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Cantilever sheet pile wall design

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Cantilever sheet pile wall

TopicsCantilever sheet pile wall in cohesionless soilCantilever sheet pile wall in cohesive soil

Cantilever sheet pile wall in cohesionless soilDesign length of sheet pile

Theory:

Calculating active earth pressure

The active and passive lateral earth pressure of soil can be written assa=qKa+2CÖKa, sp=qKp+2CÖKpWhere C is cohesion of soil and q is surcharge andKa=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 aspa = g h Ka+ q Ka,Where, g is unit weight of soil, h is the height of excavation.The lateral forces Ha1 is calculated asHa1=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 excavationside. Since passive pressure is larger than active pressure, the earth pressure on the earth side decreases. At a depth “a” below thebottom of excavation, the earth pressure is zero. The depth a can be calculated asa = 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

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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 asHa2=pa*a/2

Derive equation for depth Z from åFx = 0

Summarize lateral forces, we haveåFx = 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 betweentwo 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 asHa1+ Ha2 – p1*Y/2+ (p1+p2)*Z/2 = 0Solving 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 lineCEF 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 andoverburden pressure. When the sheet pile rotates, there are active pressure on the excavation side and passive pressure on the earthside at the bottom of sheet pile. The overburden pressure from bottom of excavation isg(a+Y), the active pressure is g Ka (a+Y). Theoverburden pressure from the top to the bottom of sheet pile on the earth side is g(h+a+Y), the passive pressureis 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 from åMo = 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 = 0OrHa1*(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 thelength 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 lateralearth pressure change from active to passive.Vmax = Ha1+Ha2Maximum 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 isMmax = Ha1*(h/3+a+y)+ Ha2*(2a/3+y)-g (Kp-Ka)*y3/6The required section modulus is S = Mmax / Fb, Fb is allowable stress of sheet pile.The sheet pile section is selected based on section modulusDesign Procedure

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



pa = g Ka h, Ha1=pa*h/22. Calculate the length a, and Ha2.

a = pa / g (Kp-Ka), Ha2=pa*a/23. Assume a trial depth Y, calculate p1and p2.

p1 = g (Kp-Ka)*Y,p2 = g Kp (h+a+Y) - g Ka (a+Y)

4. Calculate depth Z.

5. 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.

6. Calculate the length of sheet pile, L = h+1.2*D7. Calculate y = {2*(Ha1+Ha2)/[g(Kp-Ka)]}1/2.8. Calculate Mmax = Ha1*(h/3+a+y)+ Ha2*(2a/3+y)-g (Kp-Ka)*y3/69. Calculate required section modulus S= Mmax/Fb.

10. Select sheet pile section.

Example 1: Design cantilever sheet pile in cohesionless soil.

Given:Depth of excavation, h = 10 ftUnit weight of soil, g = 115 lb/ft3

Internal friction angle, f = 30 degreeAllowable 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) = 0.333Kp = tan2 (45+f/2) = 3The lateral earth pressure at bottom of excavation ispa = Ka g h = 0.333*115*10 = 383.33 psfThe active lateral force above excavationHa1 = pa*h/2 = 383.33*10/2 = 1917 lb/ftThe depth a = pa / g (Kp-Ka) = 383.3 / [115*(3-0.333)] =1.25 ftThe corresponding lateral forceHa2 = pa*a/2 = 383.33*1.25/2 = 238.6 lb/ftAssume Y = 8.79 ftp1 = g (Kp-Ka)*Y = 115*(3-0.333)*8.79 = 2696 psfp2 =g Kp (h+a+Y)-g Ka(a+Y)=115*3*(10+1.25+8.79)-115*0.333*(1.25+8.79)= 6529 psfThe depthZ = [p1*Y-2*(Ha1+Ha2)]/(p1+p2) = [2696*8.79-2*(1917+238.6)]/(2696+6529) = 2.1 ftThe valueR = 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 zeroThe embedded depth D = 1.25 + 8.79 = 10.04 ftThe 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 ftMmax = 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/ftAllowable bending stressFb=32 ksiRequired section modulusS = 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 atvarious depthDesign 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 4.21Value R (lb) 12.69 3.59 -15.11 -45.85 -91.10 -153.30Is assumed Y O.K. Y Y Y Y Y YRequired embeded depth, D (ft) 10.04 12.05 14.06 16.07 18.08 20.09Total 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 soilDetermine length of sheet piles for stability

Theory:

For cohesive soil, friction angle, f = 0, the sheet pile is supported by soil cohesion, C. Because cohesion, the soil can stands by itselfat 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 assa=qKa-2CÖKa, sp=qKp+2CÖKpWhere C is cohesion of soil and q is surcharge andKa=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+2CIf 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. Thepassive pressure is sp= 2C, since q = 0 Therefore, the net pressure isp1= 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 earthside. The active pressure is sa=gD-2C, and the passive pressure is sp=g(h+D)-2C. Therefore, the net pressure isp2= sp-sa= gD+2C-[g(h+D)-2C] = 4C+gh

Derive equation for depth z from åFx = 0

Summarize horizontal forces, we haveåFx = Ha – Hp1 + Hp2 = 0Where Ha = pa (h-d)/2, and Hp1 - Hp2 = HBCFO + HEFGSince HBCFO = p1*D, and HEFG = (p1+p2)*Z/2=8C*Z/2 =4C*ZHa – p1*D +4C*Z= 0Then,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 haveåMo = Ha*[(h-d)/3+D]- p1*D2/2+4C*Z2/3 = 0Structural 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 ybelow the bottom of excavation where shear equal to zero. Then,Ha – p1*y = 0, therefore, y = Ha/p1The maximum moment,Mmax=Ha*[(h-d)/3+y]- p1*y2/2The sheet pile section can be selected based on maximum moment and shear.Design procedure:

1. Calculate free standing height, d = 2C/g2. Calculate pa=g(h-d)3. Calculate Ha=pa*h/24. Calculate p1=4C-gh5. Assume a trial depth, D, Calculate Z=(p1*D-Ha)/(4C)6. Calculate R=Ha[(h-d)/3+D]- p1*D2/2+4CZ2/37. If R is not close to zero, assume a new D, repeat steps 5 and 68. 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*y2/211. 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 ftUnit weight of soil, g = 115 lb/ft3Cohesion of soil, C = 500 psfInternal friction angle, f = 0 degree



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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 ftThe lateral pressure at bottom of sheet pile, pa = g(h-d)=115*(10-8.7)=150 psfTotal active force, Ha=pa*h/2 = 150*10/2 = 750 lb/ftAssume D = 2.35 ft, p1=4C-gh=4*500-115*10 = 850 psfThe depth, Z=(p1*D-Ha)/(4C)= (850*2.77-750)/(4*500) = 0.624 ftR=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 zeroThe length of sheet pile, L = 10+1.3*2.35 = 13.1 ft Use 14 ftThe maximum moment occurs at y = Ha/ p1=750/850 = 0.882 ftThe 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/ftThe required section modulus, S= Mmax/Fb=0.657*12/32=0.25 in3/ftSelect sheet pile section, PS28, S = 1.9 in3/ftTopics

Cantilever sheet pile wall in cohesionless soilCantilever sheet pile wall in cohesive soil

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