How to calculate the location of resultant earth pressure on a retaining wall?

Eg: Determine the lateral earth pressure at rest /unit length of the wall. Also, find the location of the resultant earth pressure on the retaining wall as shown below. Consider 𝛄w = 10KN/m³. 

Given data:

Depth of 1st layer = H1 = 2.5m.

The angle of friction = 𝜱1 = 32° 

Unit weight = 𝛄 = 16KN/m³

Depth of 2nd layer =H2 = 3m.

The angle of friction = 𝜱2 = 34° 

Saturated unit weight = 𝛄sat = 19KN/m³.

 Unit wt. of water 𝛄w = 10KN/m³. 

Calculation:

Coefficient of earth pressure at rest for the 1st layer

Ka01 = [1-sin𝜱1]

        = [1-sin32°]

        = [1-0.5299]

Ka01 = 0.47

Coefficient of earth pressure at rest  for the 2nd layer

Ka02 = [1-sin𝜱2]

        = [1-sin34°]

        = [1-0.5591]

Ka02 = 0.4408

The intensity of pressure at layer B

p01 = [ Ka01 x 𝛄 xH1]

       = [ 0.47 x 16 x 2.5]

p01 = 18.80KN/m².

The intensity of pressure at layer C due to 1st layer

p02 =  [ Ka02 x 𝛄 xH1]

       = [ 0.4408 x 16 x 2.5]

p02 = 17.632KN/m².

The intensity of pressure at layer C due to 2nd layer

p03 =  [ Ka02 x 𝛄¹ xH2]

Where 𝛄¹ = submerged unit wt.

                = [𝛄sat -𝛄w]

                = [19 - 10] 

                = 9KN/m³.

   p03  = [ 0.4408 x 9 x 3]

   p03  = 11.016KN/m². 

Pressure due to water

 u = [𝛄w x H2]

    = [10 x 3]

 u = 30KN/m².

Note: The pr. due to water in the 1st layer will be 0, as the water table is only in the 2nd layer.

Now, let us draw the pressure distribution diagram from the above-calculated values.

Total pressure of triangle ABD

P1= [1/2 x base x height]

    = [1/2 x 18.8 x 2.5]

P1= 23.5KN.

The total pressure of rectangle BCFE

P2= [Base x height]

     = [17.632 x 3]

P2= 52.896KN.

The total pressure of triangle EFG

P3= [1/2 x 11.016 x 3]

P3= 16.524KN.

The total pressure of the triangle EGH

P4= [1/2 x 30 x 3]

P4= 45KN.

The total active earth pressure 

P = [P1 + P2 + P3 + P4]

   = [23.5 + 52.896 + 16.524 + 45]

P = 137.92KN.

Resultant active earth pressure:

The centroid of the total active earth pressure acts horizontally at a distance of Ȳ from the base of the pressure diagram.

For a triangular pressure diagram, the pressure acts at 1/3rd distance from the base of the diagram. For the rectangular diagram, the acting pressure will be at 1/2 the distance from the base.

( According to Rankine theory)

Therefore,

PȲ = [P1Ȳ1 + P2Ȳ2 + P3Ȳ3 + P4Ȳ4]

 = [{23.5 x (3m. + 1/3 x 2.5m.)} +{52.896 x (1/2 x3m )} +{16.524 x (1/3 x 3m.)} +{30 x (1/3 x 3m.)}]

 = [{23.5 x 3.833} + {52.896 x 1.5} + {16.524 x 1} + {30 x 1}]

 = [90.076 + 79.344 + 16.524 + 30]

PȲ = 215.944

Ȳ = [215.944 ➗ P]

    = [215.944 ➗ 137.92]

  Ȳ = 1.565m.

The total earth pressure of 137.92KN  acts horizontally at a distance of 1.565m. from the base of the wall.

Thank you for going through these calculation steps❤. Have a good day 😄.

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