Question

3) At a construction site, a site investigation has been carried out. At ground level (-1 m NAP) a clay layer with a thickness of 4 m is found (y = 16 kN/m”, k = 2.0x10-12 m/s, Cp = 15), followed by a sand layer of 6 m thickness (y = 20 kN/m², k = 5.0x10-7 m/s), another clay layer of 4m thickness (y = 18 kN/m3, k = 4.5x10-11 m/s, Cp = 25), and finally another sand layer (y = 21 kN/m², k = 7.0x10-8 m/s). The upper clay layer is saturated from the ground level downwards, but the water level in a standpipe in this layer was at 2 m depth (-3 m NAP). A standpipe in the upper sand layer had a water level at -3 m NAP, and another in the lower sand layer had a water level at 1 m NAP. a. Calculate the total stresses, the pore pressures and the effective stresses with depth. [10 marks] b. Determine the flow rate between the two sandy layers. [5 marks] A 2m deep excavation is undertaken, followed by the construction of a large building placing a load of 200 kPa at the excavation level. c. Determine the settlement at the excavation level at the end of consolidation of the clay layers, splitting each clay layer into two equally thick sub-layers. [10 marks]

Answer 1

solution:- the values given in the question are as follows:

thickness of first clay layer(first layer)(H1)=4 m

saturated unit weight of fierst caly layer(first layer)($\gamma$1)=16 kN/m^3

permeability of first caly layer(first layer)(k1)=2.0*10^-12 m/s

Cp1=15

thickness of first sand layer(second layer)(H2)=6 m

saturated unit weight of fierst sand layer(second layer)($\gamma$2)=20 kN/m^3

permeability of first sand layer(second layer)(k2)=5.0*10^-7 m/s

thickness of second clay layer(H3)=4 m

saturated unit weight of second caly layer(third layer)($\gamma$3)=18 kN/m^3

permeability of second caly layer(third layer)(k3)=4.5*10^-11 m/s

Cp2=25

thickness of second sand layer(fourth layer)(H4)=6 m

saturated unit weight of fierst sand layer($\gamma$4)=21 kN/m^3

permeability of second sand layer(fourth layer)(k4)=7.0*10^-8 m/s

Ground level 1 m H1=4 m clay laye Y1=16 kN/m^ k1=2.0*10^-12 m/s and Cp1=15 B sand layer Y2=20 kN/m^3 k2=4.5*10^-7 m/s H2=6 m C H3=4 m sand layer Y3=20 kN/m^3 k3=4.5*10^-7 m/s and Cp2=25 D sand layer Y4=21 kN/m^3 k4=4.5*10^-8 m/s H4=6 m E

(a)

total stress($\sigma$)=unit weight of soil($\gamma$)*depth of soil(H)

pore water pressure(U)=unit weight of water($\gamma$w)*depth of water(h)

effective stress($\bar{\sigma }$)=total stress($\sigma$)-pore water stress(U)

total stress at point A($\sigma$A)=unit weight of soil($\gamma$)*depth of soil(H)

total stress at point A($\sigma$A)=16*0=0

total stress at point A($\sigma$A)=0 kN/m^2

pore water pressure at point A(UA)=unit weight of water($\gamma$w)*depth of water(h)

pore water pressure at point A(UA)=9.81*0

pore water pressure at point A(UA)=0 kN/m^2

effective stress at point A($\bar{\sigma }$A)=total stress at point A($\sigma$)-pore water stress at point A(UA)

effective stress at point A($\bar{\sigma }$A)=0-0

effective stress at point A($\bar{\sigma }$A)=0 kN/m^2

total stress at point B($\sigma$B)=unit weight of soil layer fist($\gamma$1)*depth of soil layer fist(H1)

total stress at point A($\sigma$B)=16*4

total stress at point B($\sigma$B)=64 kN/m^2

pore water pressure at point B(UB)=unit weight of water($\gamma$w)*depth of water(h)

pore water pressure at point B(UB)=9.81*2

pore water pressure at point B(UB)=19.62 kN/m^2

effective stress at point B($\bar{\sigma }$B)=total stress at point B($\sigma$B)-pore water stress at point B(UB)

effective stress at point B($\bar{\sigma }$B)=64-19.62

effective stress at point B($\bar{\sigma }$B)=44.38 kN/m^2

total stress at point C($\sigma$C)=unit weight of soil layer fist($\gamma$1)*depth of soil layer fist(H1)+unit weight of soil layer second($\gamma$2)*depth of soil layer second(H2)

total stress at point C($\sigma$C)=16*4+20*6

total stress at point C($\sigma$C)=184 kN/m^2

pore water pressure at point C(UC)=unit weight of water($\gamma$w)*depth of water(h)

pore water pressure at point C(UC)=9.81*8

pore water pressure at point C(UC)=78.48 kN/m^2

effective stress at point C($\bar{\sigma }$C)=total stress at point C($\sigma$C)-pore water stress at point C(UC)

effective stress at point C($\bar{\sigma }$C)=184-78.48

effective stress at point C($\bar{\sigma }$C)=105.52 kN/m^2

total stress at point D($\sigma$D)=unit weight of soil layer fist($\gamma$1)*depth of soil layer fist(H1)+unit weight of soil layer second($\gamma$2)*depth of soil layer second(H2)+unit weight of soil layer third($\gamma$3)*depth of soil layer third(H3)

total stress at point D($\sigma$D)=16*4+20*6+18*4

total stress at point D($\sigma$D)=256 kN/m^2

pore water pressure at point D(UD)=unit weight of water($\gamma$w)*depth of water(h)

pore water pressure at point D(UD)=9.81*12

pore water pressure at point D(UD)=117.72 kN/m^2

effective stress at point D($\bar{\sigma }$D)=total stress at point D($\sigma$D)-pore water stress at point D(UD)

effective stress at point D($\bar{\sigma }$D)=256-117.72

effective stress at point D($\bar{\sigma }$D)=138.28 kN/m^2

total stress at point E($\sigma$E)=unit weight of soil layer fist($\gamma$1)*depth of soil layer fist(H1)+unit weight of soil layer second($\gamma$2)*depth of soil layer second(H2)+unit weight of soil layer third($\gamma$3)*depth of soil layer third(H3)+unit weight of soil fourth layer($\gamma$4)*depth of soil fourth layer(H4)

total stress at point E($\sigma$E)=16*4+20*6+18*4+21*6

total stress at point E($\sigma$E)=382 kN/m^2

pore water pressure at point E(UE)=unit weight of water($\gamma$w)*depth of water(h)

pore water pressure at point E(UE)=9.81*18

pore water pressure at point E(UE)=176.58 kN/m^2

effective stress at point E($\bar{\sigma }$E)=total stress at point E($\sigma$E)-pore water stress at point E(UE)

effective stress at point E($\bar{\sigma }$E)=382-176.58

effective stress at point E($\bar{\sigma }$E)=205.42 kN/m^2

(b)

let discharge between first sand layer is Q1 and discharge between second sand layer is Q2

discharge per(Q)=k*i*A m^3/s

where, i=head loss(H_L)/thickness of soil layer(H)

discharge per unit area(Q)=k*i m^3/s/m^2

discharge per unit area between fist sand layer(Q1)=k2*i2

k2=5.0*10^-7 m/s

i2=(water heat at C-water head at B)/thickness of fist sand layer

i2=(8-2)/6=1

discharge per unit area between fist sand layer(Q1)=k2*i2

discharge per unit area between fist sand layer(Q1)=5.0*10^-7*1

discharge per unit area between fist sand layer(Q1)=5.0*10^-7 m^3/s/m^2

discharge per unit area between second sand layer(Q2)=k4*i4

k2=7.0*10^-8 m/s

i2=(water heat at E-water head at D)/thickness of fist sand layer

i2=(18-12)/6=6/6=1

discharge per unit area between second sand layer(Q2)=k4*i4

discharge per unit area between second sand layer(Q1)=8.0*10^-8*1

discharge per unit area between second sand layer(Q2)=7.0*10^-8 m^3/s/m^2

total discharge between both sand layer(Q)=Q1+Q2

total discharge between both sand layer(Q)=5.0*10^-7+7.0*10^8

total discharge between both sand layer(Q)=5.8*10^-7 m^3/s/m^2

(c)

let let settlement at the end of claye layer(end of excavation) is $\Delta$

load(P)=200 kPa

pressure per unit area($\Delta \sigma$)=200 kPa

depth of excavation(H)=2 m

initial effective stress at excavation level(
00
)=total stress at excavation level($\sigma$)-pore water stress at excavation level(U)

total stress at excavation level($\sigma$)=2*16=32 KPa

pore water stress at excavation level(U)=9.81*0=0

pore water stress at excavation level(U)=0

initial effective stress at excavation level(
00
)=32 kPa

settlement at the end of claye layer(end of excavation) ($\Delta$)=(H0/C)*log₁₀{(
00
+$\Delta \sigma$)/$\Delta \sigma$} , [Eq-1]

where, Ho=depth of excavation=2

C=15

values put in above equation-(1) and calculate settlement at excavation level

settlement at the end of claye layer(end of excavation) ($\Delta$)=(2/15)*log₁₀{(32+200)/200}

settlement at the end of claye layer(end of excavation) ($\Delta$)=0.008594398 m

settlement at the end of claye layer(level of excavation) ($\Delta$)=8.594398 mm

settlement at the end of claye layer(level of excavation) ($\Delta$)=8.5944 mm

[Ans]