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Mechanisms of failure for shallow foundations under earthquake loading

J.A. Knappett*, S.K. Haigh, S.P.G. Madabhushi

Schofield Centre, University of Cambridge, Madingley Road, Cambridge CB3 0EL, UK

Accepted 11 November 2004

Abstract

It is widely known that the bearing capacity of a shallow foundation is reduced when the foundation is subjected to rocking moments and

horizontal loads during an earthquake event. Analytical solutions generally require an assumption to be made of the kinematic failuremechanism in the soil, when the true failure mechanism is unknown. This paper discusses a series of experiments carried out on a new 1 g

shaking table at Cambridge University in order to measure the displacements of a shallow foundation due to seismic loading and also the

development of the failure mechanism within the soil. The failure mechanism was studied using the technique of Particle Image Velocimetry

(PIV), combined with high-speed videography and photogrammetry. In this paper, the failure mechanisms observed in these experiments will

be compared with the theoretical results found from upper- and lower-bound solutions and the effects of such parameters as earthquake

magnitude, frequency and embedment ratio (and hence surcharge) will be discussed.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Bearing capacity; Photogrammetry; Shallow foundations; Failure; Earthquakes

1. Introduction

Shallow foundations are widely used in many earth-

quake-prone regions for small to medium size structures. It

is widely known that the bearing capacity of a shallow

foundation is reduced when the foundation is subjected to

rocking moments and horizontal loads during an earthquake

event. The reasons for this in terms of the variation of the

failure mechanism involved have been studied analytically

by various methods in such papers as [1–5]. It is unknown,

however, as to what the seismic failure mechanism beneath

a foundation actually looks like, making it difficult to have

confidence in the assumed failure mechanism, which is

particularly a problem for upper-bound kinematic analyses.

This paper discusses a series of experiments carried outon a new 1g shaking table at Cambridge University in order

to measure both the displacements of a shallow foundation

due to seismic loading and also the development of

the failure mechanism within the soil. The failure

mechanism was studied using the technique of Particle

Image Velocimetry (PIV), which has been widely used in

the field of fluid mechanics to measure displacements. A

series of images was recorded of the cross-section of the

model using a high-frame-rate digital camera (up to

1000 fps). Each image was then split into patches and the

best match for each of these patches in the subsequent

images of the series was found in order to generate

displacement fields.

In this paper, the failure mechanisms observed will be

compared with the theoretical results found from upper- and

lower-bound solutions and the effects of such parameters as

earthquake magnitude, frequency and embedment ratio (and

hence surcharge) will be discussed.

2. Analytical solutions

Analytical solutions to the seismic bearing capacity

problem have been presented in the literature that provide

solutions by limit equilibrium analysis [1,2], upper-bound

kinematic analysis [3,4] and lower-bound analysis by the

method of stress characteristics [5]. All of the solutions treat

the earthquake loads/accelerations as pseudostatic, with the

soil inertia being modified by accelerations of k hg

horizontally and k vg vertically, where g is earth’s gravity.

Soil Dynamics and Earthquake Engineering 26 (2006) 91–102

www.elsevier.com/locate/soildyn

0267-7261/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.soildyn.2004.11.021

* Corresponding author. Tel.: C44 1223 742 374; fax: C44 1223 460

777.

E-mail address: [email protected] (J.A. Knappett).

http://www.elsevier.com/locate/soildyn

http://www.elsevier.com/locate/soildyn



For the purposes of this work, only the horizontal

component of earthquake shaking is considered (i.e. k vZ0).

In both [1,2], the problem was approached by

assuming that an asymmetrical Prandtl failure surface

would form, as shown in Fig. 1. For the purposes of

analysis, this was idealised to a two-part Coulomb sliding

wedge mechanism in [1], while in [2] the same

mechanism was analysed but without this simplifying

assumption. In both cases, the superstructure forces

consist of the vertical load due to the weight of the

structure and horizontal load due to the inertia of the

structure during the earthquake. The footing fails by

rotation about one corner (the right-hand corner in

Fig. 1). It will be seen from Fig. 1 that as the horizontal

acceleration magnitude increases, the failure mechanism

becomes smaller, reducing the bearing capacity.

While these limit equilibrium methods were useful firstattempts, those employing the upper-bound and lower-

bound theorems may be considered more useful as they

allow the true bearing capacity to be bounded.

By optimisation of the failure mechanisms and stress fields,

the bounds will converge, giving an accurate measure of the

bearing capacity within the stated assumptions of the model.

The upper-bound kinematic solution presented in [3] is

based on an asymmetric mechanism similar to those used in

[1,2]. As before, seismic effects in this case have been

included as an additional horizontal force applied to the

foundation due to the inertia of the structure. However, as

the centre of mass for the majority of structures will be far

above the soil–foundation interface and with this being the

point of application for horizontal inertial loads, significant

moments will also be acting on the footing due to the inertial

loads. Such moments will cause the footing to rotate about a

point not at the corner, depending on the magnitude of the

moment applied, which will lead to uplift along part of the

foundation. An upper-bound kinematic analysis including

the effects of moments has been presented in [4], where the

failure mechanism, denoted Mec 2, is modified to include

uplift by using the parameter l as shown in Fig. 2.

While upper-bound solutions are intuitive and relatively

straightforward to implement, they are by their verydefinition unconservative and a much more suitable method

for design is the lower-bound method which is generally

the more conservative of the two plasticity methods.

Unfortunately, few analytical solutions exist for the stress

characteristics problem. The solution presented in [5] only

considers horizontal inertial loading, so while it is useful to

compare with the upper-bound solution given in [3], it may

Fig. 1. Prandtl failure mechanism (after [2]).

Fig. 2. Mec 2 failure mechanism (after [4]).

J.A. Knappett et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 91–10292



not ultimately be as useful as the upper-bound method in [4]

which includes moment effects, if these are important. In

addition, the mechanism in [5] assumes that there is no

triangular trapped soil wedge beneath the footing, as shown

in Fig. 3. While the resulting stress field still satisfies

equilibrium, it has a dramatic reducing effect on the soil

self-weight component of the bearing capacity ( N gE).

Fig. 4 shows the variation in the bearing capacity factors

N gE and N qE for the upper- and lower-bound methods

presented in [3–5]. The subscript ‘E’ indicates bearing

capacity factors under earthquake loading. There is a large

discrepancy in N gE for the upper- and lower-bound

mechanisms in [3,5], respectively, due to the absence of a

triangular trapped wedge in [5]. Analysis of the static case

(k hZ0) by stress characteristics in [6] suggests that the

inclusion of such a wedge in the solution from [5] would

give a curve close to the upper-bound. N qE for these two

solutions agree well. The inclusion of moment loading,however, causes a dramatic drop in both factors as shown by

comparison between the upper-bound solutions of [3,4] for

both N gE and N qE.

3. Experimental modelling

A recently commissioned 1g shaking table at Cam-

bridge University was used to apply earthquakes to a

model strip footing on dry sand. The general arrangement

of the shaking table and the soil model is presented in

Fig. 5. The shaking table works by storing angular

momentum from a motor in a flywheel, which is then

converted to horizontal motion through a crank giving a

sinusoidal input motion at a single frequency. This is the

same principle as used on the centrifuge shaking table at

the University of Cambridge, further details of which can

be found in [7]. Earthquake frequency ( f d) can be varied

between 0 and 4.7 Hz by varying the motor speed and

magnitude is controlled by varying the length of the crank

arm, giving a possible range of 0–0.9g peak horizontal

acceleration at the highest frequency. The model

container fitted to the shaking table has a plan area of

750 mm!300 mm and is 500 mm deep as shown in

Fig. 5, with a Perspex viewing window along one long

side to permit the study of the displacements of the

structure and of the soil beneath the footing, thus enabling

the failure mechanism to be observed. Fraction B silica

Fig. 3. Stress characteristic field for lower-bound analysis (after [5]).

Fig. 4. Bearing capacity factors N gE and N qE from upper- and lower-bound

solutions.

J.A. Knappett et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 91–102 93



sand beds were prepared by dry pluviation in the

container to a nominal voids ratio of 0.6 (relative density

67%) and a depth of 300 mm. It was necessary to useFraction B sand, as finer fractions were found to give

insufficient surface texture to permit tracking of the soil

by PIV image analysis. The properties of the sand are

summarised in Table 1.

A model strip foundation and structure was constructed

consisting of a stack of brass blocks 100 mm high, 50 mm

wide and 300 mm deep, thereby spanning between the back

wall of the soil container to the viewing window. The use of

Perspex at the viewing window keeps frictional boundary

stresses here to a minimum, such that plane strain conditions

are approximated in the viewing plane. This can also be seen

in Fig. 5. The purpose of the experiments detailed here was

to observe the soil displacements beneath a strip footing at

failure (i.e. once a plastic failure mechanism has formed)

and not to determine triggering conditions for failure of real

foundations. In order to ensure failure was initiated, theinitial vertical static safety factor used were generally lower

than is typical for real footings, where it is desirable to avoid

failure. The bearing pressure exerted by the model

foundation was designed to be 8.42 kPa, and the static

safety factors are shown in Table 2. The limiting bearing

capacity used to compute the static safety factors were

determined using Terzaghi’s bearing capacity equation and

the values of N gE and N qE taken from Fig. 4 at k hZ0. The

construction of the footing allowed for the height of the

centre of gravity of the structure and the bearing pressure to

be altered, though this work is not described here. Tests

were conducted with the footing on the surface to

investigate soil self-weight effects and with the footing

embedded to examine the additional effects of surcharge. As

only dry sand models were tested, the cohesive component

of bearing capacity ( N cE) is not discussed here.

Sinusoidal earthquake input motions at a single

frequency and with a duration of 3 s were then applied to

the model as detailed in Table 2. The duration of the

earthquake was fixed by the limited image storage capacity

in the camera. For tests with the foundation on the surface of

the sand, different earthquakes were fired to examine the

effects of earthquake magnitude (prefixed by A) and

earthquake frequency (prefixed by F). For the tests with

embedded foundations, the magnitude and frequency of theearthquake were kept constant, with the embedment depth

being varied (prefixed by E). Tests A1 and E1 relate to the

same test, as do A3 and F4.

4. Image analysis

Images were obtained from the soil–structure system

throughout the earthquake events using a high-frame-rate

Fig. 5. 1g shaking table.

Table 1

Properties of fraction B sand

Property Value

fcrit 368

D10 0.84 mm

D50 0.9 mm

D90 1.07 mm

emin 0.495

emax 0.82

Gs 2.65

Table 2

Summary of test runs

Test name Maximum acceleration (g) Frequency (Hz) Embedment ratio E / B Static safety factor Voids ratio

A1 0.31 3.6 0 1.21 0.58

A2 0.26 3.6 0 1.21 0.6

A3 0.21 3.6 0 1.21 0.6

A4 0.16 3.6 0 1.21 0.6

F1 0.21 2.28 0 1.21 0.59

F2 0.21 2.55 0 1.21 0.59

F3 0.21 2.95 0 1.21 0.59

F4 0.21 3.6 0 1.21 0.6

E1 0.31 3.6 0 1.21 0.58

E2 0.31 3.6 0.25 1.66 0.60

E3 0.31 3.6 0.50 2.11 0.65




Phantom V digital video camera, manufactured by Photo-

Sonics, Inc. The camera has a resolution of one megapixel

with a storage capacity of 1000 full resolution images. The

frame-rate is adjustable from 1 to 1000 fps. The images are

stored locally on the camera due to the high-data transfer

rates that are required.

Once acquired, the images were processed using theGeoPIV7 suite of software, developed at Cambridge

University. This implements the PIV technique whereby

the images are divided into a series of patches, which are

compared between subsequent images based on texture. The

texture tracking allows the location of a patch within a pre-

defined search radius to be determined to an integer pixel

offset. A bi-cubic curve fitting procedure is then used to

interpolate the ‘best fit’ location of the patch to sub-pixel

precision, with the final position being the greatest value of

the correlation curve. This enables the patch to be located in

a subsequent image to a precision better than one-tenth of a

pixel, corresponding to 0.02 mm for the experimentsdescribed here. Displacements are then computed from the

differences in position between patches, and strain fields can

then be obtained from the displacements. This analysis

method is summarised in Fig. 6 and further details can be

found in [8,9].

The resulting displacements are in image space (units of

pixels) and calibration is required to convert the displace-

ments from pixels to millimetres. This is accomplished by

using the PIV tracks of a series of black marker dots on the

Perspex window whose position is known (in mm) relative

to the container, the locations of which were obtained using

close-range photogrammetry. As the known positions of themarkers in real space do not change between images, it is

possible to obtain calibration factors to convert between the

image displacements and real displacements for the

markers. This enables the effects of camera pose, lens

distortion, CCD aspect ratio and refraction in the Perspex

viewing window to be accounted for.

In the tests, the camera was kept stationary while the

soil container (the reference frame) moved on the shaking

table. Calibration also enabled the rigid body motion of

the reference frame relative to the stationary camera to be

subtracted from the total displacements recorded, such that

the soil movements relative to the reference frame could be

determined.

The PIV analysis described herein was undertaken using

patches of 16!16 pixels, spaced at 32 pixel centres. This

provided sufficient textural detail to give good tracking of the patches. The search area used was 36!36 pixels (i.e. a

10 pixel wide border around the initial patch). The images

were captured at full resolution at a frame-rate of 200 fps for

5 s, enabling the whole 3 s earthquake to be captured. As a

result of such a high-frame-rate the maximum patch

displacements between images were 3 pixels, which is

well within the search area used. The tracking of patches on

the structure enabled settlement and the response in rocking

and sliding to be determined to check for any resonance

effects.

5. Failure mechanisms under earthquake loading

The PIV technique enables the development of the

failure mechanism during the test to be observed which was

not possible using previously available methods such as

post-test excavation and the use of coloured sand markers,

as these showed only accumulated displacements after the

entire event. The displacements occurring beneath the

foundation for test A2 during the first-cycle of

the earthquake are shown in Fig. 7. These displacements

are cumulative over the quarter-cycle indicated.

The results suggest that the dynamic behaviour of thefoundation varies between quarter-cycles for a sinusoidal

input motion. In Fig. 7(a), an asymmetric failure mechanism

has occurred to the left side of the foundation with the

structure rotating about a point roughly at the centreline. It

can be seen that there is little or no movement to the right-

hand side of the region that is failing, thereby showing

uplift. The earthquakes imparted by the shaking table begin

at a point of maximum displacement and hence maximum

acceleration. Therefore, during this first quarter-cycle,

Fig. 6. Principles of PIV analysis (after [7]).




the acceleration experienced by the structure is reducing

from its peak value at the start of the earthquake. As such themagnitude of the inertia force will initially be large, creating

a large moment on the footing and causing the soil to fail as

in Fig. 2. Similar behaviour can be seen in Fig. 7(c) during

the third quarter-cycle, which again starts with a point of

maximum acceleration, though in this case the acceleration

has changed direction and the displacements are larger.

During the second and fourth quarter-cycles, the

foundation rotates about one corner mobilising a wedge of

soil beneath the entire footing. This failure is similar to the

case shown in Fig. 1, where moments are not considered.

This region of behaviour begins with a point of zero

acceleration, meaning that the inertial force and therefore

moment acting on the foundation will be a minimum. For allof the A- and F-series tests, sliding displacements were

small relative to rocking displacements, indicating that

rocking is the dominant failure mode. Indeed, ultimate

failure in the tests was observed to occur due to toppling. As

a result, the effects of moment loading on the foundation

which will drive this rocking mode are likely to be

important for describing the failure behaviour of the soil.

It can be seen from Fig. 7 that the soil displacements are a

maximum during the third quarter-cycle, when the failure

mechanism is asymmetric with uplift. This is to be expected,

as uplift reduces the contact width of the foundation that

Fig. 7. Displacement field beneath the foundation from PIV for test A2.




carries the vertical load of the structure, and is therefore the

critical failure mechanism. A greater deficit between the

bearing capacity of the soil and the applied bearing pressure

will then lead to greater displacements occurring. This is also

shown by the results of [4], as shown in Fig. 4, where the

inclusion of uplift dramatically reduces the bearing capacity

of the soil. These results suggest that for a structure with a

high centre of mass, the bearing capacity factors presented in

Fig. 8. Strain field beneath the foundation from PIV for test A2.




[4] are most suitable for the analysis of bearing capacity and

that moment effects should be taken into account.

Fig. 8 shows the maximum shear strains beneath the

foundation for the four periods of the first-cycle of test A2

shown in Fig. 7. As before, strains are cumulative over the

quarter-cycle indicated. These results confirm those of the

displacement fields in Fig. 7. It will be seen that there iscomparatively little straining during the second and fourth

quarter-cycles, confirming that the critical failure mech-

anism occurs when the moments on the footing are

initially high. In Fig. 8(a), a region of high-shear strain

can be seen to occur in the area where the shear fan of the

failure mechanism would be expected. The higher strains

occurring in the third quarter-cycle suggests that the

shearing prior to this has reduced the shear modulus of

the soil, allowing for greater shearing to occur throughout

the mechanism.

6. Comparison of experimental and analytical bearing

capacity

Kinematically admissible failure mechanisms were

determined in accordance with the method presented in

[4] for comparison with those obtained from the tests. Two

different flow rules were considered representing the

extremes of plastic soil behaviour, based on the angle of

dilation (j). Associative flow (jZfcrit) is implicitly

assumed in [4]. Non-associative flow has here been

modelled by adjusting the value of f used in the analyses

from fcrit to f* as suggested in [10] and given by:

fcritZ tanK1 cos j sin f

1Ksin j sin f

(1)

The method involves varying the parameters a, b, m and

l, as shown in Fig. 2 in order to find the minimum

horizontal load (lowest upper-bound) that will induce

failure ( H min). For most of the cases tested here, failure

occurred indicating that the applied load H O H min. In these

cases, a failure mechanism was found to match this load. It

is assumed in [4] that the values of k h applicable to the soil

beneath the foundation and the supported structure are the

same. Due to the low ratio of driving frequency to system

natural frequencies ( f d / f nq) in these tests, it is to be

expected that the peak acceleration magnitude of the

structure is the same as that in the soil. From the lack of

relative displacement of the free-field soil over the entire

depth of the model (see Fig. 7 for example), it is also

demonstrated that the acceleration in the soil is equal to the

bedrock (input) motion. Tests A1–A4 confirmed these

points as dynamic amplification factors between the input

motion and the structure varied between 0.93 and 1.12.

The tests therefore comply with all of the assumptions of

the analytical model from [4], though it should be noted

that the comments concerning the acceleration magnitudes

to be used will not necessarily apply in the field due to

higher typical f d / f nq ratios (i.e. resonance effects). The

angle of friction at the footing–sand (brass/sand) interface

was taken as 158.

6.1. Effect of earthquake magnitude

The effect of earthquake magnitude is shown for the

critical failure mechanism (with uplift) in Fig. 9. It will be

seen that generally with the increase of k h, the mechanism

becomes more shallow but increases in lateral extent. This is

contrary to the theoretical solutions presented in the

literature (see [3] for example) in which the failure

mechanisms reduce both in depth and in extent as k hincreases. It is thought that this is due to the low confining

stresses near the surface of the sand bed for these 1g tests.

As expected, an increase in k h led to a larger total settlement

as shown in Fig. 9.

A summary of the results from the upper-boundkinematic analyses is shown in Table 3 and the resulting

mechanisms are overlaid onto the experimental results in

Fig. 9. If no solution could be found (i.e. H ! H min), the

foundation would not be expected to fail and no

mechanism has thus been plotted in Fig. 9 for these

cases. It can be seen that the experimental results match

most closely to the associative flow case which is to be

expected, as the low stress levels lead to high dilation (jO

0). Both flow rules suggest that test A4 should not have

failed, a fact confirmed by the very small soil displace-

ments in Fig. 9. A time series of the rocking motion for

tests A1–A4 is shown in Fig. 10. This demonstrates that

the cyclic soil failures detailed in Figs. 7 and 9 lead toeventual toppling of the supported structures. Test A4 did

not topple which confirms the lack of fully plastic failure

of the soil during the test.

6.2. Effect of frequency

The small strain natural frequencies of this single

degree of freedom (SDOF) soil–structure system were

calculated to investigate the theoretical frequency

response. The effective stress was found to be 8.83 kPa

at a reference point half the foundation width below the

foundation base. K 0 was taken to be 0.5 and wasestimated after [11] using:

K 0z1K sin f (2)

The small strain elastic shear modulus was then

estimated as proposed in [12] to be 27.6 MPa according to:

GmaxZ 100 ð3KeÞ2

ð1CeÞ

ffiffiffiffi p0

p (3)

The moment of inertia I q for the SDOF structure rocking

about the centre of the foundation base was calculated to be

3.66!10K2 kg m2 and the mass measured as 13.085 kg for




Fig. 9. Effect of earthquake magnitude on critical failure mechanism.




the sliding response. The natural frequency in the rocking

mode ( f nq) was then calculated from [13]

K ryZGmax B

3

1Kn 3:73

L

B

2:4

C0:27

1CE

BC

1:6

0:35C L B

4

E

B

2" #

ð4Þ

f nqZ1

2p ffiffiffiffiffiffiffiK

ry I q

s (5)

and that in the sliding mode ( f nx) also from [13]

K hxZGmax B

2Kn 6:8

L

B

0:65

C2:4

" #

1C 0:33C1:34

1C L B

E

B

0:8 ð6Þ

f nxZ1

2p

ffiffiffiffiffiffiffiffiK hx

m

r (7)

Here, L is the length of the strip footing and n is the Poisson

ratio, which was taken as 0.3. E is the embedment and B the

footing breadth as before.

The use of the method described above gives natural

frequencies that vary from 172 (E Z0)to 197 Hz (E Z B /2) in

the rocking mode and 147 (E Z0) to 171 Hz (E Z B /2) in the

sliding mode. As the maximum frequency earthquake used in

any test was 3.6 Hz, it can be seen that the range of excitation

frequencies used are well below those at which any resonant

phenomena might be expected. This matches well with tests

F1–F4, where little change in the response in rocking or

sliding was observed at the frequencies tested. Comparison

of the failuremechanisms in these tests showedthat there waslittle change in the magnitude of the displacements or the

shape and size of the failure mechanism.

6.3. Effect of embedment

The critical failure mechanisms for tests E1–E3 are

shown in Fig. 11, along with the predicted kinematically

admissible failure mechanisms suggested by [4]. A

summary of the analytical results is given in Table 4. This

shows that with increasing embedment/surcharge, the

failure mechanism becomes much deeper below foundation

level. By test E3 (maximum tested surcharge), the actual

soil displacements are closer to those predicted by the non-

associative flow rule, which may be due to suppression of

dilation due to the additional vertical total stress. In all of the

theoretical solutions to the bearing capacity problem, the

effects of surcharge and soil self-weight are considered to be

independent effects, which can be linearly superposed via

Terzaghi’s bearing capacity equation. It can be clearly seen

from the results presented here that the surcharge loading

modifies the contribution due to the soil self-weight beneaththe foundation level meaning that there is some inter-

dependence between the separate effects.

It will also be seen from Fig. 11 that the rigid wedge

outside the foundation of the right-hand side changes shape

so that at E Z B /2 it resembles less the failure mechanism

presented in [4] and more the horizontal translation of

the soil. As a result, the bearing capacity would be expected

to increase as the passive pressures restraining the motion of

the soil are likely to be higher than the restraining force

of the typical triangular wedge. The component of motion in

the sliding mode also causes lateral passive pressures to be

developed in the surcharge region itself, as will be seen from

the translation of this region in Fig. 11. A small active

wedge on the other side of the foundation can also be seen

for the cases E s0.

Fig. 12 shows a time series of the rocking motions for

tests E1–E3. This demonstrates that the footings in tests E1

and E2 toppled, confirming the results shown in Table 4,

Table 3

Summary of kinematic analysis results (tests A1–A4)

Test name Maximum acceleration (g) H ( N ) Associative Non-associative (jZ0)

l / B H O H min? l / B H O H min?

A1 0.31 39.7 0.42 Yes 0.05 Yes

A2 0.26 33.3 0.50 Yes 0.15 Yes

A3 0.21 26.9 – No 0.22 YesA4 0.16 20.5 – No 0.41 No

Fig. 10. Time series rocking motions in tests A1–A4.




where both flow rules suggest that the bearing capacity

should have been exceeded. The small rocking motions

displayed by test E3 along with the relatively large soil

displacements in Fig. 11 corroborate the earlier suggestion

that a critical failure mechanism based on rocking may not

be appropriate at high-embedment ratios. In [4], the authors

noted that generally real shallow foundations have higher

bearing capacity (and hence better performance during

earthquakes) than their upper-bound kinematic analyses

suggested. It may be that this observation is due to the

modifications to the failure mechanism due to surcharge

presented here.

Fig. 11. Effect of embedment on critical failure mechanism.




7. Conclusions

The failure mechanism corresponding to ultimate

seismic bearing capacity has been observed for a model

shallow (strip) footing on sand using high-speed video, PIV

and photogrammetry. The critical failure mechanism has

been shown to be similar in shape to that presented in [4].

Comparison shows a good match between the observed

bearing capacity mechanism and the analytical results of [4]

using an associative flow rule for the case when the

foundation is not embedded. Thus, it has been shown that

for structures where the centre of mass is significantly above

the foundation level that moment effects are important,

effecting a large reduction in the bearing capacity due to

uplift.It was observed that the failure mechanism changes

during each cycle of the earthquake, with the point of

rotation moving from the edges following points of zero

acceleration (maximum velocity) and towards the centre

following points of maximum acceleration (zero velocity).

It is in this latter case that uplift occurs.

As a result of the low stress levels in these 1g tests,

frequency effects were found to be insignificant as the f d / f nqratios were low, varying between 0.01 and 0.02. This will not

be the case for real foundations, or model tests in ageotechnical centrifuge, where resonance effects may

occur. Embedment/surcharge loading was found to alter

the shape and size of the failure mechanism below the

foundation level which questions the use of the superposition

principle in evaluating seismic bearing capacity. Further

investigation is required into this effect so that it may be

better quantified, ideally at the correct stress levels such as

within a geotechnical centrifuge.

References

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que 1993;43(1):181–7.

[3] Soubra A-H. Upper-bound solutions for bearing capacity of

foundations. ASCE J Geotech Geoenviron Eng 1999;125(1):59–66.

[4] Paolucci R, Pecker A. Seismic bearing capacity of shallow strip

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Fig. 12. Time series rocking motions in tests E1–E3.

Table 4

Summary of kinematic analysis results (tests E1–E3)

Test

name

Embedment

ratio E / B

H ( N ) Associative flow Non-associative

flow (jZ0)

l / B H O

H min?

l / B H O

H min?

E1 0 39.7 0.42 Yes 0.05 YesE2 0.25 39.7 0.70 Yes 0.41 Yes

E3 0.50 39.7 – No 0.42 Yes


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