Stability Analysis of Well Foundations | Soil Engineering

When the worst possible combination of forces is identified, the resultant of these forces is determined and its com­ponents along the three coordinate axes are determined, that is, W is the component of the resultant in the vertical direction, P is the component of the resultant across the pier in the horizontal direction, and Q is the component of the resultant along the pier in the horizontal direction.

It has been found that the force system acting in the direction of the transverse axis of the pier, that is, perpen­dicular to the flow direction is more critical for stability. The three conditions of equilibrium to be satisfied to ensure stability of the well foundation are –

The following two methods for stability analysis of well foundations are discussed below:

1. Terzaghi’s method.

2. IRC method.

1. Terzaghi’s Method:

Terzaghi’s analysis for rigid bulkheads assumes that the bulkhead is light and there is no friction at the base and the sides. The lateral pressure is computed using Rankine’s theory. This method can be applied for stability analysis of well foundations, neglecting the moments due to friction at the base and the sides. Hence, Terzaghi’s analysis, when applied to well foundations gives conservative estimates.

When a rigid well, embedded in sand, moves parallel to its original position, the soil on the front side of the well will exert passive resistance and the soil on the rear side will exert active pressure. Assuming that the shear strength of the soil is fully mobilized simultaneously on the front and rear sides, the soil on the front side will be in Rankine’s passive state and that on rear side in Rankine’s active state. The net pressure at any depth h acting on the well is given by –

p = γh(K_p – K_a) …(21.10)

where p is the net pressure at any depth h, γ is the density of the soil, K_p is Rankine’s coefficient of passive earth pressure, and K_a is Rankine’s coefficient of active earth pressure.

Figure 21.7 shows the lateral earth pressure diagram for a well foundation. The bulkhead is considered free, if the stability of the bulkhead is derived entirely from the lateral earth pressure.

Let q_max be the applied horizontal force per unit length acting on the bulkhead with an embedment depth D. The bulkhead tends to rotate about point “O,” above the lower end of the bulkhead.

Above point “O,” the bulkhead will be subjected to passive resistance on the front side and active pressure from the rear side. Below point “O,” the bulkhead will be subjected to passive resistance on the rear side and active pres­sure from the front side. The resultant lateral pressure diagram is shown in Fig. 21.7.

In the case of heavy wells, the well rotates about the base. The pressure distribution will be as shown in Fig. 21.8. Hence, we can write –

The total resisting force is given by –

Q_max – q_max x L …(21.16)

where L is the diameter or length of the well.

The allowable horizontal force will be –

Q_a = Q_max/F …

(21.17)

where F is the factor of safety.

A minimum factor of safety of 2 should be used. The unbalanced horizontal force on the well = Q – Q_a, where Q is the horizontal force applied on the well. The resulting overturning moment about the base –

M_b = (Q – Q_a) x (H + D) …(21.18)

The pressure at the base of the well foundation is given by –

p = (W/A) ± (M_b/z_b) …(21.19)

where W is the net vertical load including skin friction and buoyancy, A is the area of the base of the well founda­tion, and z_b is the section modulus of the well base.

The maximum pressure at the base obtained from Eq. (21.19), should be less than the allowable bearing pressure. The well steining is subjected to a bending moment due to horizontal load, Q, which will be minimum when the resultant shear force is zero. The depth of zero shear below the scour level is given by –

The maximum Bending moment is given by –

M_max = Q(H + x) – Q x (x/3) Þ M_max = QH + (2/3)Qx …(21.21)

2. IRC Method:

The lateral stability of a well foundation is provided by passive resistance of the soil on its sides and base. The resistance developed at the working loads is different from that at the ultimate load.

IRC – 45-1972 recommends the following two methods for stability analysis of well foundations:

i. Elastic theory to estimate the soil pressure on the sides and at the base under working loads.

ii. The ultimate soil resistance method to determine the factor of safety against shear failure.

The IRC methods are applicable only if the depth of embedment is at least equal to 0.5 times the width of the foun­dation in the direction of lateral forces.

The methods are given for the design of well foundations resting on non-cohesive soil and surrounded by the same soil below the maximum scour level. The moment distribution between the base and the sides of the well continuously change with an increase in the deformation of the soil. The Elastic theory gives the soil pressure at the side and base under design loads. The factor of safety against failure is determined by considering the ultimate soil resistance method.

i. Elastic Theory:

The Elastic theory, used to determine soil pressure below the base and on the sides, is based on the following assumptions:

a. The soil surrounding the well and below the base is perfectly elastic, homogeneous and follows Hooke’s law.

b. The unit soil reaction (p) increases linearly with an increase in lateral deflection (z). Hence, p = k_Hz, where k_H is the coefficient of horizontal sub-grade reaction at the base.

c. The coefficient of horizontal sub-grade reaction (k_H) increases linearly with depth for cohesionless soils.

d. The well is assumed to be a rigid body, subjected to an external unidirectional horizontal force H and moment M_o at the scour level.

The total horizontal soil reaction acting on the sides of the well (P) is given by –

P = 2mKθI_v/D …(21.22)

where m = K_H/K is the ratio of the horizontal to the vertical coefficient of sub-grade reactions at the base, K is the coefficient of vertical sub-grade reaction at the base, θ is the angular rotation of the well with the vertical, I_v is the moment of inertia about the horizontal axis passing through the center of gravity (C.G.) of the projected area in elevation of the soil mass offering resistance = LD³/12, L is the projected width of the soil mass offering resistance multiplied by shape factor “S,” D is the depth of the well below the scour level, and S = shape factor = 0.9 for circular wells. The moment about the base level, M is given by –

M_p = mKθI_v …(21.23)

where M_p is the moment of P about the base.

The maximum and minimum pressures at the base of the well are given by –

where W is the total weight of the well, µ is the coefficient of friction between the sides of the well and the surround­ing soil, P is the total horizontal soil reaction as given by Eq. (21.22), A is the area of the base of the well, M is the total applied external moment at the base = M_o + HD, M_o is the moment of external forces at the scour level, B is the width of the base of the well parallel to the direction of the external horizontal force –

I = I_B +mI_v (1 + 2µ’α) …(21.26)

and I_B is the moment of inertia of the base about an axis passing through the C.G. and perpendicular to the hori­zontal resultant force. For rectangular wells –

α = B/2D …(21.27)

For circular wells –

α = Diameter/2D …(21.28)

Conditions of stability for well foundations based on elastic theory are as follows:

a. The maximum soil reaction (σ_x) from the sides shall not exceed the maximum passive pressure at any depth –

The condition given by Eqs. (21.29) and (21.30) may be simplified as follows –

b. The maximum soil pressure at the base (σ₁) shall not exceed allowable pressure on soil.

c. The minimum soil pressure at the base (σ₃) shall not be negative, that is, no tension shall be allowed to develop at any point in the soil below the base.

The following is the procedure for stability analysis of well foundations as per the elastic theory method of IRC:

a. Determine the grip length, applied loads, and moment.

b. Calculate the moment of inertia of the base of the well foundation about the centroidal axis perpendicular to the direction of the resultant horizontal force by –

I_B = (π/64) B⁴ …(21.32)

where B is the width of the well or the diameter of a circular well.

c. Determine the moment of inertia about the centroidal horizontal axis of the projected area in elevation of soil mass offering resistance –

where K_h and K_v are coefficients of sub-grade reaction in the horizontal and vertical directions, respectively, L is the projected width of the well in contact with the soil that offers passive resistance multiplied by the shape factor, S, S = 0.9 for circular wells, S = 1 for square or rectangular wells, µ is the coefficient of friction between the sides of the well and the soil = tanδ ≈ tanɸ, and D is the depth of the well below the scour level.

d. Check that the point of rotation of the well lies at the base by ensuring that the frictional force at the base is ade­quate to restrain the movement of the well –

e. Check that the soil on the sides of the well remains elastic. This can be ensured by keeping the slope of the pres­sure parabola at the top below the passive pressure line, that is –

where K_p and K_a are Coulomb’s coefficients of passive and active earth pressure assuming –

f. Check the base pressures ensuring that the maximum compressive pressure is less than the allowable bearing pressure and that the minimum pressure is not tensile.

where P is the total horizontal reaction from the sides and B is the width of the base of the well in the plane of bending.

ii. Ultimate Soil Resistance Method:

The safety against ultimate failure of well foundations is considered by the ultimate soil resistance method. The mode of failure of the soil mass under the application of transverse forces on a well foundation is shown in Fig. 21.9. The soil around the base of the well slides over a circular cylindrical path with center of rotation above the base. Failure has been observed to occur at about 3° of the rotation of the well in the case of cohesionless soil.

The following are the methods to ensure a minimum level of safety against failure:

a. The applied loads are increased by a load factor using the most critical of the following combinations –

where DL is the dead load, LL is the live load including braking force, etc., BF is the buoyancy force, W_c is the water current force, E_p is the earth pressure, WL is the wind force, and SL are the seismic forces.

The applied moment (M) is calculated by considering most critical combination of the above five load combinations.

b. The total resisting moment (M_r) is reduced by a strength factor to ensure safety against failure as

0.7 (M_r) M (21.45)

M_r = M_b + M_s + M_f (21.46)

where M_r is the total resisting moment offered by the soil, M_b is the base resisting moment, M_s is the side resisting moment, and M_f is the resisting moment due to friction on the front and back faces –

M_b = QWB tanɸ …(21.47)

where Q is a constant given by Table 21.5 for rectangular wells, W is the total vertical load, B is the width or diameter of rectangular or circular wells, and <p is the angle of internal friction of the soil.

Also, we have –

M_s = 0.1 γD³ (K_p – K_a)L …(21.48)

where L is the width of the well.