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Analytical momentrotation curves for rigidfoundations based on a Winkler model

ARTICLE in SOIL DYNAMICS AND EARTHQUAKE ENGINEERING JULY 2003

Impact Factor: 1.22 DOI: 10.1016/S0267-7261(03)00034-4

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Analytical momentrotation curves for rigid foundationsbased on a Winkler model

Nii Allotey, M. Hesham El Naggar*

Department of Civil and Environmental Engineering, Faculty of Engineering, Geotechnical Research Centre,

The University of Western Ontario, London, Ont., Canada N6A 5B9

Accepted 6 January 2003

Abstract

Analytical equations for the momentrotation response of a rigid foundation on a Winkler soil model are presented. An equation is derived

for the uplift-yield condition and is combined with equations for uplift- and yield-only conditions to enable the definition of the entire static

momentrotation response. The results obtained from the developed model show that the inverse of the factor of safety, x;has a significant

effect on the momentrotation curve. The value ofx 0:5 not only determines whether uplift or yield occurs first but also defines the

condition of the maximum momentrotation response of the footing. A Winkler model is developed based on the derived equations and is

used to analyze the TRISEE experiments. The computed momentrotation response agrees well with the experimental results when the

subgrade modulus is estimated using the unload reload stiffness from static plate load deformation tests. A comparison with the

recommended NEHRP guidelines based on the FEMA 273/274 documents shows that the choice of value of the effective shear modulus

significantly affected the comparison.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords:Subgrade modulus; Foundation uplift; Soil yield; Momentrotation; Unloadreload stiffness; Winkler model; Bearing capacity; Backbone curve;

Seismic

1. Introduction

The foundation rocking behavior could greatly contrib-

ute to the response of the supported structure to seismic

loading, and in some cases it may become the governing

factor when choosing a retrofitting scheme[1]. In the past,

seismic provisions in most codes accounted approximatelyfor the effects of foundation behavior on the structural

response (usually referred to as soilstructure interaction

(SSI) effects) by adjusting the fundamental period and

damping ratio of the structure. The implementation of the

performance-based seismic design approach requires simple

and efficient cyclic load deformation models for different

structural elements [2]. Thus, the simplified approaches

used in older codes to model SSI effects are not appropriate,

and it is necessary to develop foundation models that can

capture the most important characteristics of the foundation

cyclic loaddeformation behavior. Therefore, new seismic

design guidelines such as FEMA 273/274 [3] and ATC 40

[4]require explicit modeling of foundation elements when

determining both linear and nonlinear responses of

structures.

Foundation rocking contributes significantly to the

seismic response of a foundation for both tall slender

structures and medium-rise buildings[5].The rocking modeinvolves uplift of the foundation at one side and soil

yielding at the other side of the foundation, and generally

results in the permanent settlement of the footing. Many

researchers have investigated the nonlinear foundation

rocking action using rigorous finite element and boundary

element models [68]. However, finite element and

boundary element solutions are not efficient for nonlinear

time domain analysis since they require large computational

time and effort, and thus are not practical for regular design

purposes.

The Winkler model is widely used in SSIanalysisbecause

of its simplicity and ability to incorporate different nonlinear

aspects of the behavior at a reduced computational effortcompared to other approaches. The use of the Winkler model

0267-7261/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0267-7261(03)00034-4

Soil Dynamics and Earthquake Engineering 23 (2003) 367381www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: 1-519-661-4219; fax:1-519-661-3942.

E-mail address:[email protected] (M. H. El Naggar).
http://-/?-https://www.researchgate.net/publication/222796972_Performance-based_design_in_earthquake_engineering_state_of_development_Eng_Struct?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==https://www.researchgate.net/publication/222796972_Performance-based_design_in_earthquake_engineering_state_of_development_Eng_Struct?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==http://-/?-http://-/?-https://www.researchgate.net/publication/229710172_Dynamic_response_of_tipping_core_building?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==https://www.researchgate.net/publication/229710172_Dynamic_response_of_tipping_core_building?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==http://-/?-http://www.elsevier.com/locate/soildynhttps://www.researchgate.net/publication/222796972_Performance-based_design_in_earthquake_engineering_state_of_development_Eng_Struct?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==https://www.researchgate.net/publication/229710172_Dynamic_response_of_tipping_core_building?el=1_x_8&enrichId=rgreq-e07426c7-9c11-471c-a87d-00aeee2fceed&enrichSource=Y292ZXJQYWdlOzIzOTM1Nzk2NztBUzoyMTgxOTM3NjAxMzMxMjFAMTQyOTAzMjg1NDA3Mg==http://www.elsevier.com/locate/soildynhttp://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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has been extended to dynamic SSI applications by introdu-

cing the Beam-on-Nonlinear Winkler Foundation (BNWF)

models [9]. Filiatrault et al. [10] and Chaallal and Ghlamallah

[11] have used Winkler models to account for foundation

flexibility in their numerical analyses to study the effects of

SSI on both the linear and nonlinear responses of various

structures. A schematic for a rigid foundation on a Winklersoil is shown inFig. 1.

Bartlett[12]introduced a Winkler approach to model the

cyclic response of footings on clay. It was noted from the

results that foundation uplift occurs before soil yielding

when the static factor of safety (FS) is ,2. However, he

studied the response using a numerical approach and did not

provide any general equations to predict the response under

different footing conditions. The FEMA 273/274 guidelines

[3]for modeling foundations are based mainly on results of

Bartlett [12], which are depicted schematically in Fig. 2.

The figure shows the moment expressions for the two

extreme conditions for the soil underneath the foundation:

the ideal condition of an uplifted rigid footing supported onelastic soil at only one corner; and the condition in which the

uplifted footing is supported by a fully developed plastic

block as a result of soil yielding. The moment expressions

corresponding to these two extreme conditions are easily

estimated from simple statics.

The backbone curve of the pseudo-static cyclic moment

rotation response forms an important part of the cyclic

response of a footing, and thus, has to be accurately modeled

when analyzing the seismic response of the supported

structure. Analytical solutions for the moment rotation

response of foundations are difficult to derive because of the

complex nonlinear foundation behavior. Therefore, either a

numerical technique is used for the analysis or simplifyingassumptions are introduced. Siddharthan et al. [13] presented

a set of equations to evaluate the momentrotation response

for both uplift- and yield-only conditions of a rigid

foundation on a Winkler soil model. The equations were

derived in the context of a retaining wall foundation, and as

such, they do notconstitute all thenecessary equationsfor the

complete staticresponse of a rigid foundation. Siddharthan et

al.[13,14]assumed that the ultimate moment occurs when

uplift commences after soil yield has occurred. Their

assumption may represent an acceptable approximation for

the ultimate moment capacity of thefoundation under certain

conditions but may not be valid under all conditions.

2. Objectives and scope of work

The objective of the current study is to provide acomplete analytical solution for the static momentrotation

response of a rigid foundation resting on a Winkler soil

model. The solution by Siddharthan et al.[13]is extended to

provide additional equations in order to completely define

the entire momentrotation response curve. The different

parameters governing the moment rotation response are

examined using the derived equations. The moment

rotation response curves computed using the derived

equations are compared with experimental results found in

the literature. The analytical results are also compared with

code provisions used to estimate the moment rotation

response in order to shed some light on their level of

accuracy.

3. Derivation of state equations

The following assumptions are made in the derivation

of the state equations: the axial load is constant and acts

at the center of the footing; the moment acts about the

longitudinal axis of the footing and is computed about its

center; and the length of the footing is one unit. Fig. 3

shows a schematic of the assumed stress and displacement

conditions for various footing states. State 1 represents

elastic conditions, state 2 represents the initial foundation

uplift condition (uplift-only), state 3 represents the initialsoil yield condition (yield-only) and state 4 represents the

soil yield and foundation uplift condition. These states

correspond to different segments of the momentrotation

curve shown inFig. 2and are considered herein to derive

the foundation momentrotation response curve as

follows.

3.1. Elastic condition

This stress state, represented by segment (1) inFig. 2, is

widely used in practice for the design of footings.

Considering the kinematics for this state yields (Fig. 3a)

d0 d1x d2x 1a

Fig. 1. Schematic of Winkler soil model.

N. Allotey, M. H. El Naggar / Soil Dynamics and Earthquake Engineering 23 (2003) 367381368
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This is

qx qp 2 kvu x 2B

2

1b

whereB is the width of the footing, x the distance from the

left end of the footing to spring i (in the Winkler model), qxis the pressure at pointx;d0is the vertical distance between

the footing base center after loading and its original level

(GL) before loading,d1xandd2xare vertical distances given

by qx=kv and x 2 B=Lu; respectively, u the footing

rotation and kv the subgrade modulus. Finally, the initialstress qp P=B; where P is the axial load applied to the

footing. Summing moments about the center of the footing

gives a linear relation between moment and rotation, i.e.

MkvB

3u

12 2

whereMis the moment acting on the footing. From Eq. (2),

the commencement of foundation uplift with no initiation of

soil yield (point 2 inFig. 2) occurs when

M2l PB

6 and u2l

2P

kvB2

3

If soil yield occurs before foundation uplift, the maximumstress at one end of the footing reaches the static bearing

capacity of the footing under vertical load, qu:The onset of

this situation occurs when (from Eq. (1b))

M2uB2

6 qu2qp and u2u

2

kvBqu2qp 4a

and can also be expressed as

M2uquB

2

6 2

PB

6 and u2u

2qukvB

22P

kvB2

4b

Noting that these two limiting conditions occur onopposite sides of the footing, it can be easily shown

that soil yielding would occur before foundation uplift

when [12]

qp$qu

2 5

3.2. Initial uplift condition

This stress state is represented by segment (3) in the

moment rotation curve shown inFig. 2. The lower limit of

this segment begins from the point represented by Eq. (3).

Referring toFig. 3band considering the kinematics of this

condition, Eq. (6) can be obtainedd0 d1x d2x d2v 6a

Fig. 2. Schematic of different states of momentrotation response (after FEMA 273/274).

N. Allotey, M. H. El Naggar / Soil Dynamics and Earthquake Engineering 23 (2003) 367381 369
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where d2v B12 h2 1=2u; and h is t he r at io

of footing part not in contact with soil (length of

uplift) to the footing width. Substituting into Eq. (6a)

yields

qx kvuB12 h2x 6b

Calculating the moment at the center of the footing, the

following equation is obtained

M M2l12h 7

from which the limiting value ofM for an infinitely strong

soil is PB=2. The equation of this segment is given by[13]

M M2l 32 2

ffiffiffiffiffiffiffiffiffi2P

kvB2u

s" # or

M M2l 32 2

ffiffiffiffiffiu2l

u

r" # 8

Eqs. (7) and (8) show that the moment, M; is a linearfunction of the uplift ratio, h; and a nonlinear function of

Fig. 3. Stresses and displacements for different footing states.

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the rotation, u: When foundation uplift occurs first, soil

yield is initiated when the stress at the extreme edge

reaches the yield value, i.e. beforeM 3M2l; which is the

ultimate condition for an infinitely strong soil. This

condition occurs when [13]

M3l 3M2l 22P2

3quand u3l

q2u

2Pkv9a

which can also be expressed as

M3l PB

2 2

2P2

3qu9b

3.3. Initial yield condition

Eq. (5) shows that soil yield occurs before foundation

uplift when the initial stress is greater than half the ultimate

bearing capacity of the footing. This condition is rep-

resented by segment (4) of the momentrotation curve inFig. 2and its lower limit is defined by Eq. (4). Based on the

kinematics of this condition (Fig. 3c)

d0 d1x d2x d1u d2u 10a

whered1u qu=kv;d2u Bj2 1=2uand jis the ratio of

the footing portion on yielded soil (yielded length) to thefooting width. Substituting into Eq. (10a) gives

qx qu kvuBj2x 10b

Similar to the approach used in the initial uplift condition, it

can be shown that

M M2u12j 11

Eq. (11) shows that the moment is a linear function of the

yielded length. The expression for this segment of the

momentrotation curve is given by[13]

M M2u 32 4

ffiffiffiffiffiffiffiffiffi3M2u

kvB3u

s" # or

M M2u 32 2

ffiffiffiffiffiffiu2u

u

r" # 12

Eq. (12) shows that similar to the initial uplift condition, the

moment is inversely proportional to the square root of the

rotation. Assuming that progressive soil yield without

footing uplift continues until the ultimate condition is

reached (i.e. footing failure at which j 1), from Eq. (11),

M 3M2u: However, uplift would generally occur before

the condition ofj 1 is reached. The onset of footing upliftafter initiation of soil yield can be obtained from Eqs. (10a),

(10b) and (11) as[14]

M3u M2u 3 2

24M2u

quB2

and

u3u q2uB

12kvM2u

13a

which can also be expressed as

M3u 5PB

6 2

2P2

3qu2

quB2

6 13b

3.4. Soil yield and foundation uplift condition

All previous work reported in the literature addresses

either elastic condition, uplift- and yield-only stress states. Inthis study, the state of stress of combined soil yield and

foundation uplift is considered. An expression is derived to

describe this state represented by segment (5) of the

momentrotation curve shown in Fig. 2. This segment of

the moment rotation curve represents the foundation

response beyond the states defined by Eqs. (9a), (9b), (13a),and (13b), regardless of which occurred first, uplift or yield.

Considering the kinematics of this stressstate yields(Fig. 3d)

d0 d1x d2x d1u d2u d2v 14

Substituting into Eq. (14), the following equations are

obtained

qx kvuB12 h2x 15a

qx qu kvuBj2x 15b

Equating Eqs. (15a) and (15b) gives

hj 1 2qu

kvuB16

Calculating forces and moments at the footing center gives

P quBjkvuB

2

2 12 hj2 17a

MquB

2

2 jj2 1

kvuB3

12 12 hj2212 2h4j

17b

Substituting Eq. (16) into Eqs. (17a) and (17b) and

rearranging, the equation for the momentrotation curve

for this condition can be derived as

MPB

2 2

P2

2qu2

q3u

24kvu2

18

the limiting case foru!1 yields

limu!1

MPB

2 2

P2

2qu19

Eq. (19) gives the maximum moment capacity of the

foundation, which can also be derived by considering thefoundation equilibrium (statics) when a fully plastic stress

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block is assumed. Eq. (18) shows that the moment, M; is

inversely proportional to the square of the rotation.

Substituting Eqs. (9a) and (13a) into Eq. (19) yields the

same expressions as Eqs. (9b) and (13b). This validates the

derived momentrotation expression for this stress state.

4. Properties of momentrotation curves

The equations derived above define the static moment

rotation response curve completely for any stress state.

These equations are used to evaluate the response of

different foundations under different loading conditions. To

enable a comparison between different footings under

different response conditions, nondimensional variables

(c; x; MqB) are introduced as follows

ckvB

qu20a

xP

quB20b

MqB M

quB2

20c

wherecis a soil property and represents the ratio of the soil

stiffness to its strength; x is the inverse of the foundation

bearing capacity safety factor under vertical load, FS; andMqBis a normalized (nondimensional) moment. Using these

nondimensional variables, the complete momentrotationrelation can be expressed as

Forx# 12

MqB

cu

12 0# u#

2x

c

x

6 32 2

ffiffiffiffiffi2x

cu

s ! 2x

c # u#

1

2cx

1

2x2 x

22

1

24c2u2 u$

1

2cx

8>>>>>>>>>>>>>>>:

21a

and forx# 12

MqB

cu

12 0#u#

212x

c

12x

6 322ffiffiffiffiffiffiffiffiffiffiffi212x

cus ! 212x

c #u#

1

2c12x

1

2x2x

22

1

24c2u2 u$

1

2c12x

8>>>>>>>>>>>>>>>:

21b

Eqs. (21a) and (21b) show that the normalized moment,

MqB; is a function of only c and x: Figs. 4 and 5 showthe moment rotation response curves for a range of values of

cand x:

Fig. 4 shows the momentrotation curves for x 0:2;

and a range of practical values of c (50 1200). Small

values ofc (Fig. 4a) represent foundations supported on

strong soils such as stiff clays and dense sand, where thesoil strength is high compared to its stiffness. Such

foundations will usually have a small width. On the otherhand, large values ofc (Fig. 4b) represent foundations of

Fig. 4. Computed momentrotation curves for x 0.2: (a) for small c;(b) large c:

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large width supported on a relatively low bearing capacity

soil such as soft clay or loose sand. It is noted fromFig. 4

that the rotational stiffness of the foundation (manifested

by the slope of the momentrotation curve) increases withan increase in c: The figure also reveals that the ultimate

rotation decreases as cincreases, almost linearly, i.e. the

ultimate rotation decreased by an order of magnitude as c

increased by an order of magnitude. It is worth

mentioning here that Eurocode 7 [15] specifies 6 millirad

(mrad) as the relative rotation to cause an ultimate limit

state. From Fig. 4, it can be observed that rotation of

6 mrad represents an elastic response for the case ofc

50 and represents a nonlinear response state for the case

ofc 1200:

Fig. 5 shows the effect of x on the momentrotation

response of foundations withc 200:It can be clearly seen

from the figure that the moment response increases as x

increases until it reaches 0.5, and then declines as x

continues to increase. The insert in Fig. 5 shows that the

maximum value of MqB; which does not depend upon c;

varies withxin a parabolic manner, and attains a maximum

value of 0.125 at x 0:5: This shows that x 0:5

represents a limiting condition on the moment rotation

response of a spread rigid footing based on the Winkler soil

model.

5. Discussion

The FEMA 273/274 documents and Siddharthan et al.[13]state that the significance ofxis that its value, above

or below 0.5, indicates whether uplift of the foundation or

yielding of the soil would occur first. However, the main

significance of x 0:5 is that it defines the maximum

moment rotation response possible as shown in Fig. 5.This is further illustrated in Fig. 6, which shows initiation

of different stress states for different x values. For x

0:5; the uplift-yield portion segment follows immediately

after the elastic segment (point R). This shows that for

this case x 0:5; a yield-only or uplift-only condition

does not occur. On the other hand, uplift- and yield-only

conditions occur after the elastic condition at u 1 mrad

(point P) for x 0:1 (e.g. foundations where conditions

other than bearing capacity demands govern the design)

and x 0:9 (e.g. foundations of existing structures that

need retrofitting because of increased loads as a result of

code revisions or change in the use of structure),respectively. However, the yield-uplift condition occurs,

but at a large rotation of u 25 mrad (not shown on the

graph). For x 0:3 (typical foundation design) and x

0:7 (foundation designed to mobilize its ultimate capacity

under seismic conditions), uplift (for x 0:3) and yield

(x 0:7) initiate at u 3 mrad (point Q). The onset of

the yield-uplift condition occurs at u 8 mrad (point S).

It can thus be concluded that as x approaches 0.5 from

either side, the region where uplift-only or yield-only

occurs shrinks and the region where yield and uplift occur

expands. Based on this observation, three regions of

moment rotation responses can be postulated: uplift-

dominant region; uplift-yield region; and yield-dominantregion.

Fig. 5. Computed momentrotation curves for different values ofxfor c 200:

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Based on the ensuing discussion, the following important

observations can be made1. The momentrotation curve included in the FEMA

273/274 documents (Fig. 2) presents a seemingly different

picture to some of the inferences drawn in this section. First,

the curves for the initial uplift or initial yield conditions are

shown as two separate curves (1356 and 1456,

respectively), with the initial uplift curve lying above that

for the initial yield. This implies that for the samecvalue, a

foundation design with x, 0:5 (which leads to initial

uplift) would result in a larger moment response than the

case wherex. 0:5 (which leads to initial yield). However,

both curves are similar and the momentrotation response

is rather influenced by the absolute difference betweenxandx 0:5: Secondly, Fig. 2 shows that segment (5) of the

curve, which represents the uplift-yield condition is

asymptotic to the ultimate elastic condition, M PB=2:

This is incorrect, since no yielding occurs by definition

(infinitely strong soil).

2. Siddharthan et al. [13,14] stated that the rocking

response could be grouped into either the initial uplift

condition, or the initial yield condition. It has been shown

that the equations for ultimate moment for both conditions

(Eqs. (9b) and (13b)) are the same if formulated in terms

ofx: The results presented herein show that based on the

value of x; the momentrotation response can rather be

grouped into three categories based on dominating behaviorand not two categories based on the initiation of uplift of

yield. The correct expression for the ultimate moment has

also been derived.

6. Comparison with experimental results

6.1. Foundation rocking experiments

It is important to validate and corroborate any analytical

or numerical model by experimental evidence. Bartlett[12]

and Wiessing[16] conducted rocking tests on foundations

installed in clay and sand, respectively. The experimental

results agreed, in general, with those obtained numerically

from an elastic perfectly plastic cyclic Winkler model.

The ability of the Winkler approach developed in thisstudy to evaluate the momentrotation behavior of a

foundation is verified using available experimental results.

The model developed is used to analyze the moment

rotation response of foundations subjected to rocking action

in a laboratory testing program and the results are compared

with the measured values.

The European Commission (EC) sponsored the project

TRISEE (3D Site Effects of SoilFoundation Interaction in

Earthquake and Vibration Risk Evaluation), which included

large-scale model testing to examine the response of rigid

footings to dynamic loads. The results of these tests are of

high quality and are readily available[17]. Therefore, these

tests are analyzed using the developed model and the resultsare compared with the measured values.

Fig. 6. Computed momentrotation curves showing points of change of state.

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6.2. TRISEE experiment

The experiments involved a 1 m square footing model

embedded to a depth of 1 m, in a 4.6 m 4.6 m 3 m deep

sample of saturated Ticino sand. Ticino sand is a uniform

coarse-to-medium silica sand. The properties of the sand are

as follows:D50 0:55 mm; coefficient of uniformity, Cu

1:6;specific gravity, Gs 2:684; emin 0:579;and emax

0:931[18]. Two series of tests were performed on the model

foundation installed in two different soil samples with

relative density of 45% (low density, LD) and 85% (high

density, HD).

A vertical load of 100 and 300 kN was applied to the LD

and HD samples, respectively, before the application of the

horizontal cyclic loading phase. The imposed pressures of

100 and 300 kPa represent typical design pressures forfoundations in medium to dense sands, where the design is

usually governed by admissible settlement, and not bearing

capacity requirements. The resulting static FS under vertical

load only was found to be about five in both cases. The

cyclic loading involved three phases: Phase Ithe appli-

cation of small-amplitude force-controlled cycles; Phase

IIthe application of a typical earthquake-like time history;

and Phase IIIsinusoidal displacement cycles of increasing

amplitude. Only relevant sections of the results of the tests

would be presented for comparison purposes. Further

information on the experiments can be found in Refs.

[1720].

6.3. Comparison with TRISEE experiments

A reasonably accurate estimate of the soil subgrade

modulus, kv; is required for the analytical model to

accurately evaluate the foundation rocking response during

the loading tests. The loaddeformation results obtained

during the application of the static vertical load only (similar

to a plate loading test) were used to back figure the soil

subgrade modulus. Atkinson[21]recommended that results

of plate loading tests, when expressed in terms of an

average strain given by the settlement/width ratio, ea

rs=B;are similar to the results of a triaxial test but scaled up

by a factor of 23. Briaud and Gibbens [22] Ismael [23]

made similar observations regarding the relationship of P

versusrs=Bin plate loading tests.

Fig. 7 shows the load deformation results of the

TRISEE experiments plotted in terms of ea; along with

the stiffness values (subgrade modulus) for initial, secant

and unloadreload loading conditions as evaluated from the

test results. These values of the subgrade modulus are usedin the developed Winkler model to calculate the moment

rocking response of the foundation and the results are

compared with the measured response inFig. 8. It should be

noted that the measured response represents the envelop of

the loading cycles with gradually increasing peak amplitude

(i.e. obtained by connecting the tips of the hysteretic loops)

[24,25]. Lo Priesti et al. [26] noted that because of the

different impact of plastic strains under different loading

conditions, it is impossible to obtain the same backbone

curve for both monotonic and cyclic loading. Fig. 8shows

that the results computed using the unloadreload stiffness

gives the best agreement with the experimental results. The

initial stiffness is slightly underestimated, but the overall

response is generally satisfactory. This is expected since

cyclic loading represents an unloadreload action, and the

unloadreload stiffness is more representative of the small-

strain stiffness.

Fig. 7. Loaddeformation results from TRISEE experiments: (a) for HD tests; (b) for LD tests.

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Soil nonlinearity and creep effects significantly influ-

ence the initial stiffness. For example, the experimental

results for the LD specimen showed that the creep

settlement accounted for about 40% of the observed

settlement [18]. The comparison in Fig. 8 of moment

rotation curves (rocking stiffness) calculated using differ-

ent subgrade moduli with the experimentally determined

curve shows that rocking stiffness is grossly under-

estimated when the secant subgrade modulus is used in

the calculations. This result shows that care and judgmentare required to select an appropriate subgrade modulus.

The issue of an appropriate subgrade modulus is discussed

in Section 7.

7. Comparison with code recommendations

7.1. Code recommendations

The NEHRP FEMA 273/274 provisions for SSI in a

seismic response analysis vary according to the type ofanalysis performed. For the linear static procedure (LSP),

Fig. 8. Comparison between predicted and experimental momentrotation curves.

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a simplified method given in the FEMA 302/303 documents

[27] should be used. For the linear dynamic procedure

(LDP), equivalent elastic foundation stiffnesses are used.

For the nonlinear static procedure (NSP), static force

deformation curves are used to model the foundation

response. Finally, complete cyclic force deformation

curves are used to model the foundation in the nonlinear

dynamic procedure (NDP).

The recommended forcedeformation curve is a bilinear

momentrotation relationship, and will be referred to as the

bilinear model herein. The bilinear model accounts for soil

variability and difficulty in accurately determining the

foundation loads and other factors by introducing upper and

lower bounds. The upper and lower bounds are specified as

twice and one half the best estimates of stiffness and

strength. The best stiffness estimates are based on the elastichalfspace solutions[28], the best bearing capacity estimate

is based on bearing capacity of a shallow footing under

vertical load [29] and the ultimate moment capacity is

evaluated using Eq. (19).

7.2. Discussion of factors affecting estimation of the shear

modulus

7.2.1. Mean effective pressure

The FEMA 302/303 documents state that the mean

effective pressure, sm; should be estimated using both the

overburden and applied pressures, while the FEMA 273/274

documents stipulate the use of the overburden pressure only.The assumed valuesm may have a significant effect on the

small-strain soil modulus, E0; calculated from expressions

that relate the soil modulus to the mean effective pressure,

e.g.[30]

E0 15102:172 e2

1e s0m0:53p

0:47a 22

It should be noted that Eq. (22) was developed based on

triaxial tests performed on Ticino sand that was used in the

TRISEE experiments. The variation of the calculated shear

modulus with the assumption used to calculate sm for the

soil sample used in the TRISEE experiment is shown in

Table 1. The difference between G0 (calculated from E0assuming isotropic elastic conditions) based on the two

assumptions within 1 m below the foundation (i.e. one B;

where most of stress increase and deformation occur due to

applied vertical load) is around 50% for the HD case. Since

the FEMA 273/274 specifies a lower bound based on one

half of the best estimate of stiffness and bearing capacity to

cover all sources of uncertainty, care must be exercised

when calculating the mean effective pressure.

7.2.2. Shear modulus reduction curve

Equivalent linear methods that make use of the secant

stiffness (i.e. reduced modulus) estimated from load

deformation curves [21] are usually employed whenanalyzing the nonlinear response of foundations subjected

to static loads. On the other hand, the modulus reductioncurves approach was developed mainly for the dynamic

loading case, but has been used for both static and dynamic

loading cases. This causes some confusion regarding the

definition of modulus reduction curves, which has been the

focus of some research[21,31,32].

The maximum, small-strain soil shear modulus, G0; has

the same value for static monotonic, static cyclic and

dynamic loading cases [31,32]. On the other hand, the

modulus reduction curve is the same for static cyclic and

dynamic loading cases, but different for static monotonic

loading. For static monotonic loading, the modulus

reduction curve is defined as a reduction in the secant

stiffness from the origin to a specified point on the

monotonic curve with an increase in static shear strain;

while for cyclic (dynamic) loading, it is defined as a

reduction in the peak-to-peak secant stiffness of the

unloadreload hysteretic loops with an increase in cyclic

shear strain. For dynamic analysis, G0 can thus be

evaluated using monotonic loading tests, however, modu-

lus of reduction curves must be obtained from cyclic

loading tests [33]. This further explains why the unload

reload stiffness gave a better comparison with the TRISEE

experimental results.

Fig. 9 shows the modulus reduction curves established

from the TRISEE test results during the initial static loadingphase[18]by normalizing the soil elastic modulus,E;back-

calculated from the test results using the small-strain soil

modulus, E0: Fig. 9 shows that the unloadreload elastic

modulus represents 70% of the small-strain modulus

calculated by Eq. (22). A better agreement between the

two values of the soil modulus would be expected.

However, the difference is mainly due to the assumptions

used when estimating the secant Youngs modulus, and the

effect of soil nonlinearity on the modulus obtained from

only one unloadreload cycle [31,34]. It is thus expected

that the correct value of the unloadreload modulus would

be . 0:7E0: During the cyclic phase of the experiment, an

increase in the cyclic shear strain as a result of theinteraction between the foundation and the soil (commonly

Table 1

Variation ofG0 with depth for different assumptions of the mean effective

pressure,sm

Depth

(m)

G0 overburden applied stress

(MPa)

G0 overburden only

(MPa)

Difference

(%)

HD test

0.5 91.0 26.0 71

1 71.5 30.5 57

2.5 55.0 41.5 25

LD test

0.5 43.0 20.0 53

1 37.0 23.0 38

2.5 35.0 31.0 11

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termed secondary nonlinearity) would result in a reduction

in the unload reload modulus. This phenomenon is

localized and when its effect is averaged over the depth of

the soil medium, the reduction in G is not expected to be

large. Therefore, it is assumed that the opposing effects

would cancel out and G can be taken as 0:7G0:

7.2.3. Assumptions for G

Table 2shows values of the rocking and vertical stiffness

computed using generalized uncoupled static stiffness

expressions [35] (no consideration of loading frequency)

given in FEMA 302/303 based on different assumptions for

G: The table also shows the equivalent subgrade modulus

estimated from either the rocking or vertical stiffness

(according to the approach given in the FEMA 273/274

guidelines). The FEMA 302/303 recommendations specify

the depth of influence for rocking and horizontal/vertical

motions as 1:5ruand 4ru; respectively, where ruand ru are

the equivalent radii of a circular foundation with the same

moment of inertia and area. For the square foundation, this

results in a depth of influence of 0.6 m for rocking motion

and 2.2 m for translational motion. Representative values of

the small-strain shear modulus,G0;are thus computed at 0.3

and 1 m (representing one B) below the footing for both

motion types. The different assumptions used for G are as

follows.

1. G back-calculated from the unload reload modulus

shown in Fig. 9. As discussed earlier, this value is

assumed to be representative of the best estimate ofG:

2. G G0; i.e. no reduction in the soil shear modulus,

assuming that the cyclic strain is relatively small. This

assumption is expected to result in a slight overestima-

tion of the stiffness and thus underestimates the predicted

response.

3. Gback-calculated from the rocking stiffness expression,

using the measured small displacement rocking stiffness

obtained during Phase I of the TRISEE experiment.

Fig. 9. Modulus ratio versus average strain for HD and LD tests.

Table 2

Computed foundation stiffnesses based on the different assumptions for G

Uncoupled rocking

stiffness (MN/m)

Uncoupled vertical

stiffness (MN/m)

Subgrade modulus estimated

from rocking stiffness (MN/m2)

Subgrade modulus estimated from

vertical stiffness (MN/m2)

HD test

G;taken as equal to

G0 (MPa)

22.0 252.0 270.0 252.0

G; from unloadreload

stiffness (MPa)

16.0 176.0 190.0 176.0

G;from Phase I experimental

M2 ustiffness (MPa)a40 627.0 480.0 627.0

G;from guidelineG reductionfactors (MPa) 9.0 101.0 100.0 101.0

Winkler-based on unload

reload stiffness

23.3 280.0b

LD test

G;taken as equal to

G0 (MPa)

10.5 129.5 125.0 129.5

G; from unloadreload

stiffness (MPa)

7.0 90.5 88.0 90.5

G;from Phase I experimental

M2 ustiffness (MPa)a16 251.0 192.0 251.0

G;from guidelineG reduction

factors (MPa)

4.05 52.0 50.0 52.0

Winkler-based on unload

reload stiffness

8.3 100.0b

a Gback-calculated from measured rocking stiffness using rocking stiffness equation.b Value of Winkler spring stiffness estimated using measured unloadreload stiffness.

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The estimated values ofGin this case are larger than the

calculated values of G0: This value is considered to

represent an upper bound for G:

4. G estimated using the reduction factors specified in the

FEMA 273/274 guidelines. This value is estimated using

the maximum base shear of 0:4P for Phase III of the

TRISEE tests, which leads to an effective shear modulus

ratio of 0.4. This estimate is not expected to produce a

good comparison with the measured response because

the cyclic loading is through the foundation, while the

reduction factor used is proposed for soil primary

nonlinearity associated with cyclic loading through the

soil. Nonetheless, this value could be considered as a

lower bound for G:

7.3. Comparison between Winkler approach and code

bilinear model

The force deformation curves are established employing

the bilinear model and using stiffness constants based on the

different G values as shown inTable 2for both HD and LD

tests. The results are compared in Fig. 10 with force

deformation curves computed based on the Winkler model

and assuming two different values for the subgrade modulus:

Fig. 10. Comparison of code bilinear approximation and computed curves based on the rocking and vertical stiffnesses with experimental curves.

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one established from the vertical stiffness, and another from

the rocking stiffness as described in the NEHRP FEMA 273/

274 guidelines. The forcedeformation curves measured

during the tests are also shown in the figure.

Fig. 10shows that both the code bilinear approximation

and the computed Winkler curves agree quite well with the

measured responses for HD and LD tests whenGvalues are

obtained from cases 1 or 2 (i.e. G back-calculated from the

unloadreload modulus orG G0). The prediction for case

2 was better than case 1, even though the opposite was

expected. It is interesting to note from Table 2 that the

computed stiffnesses for case 2 are very close to the

computed Winkler values, which were obtained based on

the measured unloadreload stiffness (i.e. the curves for this

case are almost identical to those shown in Fig. 8). For

almost all cases, the initial stiffness is lower than themeasured for both the bilinear and Winkler models.

However, the overall response prediction is generally

satisfactory. Using the G value for case 3 (i.e. Phase III in

the TRISEE experiment) the initial stiffness is accurately

predicted, but as expected, both models, especially the

bilinear model, overestimate the overall response. Finally,

using the code G reduction factors resulted in a large

underestimation of the initial stiffness and in general a poorprediction of the response for both models. This case

represents a lower bound for stiffness.

It may be inferred from the results that using a lower

bound stiffness estimate in combination with a best ultimate

moment estimate could lead to a significant overestimationof the yield displacement, which for bilinear models defines

the onset of nonlinear effects. In this case, the permanent

displacement of the foundation may be underestimated. It

can also be concluded that a good estimate of the value ofG

is important to accurately predict the moment rotation

response of the foundation using the code bilinear or

Winkler models.

Fig. 10 also shows that using a value of the subgrade

modulus back figured from either the vertical or rocking

uncoupled stiffness did not have a significant effect on the

computed response. For situations where the depth ofinfluence for both rocking and translation are assumed to be

the same [36,37], the vertical response would be onlyslightly larger than that of the rocking, and could be ignored

when compared to the effect of the value of G: Thus, the

selection of the value of G is more crucial regardless of

which approach is used to evaluate the subgrade modulus.

8. Conclusions

The momentrotation response of rigid spread footings

has been investigated. A solution for the uplift-yield

foundation condition has been derived. The developed

solution, along with the solutions for uplift- and yield-only

conditions enables the full definition of the entire staticmomentrotation response. The developed model was used

to analyze the TRISEE experiments and the following

conclusions are made.

1. x 0:5 represents the condition for the maximum

moment response based on the Winkler soil model.

2. The momentrotation response is rather influenced by

the absolute difference between x and x 0:5; which

dictates whether uplift or yield is dominant. Based on the

value of x; the moment rotation response can be

grouped into three categories: uplift-dominant, uplift-

yield and yield dominant.

3. The unloadreload stiffness should be used to estimate

the value of the subgrade modulus in dynamic analysis.

The secant stiffness and initial stiffness lead to under-

estimating the subgrade modulus.

4. The value of G used in the analysis has a significanteffect on the calculated response. The response calcu-

lated using G with no reduction factor in both the code

bilinear model and the Winkler model agreed well with

the measured response.

5. Using either the rocking stiffness or vertical stiffness to

estimate the value of the subgrade modulus has a

minimal effect on the calculated response.

Acknowledgements

This research was supported by financial support from

the Institute for Catastrophic Loss Reduction at theUniversity of Western Ontario to the senior author and a

graduate scholarship to the first author from the National

Science and Engineering Research Center (NSERC). The

authors would also like to thank Dr P. Negro of the

European Laboratory for Structural Assessment (ELSA) for

making available the test results of the TRISEE

experiments.

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