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## Design using free earth support method

Assumptions:

1. Sheet pile is rigid, and lateral deflection is small.
2. The lateral pressure distributes according to Rankine’s or Coulomb’s theories
3. The tie back is strong, and sheet pile rotate about the tie rod anchor point at failure.
4. Bottom of sheet pile is free to move. The embedded depth can be determined by summarizing horizontal earth pressures and moments about the anchor.

å Fx = 0                                                                      

åMo = 0                                                                      

The difficulty is that the lateral earth pressure is a function of embedded depth.  Both equations are highly nonlinear. A trial and error method has to be used to determine the root.

For structural design, the sheet pile needs to be able to withstand maximum moment and shear from lateral pressure. A structural analysis needs to be done to determine maximum moment and shear.

### Design length of sheet pile ### Theory

#### Calculating active earth pressure

The method for calculating active earth pressure is the same as that in cantilever sheet pile wall. The lateral forces Ha1 is calculated as

Ha1=g Ka h2/2+q Ka h

The depth a can be calculated as

a = pa / g (Kp-Ka)

The lateral forces Ha2 can be calculated as

Ha2=pa*a/2

#### Calculating passive earth pressure

The slope from point C to E in the figure above is g (Kp-Ka). The passive earth pressure at a depth Y below a is calculated as

Pp = g (Kp-Ka) Y

The passive lateral force

HCEF = g (Kp-Ka) Y2/2

Derive equation for Y from åMo = 0

åMo = Ha1*y1 + Ha2* y2 – HCEF* y3 = 0

Where

y1 = (2h/3-b)

y2 = (h+a/3-b)

y3 = (h+a+2Y/3)

The equation needs to be determined by a trial and error process.

Determine anchor force T from å Fx = 0

å Fx = Ha1+ Ha2– HCEF-T = 0

Then,

T = Ha1+ Ha2– HCEF

### Design size of sheet pile

The structural is the same as cantilever sheet piles in cohesionless soil.

Maximum moment locates at a distance y below T where shear stress equals to zero.

T-g Ka (y+b)2/2=0

Solve for y, we have, y = -b+Ö2*T/(g Ka)

The maximum moment is

Mmax = T y - g Ka (y+b)3/6

The required section modulus is S = Mmax / Fb

The sheet pile section is selected based on section modulus

### Design of tie rod and soldier beam The sheet pile design above is based on a unit width, foot or meter.  The tie back force T calculated from sheet pile design is force per linearly width of sheet pile.  The top of sheet pile often supported with soldier beams and tie rods at certain spacing.

Assume the spacing of tie rod is s, the tension in the rod is T times s.  The required area of tie rod is

A = T s / Ft

Where Ft is allowable tensile stress of steel and is equal to 0.6Fy in AISC ASD design.

The soil beam is designed as a continuous beam that subjected to tie back force T.  The maximum moment in the soldier beam is calculated from structural analysis.  The required section modulus is equal to S = Mmax / Fb.

### 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 HCEF.

HCEF = g (Kp-Ka) Y2/3

1. Let R = Ha1*y1 + Ha2* y2 – HCEF* y3

y1 = (2h/3-b)

y2 = (h+a/3-b)

y3 = (h+a+2Y/3)

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

If not, assume a new Y, repeat step 3 to 4.

1. Calculate the length of sheet pile, L = h+F.S.*D, FS is from 1.2 to 1.4.

2. Calculate anchored force T = Ha1+ Ha2– HCEF

3. Calculate y = -b+Ö2*T/(g Ka)

4. Calculate Mmax = T y - g Ka (y+b)3/6

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

6. Select sheet pile section.

7. Design tie rod

8. Design soldier beam.

### Example 3. Design anchored 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 = 32 ksi

Yield strength of soldier beam, Fy = 36 ksi

Location of tie rod at 2 ft below ground surface, spacing, s = 12 ft

Requirement: Design length of an anchored sheet pile, select sheet pile section, and design tie rod

Solution:

Design length of sheet pile:

Calculate lateral earth pressure coefficients:

Ka = tan (45-f/2) = 0.333

Kp = tan (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 = 2.85 ft

HCEF = g (Kp-Ka) Y2/3 = 115*(3-0.333)*2.852/3 = 830.3 lb/ft

y1 = (2h/3-b) = (2*10/3-2)=4.67 ft

y2 = (h+a/3-b) = (10+1.25/3-2)=8.42 ft

y3 = (h+a+2Y/3) = (10+1.25+2*2.85/3) = 13.15 ft

R = Ha1*y1 + Ha2* y2 – HCEF* y3  = 1917*4.67+238.6*8.42-830.3*13.15 = 42.5 lb

R closes to zero, D = 2.85+1.25 = 4.1 ft

Length of sheet pile, L = 10 + 1.2* 4.1 = 14.9 ft                       Use 15 ft

Calculate anchor force,

T = Ha1+ Ha2– HCEF = 1917+238.6-830.3 = 1326 lb/ft

Calculate location of maximum moment,

y = -b+Ö2*T/(g Ka) = -2 ft + Ö2*1326/(115*0.333) = 6.32 ft

Mmax = T y - g Ka (y+b)3/6 = 1326*6.32 – 115*0.333*(6.32+2)3/6 = 4.7 kip-ft/ft

The required section modulus S= Mmax/Fb = 4.7*12/32 = 1.8 in3/ft

Use PS28, S = 1.9 in3/ft

Design tie rod, the required cross section area,

A = T s / (0.6*Fy) = 1.326*12/(0.6*36) = 0.442 in3.

Use ¾” diameter tie rod, A = 0.442 in3.

Design soldier beam:

The maximum moment of a continuous beams with 3 or more span is

M = 0.1*T s2 = 0.1*1326*122 =19.1 kip-ft

Required section modulus, S = M / (0.6*Fy) = 19.1*12/(0.6*36) = 6.4 in3.

Use W6x15, S = 9.72 in3.

### Anchored sheet pile wall in cohesive soil. ### Theory

#### Calculating active earth pressure

Calculation of active earth pressure above excavation is the same as that of cantilever sheet pile in cohesive soil. The free-standing height of soil is d = 2C/g

The lateral earth pressure at bottom of excavation, pa = g h – 2C, where g is unit weight of soil. The resultant force Ha=pa*h/2

#### Calculating passive earth pressure

For cohesive soil, friction angle, f = 0, Ka = Kp = 1.  The earth pressure below excavation,

p1= sp-sa= 2C-(gh-2C) = 4C-gh

Assume the embedded depth is D, the resultant force below bottom of excavation is

HBCDF = p1*D

Derive equation for D from åMo = 0

åMo = Ha1*y1 – HBCDF* y3 = 0

Where

y1 = 2(h-d)/3-(b-d)

y3 = h-b+D/2

The equation can be determined with a trial and error process.

Determine anchor force T from å Fx = 0

å Fx = Ha1– HBCDF-T = 0

T = Ha1+ Ha2– HCEF

### Design size of sheet pile

Maximum moment locates at a distance y below T where shear stress equals to zero.

T-g Ka (y+b-d)2/2=0

Solve for y, we have, y = -b+d+Ö2*T/(g Ka)

The maximum moment is

Mmax = T y - g Ka (y+b-d)3/6

The required section modulus is S = Mmax / Fb

The sheet pile section is selected based on section modulus

### Design of tie rod and soldier beam

Design of tie rod and soldier beam is the same as that of anchored sheet pile in cohesionless soil.

### 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 value of D, and calculate HBCDF = p1*D

6. Calculate R= Ha*y1 – HBCDF* y3.

Where

y1 = 2(h-d)/3-(b-d)

y3 = h-b+D/2

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

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

2. Calculate anchored force T = Ha – HBCDF

3. Calculate y = -b+d+Ö2*T/g

4. Calculate Mmax = T y - g (y+b-d)3/6

5. Calculate required section modulus S= Mmax/Fb. Select sheet pile section.

6. Design tie rod

7. Design soldier beam.

### Example 4: Design anchored sheet pile in cohesive soil.

Given:

Depth of excavation, h = 15 ft

Unit weight of soil, g = 115 lb/ft3

Cohesion of soil, C = 500 psf

Internal friction angle, f = 0 degree

Allowable design strength of sheet pile = 32 ksi

Yield strength of soldier beam, Fy = 36 ksi

Location of tie rod at 2 ft below ground surface, spacing =12 ft.

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

p1=4C-gh = 4*550-115*15 = 275 psf

Assume D = 11.5 ft,

HBCDF = p1*D = 3163 lb/ft

y1 = 2(h-d)/3-(b-d) =2 (15-8.7)/3-(2-8.7) = 10.9 ft

y3 = h-b+D/2 = 15-2+11.5/2 = 18.75 ft

R= Ha*y1 – HBCDF* y3 = 5438*10.9-3163*18.75 = -36 lb     Close to zero

The length of sheet pile, L = 15 + 1.2*11.5 = 28.8 ft      Use 29 ft

Anchored force per foot of wall, T = Ha – HBCDF = 5438 – 3163 = 2275 lb/ft

Calculate location of maximum moment,

y = -b+d+Ö2*T/g = -2+8.7+Ö2*2275/115 = 13 ft

Maximum moment,

Mmax = T y - g (y+b-d)3/6 = 2275*13 – 115*(13+2-8.7)3/6 = 24770 lb-ft/ft

Required section modulus of sheet pile, S= Mmax/Fb = 22.47*12/32 = 8.4 in3/ft

Use PDA 27 section modulus 10.7 in3/ft

##### Design tie rod

Cross section of tie rod required, A = T*s/(0.6*Fy) = 2.275*12/(0.6*36) = 0.91 in2.

Diameter of tie rod, d = Ö4*A/p = 1.08 in

Use 1-1/8” diameter tie rod.

##### Design soldier beam

Maximum moment in solider beam, Mmax = 0.1*T*s2 = 0.1*2275*122 = 32760 lb-ft

Required section modulus, S= Mmax/Fb= 32.76*12/(0.6*36) = 13.1 in3.

Use W 8x18, section modulus S = 15.2 in3.