Poncelet’s Construction for Earth Pressure | Soil Engineering

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:

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.

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.

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).

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 C₁ on the ɸ line. Draw a line C₁ A₁ parallel to the ground line to intersect the back of the wall at point A₁.

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

v. Describe a semicircle on BC₁ with BC₁ as the diameter.

vi. From point A₁, draw a line A₁D₁ parallel to the Ψ line to intersect the ɸ line at point D₁.

vii. From point D₁, draw a perpendicular D₁E₁ to the ɸ line to intersect the semicircle at point E₁.

viii. Draw an arc with point B as the center and BE₁ as the radius to intersect the ɸ line at point F₁.

ix. Join the points F₁ and A₁ by a straight line.

x. From point A, draw a line AF parallel to line A₁F₁ 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 H₁. In fact, the arc also intersects the ɸ line at point H₂.

xi. Join the points G and H₁ to get ΔFGH₁. Also, join the points G and H₂ to get ΔFGH₂. These two triangles, known as force triangles, have the same area.

xii. Determine the area (A) of either of the force triangles (ΔFGH₁ or ΔFGH₂).

xiii. The active earth pressure is given by –

P_a = γ × A × 1 = (γ/2) × GH₁ × FI₂ = (γ/2) × GI₁ × FH₂

where GI₁ and FI₂ are the perpendicular lines drawn from points G and F on H₂F and GH₁, respectively, as shown in Fig. 15.47.