Chap 12 With Appendix 29 August 2012

7/30/2019 Chap 12 With Appendix 29 August 2012

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1122SOIL-FOUNDATION-

STRUCTURE-INTERACTION(SFSI)

12.1 INTRODUCTION

In chapter 11 we considered soil-structure interaction, but mainly from theclassical consideration of the influence of elastic behavior of the soil beneaththe foundation. We found, Example 11.1 a water tank supported on a shallowfoundation, that when truly elastic behavior the soil supporting the foundationis considered the contribution from soil-structure interaction is very small. Wethen extended this to represent earthquake induced nonlinear soil behavior byreducing the soil modulus in the manner recommended in Tables 11.1 and11.2. Even then we found that a very substantial reduction in soil modulus wasrequired to give significant soil-structure interaction effects. Later in thechapter we the made the model more realistic, Example 11.2, by including themass of the foundation block supporting the water tower. This additional mass

meant that the number of degrees of freedom was increased from 1 to 3 butthe size of the foundation required to satisfy ultimate limit state conditionscould be reduced because of the stabilizing effect of the additional weight.Once again we concluded that soil-structure interaction effects were not ofgreat significance. For this model there was more than one natural period (orfrequency) for the system but the equivalent system period is likely to be ofmost significance during an earthquake. An additional limitation of all the workin Chapter 11 is the implicit assumption that there is always full contactbetween the underside of the foundation and the supporting soil and that thereis always a reserve of bearing strength in hand. The purpose of this chapter isto present four separate strands of information. First we will consider data

from field testing of shallow foundations at a site in Albany. Nonlinearfoundation behaviour was an important part of these tests. Second, we will


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Earthquake Resistant Design of Foundations

10

introduce the macro-element idea which provides a means of allowing forbearing strength failure of the soil beneath the foundation (this imposes limitson possible actions that the foundation can sustain). Third, we will comparethe insight so obtained with information gained from modeling the foundation

as a bed of nonlinear springs that can detach and reattach as the foundationresponds to the earthquake excitation. Finally, an example is given of thedesign of a shallow foundation for an unreinforced masonry building followinga displacement approach.

12.2 SHALLOW FOUNDATION NONLINEAR PUSHOVER RESPONSE

Field experiments have been conducted at a site in Auckland with both shallowand deep foundations subject to cyclic loading. More details are given by Algieet al (2010) and M.Sadon et al (2010). The first batch of tests used an eccentric

mass shaking machine to excite the foundations with sinusoidal oscillations at arange of frequencies. Although successful there are limitations to this approachfor the following reasons. First a given level of excitation force cannot beobtained until the shaker frequency has been increased from zero to thefrequency required to generate the required force. Second the response of thesystem is measured under steady state excitation at a fixed frequency. In this

way what is obtained from the use of a shaking machine is not representativeof what happens during earthquake excitation.

An alternative, described here, is the use of snap-back testing. This test issimpler than using an eccentric mass shaking machine. It gives the response of

the system to one impulsive excitation instead of continuous excitation; it ismore representative of what occurs during an earthquake. An added bonus isthe static load-deflection curve obtained during pull-back phase of the test. Theinitial pull-back can generate a force of comparable magnitude to themaximum force that can be produced by the shaking machine used.

Results are presented for the nonlinear stiffness and damping of shallowfoundations from snap-back testing. Tests were done at a site with Aucklandresidual clay. A sequence of snaps from different initial loads shows how thenonlinear behaviour of the foundation develops as the applied load increases.

The site used for the tests, in Albany in the northern part of Auckland, consistsof a profile of stiff cohesive soil formed by in situ weathering from tertiary agesandstone and siltstone (it is thus a residual soil profile). There were eightshallow foundations, which support the ends of the steel frame shown inFigure 12.1a; these are reinforced concrete 2.0 m in length and 0.4 m square.

The steel frame structure is 2 m wide, 3.5 m high and 6 m long. Steel kentledgeis strapped to the top of the frame to provide the required vertical foundationload. The soil profile was investigated with 21 CPT tests between the surfaceand depth of 5 or 8 m; in some of these the shear wave velocity of the soil wasmeasured. The su values obtained from the CPT qc values are reasonablyconsistent with depth at about 100 kPa. Hand shear vane testing also done.

The shear wave velocity measurements from the seismic cone penetration testswere supplemented with WAK tests (Briaud and Lepart 1990) and SASW tests


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Chapter 12: Soil-Foundation-Structure-Interaction

11

(Stokoe et al 1994, Stokoe et al 2004). All of these indicated a reasonablyconsistent shear wave velocity for the materials at the site equivalent to a smallstrain shear modulus for the soil of about 40 MPa.

(a) Shallow foundation mounted demountable steel structure

(b) Snapbacktestarrangement,showingthereactioncrane,chainstoapplythepullbackforce,thesteelframewithsteelkentledgeinplace.

Figure 12.1 Shal low found at ion c onf igura t ion and set -up for snap -ba c ktesting


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12

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

Cone Resistance, q c MPa

Depth(m)

0.00 0.05 0.10 0.15 0.20

Sleeve friction, fc (MPa)

qc

fc

0

1

2

3

4

5

6

7

8

9

10

11

0 100 200 300 400 500

Undrained Shear Strength, su

kPa

Depth(m)

GL: 0.0m

Figure 12.2Rep resentat ive CPT da ta from the A lba ny si te.

The CPT profiles in Figure 12.2 indicate a reasonably consistent qc values forabout the upper 5 m of the soil profile. The water table is noted to be at a

depth of 5 m, this reflects the time of the year at which the CPT tests weredone (June 2009) but since the soil is cohesive it is saturated to near the groundsurface the whole year. Many of the tests were done shortly after rain so thatfree water was lying in any pockets on the ground surface (cf Figure 12.1b).

The equipment required for snap-back testing is simple. Instrumentation anddata logging equipment are, of course, the same as needed for the testing withthe eccentric mass shaking machine. To apply the snap-back force a hydraulicjack, load cell and quick release mechanism are needed. In addition a reactionforce has to be mobilized against the cable applying the snap-back force; in ourcase the crane located at the site provided this. The quick release device used

was obtained from a yacht chandler and is one of a range used for releasingspinnakers quickly. The device obtained, the largest available, had a workingforce capacity up to 100 kN. The set-up for the shallow foundation tests isshown in Figure 12.1b.

In Figure 12.3a are shown all the Test 7 static moment-rotation curvesobtained during the application of the pull-back forces. Figure 12.3b does thesame for the Test 9 pull-backs. It is apparent that there is considerablenonlinearity in the moment-rotation curves and also that the stiffness isreduced from one snap-back to the next, in particular for those tests followingthe snap-back which applies the largest moment to the system.


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0 5 10 15 20 25 30

0

20

40

60

80

100

Rotation (millirads)

Moment(kNm)

Snap 1

Snap 2

Snap 3

Snap 4

Snap 5

Snap 6

Snap 7

Snap 8

Snap 9

0 5 10 15 20

0

20

40

60

80

100


Moment(kN

m)

Snap 1Snap 2

Snap 3

Snap 4

Snap 5

Snap 6

Snap 7

Snap 8

Snap 9

Figure 12.3Mom ent - ro ta t ion c urves obta ined d uring the pu l l-ba c k p ar ts of

the snap -b a c k te sts; (a) Test 7 and (b ) Test 9.

a

b


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14

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Normalised foundation rotation

Normalisedmomentca

pacity Normalised moment capacity at applied vertical load

Best fit curve through measured moment-rotation data

Measured moment-rotation data

a

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Normalised foundation rotation

Normailsedsecantrotationalstiffness

Foundation rotational stiffness based on Gmaxb

10-5 10-4 0.001 0.01 0.1 10

0.2

0.4

0.6

0.8

1

1.2

Normalised foundation rotation (log 10 scale)Normalisedsecantrotationalstiffness

Foundation rotational stiffness based on Gmax

Measured data

c

Rotation

Moment

1

K_secant

Figure 12.4Curve- fi tted m om ent - ro ta t ion re la t ions ma tched to the rec orded

d a ta for the first three p ull-ba c ks of Test 7 a nd Test 9. (a ) m om en t-rotat ion

da ta , (b) a nd (c ) sec ant m od u lus ag a inst ro ta t ion .


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12.2.1 Interpretation of the field test data

In Figure 12.4 the data presented in Figure 12.3 for the initial three pull-backsof Test 7 and Test 9 are processed further. Each of the graphs in Figure 12.4

has the data plotted in a normalised form.

In Figure 12.4a a best-fit curve based on the stiffest test results of Test 7 and 9is shown. The reason that the stiffest response is shown is because after thefirst few snap-back tests there is degradation of the foundation stiffness,presumably because of the irrecoverable deformation of the ground beneaththe foundation. Also shown in this figure is the estimated moment capacity ofthe foundation given the applied vertical load. Figure 12.4b has plotted thenormalised rotational stiffness of the foundation. The rotational stiffness isnormalised with respect to the elastic rotational stiffness obtained from thesmall strain shear modulus of the soil (the value used to obtain the maximum

value was 30 MPa). The horizontal axis on this diagram is the normalisedfoundation rotation; the normalisation parameter being the rotation at whichthe structure would topple (for the structure tested this is 0.3 radians).However, Figure 12.4b is not particularly useful as the rotational stiffnessdecreases so rapidly at small foundation rotations. A more useful view of thisdata is given in Figure 12.4c, which is simply Figure 12.4b replotted with alogarithmic scale for the rotation axis. Also noted in Figure 12.4c is the rangeof rotations over which data was recorded. The graph shows very clearly thatthe test data covers the range of rotations over which there is a rapid reductionin rotational stiffness.

12.3 MEASURED NONLINEAR DYNAMIC EXCITATION OF SHALLOW

FOUNDATIONS

In this section the above work on the pull-back phase of snap-back tests isextended by presenting results of dynamic loading of shallow foundations the loading having been generated by the use of an eccentric mass shakingmachine and by snap-back testing. As explained above the purpose of thesefield tests is to obtain nonlinear moment-rotation data for shallow foundations.

Figures 12.5 (a) and (b) show the steady state response of one of the shallow

foundation pairs at gradually increasing frequencies generated with an eccentricmass shaker. The diagrams show that many cycles are required prior toapplying large amplitude cyclic moment. Through the origin a line is drawn

which defines the rotational stiffness of the shallow foundation based on the

small strain shear modulus of the soil ( 40 MPa) with full contact between thesoil and the underside of the foundation. Clearly all the cycles plotted indicatethat the operational soil modulus is less than 40 MPa. Also plotted in thediagram are bounds on the moment capacity defined by the bearing strength ofthe soil (Algie et al 2010) from which one can see that the maximum appliedmoment induces bearing failure in the soil beneath the foundation.

Figures 12.6 and 12.7 show snap-back results for some of the shallowfoundation tests. From Figure 12.6 the damping during the foundation rocking


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16

is evident and Figure 12.7 shows that there is energy dissipation during thefoundation rocking because of the hysteresis loops.

Figure 12.6 plots the response of snap-backs 1, 6 and 9 of Test 7. Note that the

maximum rotation for snap-back 6 is about an order of magnitude greater thanthose for the other two. The response for snap-back 6, the middle plot inFigure 12.6, shows a small amount of permanent rotation when the dynamicresponse comes to an end. The damping values, determined by logarithmicdecrement during the first half cycles, are 42% for snap-back 1, 32% for snap-back 6 and 34% for snap-back 9. These damping values are large in relation to

values usually applied in structural and foundation design. The decrease indamping value going from snap-back 1 through to 9 is a consequence of theaccumulation of permanent deformation in the soil beneath the foundation, sothe effective length of the foundation will be decreasing gradually.

Figure 12.7 presents moment-rotation information calculated from datarecorded during two of the snap-back tests. Also included in the diagrams arethe data from the initial pull-back parts of the tests. The moments werecalculated from data obtained from strain gauges attached to the legs of thesteel frame structure.

Figure 12.8 has all the damping values obtained from snap-back testing of thesteel frame on the shallow foundations. The most important feature of thisdiagram is the large values obtained for the damping parameter. The next mostsignificant feature of the diagram is the amount of scatter present. A factorcontributing to this will be the accumulation of permanent deformation beneath

the shallow foundations as the number of snap-back tests on a particularfoundation increases. This scatter will be related to the fact the moment-rotationplots in Figure 12.3, coming from the same foundation, do not all fall along thesame curve. Centrifuge tests on rocking foundations have also shown no apparentrelationship between the damping value and the amplitude of rotation, Gajan andKutter (2008).

In a famous paper, Housner (1963), a relationship was given between the periodof a rigid block rocking on a rigid surface and the initial angle of rotation. The

early part of Housners curve is plotted in Figure 12.9 (the parameter on thehorizontal axis is the angle of tilt at which the block would fall over, for the

structure shown in Figure 12.1 this angle is about 0.3 radians). Also plotted inFigure 12.9 are the half periods measured during the rocking response in the snap-back tests which are seen to fall along the Housner curve.


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-10 -8 -6 -4 -2 0 2 4 6 8 10

-60

-40

-20

0

20

40

60


Moment(kNm)

Bearing strength bound


Rotational stiffness based on

small strain modulus of the soil

-4 -3 -2 -1 0 1 2 3 4-120

-80

-40

0

40

80

120


Moment(kNm)



Rotational stiffness based on

small strain modulus of the soil

Figure 12.5Stead y-state forc ed vibrat ion response of the shal low found a t ion

struc ture; (a ) test ?, (b) te st 4.

a

b


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0 1 2 3 4 5-4

-3-2

-1

0

1

Time (secs)

Rotation(milirads)

0 1 2 3 4 5-30

-20

-10

0

10

Time (secs)

Rotation(milirads)

0 1 2 3 4 5-4-3

-2

-1

0

1

2

Time (secs)

Rotation(milirads)

Figure 12.6 Tim e h istories for sna p -b a c ks 1, 6 an d 9 of Test 7 on the sha llowfounda t ion structure. (First hal f c yc le da m ping va lues: snap -b ac k 1 = 42%,

snap -ba c k 6 = 32%, snap -ba c k 9 = 34%)

-15 -10 -5 0 5 10 15 20 25-80

-40

0

40

80

120


Moment(kNm)

-15 -10 -5 0 5 10 15 20 25-80

-40

0

40

80

120


Moment(kNm)

Figure 12.7Mom ent-rotat ion loop s for two Test 7 snap -b ac k respo nses of thesha l low founda t ion structure ( lef t: snap -b ac k 4, r ight : sna p- ba c k 5)


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10-1

100

101

102

0

10

20

30

40

50

60

Half Amplitude Rotation (millirads)

DampingRatio(%)

Test 5 east

Test 6 east

Test 6 west

Test 7 east

Test 7 west

Test 9 east

Test 9 west

Figure 12.8Dam ping va lues m ea sured dur ing the snap-b ac ks of the sha l lowfound a tion struc ture

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalised Rotation - 0/

HalfPeriod(secs)

Housner's equation

Test 6

Test 7

Test 9

Figure 12.9Housner s in it ia l rotat ion - roc king pe r iod c urve ( i s the ang le atwhic h the struc ture w ould top ple, ab out 0.3 rad ians) plot ted w ith the

m ea sured ha l f per iod s dur ing the snap -ba c k respo nses


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12.4 SHALLOW FOUNDATION MACRO-ELEMENT

In recent years several researchers have pointed out that the LRFD requirement,that no more than half the bearing strength should be mobilised during

earthquake shaking, is very conservative. Since the earthquake applies a highfrequency dynamic load it is proposed that the consequences of brief instances

when the LRFD requirement is infringed and even brief instances where thebearing strength is fully mobilized are unlikely to lead to problems. Paolucci(1997), Pecker et al (2011), Gazetas et al (2007) and (2011), Deng et al (2009),

Toh and Pender (2009), Sullivan et al (2010), Pender et al (2009), and others, allmake the point that if gains are to be realised from soil-structure interaction thennonlinear soil behaviour must be mobilised. An early contribution to this line ofthinking is the paper of Taylor et al (1981). As explained in Chapter 1 when thereis truly nonlinear behavior of the soil beneath the foundation and loss and re-establishment of contact over part of the foundation-soil interface, the term Soil-

Foundation-Structure-Interaction (SFSI) has come to replace soil-structure-interaction (SSI) which has the connotation of elastic soil behaviour.

A macro-element is a single computational entity that represents the behaviourof a shallow foundation and the adjacent soil (Figure 12.10). The approach isthe antithesis of the finite element technique which requires discretisation of asubstantial volume of soil beneath the foundation and is therefore not apractical design tool. From the macro-element, only the response at arepresentative point (the foundation centre) is obtained, but this gives a usefulapproximation to the foundation response. In this way the macro-element is apractical option for modelling soil-foundation-structure interaction.

The model used herein was developed by Toh (2008). It is a version of themacro-element model developed by Paolucci (1997). Although the model issimple compared to other shallow foundation macro-elements (for example,Cremer et al, 2001, and Chatzigogos et al, 2007), it provides results thatcompare well to those from experimental results and yielding spring models(Pender et al, 2009).

We saw in chapters 7 and 8 bearing strength surfaces define the combinationsof vertical load, horizontal shear, and moment that cause bearing failurebeneath a shallow foundation.


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Singledegree

offreedom

superstructuremodel

Rigidfoundation

block

Outerzonewithelastic

soilbehaviouronly

Macroelement

Nonlinear soil

a b

Figure 12.10 Macro-element model and the Euroc od e 8 und ra ined be ar ingstren g th surfa c e.

The macro-element model used in this chapter is based on the Eurocode 8bearing strength surface for cohesive soils (Figure 12.10b). The surface acts asa yield surface in that, when the actions locus lies inside the surface, behaviouris considered to be elastic, while, when the actions locus reaches the surface,bearing failure occurs and behaviour is considered to be perfectly plastic. Inthis manner, the macro-element is able to capture nonlinear foundation

response.

12.4.1 Example design scenario

To demonstrate the relative performance of the three alternative designapproaches considered, the macro-element was used in an example designscenario to model a bridge pier founded on clay with an undrained shearstrength of 100 kPa. The static factor of safety was varied by changing the sizeof the square, surface foundation. Three foundation approaches are compared:standard LRFD with strength reduction factor of 0.5, a variation in whichbearing strength failure is just prevented under the seismic actions, and the

third is to proportion the foundation to achieve a bearing strength factor ofsafety under static loading of 3 and allow bearing failure during earthquakeloading (this is referred to as a yielding foundation below).

The macro-element used requires minimal input parameters, all of which areeasily obtained. Table 12.1 presents the structural parameters used in themodel. The structural parameters were selected to represent a hypotheticalbridge pier that behaves as a single-degree-of-freedom oscillator in thetransverse direction. The foundation stiffness and damping values are varieddepending on the size of the foundation, and were calculated using formulaepresented by Gazetas (1991).


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Table 12.1 Structural parameters for example design scenario.___________________________________________________Height of superstructure centre of gravityabove centre of foundation 15 mEffective mass of superstructure 2 106 kgMass of foundation 5 105 kg

Rotational inertia of superstructureand foundation 5.15 106 kg.m2Horizontal stiffness of superstructure 4.94 105 kN/mHorizontal damping of superstructure 6280 kN.s/m

___________________________________________________

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Period (seconds)

Spectral

acceleration(g)

El Centro, 1940 (scaled)

NZS1170.5 design spectrum

(Christchurch, shallow soil)

Figure 12.11 Response spect rum of ear thquake record over la id onNZS1170.5 de sign sp ec trum .

An earthquake acceleration time history from the 1940 El Centro earthquake

(scaled up marginally) was used as the dynamic input to the model. Theresponse spectrum of this time history (Figure 12.11) closely represents thedesign response spectrum at a shallow soil site for an Importance Level 4structure (an essential facility or one with post-disaster function) inChristchurch, New Zealand, as specified by NZS1170.5:2004. In a real designscenario, more than one earthquake record would usually be adopted (forexample, under NZS1170.5:2004, three or more records must be used). Forclarity, only one record was used in this example scenario.

Behaviour under alternative design approaches

Figure 12.12 shows the path followed by the actions locus in [M, H] space (i.e.within a section of the bearing strength surface at a constant vertical load V)for all three design approaches considered. Overlaid are the outlines of thedifferent sized bearing strength surfaces of each design approach.

For this particular example scenario, a static factor of safety of three wasadopted for the yielding foundation design approach, although it is noted thatany foundation with a seismic factor of safety less than one will experiencesome yielding. Figure 12.12 shows how, for the yielding foundation designapproach, occasional instances of bearing failure occur during the earthquake,as the actions locus momentarily reaches the bearing strength surface.

It was found that a static factor of safety of eight was required to only just

prevent seismic bearing failure from occurring (seismic factor of safety of


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. -300,000

0

300,000

-60,000 0 60,000

M (kNm)

H (kN)

16

83

Static FoS

Figure 12.12 Bea ring st reng th surfac e and pa th fo l low ed by foundat ionac t ions loc us under va rious de sign a pp roa c hes.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14 16 18

Static factor of safety

System

naturalperiod(s)

Y

ieldingfoundation

designapproach

SeismicFoS=1

Traditionalapproach

Fixed base structural period

Figure 12.13Natural pe riod of soi l - found at ion-struc ture system und er va riousdes ign ap p roac hes.

unity). Figure 12.12 shows the path of the actions locus almost reaching, butnot touching, the bearing strength surface for a static factor of safety of eight.

Based on this, the static factor of safety required to satisfy traditional designapproach requirements is 16. Note that the factor of safety is defined in termsof vertical action, not moment or shear, so the dimensions of the bearingstrength surface in the [M, H] plane do not double when the static factor ofsafety is doubled. This is demonstrated in Figure 12.13, which shows that for astatic factor of safety of 16, the actions locus passes further than half of thedistance from the origin of the [M, H] plane to the bearing strength surface.

The definition of factor of safety in terms of vertical loads only can thereforegive a misleading impression of the reserve of bearing strength available under

lateral loading (when the actions locus moves mainly within the [M

,H

] plane).


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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Period (seconds)

Spectralacceler

ation(g)

Static FoS = 3 (Tn 1.35 s)



Figure 12.14 Na tural pe riod o f soi l - found at ion-struc ture system c om pa redwith El Centro e arthqua ke respo nse spec trum .

Figure 12.14 shows how the natural period of the soil-foundation-structuresystem varies with static factor of safety. The natural periods were obtained byanalysing response spectra. Overlaid on the plot is the fixed-base period of thesingle-degree-of-freedom superstructure. As the static factor of safetyincreases, the soil-foundation-structure system becomes stiffer, and the systemperiod decreases towards the fixed-base period of the structure. Under thetraditional design approach, the foundation is so stiff that the system period is

very close to the fixed-base period of the structure and the effect of soil-foundation-structure interaction is minimized; as we found in Example 11.1.

The consequences of the different system natural periods obtained under thethree alternative design approaches depend on the characteristics of the designearthquake spectrum. The system natural periods for the three designapproaches overlaid on the earthquake response spectrum for this designexample are shown in Figure 12.14.

For the particular earthquake acceleration time history used, the spectralacceleration at the system natural period increases as the system natural perioddecreases. Consequently, the foundation designed according to the traditionalapproach experiences the highest spectral acceleration, whereas the yieldingfoundation experiences the lowest spectral acceleration. This has importantimplications for the foundation and structural actions, as discussed in thefollowing section.

Figure 12.15 presents plots of the structural action and foundation moment fora range of static factors of safety. The factors of safety corresponding to thethree alternative design approaches are marked. Similar behaviour is observedfor foundation shear.


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25

0

5,000

10,000

15,000

Maximumstructuralaction

(kN

)

0

50,000

100,000

150,000

200,000

250,000

0 2 4 6 8 10 12 14 16 18

Static factor of safety

Maxim

umfoundation

momentt(kNm)

Yield

ingfoundation

designapproach

Seism

icFoS=1

Traditionalapproach

Figure 12.15 Found at ion a nd sup erstruc ture a c t ions unde r var ious de signapp roaches .

Table 12.2 Foundation displacements under various design approaches.___________________________________________________Design approach Traditional Seismic Yielding

FoS=1 foundation___________________________________________________

Static factor of safety 16 8 3Elastic sliding (mm) -10 to 12 -19 to 18 -17 to 17Elastic rotation (mrad) -1 to 1 -3 to 3 -7 to 7Residual sliding (mm) 0 0 2Residual rotation (mrad) 0 0 0.5Settlement (mm) 0 0 0.5___________________________________________________

As discussed above, under the traditional design approach the system issubjected to high spectral acceleration. This causes both the foundation andthe superstructure to attract more load than under the other two designapproaches. The reason why such a high static factor of safety is required toprevent any yielding during this particular earthquake is that as the foundation

size is increased, it becomes stiffer and attracts more load. This in turn requiresthe foundation to be sized even larger to prevent bearing failure fromoccurring under the higher attracted load. Significant inefficiencies can resultfrom this process of chasing ones tail.

Under the design approach where the foundation is permitted to yield duringan earthquake, the system is comparatively much less stiff, with a longer naturalperiod. It is subjected to lower spectral acceleration, and as a result, thefoundation and superstructure both attract less than half the loads of those inthe traditional design approach.

Note that behaviour different to that discussed could occur for earthquakeswith different shaped response spectra.


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Permanent foundation displacements

Although the yielding foundation design approach may have the benefit oflower foundation and structural loading when compared with the traditionaldesign approach, there is of course the consequence of accumulated permanent

foundation displacement as a result of the instances of bearing failure. Table12.2 presents the range of elastic displacements and the residual plastic(permanent) displacements for each of the three design approaches.

The permanent displacements under the yielding foundation design approachare quite modest, especially when compared with the elastic displacements.

Therefore, the question must be asked: is it time to reassess the traditionaldesign approach for earthquake resistant shallow foundations? It is proposedthat a yielding foundation that experiences some permanent displacement maybe acceptable if the displacements are modest, and if significant benefits can beobtained in terms of reduced foundation and superstructure actions.

12.4.2 Design criteria for yielding foundations

This section uses numerical modelling to demonstrate the type of foundationbehaviour that may be expected in relation to each of the performance aspectsdiscussed in the previous section. The results of the modelling are used todiscuss how criteria may be developed for each aspect. For the modelling, thefoundation from the example discussed above with a static factor of safety ofthree was used. Table 12.3 presents the soil stiffness and damping values that

were calculated for the foundation using formulae presented by Gazetas(1991). The dynamic inputs for the modelling were 20 different earthquakerecords. For the purposes of illustration, and to capture a wide range of

foundation responses, the earthquake records were selected indiscriminately,and do not represent a single type of site condition.

Foundation rotation and slidingFigure 12.16 presents scatter plots of the maximum absolute total (elastic plusplastic) rotation experienced by the foundation at any time during each of theearthquakes. The top plot presents the data against the horizontal peak groundacceleration of the earthquake, while the bottom plot presents the data againstthe spectral acceleration of the earthquake at the system natural period(approximately 1.35 seconds, Figure 12.14). Similar results were obtained forsliding. The rotation exhibits no correlation with horizontal peak ground

acceleration, but very good correlation with the earthquake spectralacceleration at the system natural period. This reinforces the idea that theentire soil-foundation-structure system should be considered in design, as eachcomponent of the system has an effect on its dynamic characteristics andnatural period. It may be possible to develop guidelines that require the systemnatural period to be sufficiently different to that of the design earthquake.


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0

10

20

30

40

0 0.5 1 1.5 2

Earthquake spectral acceleration at Tn (g)

Maximumabsolutetotal

rotation(mr

ad)

0

10

20

30

40

0 0.5 1 1.5 2

Horizontal Peak Ground Acceleration (g)

Maximumab

solutetotal

rotation

(mrad)

Figure 12.16 Residual foundat ion rotat ion versus PGA and spectra lac c e lera t ion at system natura l p er iod .

Table 12.3 Foundation stiffness and damping parameters for static factor ofsafety of three.___________________________________________________Foundation horizontal stiffness 3.3 105 kN/mFoundation rotational stiffness 1.2 107 kNm/radFoundation vertical stiffness 5.0 105 kN/mFoundation horizontal damping 1.6 104 kN.s/mFoundation rotational damping 3.5 104 kN.m.s/radFoundation vertical damping 3.5 104 kN.s/m___________________________________________________

Figure 12.17 elaborates on Figure 12.16 by separating out the differentcomponents of foundation rotation. The components considered are the

maximum absolute elastic, plastic, and total rotation experienced by thefoundation at any time during the earthquakes, and the residual plastic rotationthat remained at the end of the earthquakes. Similar results were obtained forsliding, and the points noted below are equally applicable to sliding: In this example, foundation bearing failure and associated plastic rotation

do not occur below a spectral acceleration of approximately 0.2g; during the earthquake, maximum elastic rotation is higher than maximum

plastic rotation for spectral accelerations up to approximately 0.4g to 0.5g.At such levels of spectral acceleration, performance during an earthquakemay be a more critical criterion than post-earthquake residualdisplacements, and therefore allowing a foundation to yield may not resultin performance worse than that of a non-yielding foundation;


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0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


Rotation(mrad

Maximum absolute total rotation

Maximum absolute plastic rotation

Maximum absolute elastic rotation

Absolute residual rotation

Figure 12.17 Elast ic and plast ic foundat ion rotat ion versus spectra lac c e lera t ion at system natura l p er iod .

if bearing failure occurs, the maximum elastic rotation is virtually constantregardless of the spectral acceleration. This is because the maximumfoundation actions (shear and moment) are limited by the bearing strengthof the soil;

conversely, if bearing failure occurs, maximum plastic rotation during anearthquake increases roughly linearly with spectral acceleration, which is agood indicator of foundation performance;

the maximum plastic rotation does not always equal the residual rotation, asplastic rotation can occur in both directions during an earthquake;

therefore, the residual rotation obtained from the numerical model does notreliably represent the maximum possible residual rotation of the foundation.

In many cases, the calculated residual rotation following the earthquake isless than the maximum plastic rotation that occurs during the earthquake,due to the characteristics of the particular earthquake records used;

in reality, the actual residual rotation could lie anywhere within the range ofplastic rotations that are experienced, so the calculated maximum plasticrotation during the earthquake may be a better indicator of residual rotationfor design purposes.

Foundation settlement

Figure 12.18 presents the calculated earthquake-induced foundation settlementversus spectral acceleration at the system natural period, for all of the 20

earthquake records used.


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0

50

100

150

200

0 0.5 1 1.5 2


Earthquake-inducedsettlem

ent(mm)

Elastic settlement

under static load

Figure 12.18 Earthqua ke - induc ed founda t ion set t lem ent versus spe c tra lac c e lera t ion at system natura l p er iod .

There is a non-linear (approximately parabolic) relationship between settlementand spectral acceleration. As a result, earthquake-induced settlements are zeroor negligible at low spectral accelerations. The static elastic foundationsettlement of approximately 50 mm is higher than earthquake-inducedsettlements for spectral accelerations below approximately 0.9g, which wouldusually correlate to a very strong earthquake. In fact, of the 20 earthquakesrecords used, only one caused settlement greater than the static elasticsettlement.

Design criteria could compare static and earthquake-induced settlement toassess the significance of the earthquake-induced settlement. The resultssuggest that earthquake-induced settlements are only significant under veryunfavourable conditions where strong spectral accelerations coincide with thenatural period of a soil-foundation-structure system.

12.4.3 Effect of variability of soil shear strength on macro-

element modelling

In November 2009 an international workshop was held in Auckland on Soil-Foundation-Structure-Interaction (Orense et al 2010). Some of the attendeesproposed that the beneficial effects of allowing brief instances of bearingstrength failure beneath shallow foundations during an earthquake couldoutweigh the negative effects, as discussed in the previous section. During adiscussion session the issue was raised that natural soil properties may be so

variable that this performance based design approach might not be reliable.

This section examines the implications of natural variability in the undrainedshear strength and stiffness of a cohesive soil on the earthquake response of ashallow foundation. Calculating the response of a simple structure-foundationsystem to seven different earthquake records to show the implications of soil

variability for the maximum actions imposed on a shallow foundation, the


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natural period of the soil-foundation-structure system, and permanentfoundation displacements.

The example scenario is hypothetical, but represents a simplified structural type

that has been used elsewhere to demonstrate soil-foundation-structure-interaction effects, for example by Paolucci et al (2011). The structure-foundation system was set at a Class C (shallow soil) site in Wellington. Withadequate shear strength this type of site would usually be suitable for shallowfoundations. A 1000 year return period earthquake was selected as the designearthquake, which is applicable to high value structures with a design life of50 years.

It was assumed that the site was underlain by homogeneous cohesive soil witha mean undrained shear strength of 100 kPa, which is typical of a Class C site.

A log-normal distribution was adopted to represent soil strength variability

following the examples given by Fenton & Griffiths (2008). The log-normaldistribution is commonly used when considering soil variability; in this case it isrequired to avoid negative values of undrained shear strength.

Coefficient of variation (CoV) values of 10%, 30%, and 50% (standarddeviations of 10 kPa, 30 kPa, and 50 kPa respectively) were considered (Figure12.11). Lumb (1974) indicates that typical CoV values for undrained shearstrength of clays are 20% to 50% and Baecher and Christian (2003) also give

values in the range of 20% to 50%. This means that the range of CoV valueschosen for the modelling herein covers the range of variability that can beexpected of real soil deposits.

Ultimate bearing pressure was assumed to equal 6su. The Eurocode 8, Part 5(BSI, 2005) undrained bearing strength surface shows that, for the bearingstrength mobilised under the static loading imposed by the structure-foundation system considered herein, earthquake inertial effects in the soilbeneath the foundation are negligible. Thus the Eurocode 8 surface withoutearthquake effects was used to define combinations of foundation actions(moment, shear, and vertical load) that would cause bearing failure. The macro-element model used assumed that foundation response was linear elastic whenactions are inside the bearing strength surface and perfectly plastic whenactions reached the surface (when bearing failure occurred).

The structure considered was the same bridge pier specified Table 12.1,supported by a shallow foundation, subject to horizontal earthquake loadingacting perpendicular to the bridge. A simplifying assumption was made thatstiffness was linearly related to undrained shear strength, with G = 100 su, asoil stiffness value that includes the PGA-related reduction specified in

Table 4.1 of Eurocode 8 Part 5 (BSI, 2005). Foundation elastic stiffness andradiation damping values were calculated using formulae presented by Gazetas(1991).


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0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0

T1 (s)

Recor

dsca

lefac

tor,

k1,

norma

lise

d

by

k1a

tfixe

dbaseperio

d

Arcelik

Duzce

El Centro

Hokkaido

La Union

Lucerne

Tabas

Fixedbaseperiod

Figure 12.19 Prob a b ili ty de nsity func tions for und rained she a r stren g th withlog- norma l dist ribut ion.

0.00

0.01

0.02

0.03

0.04

0.05

50 75 100 125 150 175 200

Undrained shear strength, su (kPa)

Pro

babilitydensityfunction

Coefficient of variation 10%



Meansu

Figure 12.20Rec ord sc ale fac tors for the seve n ea rthq ua ke rec ords plot tedag a inst funda m enta l system pe riod .

The seven earthquake records recommended for time history analyses for ClassC soil sites in Wellington (Oyarzo-Vera et al, 2010) were used. The earthquakerecords were scaled according to the design code NZS1170.5:2004 (StandardsNew Zealand, 2004), to approximate the design spectrum. This process

involved minimising, in a least mean square sense, the difference between theearthquake record and the design spectrum over the range of 40% to 130% ofthe fundamental period of the soil-foundation-structure system.

Because soil variability affects foundation stiffness, which in turn affectsfundamental system period, the record scale factor (k1) for each earthquake

varied with period, sometimes up to 50%. This is shown in Figure 12.19,where scale factors calculated for each earthquake record are plotted. Thisprocess was required to ensure that variability of earthquake records wasminimised as much as permitted by the design code used. The process wasrather time consuming. Because seven records were used, a family scale

factor (k2) applying to the entire suite of records was not required (because of


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the higher chance that at least one of the records exceeded the design spectrumin the period range of interest).

Calculations used all seven earthquake records, and undrained shear strengths

ranging from 50 kPa to 200 kPa (50% to 200% of the mean) at increments of1 kPa, Figure 12.20. This required a total of 1050 time history analyses, which

were reasonably quick to complete using the simple macro-element model,Toh (2008). Foundation elastic stiffness and radiation damping (and thereforesystem period), and bearing strength were calculated for each undrained shearstrength value. Earthquake record scale factors were also set on a case by casebasis, to ensure scaling appropriate to the system period.

Figure 12.21 shows the effect of soil variability on the calculated fundamentalperiod of the soil-foundation-structure system. The system period isconsiderably longer than the fixed base period, ranging from about 1 second

for an su of 50 kPa to about 1.9 seconds for an su of 100 kPa. For a low CoVof 10%, the range between which 95% of periods lie is relatively small at about0.3 seconds. For a very high CoV of 50%, this range is considerably larger atover 1 second.

Figure 12.22 presents the calculated relationship between maximum actionssupported by the foundation and soil strength. This relationship is dependentonly on the vertical foundation loads in relation to the ultimate bearingstrength. This is because, as explained above, the size of the bearing strengthsurface was not dependent on the magnitude of earthquake loading, and thefoundation was sized to achieve a static factor of safety of 3 (without

considering seismic loading). The actions increase with increasing undrainedshear strength, because the foundation can support higher loads as soilstrength increases.

The maximum plastic displacements that occurred at any time during or afterthe earthquake were considered to be the best indicator of maximum potentialpermanent foundation displacements (Toh and Pender, 2010). The calculatedresidual plastic displacement at the end of the earthquake is often less than thismaximum value, because plastic displacements can occur in both directions.

The absolute values of maximum plastic settlement and rotation of thefoundation are plotted in Figure 12.23 for all seven earthquake records. The

foundation sliding plot is not shown because it is very similar to the rotationplot.


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0%

20%

40%

60%

80%

100%

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Fundamental period of soil-foundation-structure system, T1 (s)

Cumulativepr

obability




T 1 at mean s u of 100 kPa

Fixed

base

period

Figure 12.21 Cumula t ive probab i l i t y o f fundamenta l sys tem per iod ford ifferent leve ls of soil va ria b ili ty.

0

2,000

4,000

6,000

8,000

10,000

12,000

50 75 100 125 150 175 200

Undrained shear strength, su (kPa)

Max

imum

foun

da

tion

shear

force,

H

(kN)

0

20,000

40,000

60,000

80,000

100,000

120,000

Max

imum

foun

da

tion

momen

t,M

(kNm

)

Shear

MomentMeansu

Figure 12.22 Rela t ionsh ip between maximum foundat ion act ions andund raine d shea r streng th.

The displacements are a result of complex interaction between soil strength,foundation stiffness, the natural period of the system, and the variablecharacteristics of the earthquake records. Increasing soil strength andincreasing foundation stiffness have counteracting effects. Increasing soilstrength increases the bearing strength of the foundation, which decreasesfoundation displacements. However, increasing foundation stiffness decreases

the system natural period, which (usually) leads to an increase in spectralaccelerations (i.e. moving up the right leg of the design spectrum), and anassociated increase in foundation displacements. The randomness of theearthquake records adds further complexity to the calculated displacements,and it is not possible to identify a fixed relationship between soil strength anddisplacements. The following observations are made:

The magnitudes of calculated displacements would probably be acceptablein most circumstances.

The range of calculated displacements from all seven earthquake records isfairly similar across a wide range of undrained shear strengths. The range

at the mean su value is fairly representative of the range observed at almostall other values of su. This indicates that the variability of the earthquake


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records has a more significant effect on displacements than soil strengthvariability.

There may be a slight decrease in displacements (and the range ofdisplacements) for high values of su. Further, calculated maximum elasticdisplacements during the earthquake (not shown) decreased slightly withincreasing soil strength, due to increasing foundation stiffness.

Only for high variability associated with low su values (in this case for suvalues near 50 kPa) is there an observable increase in displacements. Thesecases would have a very low static factor of safety, so there are likely to beother more significant issues (e.g. static settlement).

0

0.02

0.04

0.06

0.08

0.1

50 75 100 125 150 175 200

Se

ttlemen

t(m)

Range of calculated settlementsat mean s u of 100 kPa

0

0.01

0.02

0.03

0.04

50 75 100 125 150 175 200

Undrai ned shear strength, su (kPa)

Max.

pla

sticro

tationa

tany

time

duringo

ra

fterear

thqua

ke

(ra

d)

Arcelik Duzce El Centro Hokkaido

La Union Lucerne Tabas Average

Range of calculated rotations at

mean s u of 100 kPa

Figure 12.23 Ca lcu la ted found at ion d isp lac em ents ac ross the rang e o f so i lstreng th c onside red, for al l 7 ea rthqua ke s.


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Using the results presented in Figure 12.23, and the probability distributionsshown in Figure 12.20 and 21, foundation displacements were plotted againstthe cumulative probability of the occurrence of the undrained shear strength

values associated with the displacements. This allowed comparison between

different values of CoV. These comparisons are shown in Figure 12.24, withthe plots indicating the average displacements corresponding to each CoVconsidered. Also plotted are the minimum and maximum displacements fromall seven earthquake records. The following observations are made:

Average displacements are relatively constant for cumulative probabilitiesof su greater than about 10-20%, for all CoV values considered. Thisindicates that there is a high probability that permanent foundationdisplacements will not be significantly affected by soil strength variability.

The CoV of su has little or no observable effect on average displacements.The calculated average displacements are mostly very similar regardless ofthe CoV value. The only clear significant effect is on settlement, for veryhigh CoV values.

0%

20%

40%

60%

80%

100%

0 0.01 0.02 0.03 0.04

Max. plastic rotation at any time during or after earthquake (rad)

Cumu

lativepro

ba

bility

0%

20%

40%

60%

80%

100%

0 0.02 0.04 0.06 0.08 0.1

Settlement (m)

Cumu

lativepro

ba

bility




Average valuesMinimum

values

Maximum

values

Figure 12.24 Found at ion d isp lac em ents c om pa red wi th the c um ula t ive

probabi l i ty of the undrained shear strength values associated with thed isp lacements .


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The range between minimum and maximum displacement values for thesame CoV is much greater than the range between average values fordifferent CoVs, again demonstrating the variability of the earthquakerecords is more significant than soil strength variability.

12.5 SHALLOW FOUNDATION SPRING-BED MODELS

This section considers simplified spring-bed models for representing soil-foundation interaction; foundations that are relatively stiff are considered. Asexplained in earlier chapters on the design of shallow foundations we need toconsider vertical, horizontal and rotational stiffness. Of particular importanceis the possibility that at some stage during earthquake loading the momentsapplied to the foundation might lead to uplift of the edges of the footing withconsequent reduction in stiffness. An appealingly simple way of handling this

eventuality is to represent the soil beneath the foundation as a bed ofindependent springs which can detach and reattach as the foundation rocksback and forth. The main thrust of this section is concerned with the verticaland rotational stiffness of shallow foundations. The stiffnesses obtained usingthe spring-bed and elastic continuum models are compared. The mainconclusion is that, for a rigid rectangular foundation resting on the groundsurface, the rotational stiffness of a bed of springs is less than that of acontinuous elastic material. Furthermore, when the soil is modelled as anonlinear material the ratio of rotational to vertical stiffness gradually decreasesas bearing failure is approached.

In the above paragraph the terms relatively stiff and relatively flexible are used.These terms depend on a stiffness parameter related to the bending stiffness ofthe foundation and the elastic properties of the foundation. At this point we donot need to develop a relative stiffness parameter other than to note that theshallow foundation can usually be considered as rigid, in the sense that anybending deformation of the foundation itself will be small in relation to thedeformation of the underlying soil.

Set-out in chapter 9, are the expressions Gazetas (1990) and his co-workersdeveloped giving vertical and rotational stiffness of rigid shallow foundationsof arbitrary shape and embedment. In this chapter we consider only

foundations at the ground surface. The vertical stiffness of a rectangularfoundation, obtained by combining equations 9.18 to 9.21, is given by:

0 75

bV elastic continuum 2 2

AGLK 0 73 1 54

1 L

.

_ . . = + - n

where: KV_elastic continuum is the vertical elastic stiffness of the foundation, G is

the shear modulus of the soil, is the Poissons ratio, L is the length of thefoundation, and Ab the contact area between the underside of the foundationand the soil below.


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0 5 10 15 200

20

40

60

80

100

Footing length (m)

Ratioofrotationaltovertic

alstiffness

Elastic continuum

Discrete springs

Figure 12.25Ra tio of the rotat iona l to ve rtic a l st iffness of sq ua re rig id shallowfoot ings, for a uni form elast ic foundat ion mater ia l and the foundat ionrep resented as a b ed of disc rete spr ing s.

The rotational stiffness about an axis through the centroid of the foundation,reorganised from equations 9.28 and 9.33, is given by:

0 150 75

axiselastic continuum

3GI LK

1 B

..

_q

= - n

where: K_elastic continuum is the elastic rotational stiffness of the foundation, Iaxisis the second moment of area of the shallow foundation about the axis ofrotation, and B is the length of the axis about which the rotation occurs.

The vertical stiffness of a rectangular foundation on a bed of springs is givenby:

V springs b sK A k_ =

where: KV_springs is the vertical stiffness of the foundation on a bed of springsand ks is the spring stiffness per unit area of the foundation.

The rotational stiffness of the rectangular foundation on a bed of springs isgiven by:

( ) ( )L springs 2 L springs 2N N

2 22 3

springs fs s s

j 1 j 1

1 1K A s k 2j 1 Bs k 2j 1

2 2

_ / _ /

_q= =

= - = -

where: ks is the vertical stiffness of the bed of springs per unit area of thefoundation, s is the spring spacing (assumed to be the same in the length andbreadth directions), j is a counter, NL_springs is the number of rows of springs(assumed to be even) in the longitudinal direction of the footing (= L/s), and

Afs is the area of the foundation associated with each row of springs (=Bxs).


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In Figure 12.25 the ratio of the rotational stiffness to the vertical stiffness ofsquare foundations resting on the ground surface is plotted for both the elasticsoil and the bed of springs. These calculations were done by setting the springstiffness for the bed of springs so that, for given foundation dimensions, the

vertical stiffness was the same for both models. Figure 12.25 makes very clearthat the rotational stiffness of a shallow foundation on a bed of springs isconsiderably less than that when the foundation is on a continuous elasticmaterial.

The explanation of this difference is apparent if one considers the reactionpressure distribution beneath these foundations. For the bed of springs atevery point the reaction pressure depends only on the displacement at thatpoint, so for uniform vertical displacement of a rigid foundation there will be auniform reaction pressure. Similarly for a rotational displacement of afoundation on the bed of springs the reaction pressure distribution will be

linear as dictated by the foundation displacements. However, for a rigidfoundation on an elastic material the reaction pressure distribution is notuniform for a uniform settlement. The pressure tends to be very large at theedges and corners. The reason for this is that the strains imposed on the soilare very large at the edges of a rigid foundation. Furthermore the pressure atany point beneath the foundation influences the pressure at every other pointbeneath the foundation. The calculated pressure distribution for uniform

vertical displacement of a rigid square footing resting on an elastic layer and isplotted in Figure 12.26a. The vertical load applied to the foundation resting onsaturated clay with an undrained shear strength of 100 kPa is such that theultimate limit state LRFD inequality is satisfied (in out-of-date terminology we

would say that the bearing capacity factor of safety at this vertical load is 3).The calculation was done using the well-known solution for the verticaldisplacement of the surface of an elastic half-space when a pressure loading isapplied over a rectangular area (reproduced by Poulos and Davis 1974). In asimilar manner the pressure distribution when the foundation is subject tomoment can be calculated. The distribution when the foundation is subject tomoment superimposed on the vertical loading is plotted in Figure 12.27a. Themagnitude of the moment in this case is such that one edge of the foundationis at the point of generating negative contact pressure, that is the edge of thefoundation is about to start pulling on the soil below. Clearly this is notpossible, and the underside will start to detach from the soil.

Thus for both vertical and moment loading of rigid shallow square foundationson elastic soil the reaction pressure distribution is far from constant or linear,but concentrated towards the edges. It is the concentration of the reactiontowards the edge of the foundation that is the explanation for the rotationalstiffness of the foundation on the elastic soil being so much larger than therotational stiffness of the same sized foundation on a bed of springs (thestiffness of which is adjusted so that the vertical stiffness of the foundations isthe same). The underlying reason is that for the spring foundation there is nointeraction between the springs so that what happens at one point has nocommunication with what happens at the other points.


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If we examine the distributions of reaction pressure beneath the squarefoundation in Figure 12.26a and 12.27a with respect to the ultimate bearing

pressure (qu = 5.14x1.2xsu 617 kPa) we find that there are some locationswhere the elastic pressure exceeds this value. In Figure 12.26b and 12.27b the

pressure distributions are replotted with any values in excess of 617 kPaclipped. It is apparent that only a small number of locations near the edges ofthe footing are clipped because of local bearing failure.

So far we have found that for a rigid shallow foundation there is anincompatibility between the stiffness modelling on a continuous material andon a bed of springs. This becomes most apparent when one considers the ratioof the rotational stiffness to the vertical stiffness a bed of elastic springsgiving a smaller rotational stiffness than a continuous elastic medium.

If one is concerned only with elastic modelling this can be remedied by adding

an additional rotational spring beneath the footing, or alternatively addingadditional width to the footing and adjusting the spring stiffnesses so that the

vertical and rotational stiffnesses of the footing are the same as that given byequations 1 and 2. Both of these approaches have been tried, Wotherspoon etal 2004, Pender et al 2006. However, one can ask why go to all this trouble?Surely the correct foundation stiffness can be obtained by using separatediscrete springs for the rotational and vertical stiffness of the footing, as iscatered for in the majority of software packages for the analysis of structures.

The attraction of the bed of springs is simply that it allows modelling of theprogressive uplift of the shallow foundation under moment loading. The questto achieve a correct ratio of vertical to rotational stiffness is so that any

progressive uplift from the edges of the footing is modelled realistically.

0

2

4

6

8

10

0

2

4

6

8

10

200

200

400

400

600

600

800

800

1000

1000

X (m)

Y (m)

v (kPa)

0

2

4

6

8

10

0

2

4

6

8

10

200

200

400

400

600

600

800

800

1000

1000

X (m)

Y (m)

v (kPa)

a

b

Figure 12.26 Pressure dist ribut ion b ene ath a vert ic a l ly loa de d squ are rig idshal low founda t ion. (a) uni form elastic soi l , (b) results in (a) c l ipp ed whe nthe u l timate be ar ing pressure of the so i l exc eed ed .


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0

2

4

6

8

10

0

2

4

68

10

0

500

1000

1500

2000

X(m) Y (m

)

v(kPa)

0

2

4

6

8

10

0

2

4

68

10

0

500

1000

1500

2000

X(m) Y (

m)

v(kPa)

a b

Figure 12.27 Pressure d ist ribut ion be nea th a squ are r ig id shal low founda t ion,sub jec t to the sam e ver tica l loa d as in Figure 12.26 and m om ent ap p l iedsub seq uent ly. (a) uni form elastic soi l , (b) results in (a) c l ipp ed whe n theul tima te be ar ing pressure of the soi l exc ee de d.

In Figures 12.26b and 12.27b the pressure distributions have been clippedwhen the bearing pressures have been exceeded, these diagrams do not showthe next stage of the calculation in which the load represented by the clipped

part of the pressure distribution is redistributed so the same vertical loadcontinues to be applied to the footing.

The rotational response of a shallow foundation resting on a continuous layer,calculated with the three-dimensional finite element software Abaqus (Simula(2010)) is plotted in Figure 12.28. The soil is considered to be a uniform elasticmaterial and the foundation is resting on the surface, but the contact betweenthe foundation and the underlying material is not able to sustain tensile contactstresses so when the moment is sufficiently large uplift at one side of thefoundation occurs. As can be seen from Figure 12.28 the moment-rotationrelationship for the foundation is highly nonlinear, but, as the system is elastic,

the unloading curve follows the loading curve follows back the loading curveexactly. The plot also indicates that for small moment the rotational stiffnessobtained from Abaqus is very close to the Gazetas value given by equations9.28 to 9.33.


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-2 0 2 4 6 8 10 12 14 16 18

-40

-20

0

20

40

60

80

100

120

140


Moment(kNm)

Loading

Unloading

Figure 12.28 Mom ent-rotat ion curve for a r ig id found at ion resting on a n

ela stic lay er for whic h up lift is p ossible.

0 2 4 6 8 100

20

40

60

80

100

120

140


Moment(kNm)

Gezatas formula

Abaqus

OpenSEES - constant elastic

Ruaumoko - constant elastic

OpenSees - FEMA elastic

Figure 12.29Elast ic m om ent - ro ta t ion c urves co m pa red .

Next we need to compare the moment rotation curves obtained from Abaquswith those for a simple spring bed. This is also done in Figure 12.28. Finally,included in Figure 12.29 is the moment-rotation curve obtained if the springbed is modified in the way suggested in the FEMA 356 document where thestiffness of the springs is increased towards the edges of the foundation. Asexpected from Figure 12.25 the simple elastic spring-bed underpredicts therotation stiffness of the foundation, but the version using the FEMAprovisions matches the rotational stiffness quite well.


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We have demonstrated that when modelling the soil beneath a shallowfoundation as a bed of discrete elastic springs, with the spring stiffness set sothat the vertical stiffness of the foundation on the bed of springs is the same as

that for a footing on an elastic soil, the rotational stiffness of the foundation onthe springs is less than that of the foundation on a uniform elastic soil. Thenature of the pressure distribution beneath the foundation on an elastic soilprovides the explanation for the differing rotational stiffnesses in the case ofthe elastic soil the reactions are concentrated towards the edges of thefoundation.

Three dimensional finite element calculations with Abaqus show that theelastic rotational stiffness given by the Gazetas expressions give accuratemodelling to the elastic rotational stiffness of the foundation for smallmoments but that uplift of the foundation makes the moment-rotation curve

highly nonlinear. Also the Abaqus calculations show that the suggestion givenin the FEMA 356 document where the vertical spring stiffness is concentratedtowards the edges of the foundation gives accurate modelling of the initialmoment-rotation response.

12.6 COMPARISON OF SHALLOW FOUNDATION MODELS

Shallow foundations are, of course, feasible only in non-liquefiable soils.Liquefiable deposits will need either ground improvement to make themsuitable for shallow foundations or, more likely, mandate the use of deep

foundations. Likewise deposits of soft sedimentary clay, normally consolidatedor lightly overconsolidated, will require deep foundations. This leaves sites withstiff cohesive soils as contenders for shallow foundations during earthquakes(although deep foundations could be used here also). Residual soils andoverconsolidated sedimentary clays that would be described as stiff, or evenhard, soils are thus candidates for shallow foundations.

The conventional wisdom is that shallow foundation design is controlled bystatic settlement. This statement is undoubtedly true when the long termbehaviour of a foundation under static load is being considered, andparticularly when one allows for the increase in bearing strength caused by the

consolidation of the soil beneath from the stresses generated by the weight ofthe building and contents. Thus for larger foundations for which the staticdesign is controlled by long term settlement considerations the reserve of staticbearing strength under vertical load can be expected to be generous.

However, there are two important differences when it comes to earthquakeloading. First shear and moment are applied to the foundation during cyclicloading, consequently the bearing strength is less than that under static verticalload. Second, since we are dealing with cohesive soils, the undrained shearstrength will control the bearing strength of the foundation during theearthquake loading rather than the drained bearing strength which controls the

long term static bearing strength.


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Traditionally foundation bearing strength, in relation to the applied actions, hasbeen expressed in terms of a bearing strength factor of safety; this lumps intoone factor uncertainties associated with both the loading and the properties ofthe soil. More recent approaches have separate factors for foundation actions

and soil properties the so-called limit state design methods. These come intwo styles - Load and Resistance Factored Design (LRFD) used in NewZealand, Canada and parts of the United States and the Partial Factor approachof European origin. A typical LRFD approach requires that under the designearthquake the demand on the foundation bearing strength should not exceeda certain fraction, say 50% to 75%, of the bearing strength. In New ZealandLRFD Design is used for proportioning shallow foundations under earthquakeloading and the mobilisation of bearing strength is limited to about 50 to 60%,NZS1170.5 (Standards NZ (2004) and B1/VM4 (Department of Building andHousing (2003)). In Europe where a partial factor approach is used, theearthquake loading of a shallow foundation is restricted to mobilising about

75% of the foundation bearing strength.

Conventional SSI leads to the conclusion that for many structures the inclusionof foundation compliance effects will reduce the earthquake design actionsimposed on the structure. The concept is that if the period of the structure ison the falling branch of the response spectrum then the period of thestructure-foundation system will be longer than that for the fixed basestructure and so reduce the earthquake actions (although the opposite will betrue for a rising branch of the response spectrum). However, if one couplesthis thinking with the requirement that the bearing strength demand on thefoundation must satisfy the LRFD requirement, then the foundation size may

be limited by bearing strength considerations. This may require for tallerstructures that the plan area of the foundation needs to be larger than the planarea of the building. As the size of the foundation increases the rotationalstiffness increases rapidly and this reduces the amount of period lengtheninginduced by the SSI effects, Pender and Butterworth (2003).

The message of the above paragraph is that driven by LRFD requirements theshallow foundation size may be such that any advantage in system performancefrom soil compliance may be offset by the large foundation required. One canask, however, if brief instances of shallow foundation bearing strength failureduring the course of an earthquake are actually unacceptable. The consequence

will be some permanent displacement at the end of the earthquake. Perhaps thepermanent displacement could become the foundation design criterion in justthe same way as settlement is the controlling factor in shallow foundationdesign for static loads.

This suggestion is hardly radical, as this idea is well established in consideringthe earthquake response of earth dams and slopes (Newmark 1965) and gravityretaining structures (Richards & Elms 1979) in terms of the residual, orpermanent, displacement at the end of the earthquake. In taking this approachone admits that brief instances of failure of the system during the course of theearthquake might not be important if the permanent displacements generated

during the earthquake are modest. Perhaps one needs to extend the focus onresidual displacement to the amplitudes of the cyclic displacements during the


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earthquake as factors, for tall structures, such as interaction with adjacentstructures or damage to service connections. Even taking this extended viewthe question still remains as to whether brief instances of bearing strengthfailure during the course of an earthquake is a serious matter.

The design philosophy proposed above is not new, similar ideas have beendiscussed by Taylor et al (1981), Paolucci (1997), Cremer et al (2001), Gajan etal (2005), Ugalde et al (2007), Algie et al (2008), Anastasopoulos (2009), Penderet al (2009) and Toh and Pender (2009). All of these papers reach theconclusion that brief instances of bearing strength failure enhance theperformance of shallow foundations in that they reduce the earthquake actionsat the cost of modest residual deformations.

12.6.1 Example tower structure on a block foundation

To demonstrate that brief instances of bearing failure during cyclic loading maynot compromise a shallow foundation, this example compares the resultsobtained from dynamic tests on a shallow foundation system in the UC Daviscentrifuge with numerical calculation of the response. The foundation systemis represented using three different macro-element models.

The elastic structure rests on shallow foundations on a layer of clayconsolidated from reconstituted San Francisco Bay Mud with an undrainedshear strength of 100 kPa. The prototype dimensions of the footing are 2.67 min length and 0.63 m in width. The effective height of the mass during dynamicexcitation is 4.66 m. With these footing dimensions the static bearing strength

factor of safety is 2.8. The dynamic input applied in the centrifuge was aramped cosine wave of frequency about 1 Hertz and building to a maximumacceleration of 0.7 g over about 8 seconds. Further details of the centrifuge testand results are presented by Pender et al (2009) and Rosebrook and Kutter(2001). Figure 12.32 gives the measured response history of the centrifugemodel in terms of moment - rotation, settlement - rotation and horizontalshear force - horizontal displacement.

Below the three different elements used to models the shallow foundationstiffness and strength are described briefly; a feature of all three models is thecoupling between shear, moment and vertical loads. The elastic stiffness and

radiation damping of the foundations was calculated using the expressionsgiven by Gazetas (1991).

Foundation model 1: Bearing strength surface macro-element model

This is a version of the macro element of Paolucci (1997). A moresophisticated variant is that of Cremer et al (2001). This model is based on abearing strength surface which defines the combinations of vertical load,horizontal shear, and moment that cause bearing failure beneath a shallowfoundation. Herein the surface defined in Eurocode 8 for cohesive soils isused; it is shown in Figure 12.10. The surface shows how the amount ofmoment and shear that can be applied to a shallow foundation depends on the

vertical load. This surface acts as a yield locus, state paths inside the surface areelastic and those on the surface perfectly plastic, a non-associated plastic


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potential must also be specified. Elastic behaviour of the foundation is givenby three springs the stiffness and associated radiation damping or which arecalculated using the formulae of Gazetas (1991). Further details of the modelare given by Toh (2008) and Pender et al (2009) and Toh and Pender (2009).

Foundation model 2: Spring-bed model in the Ruaumoko software

The Ruaumoko software, Carr (2003), is capable of dynamic time historynonlinear structural analysis. One of the elements provided is a nonlinear,compression only, detachable-reattachable spring. A bed of these springsprovides the shallow foundation model which has the facility to uplift part ofthe foundation during cyclic loading and to reattach it at some stage after thedirection of motion is reversed. The springs are bilinear so yielding is possible

when the contact pressure reaches a limiting value. The details are shown inFigure 12.30. Radiation damping is included with values calculated from theGazetas expressions.

Foundation model 3: Spring-bed model in the OpenSees softwareThis spring-bed model was developed in the OpenSees software (PEER 2009)and is very similar to the bed of springs model developed in Ruaumoko, thedifference being that the springs in this model are non-linear, rather thanbilinear as in Ruaumoko. Forty eight q-z springs were used in the verticaldirection and one t-z spring in the horizontal direction. These non-linear springmodels, QzSimple1 and TzSimple1, were developed by Boulanger (2000).Backbone curves for these springs have a hyperbolic shape and hystereticdamping is generated when the direction of loading is reversed. The FEMA356 document suggests that spring-bed models should have the springsconcentrated towards the outer ends of the foundation as shown in Figure

12.31; this was how the springs were arranged for this model. The tensioncapacity of the springs was set to zero so uplift of parts of the footing could beincluded.

As the three sets of calculations were done with different software, beforeproceeding to the nonlinear calculations a check was made on the resultsobtained when the foundation was modelled with linear elastic springs onefor the vertical stiffness, one for the horizontal stiffness, and one for therotational stiffness. The stiffness values for these were estimated using therelations given by Gazetas (1991). The Youngs modulus value used inestimating the stiffness was taken as 500su, that is 50 MPa. The damping values

were also calculated from Gazetas (1991). Once it was established that all threesoftware packages gave the same elastic output the nonlinear response wascalculated.

Figure 12.33 presents the calculated response of the centrifuge model structurefor the three macro-element models. All responses are given at prototype scale.Figures 12.32 and 12.33 are presented on one page to aid comparison of theresponses. The three columns of figures on the page each have the same plot,so like outputs are found in each column. In Figure 12.33 the upper row is forthe bearing strength surface based macro-element model; the middle row in thediagram is for the Ruaumoko spring bed model; the bottom row is for the

OpenSees spring-bed model.


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Bi-linear detachable

springs Rigid beam

Excitation

RigidbeamBilineardetachable

springs

Excitation

Bi-linear detachable

springs Rigid beam

Excitation

RigidbeamBilineardetachable

springs

Excitation

Figure 12.30Ruaumo ko sp r ing -be d m ode l.

Figure 12.31 Op enSee s spr ing-b ed m od el with co nc entrat ion o f st if fness

tow a rds the o uter ed g es (after FEMA 356).

It is apparent in Figure 12.33 that all three models give a reasonablerepresentation of the observed moment-rotation and horizontal shear-horizontal deformation behaviour. Table 12.4 summarises the residualdisplacement at the completion of the centrifuge testing and calculations.

Discussion

The first model used the bearing strength surface for cohesive soils given inEurocode 8. As the vertical load on the foundation was constant the actionsare constrained to a vertical section through the bearing strength surface priorto yielding. This model produces rather boxy graphs for the moment -rotation curve and the horizontal shear horizontal displacement plots. The

reason for this is that all behaviour within the yield locus, that is the bearingstrength surface, is elastic and nonlinear behaviour occurs only when the action


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Table 12.4: Comparison of prototype scale residual displacements after the centrifuge dynamic excitation

Centrifugemeasurements Ruaumoko Macro element OpenSees

Settlement (mm) 7.1 10.2 7.7 10.1Rotation (mrad) 0.8 0.6 4.6 0.03Horizontal dis lacement mm 0.1 3.8 3.3 5.8

path reaches the bearing strength surface. It would be possible to make thestiffness within the bearing strength surface degrade with the number of cycles,or based on position within the surface as is done by Cremer et al (2001).

Another effect not accounted for in this model is uplift at the edges of the

foundation. This is the reason for the difference between the shape of thesettlement rotation plot for this model and the other two models. Upliftcould be incorporated, see for example by Cremer et al (2001), but that is anadditional complication.

The second approach used the detachable, that is no-tension, spring elementprovided in Ruaumoko. We set the maximum vertical stress on any spring (thatis the load carried by the spring divided by the tributary area) to 5.14su; whenthis pressure is reached the spring yields and from then on the vertical stiffnessis reduced. When the direction of loading reverses then the stiffness of thespring reverts to the original value. From Figures 12.32 and 12.33 it is

apparent that this model represents what was observed in the centrifugereasonably well. The moment rotation curve is plotted without the dampingcontribution. When this is included the moment rotation curve thickens inthe middle and the results is very similar to that for the OpenSees spring bedmodel.

The final point to make with regard to the Ruaumoko model relates to therotational stiffness of the bed of springs. We calculated the elastic stiffness ofour footing using the Gazetas relations. One can then determine the verticalstiffness of the bed of springs so that it is the same as that for an elastic halfspace. However, if this is done, then the rotational stiffness from the bed of

springs is considerably less than that of the half space. By adding anadditional rotational spring to the centre of the footing this deficiency iscircumvated.

In the OpenSees model, the calculated moment-rotation plot is quite similar tothe measured results. The computed settlement appears to be similar to that ofthe Ruaumoko model. On the other hand the horizontal sliding range of 27mm overpredicts the experimental sliding range of 11 mm.

Herein we have concentrated on the residual displacements at the completionof the dynamic loading. Others interested in this approach, Cremer et al (2001),

Ugalde et al (2007) and Anastasopolous (2009), have emphasized that anotherbenefit of allowing brief instances of bearing failure beneath shallowfoundations during earthquake loading is that the actions applied to thefoundation and structure will be less than those on a stiffer foundation. This inturn will lead to economies in design. However, because the foundation is lessstiff using this design approach the amplitude of the structural displacementsduring the earthquake may be as important as the residual deformations, Tohand Pender (2009).

7/30/2019 Cha

Chap 12 With Appendix 29 August 2012

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Transcript of Chap 12 With Appendix 29 August 2012