Cantilever Sheet Pile Walls In cantilever sheet pile wall construction, heavy steel sheet piles are driven into the ground prior to excavation taking place. When excavation is carried out, the soil behind the sheets is retained by their cantilever action. Cantilever sheet pile walls tend to be prone to movement and are generally only used for temporary structures or for permanent walls of low height in sands and gravels.

Figure 1 After Craig Figure 6.22

Additional Design Requirements In addition to the above, there are also two other important criteria that are adopted in practical design of sheet pile walls as follows:

1. An additional depth is allowed at the front of the wall to allow for the possibility that an accidental excavation may occur here. BS 8002 (Earth Retaining Structures) suggests that this allowance should be 10% of the retained height up to a maximum allowance of 0.5 m.

2. A surcharge load is applied behind the wall to allow for loading in this area. The applied surcharge will depend on the load that might be expected, but general practice allows for a minimum nominal surcharge of 10 kN/m2.

Design of Cantilever Walls by Moment Balance Design of cantilever walls is carried out by finding a moment balance about the assumed point of fixity, which is shown as point C in Figure 1(c). As has been discussed previously, generation of full active and passive pressure requires a certain strain to develop in the soil. Generation of full passive resistance will, in particular, require quite significant strains to develop. Several different design methods have been proposed for cantilever retaining walls depending on different assumptions of precisely what earth pressures are mobilised. The four main design methods are described below:

Gross Pressure Method (CP2 Method)

This method commences with the assumption that full active pressure will develop behind the wall and full passive pressure in front of the wall. The approach requires that there is a factor of safety against failure and this is obtained by dividing the calculated moment due to the passive earth pressure by a factor, Fp, and then equating this to the moment due to the active earth pressure plus the net moment due to any water pressure acting on the wall.

moment PressureNet Water Pressure Active fromMoment F

Pressure Passive fromMoment

p

Net Pressure Method (British Steel Method) This method was developed by British Steel and adopts the pressure distribution shown bounded by the solid lines in the diagram opposite, where the net pressures acting on the rear and front of the wall are calculated by subtracting the pressures acting on one side of the wall from those on the other side, with the effects of water pressure included in these calculations. As for the Gross Pressure method, a factor of safety is applied to the moment due to the passive pressure in order to ensure that the wall design is safe. In this case:

Water)(Earth Pressure ActiveNet fromMoment F

Water)(Earth Pressure PassiveNet fromMoment

np

As a supplier of steel sheet sections, British Steel contributed significantly to the literature on steel sheet pile design and produced a Piling Handbook, which went through a number of editions. The part of the business dealing with sheet piles was subsequently sold to Arcelor, whose web site includes a lot of data relevant to sheet pile design. In particular, see www.arcelor.com/sheetpiling under documentation, where a copy of the current Piling Handbook can be downloaded.

Net Available Passive Resistance Method (Burland and Potts) This method was developed by Burland and Potts and adopts the pressure distribution shown bounded by the solid lines in the diagram opposite. The active pressure is assumed to reach a maximum value at the level of the ground in front of the wall. This value of active pressure is then maintained to the base of the wall while the net active pressure is calculated as the gross passive pressure less the drop in active pressure below the theoretical hydrostatic value. As for the previous methods, a factor of safety is incorporated in the moment balance in order to ensure that the design is safe, as follows:

moment PressureNet Water Pressure Active Modified fromMoment F

Pressure Passive Modified fromMoment

r

Factor of Safety on Soil Strength Method This method differs from the previous three methods in that the factor of safety for the wall is applied to the soil strength rather than to the moments induced by the soil. Using this method for a granular soil and effective stress analysis, the soil strength for design will be determined as:

tan(design) = (tan measured) / Fs

The unfactored gross soil pressures based on the factored soil strength are then used in the analysis, so that:

moment PressureNet Water Pressure Active fromMoment Pressure Passive fromMoment Factors of Safety to be used in the above Analyses The appropriate factors of safety for the above analyses will depend on the circumstances of the particular problem. The following table summarises the relevant factors for a number of cases:

Method Effective Stress Analysis

20 20 < 30 > 30

Factor on Strength, Fs 1.2 1.2 1.2

Gross Pressure Factor, Fp 1.5 1.5 2.0 2

Net Pressure Factor, Fnp 2 2 2

Net Available Resistance, Burland and Potts, Fr

2 2 2

Example The Design Procedure Gross Pressure Method Design follows a set procedure in which the following steps need to be taken:

1. Adopt the simplified model shown in Figure 1(c), above 2. Find the appropriate design values of Ka and Kp 3. Determine the lateral earth pressures in front of and behind the wall, making

allowance for groundwater as necessary. Note that the design is almost always carried out on the basis of an effective stress approach

4. Use a moment balance to find the depth to the point of fixity 5. Increase the proposed embedment of the pile by 20% to provide the necessary

force, R, at the rear of the wall 6. Find R 7. Check that the increase in depth allowed in step 5 is adequate to provide R

Examples: The following two examples illustrate the procedure. Example 1 is a basic illustration and excludes the standard assumptions for overdig and surcharge. Example 2 is a more general example, including these and the presence of groundwater. This is, however, a special case where the groundwater levels in front of and behind the wall are the same, in which case the water pressures on the front and the back of the wall are equal and can be ignored in the calculation.

Example 1 Consider the case of a wall with a 2 metre retained height of fill constructed in a soil for which = 30 and bulk = 18kN/m3

The groundwater table is well below the level of the base of the wall and ignoring any allowance for accidental overdig or surcharge load but taking a factor of safety of 2.0 on the passive pressure (Gross Pressure Method)

Assuming there is no wall friction:

Ka = (1 sin(1 + sin) = 0.333

Kp = (1 + sin(1 sin) = 3.0

Taking moments about point C

Moment due to active force:

Mmt. = 0.5 Ka bulk (h+d)2 x (h+d)/3

= 0.5 x 0.333 x 18 x (2 + d)3/3

= 8 + 12 d + 6 d 2 + d 3

Moment due to passive force:

Mmt. = 0.5 Kp bulk d2 x d/3

= 0.5 x 3.0 x 18 x d3/3

= 9 d 3

Then, equating the active and passive moments and allowing for a factor of safety of 2.0 on the passive gives:

4.5 d 3 = 8 + 12 d + 6 d 2 + d 3

Which can be solved by trial and error to give d = 3.07 m

The final embedment is then taken as 3.07 x 1.2 = 3.68 metres

It is then required to carry out a force balance to calculate the required value of R for horizontal stability:

Active force = 0.5 Ka bulk (h+d)2 = 0.5 x 0.333 x 18 x 5.072 = 77.11 kN

Passive force = 0.5 Kp bulk d 2/2 = 0.5 x 3.0 x 18 x 3.072 /2 = 127.24 kN

Hence, R = 127.24 77.11 = 50.13 kN

This force must be generated in the additional length of pile below point C

The two force elements generated will be due to the passive load on the back of the wall and the active force on the front of the wall (see figure 1(b), above). In each case there will be two elements one due to the effective surcharge imposed by the fill above the level of point C and one due to the soil self weight below level C. Overall, the lateral effective stress profile will be as shown below:

Passive Pressures:

Force 1, due to overburden surcharge = Kp x bulk x (h+d) x 0.2 d = 168.1 kN

Force 2, due to soil self weight = 0.5 x Kp x bulk x (0.2 d) 2 = 10.18 kN

Active Pressures:

Force 3, due to overburden surcharge = Ka x bulk x d x 0.2 d = 11.31 kN

Force 4, due to soil self weight = 0.5 x Ka x bulk x (0.2 d) 2 = 1.13 kN

Available force = Passive Active = 168.1 + 10.18 11.31 1.13 = 165.84 kN

Required force = 50.13 kN < Available Force of 165.84 Hence, OK

R C

2 3

4

0.2 d 1

Active Pressure Passive Pressure

Example 2 A sheet pile wall is to be installed to support a 3m high cut in sand of bulk density 18kN/m3 and 30. If the saturated bulk density of the sand is 20kN/m3 and a factor of safety of 2 is to be applied to the passive earth pressure, find the required length of sheet piles required if the groundwater level is 4 m below the existing ground level. In carrying out the design, the following points should be considered:

10% of the wall height is 0.3m, so the design height will be 3.3m

A 10kN/m2 surcharge must be applied behind the wall

The effective vertical stress due to the soil below the water table can be taken as (sat water), i.e. 209.8 = 10.2kN/m3

In the absence of other data, take:

Ka = [(1-sin)/(1+sin)] = 0.333 Kp = [(1+sin)/(1-sin)] = 3.0

The required wall and the effective earth pressures acting on it will then be as shown below, where d is the depth of embedment below the water table and is what we are required to determine: Figure 2: Design Example 2 Configuration and Active and Passive Earth Pressures By considering the earth pressures shown in Figure 2 and by taking moments about point C, the following values are obtained:

3.30m

0.70m

d

1 2

3

4

5

6 7

R

Surcharge = 10kN/m2

W.T.

C

Passive Earth Pressures Active Earth Pressures Retaining Wall Configuration

Calculations all per metre width

Force (kN) Lever Arm (m) Moment (kNm)

(1) 0.333 x 10 x (d + 4.0) = 3.33d + 13.33 d/2 + 4.0/2 1.665d2 + 13.32d + 26.64

(2) 0.5 x 0.333 x 18 x 4.02 = 47.95 d + 4.0/3 47.95d + 63.94

(3) 0.333 x 18 x 4.0 x d = 23.976d d/2 11.988d2

(4) 0.5 x 0.33 x 10.2 x d2 = 1.689d2 d/3 0.566d3

(5) (0.5 x 3.0 x 18 x 0.72) / 2 = 6.62 d + 0.7/3 6.62d 1.544

(6) (3.0 x 18 x 0.7 x d) / 2 = 18.9d d/2 9.45 d2

(7) (0.5 x 3.0 x 10.2 x d2) / 2 = 7.65d2 d/3 2.547 d3

Note densities below the groundwater table are taken as (sat water) in order to obtain soil effective vertical and lateral stresses. Table 1: Design Example Calculated Bending Moments Balancing the moments then gives:

1.981 d3 + 4.203 d2 + 54.65 d + 89.036 = 0

Giving: d3 = 44.94 + 27.59 d + 2.12 d2

Which is best solved by trial and error to find d = 7.0 m

Depth of Embedment As noted above, the depth of embedment should be increased by 20%

In solving the problem, the actual depth of embedment is 0.70 m + d = 7.7 m.

i.e. actual required embedment = 1.2 x 7.7 = 9.24 metres (1.54 m increase in length)

Required Value of R R is the force generated by the passive soil pressure on the back of the wall below point C, where the wall tends to rotate back into the soil.

For horizontal equilibrium:

Total passive Force = Total Active Force + R

So R = Total Calculated Passive Force Total Calculated Active Force

Hence, R = ([(5)+(6)+(7)] [(1)+(2)+(3)+(4)])

Substituting d = 7.0 into the values calculated in Table 1, allowing for the necessary changes of sign then gives:

R = [6.62 + 132.30 + 374.85] [36.63 + 47.95 + 167.83 +83.22] = 178.13 kN This force has to be generated on the back of the sheet piles by the extra 20% length added above i.e. between point C and a point 1.54 metres below this.

The situation is shown in Figure 3, where forces 8, 9 and 10 will be passive earth pressures and 11 and 12 will be active pressures Figure 3: Design Example2 Active and Passive Earth Pressures below point of fixity

Force (kN)

(8) 3 x 10 x 1.54 = 46.2

(9) 3 x [(4.0 x 18) + (d x 10.2)] x 1.54 = 662.508

(10) 0.5 x 3 x 10.2 x 1.542 = 36.29

(11) 0.333 x [(0.7 x 18) + (d x 10.2)] x 1.54 = 43.12

(12) 0.5 x 0.333 x 10.2 x 1.542 = 4.03

Note: In this case no reduction factor is applied to the passive earth pressures as owing to where they are generated they are more reliable R available = 46.2 + 662.508 + 36.29 43.12 4.03 = 697.85 kN > 178.13 kN OK

3.30m

0.70m

d

1 2

3

4

5

6 7 R

Surcharge = 10kN/m2

W.T.

C

10 11

12

0.2 d 8 9

Eurocode 7 Check Considering design example 2 as above but adopting a Eurocode 7 analysis using a factor on strength approach.

Firstly:

Eurocode 7 does not require any allowance for surcharge

Eurocode 7 requires the same allowance for over-excavation in front of the wall

Hence, in this case there is no surcharge to be considered but the overall depth of excavation for design will again be 3.30 metres (3.0 metres + 10%)

Figure 4: Design Example 2 Configuration for Eurocode 7 Design As before:

The bulk density of the sand above the water table is 18kN/m3

The saturated bulk density of the sand is 20kN/m3

And the angle of internal friction for the sand, is 30.

We now consider the two load combinations to be considered for Eurocode 7 Design approach 1 as follows:

Load combination 1 (A1+ M1 + R1)

Load combination 2 (A2 + M2 + R1)

3.30m

0.70m

d

1

2

3

4

5 6 R

W.T.

C

Passive Earth Resistances Active Earth Pressures Retaining Wall Configuration

Load Combination 1 (A1+ M1 + R1)

Action Type Partial Factor

Case A1 The thrust due to the retained

backfill Permanent unfavourable G, dst = 1.35

Material Property Partial Factor

Case M1 Coefficient of shearing resistance (tan ) = 1.0

Weight Density = 1.0

Resistance Partial Factor

Case R1 Earth Resistance R; e = 1.0

For = 30 and = 1.0: tan design = tan 30 / 1.0 and design = 30 Hence:

Ka design = (1 sin 30) / (1 + sin 30) = 0.5 / 1.5 = 0.3333

Kp design = (1 + sin 30) / (1 - sin 30) = 1.5 / 0.5 = 3.0 For bulk density of retained fill = 18 kN/m3 design = 18 / 1.0 = 18 kN/m

3 For bulk saturated bulk density of fill = 20 kN/m3 design = 20 / 1.0 = 20 kN/m

3 Effective unit weight of soil below water table = (20 9.8) = 10.2 kN/m3 So, as = 1.0, again for this value: design = 10.2 kN/m

3 The partial load factor for the action due to the active earth pressure behind the wall, G, dst = 1.35, will be applied to the force due to the active earth pressure behind the wall. The partial factor for resistance R; e = 1.0, which will be applied to the force due to the passive earth pressure in front of the wall. The calculations are then as follows:

Calculations all per metre width

Force (kN) Lever Arm (m) Moment (kNm)

(1) 1.35 x 0.5 x 0.333 x 18 x 4.02 = 64.8 d + 4.0/3 64.8d + 86.4

(2) 1.35 x 0.333 x 18 x 4.0 x d = 32.4d d/2 16.2d2

(3) 1.35 x 0.5 x 0.333 x 10.2 x d2 = 2.295d2 d/3 0.765d3

(4) 1.0 x (0.5 x 3.0 x 18 x 0.72) = 13.23 d + 0.7/3 13.23d 3.087

(5) 1.0 x (3.0 x 18 x 0.7 x d) = 37.8d d/2 18.9 d2

(6) 1.0 x (0.5 x 3.0 x 10.2 x d2) = 15.3d2 d/3 5.1 d3

Which gives: 64.8 d + 86.4 + 16.2 d2 + 0.765 d3 = 13.23 d + 3.087 + 18.9 d2 + 5.1 d3

Which simplifies to: 83.313 + 51.57 d 2.7 d2 0.765 d3 = 0, from which d = 3.82 m

The pile penetration is then increased by 20 % to ensure stability Actual pile penetration is taken as (d + 0.7) metres = 4.52 m Required penetration length calculated from actual pile length but increased by 20% = 1.2 x 4.52 = 5.42 m Overall pile length required = 3.0 + 5.42 = 8.42 metres

Load Combination 2 (A2+ M2 + R1)

Action Type Partial Factor

Case A2 The thrust due to the retained

backfill Permanent unfavourable G, dst = 1.0

Material Property Partial Factor

Case M2 Coefficient of shearing resistance (tan ) = 1.25

Weight Density = 1.0

Resistance Partial Factor

Case R1 Earth Resistance R; e = 1.0

For = 30 and = 1.25: tan design = tan 30 / 1.25 and design = 24.8 Hence:

Ka design = (1 sin 24.8) / (1 + sin 24.8) = 0.5805 / 1.4195 = 0.409

Kp design = (1 + sin 24.8) / (1 - sin 24.8) = 1.4195 / 0.5805 = 2.445 For bulk density of retained fill = 18 kN/m3 design = 18 / 1.0 = 18 kN/m

3 For bulk saturated bulk density of fill = 20 kN/m3 design = 20 / 1.0 = 20 kN/m

3 Effective unit weight of soil below water table = (20 9.8) = 10.2 kN/m3 So, as = 1.0, again for this value: design = 10.2 kN/m

3 The partial load factor for the action due to the active earth pressure behind the wall, G, dst = 1.0, will be applied to the force due to the active earth pressure behind the wall. The partial factor for resistance R; e = 1.0, which will be applied to the force due to the passive earth pressure in front of the wall. The calculations are then as follows:

Calculations all per metre width

Force (kN) Lever Arm (m) Moment (kNm)

(1) 1.0 x 0.5 x 0.409 x 18 x 4.02 = 58.896 d + 4.0/3 58.896 d + 78.528

(2) 1.0 x 0.409 x 18 x 4.0 x d = 29.448 d d/2 14.724 d2

(3) 1.0 x 0.5 x 0.409 x 10.2 x d2 = 2.086 d2 d/3 0.695 d3

(4) 1.0 x (0.5 x 2.445 x 18 x 0.72) = 10.78 d + 0.7/3 10.78d 2.515

(5) 1.0 x (2.445 x 18 x 0.7 x d) = 30.807 d d/2 15.404 d2

(6) 1.0 x (0.5 x 2.445 x 10.2 x d2) = 12.47 d2 d/3 4.16 d3

Which gives: 58.896 d + 78.528 + 14.724 d2 + 0.695 d3 = 10.78d + 2.515 + 15.404 d2 + 4.16 d3

Which simplifies to: 76.013 + 48.116 d 0.68 d2 3.465 d3 = 0, from which d = 4.27 m

The pile penetration is then increased by 20 % to ensure stability Actual pile penetration is taken as (d + 0.7) metres = 4.97 m Required penetration length calculated from actual pile length but increased by 20% = 1.2 x 4.97 = 5.96 m Overall pile length required = 3.0 + 5.96 = 8.96 metres Hence the design is governed by Load Combination 2 and the overall required pile length will be 8.96 metres