Cantilever sheet pile wall

Topics

• Cantilever sheet pile wall in cohesionless soil

• Cantilever sheet pile wall in cohesive soil

Cantilever sheet pile wall in cohesionless soil

Design length of sheet pile

Theory:

Calculating active earth pressure

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

s_a=qK_a+2CÖK_a, s_p=qK_p+2CÖK_p

Where C is cohesion of soil and q is surcharge and

K_a=tan²(45-f/2), K_p= 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

p_a = g h K_a+ q K_a,

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

The lateral forces Ha1 is calculated as

H_a1=g K_a h²/2+q K_a 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 = p_a / g (K_p-K_a)

Where K_p 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 (K_p-K_a)

The lateral forces Ha2 can be calculated as

H_a2=p_a*a/2

Derive equation for depth Z from åFx = 0

Summarize lateral forces, we have

åF_x = H_a1+ H_a2-H_p1+H_p2=0

From the diagram, we recognize that lateral force H_p1 is area CDE and H_p2 is area DOG.  There is a common area DEFO between two areas, and

H_p1-H_p2 = 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

H_a1+ H_a2 – 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 (K_p-K_a), p₁ = g (K_p-K_a)*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 is g(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 K_p (h+a+Y).  Therefore,

p₂ = g K_p (h+a+Y) - g K_a (a+Y)

If there a surcharge, p₂ = g K_p (h+a+Y)+q K_p - g K_a (a+Y)

Derive equation for Y from åM_o = 0

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

åM_o = H_a1*(h/3+a+Y)+ H_a2*(2a/3+Y) – HCFO*Y/3+HEFG*Z/3 = 0

Or

H_a1*(h/3+a+Y)+ H_a2*(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 = H_a1+H_a2

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

(H_a1+H_a2) = g (Kp-Ka) y²/2.  Therefore,

y = {2*(H_a1+H_a2)/[g(Kp-Ka)]}^1/2

The maximum moment is

M_max = H_a1*(h/3+a+y)+ H_a2*(2a/3+y)-g (Kp-Ka)*y³/6

The required section modulus is S = M_max / F_b, F_b 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, p_a and H_a1.

p_a = g K_a h, H_a1=p_a*h/2

2. Calculate the length a, and H_a2.

a = p_a / g (K_p-K_a), H_a2=p_a*a/2

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

p₁ = g (K_p-K_a)*Y,

p₂ = g K_p (h+a+Y) - g K_a (a+Y)

4. Calculate depth Z.

5. Let R = H_a1*(h/3+a+Y)+ H_a2*(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*D

7. Calculate y = {2*(H_a1+H_a2)/[g(K_p-K_a)]}^1/2.

8. Calculate M_max = H_a1*(h/3+a+y)+ H_a2*(2a/3+y)-g (K_p-K_a)*y³/6

9. Calculate required section modulus S= M_max/F_b.

10. 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, F_b = 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:

K_a = tan² (45-f/2) = 0.333

K_p = tan² (45+f/2) = 3

The lateral earth pressure at bottom of excavation is

p_a = K_a g h = 0.333*115*10 = 383.33 psf

The active lateral force above excavation

H_a1 = p_a*h/2 = 383.33*10/2 = 1917 lb/ft

The depth a = p_a / g (K_p-K_a) = 383.3 / [115*(3-0.333)] =1.25 ft

The corresponding lateral force

H_a2 = p_a*a/2 = 383.33*1.25/2 = 238.6 lb/ft

Assume Y = 8.79 ft

p₁ = g (K_p-K_a)*Y = 115*(3-0.333)*8.79 = 2696 psf

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

The depth

Z = [p₁*Y-2*(H_a1+H_a2)]/(p₁+p₂) = [2696*8.79-2*(1917+238.6)]/(2696+6529) = 2.1 ft

The value

R = H_a1*(h/3+a+Y)+ H_a2*(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*(H_a1+H_a2)/[g(K_p-K_a)]}^1/2

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

M_max = H_a1*(h/3+a+y)+ H_a2*(2a/3+y)-g (K_p-K_a)*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

F_b=32 ksi

Required section modulus

S = M_max/F_b = 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 cohesive soil

Determine 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 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

s_a=qK_a-2CÖK_a, s_p=qK_p+2CÖK_p

Where C is cohesion of soil and q is surcharge and

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

When friction angle, f = 0, K_a = K_p = 1, and s_a=q-2C and s_p=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, p_a = 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, s_a=g *d-2C = 0, and d = 2C/g is the free-standing height of soil.  The resultant force H_a=p_a*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 s_a=gh-2C.  The passive pressure is s_p= 2C, since q = 0 Therefore, the net pressure is

p₁= s_p-s_a= 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 s_a=gD-2C, and the passive pressure is s_p=g(h+D)-2C.  Therefore, the net pressure is

p₂= s_p-s_a= gD+2C-[g(h+D)-2C] = 4C+gh

Derive equation for depth z from åFx = 0

Summarize horizontal forces, we have

åF_x = H_a – H_p1 + H_p2 = 0

Where H_a = p_a (h-d)/2, and H_p1 - H_p2 = HBCFO + HEFG

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

H_a – p₁*D +4C*Z= 0

Then,

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

Derive equation for embed depth D from åM_o = 0

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

åM_o = H_a*[(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,

H_a – p₁*y = 0, therefore, y = H_a/p₁

The maximum moment,

M_max=H_a*[(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 p_a=g(h-d)

3. Calculate H_a=p_a*h/2

4. Calculate p₁=4C-gh

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

6. Calculate R=H_a[(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 = H_a/ p₁.

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

11. Calculate required section modulus S= _Mmax/F_b.

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, F_b = 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, p_a = g(h-d)=115*(10-8.7)=150 psf

Total active force, H_a=p_a*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-H_a)/(4C)= (850*2.77-750)/(4*500) = 0.624 ft

R=H_a[(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 = H_a/ p₁=750/850 = 0.882 ft

The maximum moment,

M_max=H_a[(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= M_max/F_b=0.657*12/32=0.25 in³/ft

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

Topics

• Cantilever sheet pile wall in cohesionless soil

• Cantilever sheet pile wall in cohesive soil