The Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand

14.536 Soil Engineering

The Bearing Capacity of a Shallow Foundation, as proposed by Vesic;
The Settlement of a Shallow Foundation on Sand
Rex Radloff

Abstract: A shallow foundation must be designed not to excessively settle or reach the ultimate bearing capacity of
the subsurface. Each criterion is dependent on the footing geometry and several soil properties, which must be
accurately determined before design. Because soil properties are rather difficult to obtain, close scrutiny should be
used when interpreting laboratory or in-situ tests and the lack of doing so may lead to grossly incorrect predictions.
Once the soil properties are understood, the proper bearing capacity factors should be selected, or left out, to
calculate an accurate bearing capacity.

Several load tests were interpreted using the ASSHTO (2008) bearing capacity equation for a shallow foundation.
Results yielded a significant over prediction of bearing capacity for those footings that failed in local to punching
shear. It is believed one major contributing factor to these discrepancies resided in the addition and deduction of two
specific bearing capacity factors.

In sand, the plate load test is a good measure when predicting the ultimate bearing capacity of a shallow foundation,
though not a great deal when predicting its total settlement. However, because obtaining non-disturbed sand samples
to test in the laboratory is impractical, the plate load test needs to be reliable. In order to accurately predict the soils
behavior it is crucial to correctly interpret the raw data and make logical changes. The importance of a soils modulus
of elasticity was especially considered in respect to depth, load, and soil type.

Introduction Bearing Capacity of a
To design a shallow foundation, there must be an
Shallow Foundation
understanding of the underlying soil bearing the load.
Modes of Failure
In general, every soil will fail in the same regard i.e.
an increase in load forces settlement of the footing Depending on the depth and compressibility of the
until shear planes develop and failure occurs. soil underlying a shallow foundation, different modes
However because every subsurface and shallow of failure may arise. When a shallow foundation is
foundation is very particular, specific bearing capa- loaded, two distinct shear planes will develop directly
city factors are necessary when attempting to predict below the base and create a triangular zone (Fig. 1).
exactly how the soil will fail. As this wedge moves downward, the adjacent soil
will yield accordingly and its ultimate bearing
On the other hand, the ultimate load of a
capacity will be reached. If the soil is not
shallow foundation may lie in the degree of
compressible (not capable of filling its own voids)
settlement, and predicting this measure can be
then general shear failure will occur and the shear
especially difficult without the understanding of the
planes illustrated in Fig. 1(a) will completely develop
subsurfaces modulus of elasticity and shear modulus.
to the surface. If the soil is very compressible
If the soil is sand, the combination of the soil
(capable of filling its own voids) then volume change
properties with the addition of in-situ plate testing
is promoted and punching shear failure will occur as
can set the groundwork for settlement prediction.
illustrated in Fig. 1(c) and the additional shear planes
In both cases, too much information is will barley develop. For scenarios in between the
always welcomed; however too little information is above modes of failure, local shear failure will occur
typically is the case. Either way, it is critical to as illustrated in Fig. 1(b) and an interpolation of the
understand the reason behind settlement and bearing shear planes between general and punching shear
capacity to better predict soil strength. failure should be taken.

1


vered, and the shear planes will develop to the
ground level.

Failure of a shallow foundation resting on
incompressible soil consists of rapid settlement at its
ultimate bearing capacity. Meanwhile, the footing
may excessively tilt to one side and the soil at the
ground level will heave.

Relative Density

(1)

The relative density (Eq. 1) of a soil is directly
related to its compressibility. Soils that exhibit a high
relative density must be incompressible on the same
magnitude, and the contrary for soils with a low
relative density. For example, if a soil sample is in a
Fig.1. (a) General shear failure (b) Local shear 100% dense state (Dr = 1.0) then it would lack the
failure (c) Punching shear failure ability to be compressed. If this wasn’t the case, then
relative densities of over 1.0 would capable, which is
not feasible.

Compressible Soil Fig. 2 demonstrates an empirical relation-
ship between a soils relative density and its failure
Loose, well graded soils with soft particles tend to be mode. The graph clearly demonstrates the previously
compressible, and when semi-laterally loaded by the mentioned concepts of compressibility, relative
downward moving wedge, the soil will fill its own density, and mode of failure
voids (volume change) before applying a load to the
adjacent soil. Therefore, the shear planes witnessed in
loading an incompressible soil will cease to develop
as there is not an increase in stress of that area.

Graphical failure of a shallow foundation
resting on compressible soil consists of a few slight
deviations from the initial load-settlement curve
(~modulus of elasticity, E) as shown in Fig 1(b) and
1(c). In other words, the shallow foundation will
settle gradually until the soil acts plastically.
Typically, when a shallow foundation fails in this
regard, the ultimate bearing capacity will be taken at
a specified settlement.

Incompressible Soil

Dense, poorly graded soils with hard round particles Fig.2. Probable mode of failure for a given relative
tend to be incompressible as they have trouble filling density of the underlying soil and relative depth of
their own voids under high loads. Therefore, the the shallow foundation.
stress provided by the wedge will be transferred
throughout the subsurface, as its volume is perse-

2


Rigidity Index

The rigidity index is a means to analytically interpret
the compressibility of soil. Given as:

(2a)

In which G is the shear modulus defined as:

(2c) Fig.3. Deformation of an elastic material subjected
to a shear stress.
(2c)
where Nc, Nq, Nγ are dimensionless bearing capacity
factors, ζc,ζq,ζγ, are dimensionless shape factors,
The shear modulus in Eq. 2(c) (which is the more
ζcd,ζqd,ζγd, are dimensionless depth factors, and
desirable equation in soil mechanics) was derived
ζcc,ζqc,ζγc, are dimensionless compressibility factors.
from Eq. 2(b) and shown in Fig 3. Since the shear
modulus examines a materials resistance to a shear Bearing Capacity Factors
force, the rigidity index will yield the factor of safety
this material has against deflecting 45 degrees The following bearing capacity factors were defined
(=tan( )) when subjected to a shear stress; in this by:
case the soils shear stress at failure.
Prandtl and Reissner: (4)
Eq. 2(a) assumes a perfect elastic material
with no volume change. However, if the soil Prandtl and Reissner: ( ) (5)
undergoes a plastic deformation and a volume change
larger than 1% occurs, Eq. (2d) should be utilized Caquot and Kerisel: (6)

(2d) The factors Nc and Nq do not vary much
with φ where Nγ does significantly, hence choosing
The significance of the rigidity index lies in the correct internal friction angle is critical when
the response a material will have under a shear stress. calculating the ultimate bearing capacity. Selecting
If the response is minimal, i.e. Ir = large value, then the correct friction angle will be briefly addressed
the material will not deform under a stress, which shortly.
much imply a lack of compression. In soil, the range
Shape Factors
of the rigidity index can vary from 10 (very
compressible) to 250 and over (very incompressible). The following shape factors are defined as:

Vesic (1973) Bearing Capacity Equation
( )( ) (7)
Vesic refined the ultimate bearing capacity of a
shallow foundation resting on a cohesive-frictional ( ) (8)
(c’-φ’) soil subjected to an axial was:
( ) (9)

(3) Depending on the shape of the shallow foundation,
different modes of failure may occur which can be
traced back the ultimate bearing capacity of a soil
(De Beer)

3


Depth Factors The following compression factors are defined as

The following depth factors are defined as: If: ≤

For Df ≤ B, then: Then:

(10)

(11)
If: ≥
(12)
Then:

(17)
For Df > B, then:
,( )
(13)
* +- (18)
(14)
Where Ir = Irr (Eq. 2d) if ΔV ≥ 1.0%
( ) (15)

If: φ = 0, then:
These depth factors incorporate the shearing
ASSHTO (2008) Bearing Capacity Equation
strength of the overburden soil which increases its
ultimate bearing capacity. However, this adjustment The American Association of State Highway &
is discouraged as a shallow foundation is typically Transportation Officials (ASSHTO) has incorporated
buried with loose fill. To correct for this loss in shear the following shallow foundation bearing capacity
strength, it is suggested to use the residual frictional equation.
angle. Regardless, each case should be analyzed
separately and realistically. (19)

Compressibility Factors In most regards, Eq. 19 was based off of the Eq. 3.
However, the ASSHTO equation does not incor-
The degree a soil will compress under a given load is
porate compressibility factors and does include depth
dictated by the critical rigidity index, which is
factors, both that were respectively encouraged and
defined as:
discourages by the Vesic in Eq. (3).
, *( ) ( )+- Case Study
(16)
Table 1 presents a series of shallow foundation load
If the rigidity index is larger than the critical rigidity tests to failure. The load was applied axially and
index, then the soil will compress and the failure there was not a water table. The modes of failure for
mode will deviate away from general shear each test carried out by Muhs were determined by
respectively. interpreting the shape of the load-settlement curve
and check its consistency with the depth of the foun-

4


Table 1. – Predicted (ASSHTO 2008) versus Measured Ultimate Bearing Capacity in c-phi soils

Predicted Measured
Df B L γ φ c qf qf error
Source Case (m) (m) (m) (kPa) (deg) (kPa) Failure Type (kPa) (kPa) (%)
Muhs 1 0.0 0.50 2.00 15.69 37.0 6.40 General Shear 658 981 -33
Muhs 5 0.4 0.71 0.71 17.65 22.0 12.80 Local Shear 499 410 22
Muhs 6 0.5 0.71 0.71 17.65 25.0 14.70 Local Shear 782 550 42
Muhs 7 0.0 0.71 0.71 17.06 20.0 9.80 Punching Shear 228 220 3
Muhs 8 0.3 0.71 0.71 17.06 20.0 9.80 Punching Shear 311 260 20
Demir 9 0.0 0.30 0.30 18.00 26.0 17.00 Punching Shear 600 198 203

dation and the soils internal friction angle. For It is now possible to interpret the accuracy
example, the load settlement curve for case 2 of the ASSHTO (2008) bearing capacity equation (in
exhibited very little settlement with an increase in respect to other soil properties) on cases 1-4. Because
load and when the ultimate bearing capacity was these tests failed in general shear, the suggested
reached there was a significant amount of settlement. compressibility factors would not influence the soil,
This must indicate general shear failure which as it is relatively incompressible.
coexists with the internal friction angle of 35.3
degrees. For the load tests carried out by Demir, Table 2. – Predicted (ASSHTO 2008) versus
every failure mode was visually verified as punching Measured Ultimate Bearing Capacity in c-phi soils
shear after each test. without a depth factor.

Results show an over prediction of bearing Case Predicted Measured Error
capacity for soils that fail in punching to local shear. 2 737 1030 -32
It is speculated that this is because the ASSHTO 3 1427 2158 -35
(2008) bearing capacity equation does not incor- 4 2088 2649 3
porate compressibility factors, which influences the 5 421 410 3
failure mechanism based on the soils likelihood to 6 640 550 16
8 272 260 5
compress. However, this assessment cannot be
verified because none of the tests gave the soils
modulus of elasticity and shear modulus, and without It should be noted, that the compressibility
these soil properties the compressibility factors factors, at worse, is a conservative reduction of the
cannot be determined. projected ultimate bearing capacity. Also, the
elimination of the depth factor in table 2 cannot be
Meanwhile, the depth factors were elimi-
completely justified as the overburden soil could
nated from the ASSHTO (2008) equation and for the
have had shear strength.
applicable cases (seen in Table 2) there was a
decrease in predicted bearing capacity. For cases 5, 6,
and 8 the error seems to approach zero, while the
remaining cases yielded a larger under prediction.

5


Settlement of a Shallow Foundation [ ] (2b)
on Sand
(2c)
Principle of Consolidation √( )

When soil is subjected to an axial load a time where q = applied stress; ν = poisons ratio; z = depth
dependent pattern of settlement will occur which can of interest; R = radius of load area, and complete
be broken up and labeled as the initial, primary, and vertical stress increase as
secondary compression. The region of initial
compression is dictated by the theory of elasticity, Δσv = (Δσvc - 2νΔσhc) (3)
and is not relatively time dependent. The region of
secondary compression is a function of the rate at where Δσv = Eq. 2(a) and Δσh = Eq. 2(b) (both
which excess pore water pressure will dissipate, and conditions are for circular loads only)
depending on the soils permeability this range can
Relative strain can now be analyzed at any
vary in respect to time. As the pore water dissipates,
point throughout the entire system. However, total
the rate of settlement will coincide with the theory of
strain or more importantly total settlement cannot yet
elasticity. Finally, secondary compression will take
be determined. It is possible to estimate the
into effect and the soil will completely settle.
settlement to a specified location, but this measure is
However, this region is negligible and the following
not accurate as the stress changes with depth. Also
will not consider this range.
this method does not take into account the additional
When sand is loaded any excess pore water settlement that may occur at a further location. To
pressure will immediately dissipate and the primary consider these conditions, the following integral of
consolidation cannot be witnessed. With the lack of strain has been taken:
this range the soil will solely undergo initial cons-
ρ=∫0zεvdz→ΔqsR/E*2(1-ν2) (4a)
olidation until the final settlement has been reached.
Therefore, the rate of consolidation for sand (or any where:
granular material) is not time dependent and is a
function of the theory of elasticity. εv = 1/E (Δσvc - 2νΔσhc) (4b)

Theory of Elasticity and ρ = total settlement. The elastic theory has been
transformed to yield the total settlement of a material
A perfectly elastic material will strain when subjected
underlying the center of a uniformly circular plate.
to a stress and fully rebound when removed. The
Fig. 3(a) and 3(b) show the typical stress increase and
degree of this tendency can be defined as
settlement curve of a medium under the center of a
(1) circular plate. It should be noted, that up to this point
the subsurface was assumed to be a homogeneous,
where E = modulus of elasticity; σ = stress; and ε = isotopic, perfectly elastic material.
strain (ΔL/L). If the material being loaded is infinitely
Settlement – Plate Load Test
large in each direction then the stress will dissipate
accordingly. Fig. 1 and 2 illustrate how the developed In sand, the total settlement of a plate can be
vertical and lateral stresses will dissipate throughout immediately recorded with an increase in stress.
a medium underlying a uniform circular load. These Theoretically the absolute modulus of elasticity or
two figures are derived using the following stress poisson’s ratio can be determined with an educated
increase equations. assumption of either/or through the back calculation
of Eq. 4(a). However, when dealing with soil, this is
(2a)

6


Fig.2. Lateral stress increase throughout a medium
underlying a uniform circular load.

Change in Modulus with Initial Loading

The above principle applies; however the modulus is
a function of the present load and not depth.
Fig.1. Vertical stress increase throughout a medium
underlying a uniform circular load. Change in Modulus due to Soil Type and
Load
an overwhelming assumption as these parameters can
vary under several conditions. The following demon- Hard round uniformly granular material will maintain
strates how these values can deviate. its modulus best with an increase in load. Because its
strength and lack to fill its own void, these soils are
Change in Modulus with Depth not as susceptible to a transforming modulus.

With an increase in vertical stress, soil particles will Soft angular residual soils are highly
fill its voids and become denser as consolidation vulnerable to a change in modulus with the
takes place. Over time as soil (sedimentary) builds up application of a load. As a stress is increased, soon
on itself, the underlying subsurface will mimic this crushing will occur due the lack of strength and low
behavior and again fills its own voids. This process cross sectional area of the particle to particle contact.
will maintain the cross section of its macrostructure, This crushing will induce a large shift in the soil
but the net cross section of its microstructure has yielding a poor modulus at that load. If the loading is
increased. Therefore, the soil requires a larger load to increased the modulus will again increase until
reach the same stress to yield the same deflection. crushing between the already fractured particles
For example, the modulus of elasticity of quartz is happens again.
around 12-14 x 106 psi, yet when it is grounded and
compacted this value will drop because it is Change in Modulus with Cyclic Loading
impossible to fill every void and the net cross
sectional area is less, making a specific stress easier Hard round uniformly granular materials is not
to reach with a lesser load. It is important to note, that susceptible to cyclic loading and will yield a fairly
the absolute soil particle modulus does not change,
rather the macrostructure of the specimen.

7


loading, the theory of elasticity should be re-
evaluated to take these parameters into effect. Eq. 6
with the use of Eq. 2(a) and Eq. 2(b) models the in-
situ settlement as

∫ (6)

This equation can now approach a more realistic
prediction of how a circular shallow foundation will
settle under any load. If a rectangular shallow
foundation were to be evaluated, Eq. 2(a) and Eq.
2(b) would have to be re-derived to take these
dimensions into consideration.
(a) (b)
Methods to determine varied parameters
Fig.3. (a) Vertical and horizontal stress increase
throughout a medium directly under the center of a In-situ methods: The pressuremeter and dilator-
uniformly loaded circular plate. (b) example of meter test can measure the lateral stress ratio at any
incremental and total settlement of a soil underlying a depth, while the CPT can measure both this
uniform circular load. parameter and the modulus of elasticity which is
dependent on K0 within this method.
large rebound as the modulus is maintained. Soft
angular material will continue to crush and very little Empirical methods: A popular empirical equation
rebound will occur as the particles lost its stored by Jaky (1994) is suggested as
energy through fracturing. For the granular soil that is
(7)
found in between this range, interpolated results are
witnessed.
This relationship is very valuable due to our good
Change in Poisson’s Ratio with Depth understanding of acquiring the friction angle (ф) from
standard penetration tests. Because this is an
Poisson’s ratio is a function (given in Eq. 5) of the empirical equation, further insight to its origin should
lateral stress ratio (K0) which is a function of several be investigated and a factor of safety assigned
soil parameters. accordingly.

(5) Additional Notes

The secant modulus of the soil determined by the
Because the lateral stress ratio is influenced by the
plate load test is not adequate to use in predicting the
type of particles and initial arrangement, it can also
settlement of a shallow foundation. Because the
be a function of depth and stress these properties are;
depth of influence is much less than the shallow
similar to the modulus. Therefore, by appropriately
foundation, the stress increase at any point will
selecting the K0 in regards to depth, poisson’s ratio
deviate by the same magnitude. For example, by
becomes a function of depth.
using fig. 1 the stress increase 1 ft. under a 2 ft.
Re-evaluation of the Elastic Theory diameter plate yielding 100 lb/ft2 will be 65 lb/ft2.
The stress under the same conditions of a 6 ft.
Now that the modulus and poisson’s ratio has been diameter shallow foundation will be 95 lb/ft2. This
established as a function of its depth and current difference allows higher stresses which can change

8


the modulus between the plate and shallow References
foundation. Also, the depth of influence for the
shallow foundation is greater than the plate, allowing ASSHTO (2008). LRFD Bridge Design
deeper and possibly weaker layers to be affected by a Specifications Section 10: Foundations, p. 3.20– 3.23
stress increase.
ASTM (2003), "Specification for Plate Load
The effects of plate rigidity and the location of Testing,"
bedrock were left out of the evaluation.
Bowles, J.E., (1996). “Foundation Analysis and
Conclusion Design – Fifth Edition”

A properly designed shallow foundation should not Das, B.M., (2007). “Principles of Foundation
excessively settle or reach the ultimate bearing Engineering – Sixth Edition”
capacity of the subsurface. These measures may seem
Das, D.M., (2006). “Principles of Geotechnical
simple, but it involves the complete understanding of
Engineering – Sixth Edition”
a soils tendency to, frankly, do anything under an
applied stress. De Beer, E.E., (1965). “The Scale Effect on the
Phenomenon of Progressive Rupture in Cohesionless
When calculating the soils ultimate bearing
soils”, Proceedings of the Sixth International
capacity, it is important to predict the mode of failure
Conference on Soil Mechanics and Foundation
under the giving footing. This can be determined by
Engineering, Vol. 2A, p.13-16
making use of the soils rigidity index, which is a
function of the soils shear modulus and modulus of Demir A., M. Ornek., M. Laman., A. Yildiz., and G.
elasticity. If, by in-situ or laboratory testing, it is Misir. (2009). “Model studies of circular foundations
assumed that the soil will fail in general shear, then on soft soils”
the cohesion and internal friction angle of the soil
would be of primary concern. On the other hand, if Lambe, T.W. , and R.V. Whitman, (1969). “Soil
local or punching shear is projected to take place at Mechanics”, p.198-199, M.I.T
its ultimate bearing capacity, then the compressibility
of the soil needs to be taken into consideration. Lee J., J. Eun., and M. Prezzi, (2008). “Strain
Influence Diagrams for Settlement Estimation of
The degree of settlement a load can induce Both Isolated and Multiple Footings in sand”, Journal
on the subsurface is also a function of the soils shear of Geotech and Geoenv Engineering, April 2008,
modulus and modulus of elasticity. However, these p.417-427
properties should be determined at varied depth to
produce the most accurate results when calculating Milovic D.M. (1965). “Comparison between the
settlement. In clay, relatively undisturbed samples Calculated and Experimental Values of the Ultimate
can be taken for laboratory testing, but in the case of Bearing Capacity”,p.142-144
sand, this is not realistic. The use of in situ tests, such
as the plate load test, standard penetration test,
pressure meter, and cone penetration test, can be very
useful in determining every soil property needed to
predict total settlement.

9

The Bearing Capacity of a Shallow Foundation, as proposed by Vesic; The Settlement of a Shallow Foundation on Sand

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