The purpose of this paper is to highlight the importance of taking the most unfavourable water levels in the design of embedded and cantilevered retaining walls designed to Eurocode 7 using partial factors. This is of topical interest as ENV 1997-1, the pre-standard version of Eurocode EC 7, Part 1: Geotechnical Design, is close to the end of its trial period and the document is currently being reviewed for issuing as a Euronorm (EN) with the status of a European standard in 2000. This paper concentrates on the partial factor approach adopted in ENV 1997-1.

The paper compares the equivalent lumped factors of safety implied in the Eurocode design approach for two design situations with the values historically used for these designs. It shows that when determining the depth of embedment required for an anchored/propped embedded retaining wall, Case B gives no margin of safety and, as there is no partial factor on water forces in Case C, the overall factor of safety against rotation about the anchor/prop may be lower, in certain situations, than traditionally used when high water forces are involved. The paper also shows that the Case B and Case C partial factors may also give a lower equivalent lumped factor of safety than traditionally used when designing a cantilever retaining wall to resist horizontal siding when high water forces are involved. It is therefore important to consider the most unfavourable water levels that can occur in these design situations.

Introduction

In order to highlight the points, the paper discusses the most unfavourable design situations for some selected examples making simplified assumptions. The most general assumptions are that the soil is purely frictional with no cohesion and that the variable actions (ie loads) are insignificant in relation to the permanent actions. This latter assumption is reasonable for many retaining walls and avoids the complication of considering the possible effect of the larger partial factors which are applied to the variable actions. It is also assumed that the boxed values given in ENV 1997-1 apply, which is also reasonable as these have been adopted by most countries in their National Application Documents for ENV 1997-1.

Partial factor approach in ENV 1997-1

ENV 1997-1 requires that geotechnical structures be designed for three cases: A, B and C, and provides a different set of partial factors for use with each case. The partial factors for the three cases are applied to characteristic actions and characteristic soil parameter values. ENV 1997-1 goes to some trouble to attempt to define the characteristic value of soil parameters. This approach is to formalise an area which previously was undefined in other codes and is expected to lead to the same values of parameters as those currently used in design.

Case A is relevant in assessing buoyancy limit states and is therefore not normally relevant in the design of retaining walls.

The partial factors for Case B are unity on soil strength and 1.35 on permanent actions. This design case comes from structural considerations and is intended to ensure safety against variations or uncertainties in the applied loads. For example, in the design of a simply supported concrete beam carrying 1m of soil as shown on Figure 1, the design moment for the reinforced concrete beam is calculated by multiplying the characteristic dead load of the beam itself and the characteristic weight of the soil by 1.35. It would appear logical, therefore, that the earth and water loads should also be multiplied by this partial factor when calculating the design moments in earth retaining structures. However in the case of embedded walls it is not always clear when the earth pressure is a load and when it is a resistance. In order to avoid unrealistic design moments that would result if only the active side were multiplied by 1.35, ENV 1997-1 requires that, for Case B, all characteristic earth pressures from the same source be multiplied by this factor, ie increased if the combined effect is unfavourable, but multiplied by a factor of unity, ie not increased, if the combined effect is favourable. This terminology does, however, lead to ambiguity in certain situations as will be demonstrated below.

Case C arises from geotechnical considerations and is intended to ensure safety against variations or uncertainty in the soil parameter values. Hence, as is normal practice in geotechnical engineering design, safety is ensured by factoring the soil parameters but not the permanent loads, such as the self weight of the soil. The partial factors for Case C are 1.25 on tank, the characteristic soil shear strength parameter for cohesionless soils, and unity on the permanent loads. These partial factor values were essentially selected to give similar safety levels in slope stability analyses as the more traditional lumped safety factor methods.

Various codes, and ENV 1997-1 is no exception, introduce allowances for uncertainty in the geometry. An example given in ENV 1997-1 is that in the case of an embedded retaining wall, which depends on the passive resistance of the ground in front of the wall for its stability, the ground level in front of the wall should be lowered by an amount a equal to 10% of its height, up to a maximum of 0.5m. Such requirements are introduced to allow for site or construction conditions which are not covered by the partial factors on the loads or the soil parameters. In this paper it is assumed that there is no uncertainty in the geometry so that a is equal to zero. The examples below investigate how the values chosen in calculations to ENV 1997-1 for the material and load factors affect the level of safety obtained.

Retaining wall design

The basic equations to be satisfied in the design of earth retaining structures are the equilibrium conditions:-

Disturbing moments Resisting moments(1)

Horizontal force Horizontal resistance(2)

Vertical force Vertical resistance(3)

Example 1 - embedded retaining wall

One of the critical dimensions in the design of the anchored embedded retaining wall shown in Figure 2 is the required depth of embedment, 'd' to give a satisfactory safety margin against rotational failure about the anchor support. This example will show that Case B cannot give any margin of safety against this failure mechanism, and that the margin of safety in Case C can be lower than the value previously used in design to other codes. These comments also apply to cantilevered embedded retaining walls. The 'free-earth' method of analysis as described in Clayton et al (1996) is used.

The overall factor of safety against failure by rotation about the anchor, F, is defined as:

F = Resisting moment/Overturning moment

where the resisting and overturning moments are obtained using the characteristic, ie unfactored, loads and material parameters to determine the forces on the wall. This overall factor of safety is equivalent to a lumped factor of safety on the passive resistance. The forces on the wall, shown in Figure 3, are the characteristic active pressure force, Pa, the characteristic passive pressure force, Pp and the characteristic water pressure force, U. The distances of the lines of action of these forces from the anchor are la, lu and lp respectively. The characteristic water pressure force is obtained in this case from the net water pressure acting on the wall.

Two values of 'd' are obtained when designing the wall for rotational equilibrium according to ENV 1997-1: dB is obtained using the Case B partial factors and dC is obtained using the Case C partial factors.

For Case B, the partial factor on the soil strength is unity so that the partial factor on the actions is applied to the characteristic earth and water pressure forces. Since the active and passive earth pressures are considered to be from the same source, as are the water pressures acting on both sides of the wall, they should be multiplied by the same partial factor according to ENV 1997-1. Hence the moment equilibrium equation for Case B is:

1.35Pala + 1.35Ulu 1.35Pplp(4)

It is clear from this equation that exactly the same value of dB is obtained whatever partial factor is applied to the forces, whether 1.35 or unity, and hence Case B provides no margin of safety, ie F = 1 in the case of embedded retaining walls subjected to just permanent loads.

Case C requires the soil strength parameters to be factored such that tand = (tank)/1.25 where d is the design value for the angle of shearing resistance. The partial factor on the permanent actions is unity. Thus the safety margin will come from the alteration in the earth pressure coefficients by using a lower strength parameter. If Kak is the active pressure coefficient determined using k (ie no partial factor) and Kad is the value determined using d, then the active pressure at each depth in Case C is increased by the ratio Kad/Kak. Similarly, the passive pressures is reduced by the ratio Kpd/Kpk. Using the same symbols as before for the earth pressure forces based on k,and since the lever arms remain the same, the depth dc for Case C is obtained by taking equilibrium about the anchor:

(Kad/Kak)Pa la + Ulu = (Kpd/Kpk)Pplp(5)

The influence of the water forces can more clearly be illustrated by relating the moment of the net water pressure to the active pressure, ie:

Ulu = X Pala (6)

Substituting for Ulu in Equation 5 gives:

((Kad/Kak) + X) Pa la = (Kpd/Kpk)Pplp

from which Pplp = (Kpk/Kpd)[(Kad/Kak) + X] Pa la(7)

The overall factor of safety, F on the moments about the anchor, which is calculated using the characteristic or unfactored soil strengths, is therefore:

F=resisting moment= Pplp(8)

overturning moment[1 + X] Pala

Substituting for PPlP from Equation 7 gives:

F=(Kpk/Kpd)[(Kad/Kak) + X] Pa la(9)

[1 + X] Pala

Inspection of Equation 9 shows that, where water pressures are the same on both sides of the wall,

F=(Kpk/Kpd)(Kad/Kak)(10)

The value of F varies with k and with the angle of friction, between the soil and the wall. Graphs showing the variation in F with k for

a) /k =0 on both sides of the wall, b) for /k equal to 0.5 on the active side and 0.67 on the passive side, and c) for /k equal to 0.67 on the active side and 0.5 on the passive side are plotted in Figure 4. The graphs in this figure show that, for k = 25degrees and /k = 0, F is about 1.4 and increases with increase in both k and /kup to a value of about 2.2 for k = 40degrees and /k= 0.5 on the active side and 0.67 on the passive side.

Taking an extreme case where the net water pressure is significantly greater than the active pressure, ie X Kad/Kak , then the term

[(Kad/Kak) + X] Pa la 1(11)

[1 + X] Pala

in which case F approaches Kpk/Kpd. The variation in F for this condition for various values of k and /k is plotted in Figure 5 which shows that F is as low as 1.18 for k=25degrees when /k= 0. This is considerably less than the factor of about 1.5 normally adopted in traditional designs for failure due to rotation about the anchor (Clayton et al, 1996).

In order to illustrate this point about the significance of the net water pressure, the propped retaining wall shown in Figure 6 was analysed for Case C (as discussed above, Case B is not relevant as it gives a shorter depth of embedment). The wall retains 10m of water and the soil properties are ck = 0 and k = 25degrees with /k = 0 on both the active and passive sides of the wall. The net differential pore water pressure head was assumed to vary linearly along the embedded length. The required depth of embedment was found to be 14.2m.

The value of F for this situation, using the definition given in Equation 8 and the unfactored active and passive pressures, is 1.27 which is in line with that expected from the above study.

The factor of safety, F used above to illustrate the effect of high water forces is based on the moment of the net water pressure. A different, and even lower, factor of safety would be computed if the water pressures on the active and passive sides were treated separately and the factor of safety were defined as

Fs = Pplp + moment of water pressure on the passive side (12)

Pala + moment of water pressure on the active side

This definition gives a factor of safety of 1.13 for the above example.

The same approach as that adopted above for the anchored embedded retaining wall can be used to show that the Case B offers no safety margin on the depth of embedment of a cantilevered embedded retaining wall. Furthermore the equation for the factor of safety against rotational failure of a cantilevered embedded retaining wall for Case C is similar to that determined above for the rotation of an anchored embedded retaining wall about its anchor.

Example 2 - cantilever retaining wall

In this example the margin of safety against sliding failure of the cantilever retaining wall shown in Figure 7 is examined. The soil is taken to be cohesionless and the active and passive pressures are taken to act horizontally. The critical dimension for this mechanism is the length 'b' of the foundation of the wall. The forces on the wall are as follows:

Wc = weight of concrete in the retaining wall

Ws = weight of soil above the heel less the weight of water above the heel

Us = weight of water in the soil above the heel

Pa = active earth pressure force

Pp = passive pressure force on front of toe of wall

Ua = water force on rear of imaginary vertical plane through the rear of the heel

UP = water force on front of toe of wall

Ub = uplift water force on base

V = vertical effective force on base

H = horizontal reaction

The active earth pressure force Pa and the water pressure force Ua behind the wall act horizontally on the vertical effective back of the wall above the rear of the heel of the wall. The passive earth pressure force Pp also acts horizontally.

It is assumed that the soil is cohesionless. The above example poses the interesting question as to the treatment of the forces above the heel of the retaining wall and the uplift pressure force Ub on the base. In order to analyse the factor of safety against sliding it is necessary to determine the vertical force on the horizontal plane of sliding on the bottom of the foundation of the wall. When checking the wall for horizontal stability for Case B, it could be argued that the contribution to the vertical effective force V from the soil and water above the heel are from the same source as the horizontal earth and water pressure forces and so should be multiplied by 1.35. However, logically, it could be argued that the soil on top of the heel is part of the structure and should not be treated as 'from the same source'. In that case it should be treated as a favourable action with regard to horizontal equilibrium. Hence two possible combinations of factored forces are possible when assessing the horizontal equilibrium of the retaining wall for Case B:

Sum of horizontal forces Vtan k (13)

The first option, treating the earth and water forces as all coming from the same source, gives:

1.35(Pa + Ua - Pp-UP) (Wc + 1.35Ws + 1.35Us - 1.35Ub) tan k (14)

and the second option, treating the earth and water forces above the heel as part of the structure, gives:

1.35(Pa + Ua - Pp-UP) (Wc + Ws + Us -1.35Ub) tan k (15)

Equation 15 could lead to unrealistic conditions where the water force is high and the weight forces of the wall and soil on its heel are low. For example, if the soil on top of the heel were peat with a unit weight () of about 12kN/m3, and the water level behind the wall was at the top of wall, (Wc + Ws + Us -1.35Ub) could result in a negative value of V.

The traditional factor of safety would give

F = Resistance available/Actual horizontal force

= (Wc + Ws +Us - Ub )tank/(Pa+Ua-Pb-UP) (16)

Rearranging Equation 14 gives:

(Pa + Ua - Pp-UP) = (Wc + 1.35Ws + 1.35Us - 1.35Ub) tan k /1.35

and inserting this into Equation 16 and rearranging gives:

F1= (Wc + Ws +Us - Ub ) /(Wc/1.35 + Ws +Us -Ub) (17)

Similarly, using Equation 15:

F2 = 1.35 (Wc + Ws +Us - Ub ) /(Wc + Ws +Us -1.35Ub) (18)

which gives F2 equal to or less than 1.35

Treating the forces as from the same source, F1 gives the lower factor of safety which is less than 1.35. In an extreme case, where Wc, the weight of the concrete wall, is considerably less than the force from the soil and the water above the heel, the factor of safety would approach unity for this Case B.

Analysing the safety against sliding for Case C, adopting the same approach as used previously for Case B, the active earth pressure force is increased while the passive pressure force is decreased in relation to the change in the active and passive earth pressure coefficients. Thus

(Kad/Kak )Pa + Ua - (Kpd/Kpk)Pp -Up (Wc + Ws +Us - Ub )tand ( 19)

As tan d= tank/1.25

(Kad/Kak )Pa+Ua- (Kpd/Kpk)Pp -Up (Wc+Ws+Us - Ub)tank/1.25 (20)

For the case where Pp and Up are small, or can be ignored, and

putting Ua = XPa

(Kad/Kak )Pa + X Pa (Wc+ Ws +Us - Ub )tank/1.25 (21 )

therefore

1.25 ((Kad/Kak ) + X )Pa (Wc + Ws +Us - Ub )tank (22 )

The traditional factor of safety, defined by Equation 16, gives:

F = (Wc+Ws +Us - Ub )tank/ (Pa+ Ua)

= 1.25{(Kad/Kak +X)}/ (1+ X) (23)

Where there is no water force, X = 0 and the factor of safety is 1.25(Kad/Kak) which varies from about 1.5 to 2 for values of 25degrees to 40degrees and /k of 0 to 2/3. Generally the maximum value of X would be dictated by the height of the wall (h), hence the maximum value would be with the water at the top of the wall, ie

X = Ua/Pa= (1/2 9.81h2 )/( 1/2 Ka h2)

where is the submerged unit weight of the soil .

Putting = 10kN/m3, which is reasonable for most soils, then the value of X is about 1/Kak. Equation 23 then becomes

F 1.25{(Kad+1)/(Kak+1)} (24)

which is relatively constant at about 1.32 for values of k from 25degrees to 40degrees and for values of /kbetween 0 to 2/3 .

Thus neither Case B nor Case C give a factor of safety of 1.5 which has traditionally been used for this mechanism. The lower factor of safety could, however, be justified in this example as the water level cannot get any higher than the top of the wall. This may not always be the case; for example, if a wall were constructed with a parapet which allowed the retention of flood water behind the wall to a level above the top of the wall, then the factor of safety could be even lower.

Discussion and conclusions

The use of the partial factor approach in Eurocode 7 results in margins of safety which are different to those traditionally used in the design of retaining walls. The actual safety level of these structures is difficult to determine as the analysis depends on the soil structure interaction. This paper compares the overall factor of safety implied by the partial factors in the Eurocode with those obtained using the more traditional lumped factors of safety which have been in use in the profession and which have stood the test of time. While these lumped safety factors are not necessarily correct, it is important to recognise and discuss any deviations from previous practice.

The study in this paper has shown that, when using the partial factors in ENV 1997-1, the overall factor of safety against rotation of an anchored/propped embedded retaining wall implied by

the partial factors in ENV 1997-1 may be low for low values of k and when high water forces are involved. The factor of safety drops to below 1.3 in certain circumstances. These comments also apply to the factor of safety against rotation of cantilevered embedded walls. ENV 1997-1 does state that the most unfavourable water pressure condition must be taken into account in the design of

retaining structure and this study highlights the importance of this requirement.

The study has also shown that the factor of safety against sliding of a cantilevered retaining wall subjected principally to water pressure would be less than the 1.5 which has been used in the

past. However the use of a lower safety factor would be justified in this case provided the water forces cannot rise any higher than the top of the wall.

The requirement that the same partial factor be applied to all forces from the same source in Case B can be confusing and could lead to unsafe designs. This requires further clarification in the revised Eurocode.

References

Clayton CRI, Milititsky J, and Woods RI. Earth pressure and earth-retaining structures, Blackie Academic Press, 1996

ENV 1997-1:1994E Eurocode 7:Geotechnical design - Part 1:General Rules, CEN (European Committee for Standardisation), Brussels

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