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In this article we will discuss how by using Poncelet’s graphical solutions we can determine active and passive earth pressure.

**Poncelet (or Rebhan’s) Construction for Active Earth Pressure: **

Poncelet and Rebhan have independently given graphical solutions for determination of active earth pressure.

**The procedure for determination of active earth pressure in Rebhan’s or Poncelet graphical method (refer Fig. 15.44) is as follows: **

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i. Draw the cross section of the retaining wall (AB) to some scale.

ii. Draw the ground line, AC, at an angle of β with the horizontal and the ɸ line, BC, at an angle of ɸ with the horizontal.

iii. Draw the Ψ line at an angle of Ψ = θ – Δ from the ɸ line at point B.

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iv. Describe a semicircle on BC with BC as the diameter.

v. From the crest of the wall A, draw a line AD parallel to the Ψ line to intersect the ɸ line at point D.

vi. From point D, draw a perpendicular DE to the ɸ line to intersect the semicircle at point E.

vii. Draw an arc with point B as the center and BE as the radius to intersect the ɸ line at point F.

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viii. From point F, draw a line FG, parallel to the Ψ line to intersect the ground line AC at point G.

ix. Draw an arc with point F as the center and FG as the radius to intersect the ɸ line at point H.

x. Join the points G and H to get ΔFGH. This triangle is known as force triangle.

xi. Determine the area (A) of the force triangle (ΔFGH).

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xii. The active earth pressure is given by –

P_{p} = γ × A × 1 = (γ/2) × FH × GI

where GI is the normal distance of point G on FH, as shown in Fig. 15.44. BG is the failure plane.

**Special Case ****–**** Angles ****β**** and ****ɸ**** are Nearly Equal: **

When angles βand ɸ are nearly equal, the ground line and the ɸ line do not meet within the limits of the paper.

**In such a case, the following procedure may be adopted (refer to Fig. 15.45): **

i. Draw the cross section of the retaining wall (AB) to some scale.

ii. Draw the ground line, AC’, at an angle of β with the horizontal and the ɸ line, BC”, at an angle of ɸ with the horizontal.

iii. Select any suitable point C_{1 }on the ɸ line. Draw a line C_{1 }A_{1} parallel to the ground line to intersect the back of the wall at point A_{1}.

iv. Draw the Ψ line at an angle of Ψ = θ – Δ from the ɸ line at point B.

v. Describe a semicircle on BC_{1} with BC_{1 }as the diameter.

vi. From point A_{1}, draw a line A_{1}D_{1} parallel to the Ψ line to intersect the ɸ line at point D_{1}.

vii. From point D_{1}, draw a perpendicular D_{1}E_{1 }to the ɸ line to intersect the semicircle at point E_{1}.

viii. Draw an arc with point B as the center and BE_{1} as the radius to intersect the ɸ line at point F_{1}.

ix. Join the points F_{1} and A_{1} by a straight line.

x. From point A, draw a line AF parallel to line A_{1}F_{1} to intersect the ɸ line BC” at point F.

xi. From point F, draw a line FG parallel to the Ψ line to intersect the ground line AC’ at point G.

xii. Draw an arc with point F as the center and FG as the radius to intersect the ɸ line at point H.

xiii. Join the points G and H to get ΔFGH. This triangle is known as force triangle.

xiv. Determine the area (A) of the force triangle (ΔFGH).

xv. The active earth pressure is given by –

P_{a} = γ × A × 1 = (y/2) × FH × GI

where GI is the normal distance of point G on FH.

**Special Case 2 – β =**** ɸ:**

When β = ɸ, the ground line and the ɸ line will be parallel.

**In such a case, the following procedure may be adopted (refer to Fig. 15.46): **

i. Draw the cross section of the retaining wall (AB) to some scale.

ii. Draw the ground line, AC’, and the ɸ line, BC”, at an angle of β = ɸ with the horizontal.

iii. Draw the Ψ line at an angle of Ψ = θ – Δ from the ɸ line at point B.

iv. Select any suitable point F on the ɸ line. From point F, draw a line FG parallel to the Ψ line to intersect the ground line AC’ at point G.

v. Draw an arc with point F as the center and FG as the radius to intersect the ɸ line at point H.

vi. Join the points G and H to get ΔFGH. This triangle is known as force triangle.

vii. Determine the area (a) of the force triangle (ΔFGH).

viii. The active earth pressure is given by –

P_{a} = γ × A × 1 = (γ/2) × FH × GI

where GI is the normal distance of point G on FH.

**Poncelet (or Rebhan’s) Construction for Passive Earth Pressure****:**

**The following procedure may be adopted to determine passive earth pressure using Poncelet’s (Rebhan’s) construction (refer to Fig. 15.47):**

i. Draw the cross section of the retaining wall (AB) to some scale.

ii. Draw the ground line, AC’, at an angle of β with the horizontal and the ɸ line, BC”, at an angle of ɸ below the horizontal as shown in Fig. 15.47.

iii. Extend the ground line and the ɸ line backward to intersect at point C on the front side of the wall.

iv. Draw the Ψ line at an angle of Ψ = θ – Δ from the ɸ line at point B.

v. Describe a semicircle on BC with BC as the diameter.

vi. From the crest of the wall A, draw a line AD parallel to the Ψ line to intersect the ɸ line at point D.

vii. From point D, draw a perpendicular DE to the ɸ line to intersect the semicircle at point E.

viii. Draw an arc with point B as the center and BE as the radius to intersect the ɸ line at point F.

ix. From point F, draw a line FG, parallel to the Ψ line to intersect the ground line AC at point G.

x. Draw an arc with point F as the center and FG as the radius to intersect the ɸ line at point H_{1}. In fact, the arc also intersects the ɸ line at point H_{2}.

xi. Join the points G and H_{1} to get ΔFGH_{1}. Also, join the points G and H_{2} to get ΔFGH_{2}. These two triangles, known as force triangles, have the same area.

xii. Determine the area (A) of either of the force triangles (ΔFGH_{1 }or ΔFGH_{2}).

xiii. The active earth pressure is given by –

P_{a} = γ × A × 1 = (γ/2) × GH_{1 }× FI_{2 }= (γ/2) × GI_{1 }× FH_{2}

where GI_{1} and FI_{2} are the perpendicular lines drawn from points G and F on H_{2}F and GH_{1}, respectively, as shown in Fig. 15.47.

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