believed to be a small source of error in most applications since the number of strips is usually large.

A lot has been written on the general subject of reinforced earth. On the topic of strip slip, however, most of the work has been of an experimental nature providing criteria for safe design. Numerical models of reinforced earth have generally assumed no slip between the strips and the soil matrix. These can be divided into ‘unit cell’ models in which the stiffness of the strips is superimposed on the stiffness of the soil to give a homogeneous anisotropic

* Lecturer. t Post-graduate student.

0363-0961/78/0402-0343$01.00 @ 1978 by John Wiley & Sons, Ltd.

Received 4 August 1977 Revised 12 December 1977

343

344 D. J . NAYLOR A N D H. RICHARDS

material,2.s.’2 and models in which the strips are treated as discrete elements. 1.6.12 An unusual ‘mixed’ finite element formulation is given by Smith.’’

Reinforced earth has some similarity to reinforced concrete and it is useful to note the work done in this field. Phillips and Zienkiewicz” model reinforced concrete as a homogeneous anisotropic material. Cracking and non-linear compression characteristics are provided in the concrete, and the steel can yield. However, it cannot slip. Ngo and Scordelis’ as early as 1967 proposed a slipping finite element model for reinforced concrete, in which linear springs connected bars to adjacent concrete nodes. Nilson” followed this up with non-linear springs.

A recent paper by Herrmann6 has dealt with the subject of strip slip. As did Ngo and Scordelis’, he uses non-linear springs to connect strip elements to the soil matrix. Thcse springs are removed and replaced by equal and opposite forces applied to the strip and adjacent soil nodes when the bond stress exceeds a Mohr-Coulomb criterion. He also looks at the local effects near the ends of the strips and infers that these are important. This paper differs from Herrmann’s in that it treats the reinforced earth as homogeneous. it also differs in another important respect. It takes into account the fact that the reinforcement is in the form of strips and not sheets. It is shown that the assumption of an equivalent sheet to replace a layer of strips will cause serious error since it interrupts the vertical transfer of shear stress through the soil.

A plane strain formulation is given here as being appropriate to nearly all engineering applications.

BASIC CONCEPTS The equivalent material

The first step in the derivation of the model is the topological transformation illustrated in Figure 1. The material of Figure l(b) in which the strip is taken outside the soil and connected to it by a conceptual shear zone is assumed equivalent to the actual material of Figure l(a).

The proportion of the cross-section occupied by the strips is denoted by

bt a , BT BT

a = - = -

The equivalent material must properly take into account the surface contact area of the strips. This quantity must be considered separately from the cross-section area. Thus in two sections of reinforced earth, one containing, say, one strip and the other containing a large number of small strips but both having the same value of ‘a’ , that containing the large number of small strips will have the larger contact area. Figure 2 shows how account is taken of this in the model. The contact length, C, in the T direction is made equal to the periphery of the strip, i.e., 2(b + 1) . The dimensionless parameter p is now defined as

C 2 ( b + r ) P=;f=T

Since t is usually much less than b, the contribution of the edges can be neglected and (2) can be written

2b p “ - T

By assigning a suitable stiffness to the conceptual shear zone the strips can be made to deform a negligible amount relative to the soil matrix-the ‘no-slip’ case, or a large amount-

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346 D. J . NAYL-OR AND H. RICHARDS

( a ) C < T ( p e l ) ( b ) C > T ( p > l )

Figure 2. Section through equivalent material to illustrate.modehg of contact area

the ‘slip’ case. The former involves an elastic analysis, the latter a non-linear analysis. The term ‘no-slip’ is used to describe the elastic analysis throughout this paper, it being understood that there is some relative displacement but that it is small enough to have no physical significance. A Mohr-Coulomb criterion is used in the slip analysis to limit the shear stress on the strips (i.e., in the conceptual shear zone).

The thickness, e, of the conceptual shear zone is arbitrary..It is convenient to take it as unity.

Strain -displacement relationships

Let Po in Figure 3 represent the initial position of a point in the material. After distortion it moves to P in the soil and to P* in the strips. SS represents the strip direction. Since the model only allows the strips to move relative to the soil in the strip direction, PP* will be approxi- mately parallel to SS. The approximation arises due to geometric changes caused by the strains. The common assumption that these are small will be made so that PP* and SS can be taken as parallel.

P’ /

X

Figure 3. Movement of a point

SLIPPING STRIP ANALYSIS OF REINFORCED EARTH 33 7

The longitudinal strain in the strips, E ~ , is the gradient along SS of the total displacement, w, of a point on the strip. From the geometry of Figure 3.

w = ua, + ua, + p

where

a, = c o s 8 and ay = s i n 8

Therefore dw du dy dp

E =-= a,-+ay-+- ds ds ds ds

Since the spatial variation of w, u, u and p is considered, the derivations of these quantities can be expanded by the chain rule as follows. Let S = 6(x, y) stand for u, u or p. Then

dS a s d x a6 dy ds ax ds a y ds -=--+-*-

i.e.,

dS as as - = a,- + a),- ds ax ay

Substituting u, u and p in turn for S in the expression for E~ gives

The shear strain in the conceptual shear zone is given simply by

P Ys = - e

The soil displacement components are related to u, u by the well-known relations

FINITE ELEMENT FORMULATION-NO-SLIP ANALYSIS

In addition to u and u the relative displacement of strip to soil, p, is chosen as a nodal variable. The finite element formulation is obtained by discretizing the strain equations (3), (4) and

(5) and using the principle of virtual work in the conventional manner to obtain the element stiffness matrix.

In the following derivation no restriction is placed on element type. In practice, however, i t is necessary to be discerning about the element type and the integration method used in the stiffness calculation. Preliminary investigations of 4-, 6- and 8-noded quadrilateral

348 D. J . NAYLOR AND H. RICHARDS

isoparametric elements'" suggested that a 6-noded quadrilateral with parabolic displacement variation in the general direction of the strips and linear displacement variation across the strips was the best. These elements are used in the examples presented.

Element stiffness matrix

This is made up of two parts-the contribution from the soil, and the contribution from the strips and conceptual shear zone. Denoting the number of nodes in an element by n. the composite stiffness matrix will be of order 3n. So that they can be added together the component matrices will also be made of order 3n. The composite element stiffness matrix may then be expressed as

in which K1 and Kz are the soil and strip component matrices, respectively.

form

K = K i + K z (6)

By means of the virtual-work principle both component matrices can be expressed in the

K I = B:D,BldR (7) I in which I = 1 or 2 identifies soil or strips, B is a matrix dependent on the element geometry and D is a modulus matrix. Their form is given below as is also the nature of the volume integral d o .

The soil contribution

The matrix B, is obtained by discretizing equation ( 5 ) . It takes the standard form for plane strain problemsi6 except that a column of zeros is inserted at every third column so that the matrix is of order 3 x 3n. B1 is thus composed of a row of submatrices Bl i , ( i = 1 , . . . , n ) of the form

in which N, is the shape function for node i . DI is the conventional 3 x 3 plane strain modulus matrix. It need not be linear. Integration is carried out over the volume of the element which has thickness B.

The strip and shear zone contributions

The matrix B2 relates and ys to the nodal variables. It is obtained by discretizing equations (3) and (4). The submatrix for node i is

in which

SLIPPING STRIP ANALYSIS OF REINFORCED E A R T H 349

The modulus matrix D2 relates the strip direct stress, u5, and the shear stress on the strips, T,, to e , and ys. In order to allow the integration of BFD2B2 to be carried out over the soil volume, Es is multiplied by a and G, by pe/B. This results in the integration for the longitudinal stiffness contribution from the strip being carried out over the strip volume and that for the shear stiffness contribution over the volume of the conceptual shear zone. Thus Dz is defined as

As noted above, G, will be large (strictly peGJB >>aE,) if the 'no-slip' condition is to be imposed. If slipping occurs it will be appropriately reduced if a tangential (as here) or secant modulus approach is used.

The 3 x 3 i, jth submatrix of Kr is then obtained explicitly by substituting equations (9) and (10) into (7), to give

a: a,a, a, K21, = aEsQiQ, [ a z x ;: ;;,J (1 1)

in which

Computation of stresses

The total element stiffness having been obtained by adding the strip and conceptual shear zone contribution (equation (1 1)) to the soil, a conventional finite element analysis is carried out. The soil strains are then calculated from element nodal displacements as

~ l = B 1 6

in which 'r T

= [ E ~ , E ~ , y ] and 6 = [. . . ui. u,, p i . . .] i = 1, . . . , n

and the strip strains as

€ 2 = B&

in which

E2 = [Es, %IT The soil stresses are then given by

u=DIEI

and the strip stresses by

us = E , E ~


Comparison with joint elements

The method of connecting the strips to the soil matrix through a conceptual shear zone has something in common with the use of joint elements in rock as described by Goodman et aL4 Relative movement rather than absolute is taken as a nodal variable. In the present work, however, only movement in the plane of the shear zone (i.e., p ) is of interest, whereas in the rock joint elements movements normal to the joint plane are included. Another difference is that the rock joints are sheets separating one zone of rock from another. Here, by the topological transformation already explained, the shear zone does not intersect the soil.

ANALYSIS O F SLIP

It is required to satisfy the Mohr-Coulomb criteria by restricting )r,( to the limiting value

(15)

in which CT, is the stress normal to the strip faces. (Note tension positive sign convention.) It is assumed that LT, is the same as the component of the soil stress tensor in the direction

normal to the strip face. This assumption is valid if the strips are thin, flexible and straight. (They are in practice thin and flexible, and are usually intended to be straight.)

A number of techniques are available for limiting 1 ~ ~ 1 to rf. These include the initial stress, tangential modulus and secant modulus The initial stress method was tried, but even when accelerated was found to converge so slowly that it was not practical. This is thought to be due to the very large ‘equivalent load’ needed to increase the relative slip of the strip and soil from the negligible ‘no-slip’ value to a value which satisfies the Mohr-Coulomb criterion. The secant modulus method was found to offer an improvement but was not as efficient as the tangential method. This method is used in a modified form to that usually given’ so will be described.

f rS = c, - LT,, tan ds

The tangential modulus method

This is an incremental method without iterations. In its usual form the stiffness matrix for each increment is calculated from the tangential modulus corresponding to the (known) stress/strain state at the start of the increment. This accumulates error, the error increasing with increment size. It is clearly better to base the stiffness on the average modulus over the increment. An estimate may be made of this by estimating the new point on the stress-strain curve.

Loading must be ‘proportional’, i.e., the nodal force vector applied for increment i must be related to the total load vector P, be a scalar parameter A , according to

APl = A,P, (16)

The procedure is illustrated in Figure 4 which shows the stress-strain relation for a typical point in the conceptual shear zone for an increment i in which the yield stress is reached. The start of the increment is represented by A, while B represents the correct final point-as yet unknown.

At the start of the increment the new strain increment is estimated by extrapolation from the previous increment as

A A i - 1

Ay; =I A Y S ( ~ - I )

The asterisk indicates an estimate.

SLIPPING STRIP ANALYSIS OF REINFORCED EARTH 35 I

T Ideal ly coincide

’ Ys

B and B*

Figure 4. Shear zone stress-strain curve

The corresponding elastic shear stress increment AT: = G,Ay:* is added to the previously accumulated shear stress and the total compared with 71, the latter being calculated by means of equation (15) with mn extrapolated to the end of the increment. If yield occurs, as illustrated in Figure 4, the stress increment is reduced to AT,(+ An approximate tangential shear modulus G,* for use in the element stiffness calculation for increment i is then given by

G,* will be the correct tangential modulus if j~ has a value such that point B* caincides with B, the correct end-of-increment point. p is selected empirically, and in general this coin- cidence will not be achieved. If, however, the increments are reasonably small the value assigned to p will not be critical. P = 1 has been used in the examples presented here.

Stress smoothing

Preliminary ‘no-slip’ analyses of a simple test problem produced results which were in agreement with independent calculations except that the strip shear stress, T ~ , ‘oscillated’ severely across the elements. If the values calculated at the four Gauss points (2 X 2 reduced integration was used”) were averaged satisfactory T~ values at the element centres were obtained.

For the non-linear analyses, it was not sufficient simply to average the results, since convergence as the increment size was reduced could not always be obtained. The problem has been overcome by averaging the shear zone strain increments, after they have been calculated from the nodal displacements for increment i. The actual values at the four Gauss points are then replaced by the average, thus imposing uniform shear across each element in the conceptual shear zone. As a consequence of this, the stress oscillations are suppressed and convergence with reduced increment size (and also with reduced element size) is obtained.

The severe shear stress oscillations are believed to be a quite normal manifestation of finite elements which occur when steep stress gradients are encountered. For linear analysis it is generally safe to average these and interpret the results as applying at the element centres.’ For the highly non-linear behaviour encountered here it was necessary to average the shear stress values before applying the test for yield. Otherwise numerical chaos was likely to ensue

3 5 2 D. J . NAYLOR AND H. RICHARDS

due to slip occurring in alternate directions (as 7, reversed sign) across elements. This procedure appears to be justified o n the basis of validation checks described in the next section.

VALIDATION

A test problem representative of a horizontal layer from the middle of the reinforced wall example studied below is used to check the theory. The problem is illustrated in Figure 5. The

Smooth’ Test problem (init ial 0, = -50 kPa

Mesh

5 element - 10 element l l ~ ! z l = J J + l z ~ = 1 ~ 1 20 element

1

I00 -

0 s (MPa 1

50 -

‘ t S ( k P a )

0

Node and element subdivisions

End of reinforcement

Exact 0s ‘ts- il I

I 0 10 .I

20 -

Exact ‘ts 7 I I ,

0 2 L 6

18.67

I Exact I 18 67 1 Wall displacement

--c x ( m )

0s and TS distribution

Figure 5. Test problem-omparison of finite element no-slip solution with exact

SLIPPING STRIP ANALYSIS OF REINFORCED EARTH 353

left-hand 8 m are reinforced with horizontal strips which are linked to the soil by specifying zero relative displacement (p) at the left-hand face. The soil is assumed,to have an initial horizontal compressive stress (-uX") of 50 kPa. 'Loading' is achieved by removing this by the application of a horizontal tension of 50 kPa to the left-hand wall. The problem is made one-dimensional by setting the soil Poisson's ratio, v, zero. The other parameters are given in Table I.

Table 1. Parameters for reinforced earth (stresses in kPa)

Stiffness Strength+ Geometry - - _ _ ~ - - - _ _

E = 25,000* C = O u = 1 /3000, p = 0.208 v = 0 . 2 5 4 d = 30" a,=1, a,=O

E, = 200 x lo6 c ,=o ~ = l m , B = l m G , = 2 x loh & = 2 1.8

8 Except test problem for which v = 0 t Since soil is assumed elastic, c and 4 are not used except in calculation of

over-stress ratio Same applies to c, and 6% for no-slip analycis

An exact solution to this problem can be obtained for both the no-slip and the slip cases. By means of it displacements at selected points and the distributions of T, and u, along the strips have been obtained. See Appendix I.

Three meshes having 5, 10 and 20 elements were used to model the test problem (see Figure 5). No-slip analyses were carried out first. Comparisons with the exact solution are given in Figure 5. Slip analyses were then carried out. There were done for each mesh using a very small load step (48 increments) so that error from this cause would be negligible. Comparisons with the exact solution are given in the upper part of Figure 6. To assess the error associated with the size of the load step the 10-element mesh was also analysed for 24, 12 and 6 load increments. The results of this study are presented in the lower part of Figure 6.

It can be seen that very close agreement with theory is obtained using the 20-element mesh both for the no-slip analysis and for the slip analysis when 48 increments were used. With the coarser meshes and using fewer increments in the slip analysis, significant divergence of 75 and to a lesser extent us from the exact values is evident. The displacements, however, remained quite accurate.

From these studies it was concluded that a prototype reinforced earth wall mesh based o n the 10-element subdivision with 24 load increments for slip analyses would keep discretization errors to acceptable levels. A 5-element subdivision using, say, 12 load increments for slip analyses would be suitable for rough checking purposes.

REINFORCED EARTH WALL EXAMPLE

The idealized wall shown in Figure 7 has been selected for this study. Material properties and other parameters are given in Table I. Values are representative of current practice.

For the purposes of analysis it is assumed that the wall is constructed under 'KO' conditions, i.e., as if temporary support were provided to prevent lateral yielding during construction. Loading is then achieved by removing this support. This involves applying a horizontal traction to the face equal to Koyh. KO is taken as 0.5 and y as 20 kN/m3. The differences between this single-step analysis and a multi-step layered analysis is outside the scope of this paper. I t seems


50

= S ( k P a )

0

I L End of reinforcement

Symbols as in Figure 5

k0 I

Symbols as in Figure 5

I .- m*+m"..Dc

1 - I I 0 2 L 6 8 x ( m ) 0 2 L 6 8

50 1

14;;cr 1 ( m j ) 1 analyses - U A -Pe

20el 21.20 3 .90 Exact 21 55 L.30

19-64 1.75 20.83 3.39

Displacement at A and relative displacement at B (Figure 5 )

Effect of mesh ( L 8 incr. analysis)

Displacement at A and relative d I splacement at B

c 0 2 L 6

Effect of number of increments -10 element mesh

Figure 6 . Test problem-omparison of finite element slip solution with exact

on the basis of preliminary work that, whereas there will be significant differences in the computed displacements, the difference in stresses is likely to be small.

a was selected to give a minimum factor of safety against breakage, Fb, of 1.5. This factor was calculated on the assumption that breaking occurs when the steel stress induced by a Rankine active soil pressure ( K , y h ) on the inside of the wall reaches its limiting value, 71, i.e.,

f - auf us btuf us BTK,yh K,yh

-- Fb=-=


I

Smooth

Figure 7. Reinforced earth wall example

whence

a is made the same at all levels. Consequently the minimum factor of safety occurs at the bottom of the wall. Substituting K , = 4.6 y = 20 kN/m3, h = H = 10 m, Fb = 1.5 and m5 = 200 MPa gives a = 1/3000.

p was selected to give a minimum factor of safety against pull-out, Fp, of 2.0. This factor is calculated on the assumption that the whole length, L, of the strip resists pull-out. This gives

f

2hL tan & - pL tan BTK, BK,

Fp = -

whence

Putting in values we obtain p = 0.208. (This approach is attributed to the Reinforced Earth Company.' It appears that the rather unconservative assumption about the pull-out length is compensated by the use of a relatively large safety factor.)

Results for two analyses using an 80-element mesh comprising 8 rows of 10 elements distributed as in the 10-element test mesh are presented here. One is a no-slip analysis and the other a slip analysis in which the load was applied in 24 equal increments.

Figure 8 shows the displacements of the wall face and on a vertical section at the ends of the strips. The strip slip on the wall face is, of course, zero because it has been constrained to be so. The maximum slip occurs at the ends of the strips. Those shown are therefore the maximum. It will be noted that a small amount of relative displacement (up to mm) occurs in the no-slip analysis. This is the negligible elastic distortion of the conceptual shear zone mentioned earlier.

Stress results are presented in Figures 9, 10 and 11 as dimensionless over-stress ratios. These relate to strip tension, strip slip and soil yield. They are denoted by RT, R S and RM,

I Consistent with 4 = 30" for the soil

3.56 D. J . NAYLOR A N D H. RICHARDS

0 5 m m

0

Vector scale - - h z 1 0 r n - -

Wall face 8 m from face

Slip analysis - _ _ - No - slip analysis

Figure 8. Soil displacement vectors on two vertical sections and relative displacement at strip ends

k k tensl'on /End of

reinforce men t

No-slip analysis Slip analysis

Figure 9. Strip tension over-stress ratio (RT) contours

SLIPPING STRIP ANALYSIS O F REINFORCED EARTH

End of

Slip analysis No-slip analysis

Figure 10. Bond stress over-stress ratio (R,) contours

No-slip analysis

Slip analysis

Figure 1 1. Soil over-stress ratio (RM) contours

358 D. J . NAYLOR AND H . RICHARDS

respectively, and are defined as follows:

lR( :> 1 indicates potential local yield. I f t h e analysis incorporates the appropriate yield criterion, IRI will not be allowed to exceed 1. Clearly, since the soil and the strips are here assumed elastic RT and RM can exceed 1, so also can Rs in the no-slip analysis. In the slip analysis, however, lRsl is restricted to 1. The element average values used to draw the contours in Figures 9, 10 and 11 are tabulated in Appendix 11.

DISCUSSION AND CONCLUSIONS

A conceptual model of reinforced earth which allows strip slipping to occur, and which is applicable to reinforced earth having sufficient strips to be treated as homogeneous, has been described. A plane strain formulation for the finite element displacement method in which the relative displacement between soil and strips is included as a nodal variable has been presented.

The method has been validated by comparing the finite element results for a test problem against exact solutions. This confirms that the finite element formulation correctly incorporates the model. It also gives insight into the extent of the discretization error. Thus is can be seen (Figures S and 6) that accurate displacements may be obtained with a relatively coarse mesh, whereas to obtain an accurate computation of the bond stress ( T ~ ) a fine mesh is required. The 1 0-element horizontal subdivision provides a suitable compromise for engineering purposes. It should be understood that the term ‘accurate’ refers here to the discretization error rather than to the physical accuracy.

The reinforced earth wall example was included to show the feasibility of the method and to highlight the significance of slip. The method is not only feasible but is also potentially useful. The elements need not match the spacing of the strip elements. If the wall face is flexible (as in the ‘bellows’ type) no special facing elements are needed. Zero relative displacement (p) is simply assigned to the nodes on the face. Significant differences between the slip and no-slip analyses have been revealed as follows.

1. The slip analysis reveals slipping over a significant length of the strips (see Figure 10). 2. The slip analysis predicts wall face displacements a little greater than those predicted by

the no-slip analyses (1 7 per cent greater at the top, less lower, see Figure 8). 3. The locus of the point of maximum stress in the strips is significantly different between

the two analyses (see Figure 9). The shape for the slip analysis is more like the concave upwards curve obtained by Schlosser and Long.I3

4. The maximum tension in the strips is unaffected. I t is interesting to note that this maximum value, which occurs at the face &ths of the wall height below the top, is between 10 and 15 per cent in excess of the Rankine value (i.e., K,yh/a).

5 . The soil over-stress ratio, R M , is similar in the two analyses. The tension and yielding zones extend nearer to the wall in the slip analysis.

SLIPPJNG STRIP ANALYSIS OF REINFORCED EARTH 359

With regard to Point 3 above, the position of the peak stress point can be much affected by the fixity conditions at the wall face. Thus if there is some ‘give’ in the connections or if the soil is less stiff near the face (due perhaps to lower compaction in this area), the peak stress point is moved inwards. It should be noted that the peak stress locus is also the ‘I, = 0 contour. The shear stresses on the strips change sign at this point. (It is easy to see that this has to be so by considering the equilibrium of a short length of strip.)

It might be questioned whether it is necessary to have a model quite as sophisticated as the one presented here. Would not a model in which the strips are modelled as sheets extending over the full plan area suffice? The answer is ‘no’. Such a model could correctly reproduce the frictional characteristics of the strips by assigning to the sheets a reduced skin friction 6, given by

b B

tan 6, = - tan ds

but it could not reproduce the resistance of the reinforced earth to shear transmitted in the vertical plane. This would be drastically reduced, e.g., with b / B = 0.1, tan q5/tan 4, = 1.5, the reduction factor is 15. In practice this resistance is little affected by the strips. A reinforced earth wall modelled analytically in this way would behave like a tilted chest of sliding drawers.

The next stage in this research is to incorporate an elastic-plastic soil so that the effect of soil yielding can be investigated (note the potential yield and cracked zones in Figure 11 .) Then there is a need for parameter studies which bracket field values.

NOT AT1 ON

a = strip cross-section per unit total area a, = cross-section area of one strip b =breadth of strip B =horizontal spacing of strips B = matrix of shape function derivatives C = length of conceptual shear zone in vertical (T) direction c = soil cohesion c, = strip surface cohesion D = modulus matrix e =thickness of conceptual shear zone E = soil Young’s modulus E, = strip Young’s modulus Fb = factor of safety against strip breaking F,, = factor of safety against strip pull out G, = shear modulus of conceptual shear zone h = depth below ground surface

H =wall height KO = coefficient of earth pressure at rest K , = coefficient of active earth pressure K = stiffness matrix L = strip length

N, = shape function for node i p = strip contact area ratio R = over-stress ratio


s = distance along strip t =thickness of strip

T = vertical strip spacing (i.e., normal to their flat faces) u = x component of soil displacement u = y component of soil displacement

w = absolute displacement of strip in s direction x, y = Cartesian co-ordinates

a,av = direction cosines defining strip inclination y =engineer’s shear strain in soil, also soil unit weight

ys = shear strain in conceptual shear zone

F~ = longitudinal strain in strips 8 = inclination of strip to x direction

A, = proportion of load applied in increment i p = empirical constant in tangential modulus calculation v = soil Poisson’s ratio p =displacement of strip relative to soil in s direction

E ~ , E , = x and y components of soil strain

a,. ay = x and y components of soil stress al, a 3 = major and minor components of soil stress

us = longitudinal stress in strip T =soil shear stress

T~ = shear stress in conceptual shear zone (bond stress) 4 = soil friction angle & = friction angle for strip surface

An extension positive strain and tension positive stress convention is used.

APPENDIX I

Analytical solution for slice from wall

Figure 12 relates to the 8 m-long reinforced section. The problem is to find the distribution of horizontal stress vS in the strips and of bond stress T , on the strips due to removal of the initial stress axe. The strips are fastened to the soil at A. For convenience A is treated as fixed in space. In the finite element example A moves to the left, the fixed point being 7 m to the right of B. The actual displacement can be obtained by imposing a rigid body motion to the left. The strips are initially unstressed.

The unloading causes a strain in the strips

au ap E 5 = - + -

ax ax

au p = y s e and E~ =-

ax

Therefore d Y s ax

F , = + e -

SLIPPING STRIP ANALYSIS OF REINFORCED EARTH 36 1

A L 1 H

0 0 Area a- i Strip’

I In i t ia l ly I

P

x - x

--I& X 4 U ’ P

- Pe +Y

Finally Displacement convention The views in this figure (except X - X 1 can be interpreted as plan views

Figure 12. Conceptual model

The change in stress in the soil and strip are, respectively, a, -axo and us, and since Poisson’s ratio is zero in both (this makes the problem one-dimensional)

EE, = a, -axxo (A2)

and

also

Using these three equations to eliminate the strains from equation ( A l )

as ax a , ~ e dr, E, E E G, dx

+- - -=---

Considering the overall equilibrium of strips and soil (Figure 13(a))

u,+au,=O

and of the strips only (Figure 13(b))

r ,p dx = a da,

or

a da , r s = - -

P dx

Eliminating ax from equation (A5) by (A6) and dr,/dx by differentiating (A7) gives


(a1 Strips a n d soil ( b ) Strips only

Figure 13. Forces on the slice

in which

PG and p = - - f - ~ ~ ~ ~ aeE

The general solution to equation (A8) is 2 us = A exp ( a x ) + B exp (- ax) -@/a

in which A and B are determined from boundary conditions. Putting in values (see Table I) gives

a = 4.7833 m-’ and p = - 2,496,000 kPa/m’

The boundary conditions are us = 0 at x = 8 and T, = 0 at x = 0. This second condition gives, by equation (A7), du,/dx = 0 when x = 0. Putting in these boundary conditions gives

A = B = - 2.6235 x lo-’* kPa

The variation of us and 7, with x may now be obtained from equations (A9) and (A7). Some values are given in Table 11.

Table 11. Strip direct and bond stress values

X m* - 7, (m) (MPa) @Pa)

0 109.1 0.0 2 109.1 0.0 4 109.1 0.0 6 109.1 0.1 7 108.2 7.0 7.5 99.1 76.5 7.8 67.2 32 1 8 0 836

- ~. -. -- .~ -

The displacement of the soil at B relative to A is obtained by integrating over AB. Equation (A2) is used to express E % in terms of the soil stress change (ox -axo), then equation


(A6) is used to express u, in terms of us and lastly us is expressed in terms of x by equation (A9). This results in a displacement of 4.67 mm. To this must be added the extension of the uniformly strained 7 m to the right of B, i.e., - 7u,o/E = 0.014 m or 14 mm. Thus the total extension of the 15-m long layer is 18.67 mm. This is the negative displacement (- u,) of the wall when the right-hand end is fixed.

Extension to slip cuse

x = r. Then if x < r, IT,^ C Y and conditions are elastic. If x 3 r, IT,[ = Y (see Figure 14).

(since us is decreasing) we have

Slipping is now assumed to occur so that T, is cut off at a constant value Y. Let this occur at

The equilibrium equation, (A7), applies for all x. For x 31 and noting that T, is negative

-=, I

r 8 X

Figure 14. Bond stress distribution with slips

Integrating and noting that us = 0, x = 8

( A l l ) PY a

us=-(8-x) ( x s r )

At x = r, us is given by both equations (A1 1) and (A9) except that the constants A and B in (A9) will have different values from those obtained for the ‘no-slip’ case:

us(x = r )= pa (8 - r ) = A exp (ar )+B exp ( - a r ) - p / a (A121 Y 2

Using equation (A10) with the differential of equation (A9) we have

aAa a P P

T,(X = r ) = - Y = - exp (ar ) - -B exp ( - a r )

The x = 0, T~ = 0 boundary condition still applies, whence A = B. Adding p/a2 to both sides of equation (A12), dividing by equation (A13) and rearranging gives

1 exp ( o r ) + exp (- a r ) exp (ar) - exp (- ar)

Substituting Y = 40 kPa (the value chosen for the analysis) and the a, /3 values derived above gives r = 3.8384 m. (Note that the value of the square bracket in equation (A14) is almost

3 64 D. J. NAYLOR A N D H. RICHARDS

exactly 1.) A (= B) is then found from equation (A12) or (A13) as -5.5426x variation of as and 7, may now be obtained. Some values are given in Table 111.

kPa. The

Table 111. us and 7’ values for slip case

2 3

109.1 109.0 0.7 Elastic

3.5 108.1 7.9 1 3.84 103.9 40.0) Slip occurs 8 0 40.0 linear us variation

Displacements are again obtained by integrating E~ over the length. This time the integration must be done in three parts: x = 0 to r using the no-slip procedure except that ‘A’ in equation (A9) has the value -5.5426x x = r to 8 with a, given by equation ( A l l ) , and lastly x = 8 to 15 exactly as before (i.e., to give a displacement of 14 mm). Carrying this out gives the total extension ( - u A ) = 21.55 mm.

The relative displacement of strip to soil at B is also required. This is obtained by integrating the strip strain ( E , ) over the 8-m length and subtracting from the result the soil extension over AB. This results in a relative displacement ( p ~ ) of 4.38 mm.

APPENDIX I1

Element average values of over-stress ratios

analysis, the lower to the no-slip analysis. Values are assigned to element centres. The upper figure of each pair relates to the slip

Table IV. Strip tension and bond stress over-stress ratios

0;

U S

Strip tension 0.s.r.. RT = 7 I S Bond stress o.s.r., R. = ___

un tan 4% Distance from face (m) Distance from face (m)

Height ~- - m 0.8 2.4 4.0 5.4 6.5 7.5 0.8 2.4 4.0 5.4 6.5 7.5

9.375

8.125

6-875

5.625

4.375

3.125

1.875

0-625

0.04 0.05 0.13 0.13 0.2 1 0.2 1 0.29 0.29 0.38 0.37 0.46 0.46 0.58 0.58 0-33 0.33

0.06 0.07 0.15 0.15 0.2 1 0.22 0.29 0.29 0.37 0.36 0.43 0.42 0.42 0.42 0.19 0.19

0.07 0.04 0.10 0.13 0.15 0.10 0.19 0.16 0.21 0.17 0.21 0.20 0.27 0.23 0.27 0.25 0.32 0.27 0.32 0.28 0.35 0.28 0.35 0.29 0.31 0.24 0.31 0.24 0.13 0.10 0-13 0.10

0.02 0.14 0.06 0.16 0.10 0.20 0.14 0.22 0.18 0.25 0.2 1 0.24 0-19 0.19 0.08 0.08

0.01 1.01 1-02 1.01 1.00 0.12 -1.19 -2.75 -2.69 -2.03 0.03 -0.29 -0.04 1.00 1.00 0.13 -0.25 -0.24 -0.12 -0.05 0.05 -0.08 0.06 0.29 1.00 0.16 -0.12 -0.00 0.10 0.13 0.07 -0.11 0.15 0.32 1.00 0.18 -0.12 0.12 0.22 0.24 0.08 -0.04 0.25 0.35 0.67 0.18 -0.05 0.23 0.30 0.31 0.10 -0.13 0.42 0.41 0-50 0.17 -0.13 0.41 0.39 0.36 0.11 0.71 0.59 0.41 0.34 0-13 0.70 0-59 0.41 0.32 0.05 0.64 0.31 0.18 0.13 0.05 0.64 0.31 0.18 0-12

1.00 0.22 1 *oo 0.3 1 1.00 0.36 1.00 0.42 1.00 0.44 1.00 0.44 0.60 0.35 0.13 0.13

1.01 17.22 1.00 4.70 1 .oo 3-36 1.00 2.74 1.00 2.30 1.00 1.83 1.01 1.23 0.47 0-43


Table V. Soil over-stress ratios--element average values

- IUI - u 3 1

(a, + 0 3 ) sin q5 R M =

Horizontal distance from face (m) Height . - ~ _ _ ~ ~ _ _ . _ _ _ _ ~ ~ __ - .~ .-

m 0.8 2.4 4.0 5.4 6 5 7.5 8.6 10.0 11.8 13.9

9,375 1.13 1.02

8.125 0.96 0.96

6.875 0.94 0.93

5.625 0.93 0.92

4.375 0.93 0.92

3.12s 0.93 0.93

1.875 0.98 0.98

0.625 0.96 0.95

T means tension.

2.19 I .53 1.04 1.03 0.96 0.96 0.93 0.93 0.92 0.92 0.93 0.92 0.93 0.92 0.88 0.88 __

47.7 2.27 1 .09 1 .05 0.96 0.95 0.9 1 0.9 1 0.90 0.89 0.89 0.89 0.88 0.87 0.82 0.82

T 3.23 1.46 I .04 1.06 0.93 0.93 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.78 0.78

T 4.6 I 1.83 I 4 5 1.27 0.93 1.08 0.88 0.95 0.85 0.87 0.83 0.82 0.81 0.76 0-76

T T T T

2.20 2.60 1.28 2.76 1.46 1.64 1.05 1.67 1.22 1.34 0.96 1.36 1.07 1.17 0.90 1.18 0.95 1.03 0.86 1.04 0-84 0.90 0.81 0.89 0.75 0.76 0.7s 0.75

T 1'

2.6 1 2.74 1.58 1.62 1.27 I .29 1.10 1.06 0.96 0.96 0.84 0.84 0.73 0.72

T T

2.62 2.73 1.52 1.56 1.20 1.22 1.03 1.04 0.90 0.90 0.80 0.80 0.70 0.70

T T

2.62 2.73 1.49 1.52 1.16 1.18 0.99 1 .oo 0.87 0.87 0.77 0.77 0.69 0.69

1.

2.

3. 4.

5 .

6.

7.

8.

9.

10.

11.

12.

13.

14

15.

REFERENCES

P. K. Ranerjee, 'Principles of analysis and design of reinforced earth retaining walls'. J. Inst. Highway Eng., 22. No. 1. 13-18 (1975). J. C. Chang and R. A. Forsyth, 'Finite element analysis of reinforced earth wall', Proc. Am. Soc. c i v . Eng., J . Geotech. Engng Div.. 103. 71 1-724 (1977). C. S. Desai and J. F. Abel. Introduction to the Finite Element Method, Van Nostrand Rheinhold. New York, 1972. R. E. Goodman, R. L. Taylor and T. L. Brekke. 'A model for the mechanics of jointed rock', froc. Am. Soc. Civ. Eng.. 1. Soil Mech. Found. Div., 94. 637659 (1968). W. J. Harrison and C. M. Gerrard, 'Elastic theory applied to reinforced earth', froc. Am. Soc. Civ. Eng., J. Soil Mech. Found. Div.. 98. 1325-1345 (1972). L. R. Herrmann, 'Nonlinear finite element analysis of frictional systems', Int. Conf. on Finite Element3 in Nonlinear Solid and Srructural Mechanics, Geilo. Noway. 2. Section G (August 1977). N. W. M. John, 'An investigation in the behaviour of reinforced earth retaining walls', Internal Report, Depart- ment of Civil Engineering, Portsmouth Polytechnic (1977). D. J. Naylor, 'Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressure', lnr. J. num. Merh. Engng, 8, 443-460 (1974). D. Ngo and A. C. Scordelis, 'Finite element analysis of reinforced concrete beams', 1. Am. Concr. Insr.. 64, 152-163 (1967). A. H. Nilson, "on-linear analysis of reinforced concrete by the finite element method', J. Am. Concr. Inst., 65, 757-766 (1968). D. V. Phillips and 0. C. Zienkiewicz. 'Finite element non-linear analysis of concrete structure', Proc. Insr. Civ. Eng.. Pr. 2.61, 59-88 (1976). K. M. Romstad, L. R. Herrmann and C. K. Shen. 'Integrated study of reinforced earth-I: Theoretical formulation', Proc. Am. Soc. Civ. Eng.. 1. Georech. Eng. Div., 102,457-471 (1976). F. Schlosser and N. T. Long. 'Recent results in French research on reinforced earth', Proc. Am. Soc. Civ. Eng., J . Consn. Div.. 100, 223-237 (1974). C. K. Shen, K. M. Romstad and L. R. Herrmann, 'Integrated study of reinforced earth-11: Behavior and design'. Proc. Am. Soc. Ciu. Eng.. 1. Georech. Eng Div.. 102. 577-590 (1976). A. K. C. Smith, 'Experimental and computational investigation of model reinforced earth retaining walls, Ph. D. thesis. University of Cambridge (1977).


16. 0. C. Zirnkiewicz. The Finire Element Merhod in Engineering Science. 2nd edition, McGraw-Hill. London, 1971. 17. 0. C. Zienkiewicz, R. L. Taylor and J . H. Too, ‘Reduced integration, technique in general analysis of plates and

18. 0. C. Zienkiewicz, S. Valliapan and I . P. King, ‘Elasto-plastic solutions of engineering problems “initial stress”, shells’, Inr. J. num. Merh. Engng, 3, 275-290 (1971).

finite element approach’. Inr. J. num. Merh. Engng. 1. 7.5-100 (1969).

slipping strip analysis of reinforced earth

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