Theoretical notes

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The bearing capacity of a shallow foundation can be defined as the maximum value of the load applied, for which no point of the subsoil reaches failure point (Frolich method) or else for which failure extends to a considerable volume of soil (Prandtl method and successive).

Experimental observations have shown that the soil can get to failure through three mechanisms (see image below):

General failure is characterized by the formation of well-defined sliding surfaces that start from the foundation and reach the ground level, and a swelling of the soil at the sides of the foundation.

Punching failure  when the lowering of the foundation is made ​​possible by the formation of vertical shear planes along the perimeter, without generating sliding surfaces.

Local failure corresponds to the formation of a clear sliding surface below the foundation, which is dispersed in the adjacent soil, it shows a tendency to swelling timid side of the ground.

 

Types of soil failure

 

The solutions available for the calculation of the bearing capacity are based on the assumption of rigid-plastic behavior of the soil and are therefore, strictly speaking, only applicable to the case of general failure.

It can be shown that the bearing capacity of a soil is the sum of three factors: soil weight γ', q' overload and cohesion c'; solutions available today have been obtained by the superposition of individual independent problems.

 

Prandtl (1921) has studied the problem of failure of an elastic half-space due to a load applied on its surface with reference to steel, characterizing the resistance to failure with a law of the type:

valid even for soils.

 

Prandtl assumes:

 

Upon failure, the yield of the material within the space GFBCD is verified (see image below).

Within the triangle AEB failure occurs according to two families of straight segments inclined by 45°+φ/2 to the horizontal.

Within zones ABF and EBC failure occurs along two families of lines, the ones made up of straight lines passing through points A and E, and the other consisting of arcs of logarithmic spirals. The poles of these are points A and E. In the triangles AFG and ECD failure occurs along segments inclined at  ±(45°+ φ/2 )  to the vertical.

 

Solution of Prandtl

 

Having thus identified the soil tending to failure by application of the ultimate load, this can be calculated expressing the equilibrium between the forces acting in any volume of soil whose base is delimited by whichever slip surface.

Thus we reach the equation:

where the coefficient B depends only upon the soil’s angle of friction φ'. For φ'≠0 the factor B = 5,14.

 

In the alternate case, namely that where the soil is cohesionless (c=0, γ'≠0) q'=0 so that according to Prandtl it would not be possible to apply any load to cohesionless soils.

 

This theory, even if not practically applicable, has been the base for all the following researches and computation methods.

 

Caquot proceeds from the same premises as Prandtl excepting that the load strip is no longer placed on the surface but at a depth h, where h ≤ 2b; the soil between surface and the depth h has the following characteristics: γ'=0, φ'≠0, c'=0 and is a material with weight attribute but with no resistance.

The equilibrium equations thus resolve to:

 

which is certainly a step forward but hardly reflects reality.

 

Terzaghi (1955)

Terzaghi, continues on the same lines as Caquot but adds modifications to take into account of the real characteristics of the foundation-soil system.

Under the action of the load transmitted by the foundation, the soil at the contact with the foundation tends to move laterally, but is restrained in this by the tangential resistances that develop between the soil and the foundation. This results in a change of the stress state in the ground placed directly below the foundation.

Terzaghi assigns to the sides AB and EB of Prandtl’s wedge, an inclination Ψ to the horizontal, assigning to this a value as a function of the mechanical characteristics of the soil at the contact soil-foundation.

Thus γ' =0 for soil below the foundation is reviewed assuming that the failure surfaces remain unaltered, the expression for ultimate load becomes:

 

 

where:

b        is the half width of the strip

 

Further on the basis of experimental data, Terzaghi introduces factors due to the shape of the foundation.

Again Terzaghi refines the original hypothesis of Prandtl who considered the behaviour of soil as rigid–plastic.

Terzaghi instead assigns such behaviour only to very compact soils. In these soils the curve loads/settlements is at first linear, followed by a short curved segment (elastic-plastic behavior). Failure is instantaneous and the value of the ultimate load is easily identifiable (general failure).

In a very loose soil however the relation loads/settlements has an accentuated curved line even at low levels of load due to a progressive failure of the soil (local failure) and thus the identification of ultimate load is not so clear like for compact soils..

For very loose soils therefore Terzaghi introduces in the previous formula the reduced values for the mechanical properties of the soil:

 

Thus Terzaghi’s formula becomes:

where:

 

 

 

 

 

 

 

Meyerhof (1963)

Meyerhof proposed a formula for calculation of bearing capacity similar to the one proposed by Terzaghi but introducing further foundation shape coefficients.

He introduced a coefficient sq' that multiplies the Nq factor; depth factors di and inclination factors ii for the cases where the load line is inclined to the vertical.

Meyerhof obtained the N factors by making trials on a number of BF arcs (see Prandtl mechanism) whilst shear along AF was given approximate values.

The N factors  are given below together with the complete formula:

 

 

 

Factors of form, depth and inclination that appear in the formula of Meyerhof

 

Hansen (1970)

Hansen’s formula is a further extension on Meyerhof’s. The additions consists in the introduction of bi that accounts for the possible inclination of the bearing surface to the horizontal and a factor gi for inclined soil surface.

Hansen’s formula is valid for whatever ratio D/B and therefore for both shallow and deep foundations. However the author introduces coefficients to compensate for the otherwise excessive increment in bearing pressure with increased depth.

 

 

Factors proposed by Hansen for the computation of qult

 

 

Vesic (1975)

Vesic  proposes a formula that is analogous to Hansen's with Nq and Nc as for Meyerhof's formula and Ng as below:

 

 

Shape and depth factors are the same as Hansen’s but there are differences in load inclination factors, ground inclination factors (foundation on slope) and bearing surface inclination factors (inclined base).

 

 

Brich-Hansen (EC 7 - EC 8)

In order that a foundation may safely sustain the design load in regard to general failure for all combinations of load relative to the ultimate limit state, the following must be satisfied:

 

where

 

When estimating Rd for fine grained soils short and long term situations should be considered.

Bearing capacity in drained conditions is calculated by:

 

where

 

 

 

Design bearing capacity in drained conditions is calculated as follows:

 

where:

 

 

Correction factors proposed by Brinch-Hansen for the computation of qult

 

In addition to the correction factors reported in the table above will also be considered the ones complementary to the depth of the bearing surface and to the inclination of the bearing surface and ground surface (Hansen).

 

 

Richards et. al.

Richards, Helm and Budhu (1993) developed a procedure that allows, under seismic conditions, to evaluate both the bearing capacity and induced settlements, therefore to proceed to the verification of both limit states (ultimate and damage). The calculation of the bearing capacity is pursued by a simple extension of the bearing capacity problem to the case of the presence of inertial forces in the foundation soil due to earthquake, while the computation of settlements is obtained through an approach type Newmark. The authors have extended the classical formula of the  bearing capacity:

 

 

Where the bearing capacity factors are calculated using the following formulas:

 

 

 

Examining with a limit equilibrium approach, a mechanism type Coulomb and bringing into account the forces of inertia acting on the terrain volume at failure. In the static field, the classical Prandtl mechanism may in fact be approximated as shown in the figure below, eliminating the transition zone (range of Prandtl) reduced to one line AC, which is taken into consideration as an ideal wall in equilibrium under the action of active and passive thrust that it receives from I and III wedges:

 

Computation of bearing capacity qlim

 

The authors have obtained the expressions of ρA and ρP angles that define the areas of active and passive thrust and the active and passive thrust coefficients KA and KP as a function of the angle of internal friction φ of the ground and the angle of friction δ ground - ideal wall:

 

 

It is still to be observed that the use of previous equations assuming φ = 0.5δ, leads to the value of the bearing capacity factors very close to those based on an analysis of Prandtl. Richards et. Al have extended the application of the Coulomb mechanism in seismic case, taking into account the inertia forces acting on the ground volume in failure. These mass forces, due to acceleration kh g and kv g, respectively acting in horizontal and vertical direction, are equal to kh g and kv g. Were thus obtained the extensions of the expressions of ρA and ρP, as well as KA and KP, respectively indicated as ρAE and ρPE, and  KAE and KPE to denote the seismic conditions:

 

 

 

The values ​​of Nq and Ng can also be determined using the above formulas, of course, using the expressions of the ρA and ρP angles and of KAE and KPE coefficients related to the seismic case. In these expressions appears the angle θ defined as:

 

The following table shows the bearing capacity factors calculated for the following parameter values:

 

φ = 30°
δ = 15°

 

And for several values ​​of the seismic thrust coefficients:

 

Table of bearing capacity factors for φ=30°

Bearing capacity for foundations on rock

Where foundations rest on rock, it is appropriate to take into consideration certain other significant parameters such as the geologic characteristics, type of rock and its quality measured as RQD. It is the practice to use very high values of safety factor for bearing capacity of rock and correlated in some way with the value of RQD (Rock quality designator). For example for a rock whose RQD is up to a maximum of 0.75 the safety factor oscillates between 6 and 10. Terzaghi’s formula can be used in calculation of rock bearing capacity using friction angle and cohesion of the rock or those proposed by Stagg and Zienkiewicz (1968) according to which the coefficients of the bearing capacity are:

 

 

These coefficients should be used with form factors from the formula of Terzaghi.

Ultimate bearing capacity is a function of RQD as follows:

 

 

If rock coring does not render whole pieces (RQD tends to 0) the rock is treated as a soil estimating as best the factors: c and φ.

 

 

Sliding check

The stability of a foundation should be verified with reference to collapse due to sliding as well as to general failure. For collapse due to sliding, the resistance is calculated as the sum of the adhesion component and  the soil-foundation friction component. Lateral resistance arising from passive thrust of the soil can be taken into account using a percentage supplied by the user.
Resistance due to friction and adhesion is calculated with the expression:

 

where

 

Computation of settlements

The application of a load of finite size on a cohesive soil generates a series of phenomena which can be summarized as illustrated in image below:

 

Settlement types

 

 

 

Elastic settlements

The settlement of a rectangular foundation of size B' × L' placed on the surface of an elastic support may be calculated by use of an equation based on the elasticity theory (Timoshenko and Goodier (1951)):

 

(1)

 

where:

Ii =f(L'/B', H, μ, D) = Influence factors that depend on: L'/B', thickness of layer H, Poisson's ratio μ, bearing surface depth D

Coefficients I1 & I2 may be calculated using the equations of Steinbrenner (1934) (Bowles), as functions of the relation M=L'/B' and N=H/B, using B'=B/2 e L'=L/2  for coefficients relative to the center and  B'=B & L'=L for coefficients Ii at the edge.

 

Influence factor IF is due to Fox (1948), and suggests that settlement is reduced with depth, depending on Poisson’s ratio μ, and of the ratio L/B.

 

In order to simplify the equation (1) the coefficient IS is introduced:

 

Relationship (1) can be written as:

 

The equation can be applied to flexible or rigid foundations with appropriate changes in the value of Is.

The author, analyzing a number of cases, has concluded that the equation formulated previously, in order to to provide good results must, be applied as follows:

 

Influence factor IF for a foundation calculated at a depth D

 

Oedometric settlements

The computation of settlement with the oerdometric approach allows the evaluation of monodimensional settlement (Terzaghi-1943), produced by stresses induced by the application of a load in conditions of inhibited lateral expansion.

However the computation with this method is to be considered empiric rather than theoretic. Nonetheless the ease of use and of controlling the influence of the various parameters involved make of this a very widespread method.

With reference to the image below the settlement ΔH of a layer of an initial thickness H0 is given by:

 

 

Oedometric settlement

 

 

The process of settlement calculation goes through two phases:

 

With reference to the results of the oedometer test, the settlement is evaluated as:

 

 

If the soil is overconsolidated (OCR>1), that is if the increment of stresses due to the application of load does not cause preconsolidation pressure to exceed σ'p (σ'v0 + Δσv < σ'p)

If on the other hand the soil is normally consolidated (σ'v0=σ'p), deformation occurs in the compression interval and the settlement is calculated as:

where:

 

As an alternative to parameters RR and CR one can refer to the oedometric modulus M, however, in such case, it will be necessary to use judgment in selecting the value to use taking account the stress interval (σ'v0 + Δσv) significant for the problem in question.

 

The correct application of this approach requires:

 

Many use the formulas above to calculate settlement both for clays and sands with fine to medium granularity as the elasticity modulus is derived directly from consolidation tests. However for soils with a coarser grain, the dimensions of oedometric testers are not very significant for the global behaviour of the layer and for sands it is advisable to use penetration tests either static or dynamic.

 

 

Secondary settlement

Secondary settlement is calculated by the expression:

 

where:

 

The assumptions at the base of this method are:

 

Schmertmann settlements

Schmertmann (1970) proposed an alternative method of calculating settlement as related to the variation to the pressure bulb on deformation.

Thus Schmertmann  proposes a triangular deformation diagram where the depth at which significant deformation occurs is 4B for strip foundations and 2B for square or circular foundations.

 

Variation of the influence factor with depth

 

 

With this approach  settlement is expressed by the following expression:

Where:

where

 

Modulus Ei is assumed as 2.5 qc for square/round foundations and 3.5 qc for strip foundations. For intermediate cases the value is interpolated dependent on the value of L/B.

The term qc in the determination of Ei is the CPT tip resistance.

The expressions for C1 and C2 are:

 

 

Where t represents the time in years, after completion of the structure, for which settlement is calculated.

 

 

Burland and Burbidge settlements

There where dynamic penetration test results are available, it is possible to rely on Burland and Burbidge (1985) method for settlement computation for which an index of compressibility Ic is correlated to the result NSPT of the dynamic penetration test. The formula proposed by the authors is:

 

Where:

q'                gross effective pressure;

σ’v0                effective vertical stress at footing depth;

B                width of the foundation;

Ic                compressibility index;

 

Ic, compressibility index is related to the average value NAV of NSPT at a significant depth zi:

 

 

To calculate the value of zi it is used the following relationship:

 

 

Trend of the significant depth as a function of the foundation base

 

As regards the NSPT values to use in calculating the average NAV, it is opportune to remember that values should be corrected for sands with silt content under the water table and NSPT >15 as indicated by Terzaghi & Peck (1948):

 

 

Where Nc is the corrected value to use in calculation.

 

For gravel or gravelly sandy deposits, the corrected value is:

 

 

For corrective factors fS, fH ed ft the expressions are:

 

 

Where

t = time in years > 3;

R3 = a constant of value 0.3 for static loads and 0.7 for dynamic loads;

R = a constant of value 0.2 for static loads and 0.8 for dynamic loads.

 

 

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