5.2 Shallow and Deep Foundation Design

Foundations Overview

Foundations act as the transition structural element between the super-structure and the underlying soil mass. They are designed to transmit loads to the ground without causing soil shear failure and within tolerable settlement limits. Foundations are broadly classified into two categories:

• Shallow foundations: These structures transfer loads to soil layers close to the surface. They typically have an embedment depth (D_f) less than or equal to their width (B). Common types include spread footings, strip footings, and mat (raft) foundations.
• Deep foundations: When surface soil layers are too weak or highly compressible to support the structure, loads must be transferred to deeper, more competent strata. These foundations typically have an embedment depth much larger than their width (D_f >> B). Common types include pile foundations and drilled shafts.

Bearing Capacity of Shallow Foundations

The bearing capacity of a foundation is its ability to support loads without undergoing shear failure or excessive settlement. The maximum pressure that the soil can support before failing in shear is the ultimate bearing capacity (q_ult).

Terzaghi's Bearing Capacity Equations

In 1943, Karl Terzaghi proposed the first comprehensive theory for evaluating the ultimate bearing capacity of shallow foundations. Terzaghi assumed a general shear failure mode, representing the soil failure wedge beneath a strip footing as three distinct zones: a triangular wedge directly under the base, radial shear zones, and linear passive shear zones.

The ultimate bearing capacity for a continuous (strip) footing is expressed as:

q_ult = c' * N_c + q * N_q + 0.5 * gamma * B * N_gamma

where:

• c' is the effective cohesion of the soil.
• q is the effective surcharge at the level of the foundation base (q = gamma * D_f, where D_f is depth of embedment).
• gamma is the unit weight of the soil.
• B is the width of the footing.
• N_c, N_q, N_gamma are dimensionless bearing capacity factors that depend solely on the soil's effective friction angle (phi').

To adapt this continuous footing equation to other shapes, Terzaghi introduced shape factors (s_c, s_q, s_gamma). The modified equation is:

q_ult = c' * N_c * s_c + q * N_q * s_q + 0.5 * gamma * B * N_gamma * s_gamma

The standard shape factors for different footing geometries are summarized in the table below:

Factor of Safety

To determine the allowable bearing capacity (q_all) that can be safely applied to the footing, a factor of safety (FS, typically between 3.0 and 4.0) is applied to the ultimate capacity:

q_all = q_ult / FS

In some design scenarios, the net allowable bearing capacity (q_net(all)) is preferred, which subtracts the initial surcharge stress:

q_net(all) = (q_ult - q) / FS

Water Table Corrections

The presence of a groundwater table reduces the effective stress of the soil due to buoyancy, which significantly decreases the soil's bearing capacity. The position of the water table (D_w) relative to the ground surface and foundation base determines which terms in Terzaghi's equation must be adjusted:

1. Case 1: Water table is located between the ground surface and the foundation base (0 <= D_w <= D_f): The surcharge term (q) must be modified to use effective stress: q = D_w * gamma_d + (D_f - D_w) * gamma' where gamma_d is the dry or moist unit weight of the soil and gamma' is the submerged (buoyant) unit weight (gamma' = gamma_sat - gamma_w). The unit weight in the third term (gamma) is replaced entirely by gamma'.

2. Case 2: Water table is located at a depth d below the foundation base (0 <= d <= B): The surcharge term remains unaffected (q = gamma * D_f). However, the unit weight in the third term is replaced by an average effective unit weight: gamma_avg = gamma' + (d / B) * (gamma - gamma')

3. Case 3: Water table is deep below the foundation base (d > B): Groundwater does not affect the bearing capacity. Use the moist unit weight gamma in all terms.

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Settlement Analysis

In addition to shear strength failure, foundations must be designed to prevent excessive settlement, which can cause structural distress or serviceability issues. Total settlement (S_T) consists of three components:

S_T = S_e + S_c + S_s

1. Immediate (Elastic) Settlement (S_e)

This occurs immediately upon load application due to elastic deformation of the soil skeleton without any change in water content. It is calculated using elastic theory:

S_e = q_0 * B * (1 - mu²) * I_p / E_s

where q_0 is the applied contact pressure, mu is the Poisson's ratio, E_s is the soil modulus of elasticity, and I_p is an influence factor depending on the footing's shape and rigidity.

2. Primary Consolidation Settlement (S_c)

Consolidation is a time-dependent process that occurs in saturated, low-permeability cohesive soils (clays) as pore water is slowly squeezed out under load.

The stress increase (delta_sigma') at depth is commonly calculated using the 2:1 method, which assumes that the vertical load P distributes over an area that increases with depth at a 2 (vertical) to 1 (horizontal) ratio:

delta_sigma'_z = P / [(B + z) * (L + z)]

Consolidation settlement equations depend on whether the clay is normally consolidated or overconsolidated:

• Normally consolidated clay (OCR = 1): The initial effective stress sigma'_0 is the maximum stress the soil has ever experienced. S_c = [C_c * H_c / (1 + e_0)] * log((sigma'_0 + delta_sigma') / sigma'_0)
• Overconsolidated clay (OCR > 1): The preconsolidation pressure (sigma'_c) is greater than the current effective stress (sigma'_0).
  • If sigma'_0 + delta_sigma' <= sigma'_c: S_c = [C_r * H_c / (1 + e_0)] * log((sigma'_0 + delta_sigma') / sigma'_0)
  • If sigma'_0 + delta_sigma' > sigma'_c: S_c = [C_r * H_c / (1 + e_0)] * log(sigma'_c / sigma'_0) + [C_c * H_c / (1 + e_0)] * log((sigma'_0 + delta_sigma') / sigma'_c) where:
  • C_c is the compression index.
  • C_r is the recompression index (or swell index, C_s).
  • e_0 is the initial void ratio.
  • H_c is the thickness of the clay layer.
  • sigma'_0 is the initial vertical effective stress at the mid-height of the clay layer.
  • delta_sigma' is the average stress increase at the mid-height of the clay layer.

3. Secondary Compression Settlement (S_s)

Also known as creep, secondary compression occurs at constant effective stress after primary consolidation is complete due to plastic adjustment of the soil fabric.

S_s = C'_alpha * H_c * log(t / t_p)

where C'_alpha is the modified secondary compression index, t_p is the time to complete primary consolidation, and t is the target time.

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Deep Foundations: Piles and Drilled Shafts

When shallow soils are weak, deep foundations bypass them to transfer structural loads to competent deeper strata. The total axial capacity (Q_ult) of a deep foundation is the sum of its base resistance (end bearing, Q_p) and shaft resistance (skin friction, Q_s):

Q_ult = Q_p + Q_s = q_p * A_p + Sum(f_s * A_s)

where A_p is the area of the pile tip, f_s is the unit skin friction along the shaft, and A_s is the surface area of the shaft.

Cohesive Soils (Clay)

For deep foundations in clay, the undrained shear strength (c_u) governs capacity:

1. End Bearing (Q_p): Under undrained conditions, the unit tip resistance is: q_p = N_c * c_u ≈ 9 * c_u Thus, Q_p = 9 * c_u * A_p.
2. Skin Friction (Q_s): The total shaft capacity is determined using the alpha (alpha) method: f_s = alpha * c_u where alpha is an empirical adhesion factor (ranging from 0.3 to 1.0 depending on c_u and pile installation method).

Cohesionless Soils (Sand)

For piles in sand, capacity depends on effective stresses:

1. End Bearing (Q_p): The unit tip resistance is: q_p = sigma'_v * N'_q where sigma'_v is the effective vertical stress at the pile tip, and N'_q is a bearing capacity factor. Note that q_p is often capped at a critical depth (typically 15 * B to 20 * B).
2. Skin Friction (Q_s): Calculated using the beta (beta) method: f_s = beta * sigma'_v = K * sigma'_v * tan(delta) where K is the earth pressure coefficient, sigma'_v is the effective stress at the depth of the section, and delta is the soil-pile friction angle.

Pile Groups

Piles are rarely installed individually; they are usually placed in clusters joined by a concrete pile cap. The structural capacity of a pile group is not always the sum of the individual pile capacities. The group efficiency (eta) is the ratio of the pile group capacity to the sum of individual capacities:

Q_group = eta * Sum(Q_individual)

For cohesive soils, eta can be less than 1.0 because stress fields overlap, whereas for compact cohesionless soils, installation vibration can densify the soil, sometimes leading to eta > 1.0.