The top frame is assumed to be 1m below the ground surface. We will assume in this example that the lower frame is positioned 5.5m below the ground surface. This assumption is based on experience to provide clearance for construction of the base slab and wall kicker.
The excavation depth required to install the lower frame, providing sufficient working space, is therefore 6.1m.
Ppat 6.1m below ground level in sand and gravel
= 6.493 x 0.00 = 0kN/m2
Ppat 11m below ground level in sand and gravel
= (6.493 x 52.87) + 48.07 = 391.35 kN/m2 Ppat 11m below ground level in firm clay
= 1.00 x 100.94 + (2.45 x 65/1.5) = 207.11 kN/m2 Ppat 16m below ground level in firm clay
= 1.00 x 193.94 + (2.45 x 65/1.5) = 300.11 kN/m2 PRESSURE DIAGRAM Stage 1 kN/m2
121.65 58.01
41.97 3.17
8.76 24.07
9.73
73.37 63.21 Fig 7.9.2
Cofferdams
Chapter 7/11 The pressure diagram for this condition is given below:
PRESSURE DIAGRAM Fixed Earth Stage 2 kN/m2 4.252m
300.11 201.97
207.11 391.35
6.10m 4.523m 1.20m
Y
0.935m
74.68 Y Zero Shear
1.00m 1.20m
8.76
108.97 123.39
63.21 73.37
9.73 24.07
3.17
3.70m 46.25
Fig 7.9.3
Fixed Earth Support Option
At this stage it may be appropriate to consider a fixed earth support condition as the pile length needed in the final stage may well be long enough to provide fixity at the toe for this lesser excavation depth. Since the simplified method assumes that the point of contraflexure in the bending moment diagram occurs at the level where the active pressure equals the passive pressure the frame load can be calculated by taking moments about this level (Y-Y).
Let Y-Y be y below –6.10m level where Pa = Pp Then 63.21 +60.18
4.9 x y =391.35 4.9 x y hence y = 63.21 x 4.9/331.17 = 0.935m Take moments about Y-Y
Force Moment abt Y-Y
kN/m kNm/m
3.17 x 1.200 x1/2 = 1.90 x 6.635 = 12.62 8.76 x 1.200 x1/2 = 5.26 x 6.235 = 32.77 8.76 x 1.200 x1/2 = 5.26 x 5.435 = 28.57 24.07 x 1.200 x1/2 = 14.44 x 5.035 = 72.72 9.73 x 3.700 x1/2 = 18.00 x 3.402 = 61.24 73.37 x 3.700 x1/2 = 135.73 x 2.168 = 294.27 63.21 x 0.935 x1/2 = 29.55 x 0.623 = 18.41 74.68 x 0.935 x1/2 = 34.91 x 0.312 = 10.89 -74.68 x 0.935 x1/2 = -34.91 x 0.312 = -10.89
210.14 520.60
PilingHandbook, 8th edition (revised 2008)
Cofferdams
Chapter 7/12
Then frame load = 520.60 = 86.26kN/m 6.035
The length of pile is found by taking moments about an assumed toe level such that the moments of all the forces are in equilibrium From the calculation above
total active force above –6.10m = 180.59 kN/m total active moment above –6.10m = 502.19 kNm/m This force acts at502.19= 2.781m above Y-Y
180.59 or 1.846m above –6.10m
Assume the pile penetration below –6.10m is d Then taking moments about the toe
clockwise moments =
180.59 x (1.846+d) +63.21x2d2+ (63.21+60.18xd) xd2
2 3 4.9 6
anticlockwise moment =391.35xd3+ 86.26 x (5.10+d)
4.9 6
Equating clockwise moments to anticlockwise moments produces a cubic equation which is solved by successive approximations giving d= 4.252m.
To correct the error caused by the use of the simplified method the depth below the point of contraflexure is increased by 20% to give the pile penetration.
Hence the required pile length
= 6.10 + 0.935 + 1.2 x (4.252 – 0.935) = 11.015m say 11.02m Zero shear occurs at 4.523m below ground level. (Where the area of the active pressure diagram above the level equals the top frame load).
Take moments about and above the level of zero shear:
kNm/m
3.17 x 1.200 x1/2 x 4.123 = 7.84
8.76 x 1.200 x1/2 x 3.723 = 19.57
8.76 x 1.200 x1/2 x 2.923 = 15.36
24.07 x 1.200 x1/2 x 2.523 = 36.44
9.73 x 2.123 x1/2 x 1.415 = 14.61
46.25 x 2.123 x1/2 x 0.708 = 34.76
-86.26 x 3.523 = -303.89
-175.31
Maximum bending moment with fixed earth support = 175.3 kNm/m
Cofferdams
Chapter 7/13 Free Earth Support Option
However if the pile length for the final stage is shorter than 11.02m, the design for this intermediate stage should be considered as free earth. The passive pressures are as before and the pressure diagram is shown below:
The length of pile required to provide stability (i.e. Active moment
= Passive moment) for rotation about the top frame is found by trial and error to be 8.788m
96.22
PRESSURE DIAGRAM Free Earth Stage 2 kN/m2 2.688m
300.11 201.97
207.11 391.35
6.10m 4.926m 1.20m
Zero Shear 1.00m 1.20m
8.76
108.97
201.97
123.39 63.21 73.37
9.73 24.07
3.17
3.70m
214.68 53.18
Fig 7.9.4
Taking moments of the active pressures about the top frame:
Active Force Active Moment
kN/m kNm/m
3.17 x 1.200 x1/2 = 1.90 x -0.600 = -1.14 8.76 x 1.200 x1/2 = 5.26 x -0.200 = -1.05 8.76 x 1.200 x1/2 = 5.26 x +0.600 = 3.15 24.07 x 1.200 x1/2 = 14.44 x +1.000 = 14.44 9.73 x 3.700 x1/2 = 18.00 x +2.633 = 47.40 73.37 x 3.700 x1/2 = 135.73 x +3.867 = 524.89 63.21 x 2.688 x1/2 = 84.95 x +5.996 = 509.39 96.22 x 2.688 x1/2 = 129.32 x +6.892 = 891.27
394.86 1988.35
PilingHandbook, 8th edition (revised 2008)
Cofferdams
Chapter 7/14
Taking moments of the passive pressure about the top frame:
Passive force = 214.68 x 2.688 x1/2 = 288.53kN/m Passive moment = 288.53 x 6.892 = 1988.55 kNm/m [equal to Active Moment therefore OK]
Top frame load = 394.86 – 288.53 = 106.33 kN/m
Zero shear occurs at 4.926m below ground level. (Where the area of the active pressure diagram above the level equals the top frame load).
Take moments about and above the level of zero shear:
kNm/m
3.17 x 1.200 x1/2x 4.526 = 8.61
8.76 x 1.200 x1/2x 4.126 = 21.69
8.76 x 1.200 x1/2x 3.326 = 17.48
24.07 x 1.200 x1/2x 2.926 = 42.26
9.73 x 2.526 x1/2x 1.684 = 20.69
53.18 x 2.526 x1/2x 0.842 = 56.55
-106.33 x 3.926 = -417.45
-250.17 Maximum bending moment with free earth
support = 250.2 kNm/m
Final Stage : Excavate to Formation Level
With the lower frame installed at 5.5m below ground level excavation is carried out to final level. The design is to include for 0.20m of unplanned excavation so the passive pressures are calculated for an excavation depth of 8.10m
Ppat 8.1m below ground level in sand and gravel
= 6.493 x 0.00 = 0 kN/m2
Ppat 11m below ground level in sand and gravel
= (6.493 x 31.29) + 28.45 = 231.62 kN/m2 Ppat 11m below ground level in firm clay
= 1.00 x 59.74 + (2.45 x 65/1.5) = 165.91 kN/m2 Ppat 16m below ground level in firm clay
= 1.00 x 152.74 + (2.45 x 65/1.5) = 258.91 kN/m2
Cofferdams
Chapter 7/15
PRESSURE DIAGRAM Final stage kN/m2 4.90m
11.016m
7.743m 8.10m
166.21 Zero Shear
1.20m 1.00m
1.20m
3.70m
5.50m Zero Shear
3.661m
201.97
108.97 123.39
24.07
63.21 73.37
9.73 8.76
3.17
258.91
165.91 231.62
109.27 83.39
31.42
63.05
The pressure diagram for this condition is given below:
Fig 7.9.5
The depth of cut off is found by considering the piles to be simply supported at the bottom frame position due to the assumption of a hinge at the support position.
Consider first the lower span. The pile length for stability is found, by trial and error, to be 11.016m.
Taking moments of the active pressures about the bottom frame:
Active Force Active Moment
kN/m kNm/m
63.05 x 0.600 x1/2 = 18.92 x 0.200 = 3.78 73.37 x 0.600 x1/2 = 22.01 x 0.400 = 8.80 63.21 x 4.900 x1/2 = 154.86 x 2.233 = 345.81 123.39x 4.900 x1/2 = 302.31 x 3.867 = 1169.02 108.97x 0.016 x1/2 = 0.87 x 5.505 = 4.80 109.27x 0.016 x1/2 = 0.87 x 5.511 = 4.82
499.84 1537.03
PilingHandbook, 8th edition (revised 2008)
Cofferdams
Chapter 7/16
Taking moments of the passive pressure about the bottom frame:
Passive Force Active Moment
kN/m kNm/m
231.62x 2.900 x1/2 = 335.85 x 4.533 = 1522.40 165.91x 0.016 x1/2 = 1.33 x 5.505 = 7.31 166.17x 0.016 x1/2 = 1.33 x 5.511 = 7.33
338.51 1537.04
(Active Moment and Passive Moment are equal therefore OK) For the lower span bottom frame load
= 499.84 – 338.51 = 161.33kN/m
Zero shear occurs at 7.743m below ground level. (This is where the area of the active pressure diagram below the bottom frame and above the zero shear level equals the bottom frame load calculated above).
Take moments about and above the level of zero shear (for the lower span):
kNm/m
83.39 x 1.643 x1/2x 0.548 = 37.54
63.21 x 1.643 x1/2x 1.095 = 56.86
73.37 x 0.600 x1/2x 1.843 = 40.57
63.05 x 0.600 x1/2x 2.043 = 38.64
-161.33 x 2.243 = -361.86
-188.25 Now consider the upper span
Taking moments of the active pressures about the bottom frame:
Active Force Active Moment
kN/m kNm/m
3.17 x 1.200 x1/2 = 1.90 x 5.100 = 9.70
8.76 x 1.200 x1/2 = 5.26 x 4.700 = 24.70
8.76 x 1.200 x1/2 = 5.26 x 3.900 = 20.50
24.07 x 1.200 x1/2 = 14.44 x 3.500 = 50.55
9.73 x 3.100 x1/2 = 15.08 x 2.067 = 31.17
63.05 x 3.100 x1/2 = 97.73 x 1.033 = 100.95
139.67 237.57
Then top frame load = 237.57/4.500 = 52.79 kN/m And hence total load in bottom frame
= (139.67 – 52.79) + 161.33 = 248.21 kN/m Zero shear occurs at 3.661m below ground level.
Cofferdams
Chapter 7/17 Take moments about and below the level of zero shear:
kNm/m
31.42 x 1.839 x1/2x 0.613 = 17.71
63.05 x 1.839 x1/2x 1.226 = 71.08
-86.88 x 1.839 = -159.77
-70.98