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Cantilever sheet pile wall in cohesionless soil

Theory:

Calculating active earth pressure

The active and passive lateral earth pressure of soil can be written as

sa=qKa+2CÖKa, sp=qKp+2CÖKp

Where C is cohesion of soil and q is surcharge and

Ka=tan2(45-f/2), Kp= tan2(45+f/2) are active and passive lateral earth pressure, and f is internal friction angle.

In cohesionless soil, C is zero. The active pressure at bottom of excavation can be calculated as

pa = g h Ka+ q Ka,

Where, g is unit weight of soil, h is the height of excavation.

The lateral forces Ha1 is calculated as

Ha1=g Ka h2/2+q Ka h

Below the bottom of excavation, the sheet pile is subjected to active pressure on the earth side and passive pressure on the excavation side.  Since passive pressure is larger than active pressure, the earth pressure on the earth side decreases.  At a depth “a” below the bottom of excavation, the earth pressure is zero.  The depth a can be calculated as

a = pa / g (Kp-Ka)

Where Kp is passive earth pressure coefficient.  When the sheet pile rotates away from the earth side, there are active pressure on the earth side and passive pressure on the excavation side.  Therefore, the slope of BC is equal to g (Kp-Ka)

The lateral forces Ha2 can be calculated as

Ha2=pa*a/2

Derive equation for depth Z fromåFx= 0

Summarize lateral forces, we have

åFx = Ha1+ Ha2-Hp1+Hp2=0

From the diagram, we recognize that lateral force Hp1 is area CDE and Hp2 is area DOG.  There is a common area DEFO between two areas, and

Hp1-Hp2 = triangle CDE – triangle DOG = triangle CFO – triangle EFG = HCFO-HEFG

Where HCFO = p1*Y/2, and HEFG = (p1+p2)*Z/2

Therefore the equation can be written as

Ha1+ Ha2  p1*Y/2+ (p1+p2)*Z/2 = 0

Solving the equation for Z, we have

The pressure at bottom of sheet pile on the excavation side p1 can be determined from the slope of line CEF. Since the slope of line CEF is g (Kp-Ka), p1 = g (Kp-Ka)*Y

The pressure at the bottom of sheet pile on the earth side p2 can be determined from active and passive earth pressure coefficient and overburden pressure.  When the sheet pile rotates, there are active pressure on the excavation side and passive pressure on the earth side at the bottom of sheet pile.  The overburden pressure from bottom of excavation isg(a+Y), the active pressure is g Ka (a+Y).  The overburden pressure from the top to the bottom of sheet pile on the earth side is g(h+a+Y), the passive pressure is g Kp (h+a+Y). Therefore,

p2 = g Kp (h+a+Y) - g Ka (a+Y)

If there a surcharge, p2 = g Kp (h+a+Y)+q Kp - g Ka (a+Y)

Derive equation for Y fromåMo= 0

Both p1 and p2 are function of Y, to determine Y, we can take moment about bottom of sheet pile O.  We have

åMo = Ha1*(h/3+a+Y)+ Ha2*(2a/3+Y) – HCFO*Y/3+HEFG*Z/3 = 0

Or

Ha1*(h/3+a+Y)+ Ha2*(2a/3+Y) – p1*Y2/6+(p1+p2)*Z2/3 = 0

The depth Y can be determined from a trial and error process.

Calculating embed depth D

Once Y is determined, the minimum embedded depth D is equal to Y+a.  Usually a factor of safety of 1.2 is applied to D, and the length of sheet pile L is equal to h+D*FS.  FS is factor of safety from 1.2 to 1.4.

Selection of sheet pile section

The size of sheet pile is selected based on maximum moment and shear.  Maximum shear force is usually located at D where lateral earth pressure change from active to passive.

Vmax = Ha1+Ha2

Maximum moment locates at where shear stress equals to zero between C and D.

Assume that maximum moment located at a distance y below point C, then

(Ha1+Ha2) = g (Kp-Ka) y2/2.  Therefore,

y = {2*(Ha1+Ha2)/[g(Kp-Ka)]}1/2

The maximum moment is

Mmax = Ha1*(h/3+a+y)+ Ha2*(2a/3+y)-g (Kp-Ka)*y3/6

The required section modulus is S = Mmax / Fb, Fb is allowable stress of sheet pile.

The sheet pile section is selected based on section modulus

Design Procedure

1. Calculate lateral earth pressure at bottom of excavation, pa and Ha1.

pa = g Ka h, Ha1=pa*h/2

1. Calculate the length a, and Ha2.

a = pa / g (Kp-Ka), Ha2=pa*a/2

1. Assume a trial depth Y, calculate p1and p2.

p1 = g (Kp-Ka)*Y,

p2 = g Kp (h+a+Y) - g Ka (a+Y)

1. Calculate depth Z.

1. Let R = Ha1*(h/3+a+Y)+ Ha2*(2a/3+Y) – p1*Y2/6+(p1+p2)*Z2/3

Substitute Y and Z into R, if R = 0, the embedded depth, D = Y + a.

If not, assume a new Y, repeat step 3 to 5.

1. Calculate the length of sheet pile, L = h+1.2*D

2. Calculate y = {2*(Ha1+Ha2)/[g(Kp-Ka)]}1/2.

3. Calculate Mmax = Ha1*(h/3+a+y)+ Ha2*(2a/3+y)-g (Kp-Ka)*y3/6

4. Calculate required section modulus S= Mmax/Fb.

5. Select sheet pile section.

Example 1: Design cantilever sheet pile in cohesionless soil.

Given:

Depth of excavation, h = 10 ft

Unit weight of soil, g = 115 lb/ft3

Internal friction angle, f = 30 degree

Allowable design stress of sheet pile, Fb = 32 ksi

Requirement: Design length of a cantilever sheet pile and select sheet pile section

Solution:

Design length of sheet pile:

Calculate lateral earth pressure coefficients:

Ka = tan2 (45-f/2) = 0.333

Kp = tan2 (45+f/2) = 3

The lateral earth pressure at bottom of excavation is

pa = Ka g h = 0.333*115*10 = 383.33 psf

The active lateral force above excavation

Ha1 = pa*h/2 = 383.33*10/2 = 1917 lb/ft

The depth a = pa / g (Kp-Ka) = 383.3 / [115*(3-0.333)] =1.25 ft

The corresponding lateral force

Ha2 = pa*a/2 = 383.33*1.25/2 = 238.6 lb/ft

Assume Y = 8.79 ft

p1 = g (Kp-Ka)*Y = 115*(3-0.333)*8.79 = 2696 psf

p2 =g Kp (h+a+Y)-g Ka(a+Y)=115*3*(10+1.25+8.79)-115*0.333*(1.25+8.79)= 6529 psf

The depth

Z = [p1*Y-2*(Ha1+Ha2)]/(p1+p2) = [2696*8.79-2*(1917+238.6)]/(2696+6529) = 2.1 ft

The value

R = Ha1*(h/3+a+Y)+ Ha2*(2*a/3+Y)-p1*Y2/6+(p1+p2)*Z2/6

=1917*(10/3+1.25+8.79)+238.6*(2*1.25/3+8.79)–2696*8.792/6 + (2696+6529)*2.12/6

=12.9 lb    close to zero

The embedded depth D = 1.25 + 8.79 = 10.04 ft

The design length of sheet pile, L = 10 + 1.2*10.04 = 22.05 ft       Use 22 ft

Select sheet pile section:

y = {2*(Ha1+Ha2)/[g(Kp-Ka)]}1/2

={2*(1917+238.6)/[115*(3-0.333)]}1/2 = 3.75 ft

Mmax = Ha1*(h/3+a+y)+ Ha2*(2a/3+y)-g (Kp-Ka)*y3/6

=1917*(10/3+1.25+3.75)+238.6*(2*1.25/3+3.75)-115*(3-0.333)*3.753/6 = 14375 ft-lb/ft

Allowable bending stress

Fb=32 ksi

Required section modulus

S = Mmax/Fb = 11680*12/32000= 5.39 in3/ft

Select PMA22 section modulus per foot of wall, S = 5.4 in3/ft

 Cantilever sheet pile wall in cohesionless soil at various depth Design Data: Depth of excavation, h (ft) 10 12 14 16 18 20 Unit weight of soil, g (lb/ft^3) 115 115 115 115 115 115 Internal friction angle, F (degree) 30 30 30 30 30 30 Lateral earth pressure coefficient: Active earth pressure coefficient, Ka 0.333 0.333 0.333 0.333 0.333 0.333 Passive earth pressure coefficient, Kp 3.000 3.000 3.000 3.000 3.000 3.000 Earth pressure above excavation: Earth pressure at bottom of excavation, pa (psf) 383.34 460.00 536.67 613.34 690.00 766.67 Active lateral force above excavation, Ha1 (lb/ft) 1916.68 2760.02 3756.69 4906.69 6210.04 7666.71 Depth a (ft) 1.25 1.50 1.75 2.00 2.25 2.50 Lateral force Ha2 (lb/ft) 239.58 345.00 469.58 613.33 776.25 958.33 Determine embedment depth: Depth Y (ft) 8.79 10.55 12.31 14.07 15.83 17.59 Pressure p1 (psf) 2695.63 3235.37 3775.11 4314.85 4854.59 5394.33 Pressure p2 (psf) 6529.01 7835.42 9141.84 10448.25 11754.67 13061.08 Depth Z (ft) 2.10 2.52 2.94 3.36 3.79 4.21 Value R (lb) 12.69 3.59 -15.11 -45.85 -91.10 -153.30 Is assumed Y O.K. Y Y Y Y Y Y Required embeded depth, D (ft) 10.04 12.05 14.06 16.07 18.08 20.09 Total length of sheet pile, L (ft) 20.04 24.05 28.06 32.07 36.08 40.09 Total length of sheet pile, L (ft) with 1.2 SF 22.05 26.46 30.87 35.28 39.70 44.11

Cantilever sheet pile wall in cohesive soil

Theory:

For cohesive soil, friction angle, f = 0, the sheet pile is supported by soil cohesion, C.  Because cohesion, the soil can stands by itself at certain height without sheet pile.  Since f= 0, lateral earth pressure distributes uniformly below excavation.

Calculating active earth pressure

The active and passive lateral earth pressure of soil can be written as

sa=qKa-2CÖKa, sp=qKp+2CÖKp

Where C is cohesion of soil and q is surcharge and

Ka=tan2(45-f/2), Kp= tan2(45+f/2) are active and passive lateral earth pressure, and f is internal friction angle.

When friction angle, f = 0, Ka = Kp = 1, and sa=q-2C and sp=q+2C

If the unit weight of soil is g, the surcharge q at bottom of excavation on the earth side is g*h, then, the lateral earth pressure, pa = g h – 2C

The lateral pressure at top of excavation will be –2C.  At a distance, d, below the top of excavation, the lateral pressure, sa=g *d-2C = 0, and d = 2C/g is the free-standing height of soil.  The resultant force Ha=pa*h/2

Determine lateral earth pressure below excavation

Below the bottom of excavation, the sheet pile is subjected to both active and passive pressure.  The active pressure is sa=gh-2C.  The passive pressure is sp= 2C, since q = 0 Therefore, the net pressure is

p1= sp-sa= 2C-(gh-2C) = 4C-gh

At the bottom of sheet pile, the sheet pile is subjected to active pressure on the excavation side, and passive pressure on the earth side.  The active pressure is sa=gD-2C, and the passive pressure is sp=g(h+D)-2C.  Therefore, the net pressure is

p2= sp-sa= gD+2C-[g(h+D)-2C] = 4C+gh

Derive equation for depth z from åFx = 0

Summarize horizontal forces, we have

åFx = Ha – Hp1 + Hp2 = 0

Where Ha = pa (h-d)/2, and Hp1 - Hp2 = HBCFO + HEFG

Since HBCFO = p1*D, and HEFG = (p1+p2)*Z/2=8C*Z/2 =4C*Z

Ha  p1*D +4C*Z= 0

Then,

Z= (p1*D- Ha)/4C                (indicate revision)

Derive equation for embed depth D from åMo = 0

Taking moment about point O at bottom of sheet pile, we have

åMo = Ha*[(h-d)/3+D]- p1*D2/2+4C*Z2/3 = 0

Structural design

The maximum shear occurs at point B, at the bottom of excavation and or at point D. The maximum moment occurs at a distance y below the bottom of excavation where shear equal to zero. Then,

Ha  p1*y = 0, therefore, y = Ha/p1

The maximum moment,

Mmax=Ha*[(h-d)/3+y]- p1*y2/2

The sheet pile section can be selected based on maximum moment and shear.

Design procedure:

1. Calculate free standing height, d = 2C/g

2. Calculate pa=g(h-d)

3. Calculate Ha=pa*h/2

4. Calculate p1=4C-gh

5. Assume a trial depth, D, Calculate Z=(p1*D-Ha)/(4C)

6. Calculate R=Ha[(h-d)/3+D]- p1*D2/2+4CZ2/3

7. If R is not close to zero, assume a new D, repeat steps 5 and 6

8. The design length of sheet pile is L=h+D*FS, FS=1.2 to 1.4.

9. Calculate y = Ha/ p1.

10. Calculate Mmax=Ha[(h-d)/3+y]- p1*y2/2

11. Calculate required section modulus S= Mmax/Fb.

12. Select sheet pile section.

Example 2: Design Cantilever sheet pile in cohesive soil.

Given:

Depth of excavation, h = 10 ft

Unit weight of soil, g = 115 lb/ft3

Cohesion of soil, C = 500 psf

Internal friction angle, f = 0 degree

Allowable design stress of sheet pile, Fb = 32 ksi

Requirement: Design length of sheet pile and select sheet pile section

Solution:

Design length of sheet pile:

The free standing height, d = 2C/g = 2*500/115 = 8.7 ft

The lateral pressure at bottom of sheet pile, pa = g(h-d)=115*(10-8.7)=150 psf

Total active force, Ha=pa*h/2 = 150*10/2 = 750 lb/ft

Assume D = 2.35 ft, p1=4C-gh=4*500-115*10 = 850 psf

The depth, Z=(p1*D-Ha)/(4C)= (850*2.77-750)/(4*500) = 0.624 ft

R=Ha[(h-d)/3+D]- p1*D2/2+4CZ2/3

=750*[(10-8.7)/3+2.35]-850*2.352/2+2*500*0.6242/2 = 0.9      Close to zero

The length of sheet pile, L = 10+1.3*2.35 = 13.1 ft   Use 14 ft

The maximum moment occurs at y = Ha/ p1=750/850 = 0.882 ft

The maximum moment,

Mmax=Ha[(h-d)/3+y]- p1*y2/2 = 750*[(10-8.7)/3+0.882]-750*0.8822/2=0.657 kip-ft/ft

The required section modulus, S= Mmax/Fb=0.657*12/32=0.25 in3/ft

Select sheet pile section, PS28, S = 1.9 in3/ft