B1IT FILE COPN ,CL4,r

Defense Nuclear AgencyAlexandria, VA 22310-3398

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INDNA-TR-88-212

Role of Soil Hysteresis in Impedance Testing as Applied toBuried Arches

R. S. AtkatshWeidlinger Associates, Inc.333 Seventh AvenueNew York, NY 10001

February 1990 DTICELECTEFq, EB 16 1990 U

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62715H I I C DHOO9 196I i. TITLE (Ocko Sauffy Ckadwft)

Role of Soil Hysteresis in Impedance Testing as Applied to Buried Arches

12. PERSONAL AUTHORS)

Atkatsh, R. S.

16. SUPPL.EMENTARY NOTATION

This work was sponsored by the Defense Nuclear Agency under RDT&E RMC CodeB-1"085466 S C 00104 25904D.

17, COSATI CODES -4IL SUBJECT TERMS (CaOm an ,mw.. Mfninaiw iidldawi by *i* meal~

FIELD GROUP SUS.GROUP I s tructure-Medium Interaction . Hysteretic Dampingf187 Finite Element Simulation)

IS. ASSTRACT h.5.aiVW.N.mpMdMhS wIu

The effect of backfill soil hysteretic damping upon the suppression of resonance for buried arches is examined. Finiteelement simulation of a buried arch structure shows that the backfill soil hysteretic behavoir is a dominant factor inexplaining the absence of resonance in arch tests. ~

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Bennie F. Maddox (703) 325-7042 DNA/CSTT

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CONVERSION TABLE

Conversion factors for U.S. customaryto metric (SI) units of measurement

To Convert From TO Multiply By

anywtrm meters (in) 1.00OOOOX B-10atmosphere(normal) kilo pssal (kPa) I103 25 XE +2

bar kilo pascal (kPa) 1.000 000 X E +2

barn meler1 (in2 ) 1.00000XE -28British Thermal unit (thernochernical) joule (J) 1.054 350 X E +3

calorie (therochemnical) joule () 4.194000

Cal (thermochem;cal )/m 2 mega joule/in (Mihn2) 4.184 000 X E -2

curie gig* becquerel (GBq)* 3.700 000 XEI + I

degree (agie) radian (rad) 1.745 329 X E-2

deg=e Fahrenheit degree kelvin (K) TK'(tf + 459.67)/I .8

electron volt joule (J) 1.60219 X E- 19

erg joule (J) 1.000O000X E-7

erg/second! watt (W) 1.0O0000X E-7

foot meter (M) 3.049 000X E- Ifoot-pound-force joule () 1.355 818gallon (U.S. liquid) meter 3 (i) 3.785 412 X E -3inch meter (in) 2-W4000X E -2

jerk joule () 1.000 000 X E+9

joulejltlogrun (i/kg) (radiation doseabsorbed) Gray (G)" 1.000000kilotons terajoules 4.183

kip (IWO Ibf) newton (N) 4.448 222 X E +3kip/inch2 (kai) kilo pascal (kPa) 6.894 757 X E +3ktsp, newton-second/n 2 (N-s/rn 2) 1.000000XE +2Micron meter (in) ISM 0000X E -6mil meter W) 2.540 000 XE -5mile (international) meier (Wn 1.609 344 X E +3

ounce kilogram (kg) 2.834 952 X E -2pound-force (Ibf avoirdupois) newton (N) 4.448222

pound-force Inch newton-meter(Wnm) 1. 129 948 XE -pound-force/nch newton/meter (N/rn) 1.751 268 X E +2

pound-force/loot kilo pascal (kPa) 4.788 026 X E -2pound-forcefinch2 (psi) kilo pascai W 'a) 6.894 757pound-mass (ibm avoirdupois) kilogram (kg) 4.535 924 X E -Ipound-maas-foo3 (moment of inertia) kilogramt-meter2 (kg-m2) 4.2140Oil X E 2

pound-mass/foot3 kilogramfneter3 (kg/rn3) 1.601 846 XE F ol

rad (radiation dose absorbed) Gray (Gy)** 1.000 MYC X E -2

roentgen coulomb/kilogram (C/kg) 2.579, 60 X E -4shake second(st) 1 -000 000X E-8slug kilogram (kg) 1.459 390X E+ I

torr (mm Hs,00C) kilo pascal (kPa) 1.3 3322 X E-I

*'Mie becquetel (Dq) is the SI unit of radioactivity; Dq -I eventia.

"Trhe Gray (0y) is the SI unit of absorbed radiation.

TABLE OF CONTENTS

Section Page

COVERSION TABLE ........................................ iii

FIG URES .................................................. v

I INTRODUCTION ........................................... 1

1.1 BACKGROUND ...................................... 1

1.2 SCOPE AND OBJECTIVES ............................. 1

1.3 IMPEDANCE TESTING: METHODOLOGY AND RESULTS 2

2 SIMULATION OF BURIED ARCH STRUCTURES ................ 4

2.1 FINITE ELEMENT METHODOLOGY .................... 4

2.2 VISCOELASTIC CAP MODEL .......................... 5

3 RESULTS OF BURIED ARCH SIMULATION STUDIES ........... 8

3.1 DISCUSSION OF RESULTS ............................ 8

4 CONCLUSIONS AND RECOMMENDATIONS ................... 11

4.1 CONCLUSIONS ...................................... 11

4.2 RECOMMENDATIONS ................................ 11

5 REFERENCES ............................................. 12

NOTATION ................................................ 25

iv

FIGURES

Figure Page

1 Geometry and instrumentation for arch structure ...................... 132 Finite element model of buried arch ................................ 143 Uniaxial stress-strain properties of compacted native backfill

ESSEX V arch ................................................. 154 ARCH structure 3A, analytical response at crown-single vibration test,

FI = 500 lbs .................................................. 165 Elastic soil cap model-ARCH structure 3A, analytical response at

crown-single vibration test, Fl = 500 lbs ........................... 176 Hysteretic soil cap model-ARCH structure 3A, analytical responses at

crown-single vibration test, Fl = 500 lbs ........................... 187 ARCH structure 3A, analytical response at gage location A3, single

vibration test, Fl = 500 lbs ....................................... 198 Elastic soil cap model-ARCH structure 3A, response data at gage

A3--two vibrators in phase, F2 = F3 = 500 lbs ....................... 209 Hysteretic soil cap model-ARCH structure 3A, response data at gage

A3--two vibrators in phase, F2= F3= 500 lbs ........................ 2110 ARCH structure 3A, analytical response at gage location A3, two vibrators

out of phase, F2 = F3 = 500 lbs ................................... 2211 Elastic soil cap model-ARCH structure 3A, response data at gage

A3-two vibrators out of phase test, F2 = F3 = 500 lbs ................. 2312 Hysteretic soil cap model-ARCH structure 3A, response data at gage

A3--two vibrators out of phase test, F2 = F3 = 500 lbs ................. 24

Accesiorti For

NTIS CRA&IOTIC TABUnannog,,cd

Justific.!j

By

Ol , , or- l ..> ,c:dI

V

SECTION 1

INTRODUCTION

1.1 BACKGROUND. The test results showed that for the rectangularstructures, resonance was observed in the

The use of impedance testing and its applica- shallow buried structures at the same frequen-tion to buried arch response is discussed in this cies as in the uncovered box structure. For thereport. A goal of impedance testing is to arch structure, resonance was not experimen-produce functional relations which character- tally observed in the shallow buried arches.ize the shock input levels at points of equipment In an attempt to explain the differences in theattachment to the structure. The experimentally experimental results of the boxes and thedetermined impedances of the structure are eeriear re ot bx an d teused to develop a transfer function between the arches, linear time domain analyses weregeneralized input loads upon the structure and previously performed to simulate the behaviorof these structures (Ref. 2). These analysesthe dynamic response velocities at the attach- examined what levels of radiation dampingment points of equipment. The restriction of would be needed to supress resonance. It wasthis methodology is that linear behavior of both determined that for the box structure, radiationthe structure and the soil must exist. The reality damping of the order of 10% to 20% wasis that some degree of nonlinearity occurs at ded o the reof Fr to asany significant pressure level with the onset of su pres rsne. th archsoilnonineaitypreedin th onet o stuc- structure, it was determined that a radiationsoil nonlinearity preceding the onset of struc- damping level of 50% would be needed totural nonlineanty. supress damping. Since measured values of

radiation damping are in the range of 10 to 15%,The particular impedance tests studied in this it was concluded that radiation damping alone

report are the forced vibration tests performed could not account for the supression of damp-

on uncovered and shallow buried arch and cn for the d hes

rectangular structures in the ESSEX V Phase 3 ing for the covered arches.

test series. (Ref. 1) Identified in Ref. 2 as an additional dampingsource was hysteretic behavior in the soil

For these tests, the original backfill was exca- adjacent to the structure. Hysteresis in soilvated and the uncovered structure was sub- stress-strain relations describes the energyjected to forced vibration tests. The sand losses produced by the load/unload and reloadbackfill was then replaced by raining it from a cycles. These cycles are produced in the soil byheight of five feet and the forced vibration tests the driving oscillations of the buried structure.were repeated. Figure I shows the geometry of The contributing effect of the soil hysteresisthe arch structure and the backfill and instru- upon the experimentally observed supressionmentation layout. of arch resonance will be further discussed in

this report.Test results include impedance (Fmax/Vmax) 1.2 SCOPE AND OBJECTIVES.

versus frequency relationships. Resonance isdefined to occur when the driving frequency This report discusses the development andapproaches a particular natural frequency of the validation of analytical techniques for hardnessstructure. This is exhibited by a valley in the assessment of shallow buried arches. It repre-impedance versus frequency curve. sents an extension of previous studies done by

I

Weidlinger Associates under DNA Sponsor- generalized input loads upon the structure andship to simulate the forced vibration testing of the dynamic response velocities at the attach-uncovered and covered arches and boxes in ment points of equipment. The restriction onProject ESSEX. Those studies indicated that this methodology is that linear behavior of bothmaterial and geometrical non-linearities need- the structure and the soil must exist. The realityed to be considered when modeling shallow is that some degree of nonlinearity occurs atburied arches. any significant pressure level with the onset of

soil nonlinearity preceding the onset of struc-The role of the various component soil material tural nonlinearity.non-linearities and their interplay with the archstructure is examined in this report. Particular Impedances are measured by applying an inputattention is drawn to the responsibility of soil forcing function to the structure using eithernon-linearities in helping to describe the appre- electromagnetic (low force and high frequen-ciable difference between the shallow buried cy) or electro hydraulic (high force, lowarch and box behavior, frequency) vibrators. A variable sine sweep

rate is used when driving the structure. TheTime marching simulation using non-linear impedance is determined using a measurementfinite element analysis is performed to over- point and a series of drive points on the

come the theoretical obstacle presented by the struture. Theres of te mesuentnonlinar oilstrctue sste. Vrios aa- structure. The response of the measurement

non-linear soil-structure system. Various ana- point to different driving frequencies is foundlytical models of the soil media are used to by summing the products of experimentallyisolate the influences of soil behavior upon the determined impedance functions and driveresponse of buried arches to impedance testing. functions in the frequency domain.

The examination of soil response is within thescope of the study of impedance testing as The particular impedance tests studied in thisapplied to buried arches. Determining the report are the forced vibration tests performedextent of non-linear soil response will greatly on uncovered and shallow buried arch andaid the investigation to determine the extent rectangular structures in the ESSEX V Phase 3non-linearities render the impedance technique test series. Constant force sinusoidal frequencyimpractical for shallow buried arches. This sweeps were run at three different force levels.study will also assist in the development of Mechanical impedance (Fmax/Vmax) versusanalytical techniques which utilize the results frequency plots were made. Resonant frequen-of impedance testing to develop shock input cies were determined from such system curves.levels at equipment attachment points. An outline of the test plan for ARCH 3A

(Covered and Uncovered) is below.

1.3 IMPEDANCE TESTING: METHOD-OLO(;Y AND RESULTS. The test results obtained showed that for the

rectangular structures, resonance was observedThe use of impedance testing and its applica- in the shallow buried structures at the sametion to buried arch response is discussed in this frequencies as in the uncovered box structures.report. A goal of impedance testing is to For the arch structures, resonance was notproduce functional relations which character- experimentally observed in the shallow buriedize the shock input levels at points of equipment arch structures. In an attempt to explain theattachment to the structure. The experimentally differences in the experimental results of thedetermined impedances of the structure are boxes and the arches, analyses were performedused to develop a transfer function between the to simulate the behavior of certain types of

2

shallow buried structures subjected to forcedvibration testing. (Refs. 2,3 and 4).

T DESCREfl DRIV POINT FORCL EIJNCfTlN

SINGLE VIBRATOR FL FREQUENCY SWEEP

DUAL VIBRATORS F2 and F3 FREQUENCY SWEEPIN PHASE IN PHASE

DUAL VIBRATORS F2 and F3 FREQUENCY SWEEPOUT OF PHASE OUT OF PHASE

I3

SECTION 2

SIMULATION OF BURIED ARCH STRUCTURES

2.1 FINITE ELEMENT adequate to represent the arches compressiveMETHODOLOGY. circumferential behavior.

The finite element mesh is extended in space soThe finite element method together with an as to maximize the simulation time before anyexplicit central difference integration scheme reflected waves from the finite element bound-is utilized in the analysis of shallow buried aries reach the shallow buried arch. Dashpotsarches subjected to dynamic loading. Both are placed at the boundaries to absorb anymaterial and geometrical nonlinearities are waves which are generated there.considered in the analysis.

For each test of the series, forced loading of the

The plane strain finite element program SAD- form P = Pmax sin(0t) is applied at the drive

NESS (SEISMIC ANALYSIS for DYNAMIC point.

NONLINEAR SLOPE STABILITY) (Ref. 5) Direct integration in time of the equations ofis used. The SADNESS program was devel- motion was performed using a time stepoped to examine the earthquake induced failure interval of 60 microseconds. A total of 0.24of earth slopes. Its range of applicability seconds of real time response was simulated.includes the non-linear response of buriedstructures subjected to vibrational loadings. The use of the advanced soil plasticity CAPBoth geometrical and material nonlinearities model, which satisfies the requirements ofare accounted for. Geometrical nonlinearities continuity and uniqueness of solution and hasare incorporated through the use of an incre- the capability of simulating a wide range of realmentally updated Lagrangian approach in con- effects in soils was deemed essential for thesejunction with a constitutive equation relating analyses (Refs. 6, 7 and 8).the co-rotational stress rate to the rate of The CAP model is based on the classicaldeformation tensor. The theory is valid for largedeformations (strains) as well as for large incremental plasticity theory and is capable ofdeflections and rotations. Material nonlineari- representing the mechanical behavior of soilsties are incorporated through the use of a cap and rocks while satisfying all theoretical re-model constitutive model extended to improve quirements for properly posed initial/boundaryits apailit torepesen cylic ehaior value dynamic problems. The proper use of theits capability to represent cyclic behavior CAP model assures existence and uniquenesscharacteristic of sinusoidal ground motions. of solution as well as continuous dependence

on the initial and boundary data. The CAPThe SADNESS model of the buried arch model has a non-softening convex yield sur-(Figure 2) consisted of 455 elements and 267 face. For stress points on the yield surface, thenodal points. Constant strain triangles are used plastic strain rate vector is outwardly directedthroughout. Five (5) different materials were and normal to the yield surface in stress space.modelled including the concrete arch, the The CAP model permits inelastic hardening inbackfill and three subsequent soil layers. The hydrostatic loading, limits the amount of dilat-finite element mesh of the concrete arch is ancy during shear failure and provides a goodcomposed of 64 elements. This discretization is fit to material property data.

4

To examine the interplay of the various compo- Linear viscous damping is included to modelnents of the soil material nonlinearities, two stress paths within the yield surface of the CAPdifferent soil CAP models are used. They are; model. A standard solid is chosen to represent

the shear behavior for stress paths within the1. Elasto-Plastic soil behavior (CAP Mod- current yield surface. This standard solid model

el) requires three material parameters to describethe deviatoric viscoelastic response. These

2. Viscoelasto-Plastic soi behavior (VICAP parameters are an instantaneous modulus Gf, acoelastic CAP) long term modulus Gs and a relaxation frequen-

These models will serve to isolate the role of cy 0).

soil non-linearities in determining the response The viscoelastic portion of the model is de-of the arch structure in forced vibration testing. scribed by

2.2 VISCOELASTIC CAP MODEL.+os=2' eE +2wG

Certain amendments to the CAP model weremade to improve its capability to representcyclic behavior characteristic of sinusoidalground motions. Much of the previous use of where s and eE are the stress and elastic strainthe CAP model has been for ground shock deviators, is a relaxation frequency and Gf andcalculations involving much higher stress lev- G, represent the shear moduli under fast andels than are applicable to low amplitude slow loading respectively.sinusoidally induced motions. Additionally, The volumetric behavior is elasticblast-induced loadings generally include asingle peak compressive load followed byseveral smaller peaks. For theses cases, any J = 3KEE (2)hersteresis in cyclic loading after the initial

pulse is generally viewed as unimportant.

In vibration induced sinusoidal motion where where JI and 4 are the traces of the stress andcyclic shear is a predominant effect, hysteresis strain tensors and K is the bulk modulus.behavior caused by cyclic loading must beconsidered. In order to model this energy To obtain numerical values for these parame-dissipation process, a viscoelastic stress-strain ters an experimentally justified approximationrelation is incorporated into the CAP model to is made that both the hysteresis and the averagesimulate the effects of low amplitude cyclic modului are relatively frequency independent.hysteresis within the yield surface. This allows The two parameters G, and Gf are related to anfor a controlled amount of energy dissipation in average modulus over a loading cycle and to thecyclic loading and can be fine tuned to provide energy loss per cycle at a central frequency ofthe correct damping at the frequencies asso- interest. The constants G, and Gf can beciated with the forcing functions. obtained from a single load - unload curve of

the compacted native backfill. The parametersExperimental results indicate that the amount G. and Gf are related to the average modulusof hysteresis for soils is relatively rate indepen- over the cycle and to the energy loss per cycle.dent. Because of this experimental observation, The parameter 0o is introduced, where is thethe following approach, as discussed in detail in circular frequency at which the energy loss perReference (9) is taken; cycle of a standard solid reaches a maximum.

5

The parameter o) is chosen to correspond to the at t = tn + At:central frequency of interest, that is the forcingfunction frequency. Thus for the main frequen-cy of interest, both the hysteresis and theaverage modului are relatively frequency inde- SR+1 = S"' + 2GseE-(l - e--)

pendent. + 2[G, + Gf ( - e-wA&)]AeE

The viscoelastic algorithm begins by comput- WAting a viscoelastic "trial value" for the stresses, (6)i.e., a value based on the assumption that nofailure or cap plasticity occurs. If the assump- which is the viscoelastic trial deviatoriction is found to be false, the stresses are revised stresses.by means of a "plastic correction" (Ref. 10).

For reasons of accuracy under arbitrary loading The viscoelastic trial stresses are then checkedpaths, the viscoelastic trial values are obtained against the failure envelope and cap. If either orfor a strain increment AE during a time step At both of these surfaces is exceeded, the stressas the exact solution for linear straining during point is corrected so as to satisfy the appropriatethe time step tn < t < tn + At, yield condition and normal flow rule.

eE(t) t A t (3) The viscoelastic model was fit to the availableet) eE + AeE (') (3) data for the backfill soil (Figure (3).

The bulk modulus K was estimated from thewhere eEt is the viscoelastic deviatoric strain uniaxial stress-strain properties of the com-at tN. Substituting equation 3 into equation I pacted native backfill.produces

s = 'G AThe "fast" and "slow" shear moduli, Gf and G,s+-s=2G + 2wG, [eE- + - (t-tf)] respectively were related to the size of the

AtAt elliptic hysteresis loop and are dependent upon

(4) the amount of energy dissipated through vis-cous damping in the soil material.

which may be solved using s(t) = s", to give

The viscoelastic relaxation parameter (o isS = + taken to be equal to the forced frequency of

S(t) = Sn + 2G, (t- t') excitation of the structure.

- [s" 2 GIM 2- Gf GJ - 1- e"wt-At For the backfill soil material, the properties

used to fit the viscoelastic CAP model are

estimated from the uniaxial stress-strain prop-(5) erties of the compacted native backfill (Fig. 3)

6

exhibiting a confined modulus (K + 4/3 G) of CB = 0.2 (1/ksi)26000 psi and are as follows:

CC = 1.0OksiK = 18000. psi

Gs =6000. psi CD = 1.5 (1/ksi)

Gf =9000. piEL = 0.0 ksi

CW .05 LTYPE = 1

CR 3.0 TCUT = .02 ksi

CA= 1.02 ksi FCUT = .02 ksi

SECTION 3

RESULTS OF BURIED ARCH SIMULATION STUDIES

3.1 DISCUSSION OF RESULTS. which resonance occurs and represents a natu-ral frequency of the buried structure.

The physical results obtained from the ESSEXV buried structure tests differed appreciably in Analytical impedance results were obtained forform from experimental predictions. In an the three test series which used a 500 poundattempt to explain the reasons for this differ- (2224 N) forcing function. These series are;ence, various damping mechanisms have beenidentified as possible causes of discrepancybetween experimental and analytical results 2. Two vibrators in phase(Ref. 2). These damping mechanisms include:

3. Two vibrators out of phase1. Radiation damping

For each test series, a frequency sweep of 50 to2. Interface friction between the soil and the 1000 hertz was experimentally performed.

structureThe analytical simulation studies were per-

3. Hysteresis in the soil backfill adjacent to formed for a frequency sweep from 50 to 250the structure hertz in intervals of 10 hertz. For each frequen-

cy in the interval, a direct integration in time4. Structural damping was performed. That is, 21 separate time

integration analyses were performed to obtainPrevious work (Ref. 2) focused on the mecha- the frequency sweep. A more limited frequencynisms of radiation damping and friction, as they range was used in the analytical studies versuswere previously believed to be capable of being the experimental studies because of limitationsthe major contributors to the total effective in the analytical procedure at both low (lessdamping. than 50 hertz) and high (greater than 250 hertz)

frequencies. The reasons for this are that in theThe prese it study focuses on the role of soil low frequency range, the use of absorbinghysteresis in contributing to the total effective boundaries along the finite element meshdamping needed to supress resonance. Soil boundaries restricts the rigid body motionhysteresis causes energy to be absorbed during prevalent for these low frequencies, therebythe cycles of load-unload-reload produced by invalidating the analytical procedure and forthe driving force oscillations. The actual the high frequency range, the size of the finiteamount of effective hysteresis damping is a element mesh restricts the frequency content offunction of the amount of energy absorbed in the solution, thereby invalidating the analyticaleach cycle. procedure.

Mechanical impedance data obtained from the The simulation study considers a cross-sectionESSEX V test series showed the variation in of the arch with length effects assumed toimpedance levels (FIV) due to changes in the satisfy plane strain kinematics. The forcingforcing function frequency. Each trough in the function for the plane strain analyses is forceimpedance curve represents a frequency at per unit length of the arch (F/L). Therefore. the

8

analytically obtained impedance values differ impedance than gage CL6 because it is furtherfrom those measured. The analytical imped- removed from the force drive point. For theance values are obtained from (F/L)V. The elastic backfill soil model, the computed reso-analytical impedance values differ in units nance at 180 hertz is not observed in thefrom the experimental values by a inverse experiment at either of the measured gages. Forlength unit. This discrepancy can be eliminated the viscoelastic (hysteretic) backfill soil model,by assuming a characteristic length L* for the computed response shows a resonance at 90which the plane strain analyses impedance can hertz. A resonance of 80 hertz is exhibited bycompare with the full three dimensional re- gage CL8. Note that the magnitude of thesponse. A possible value for L* could be the calculated response differs from the magnitudearch's radius. Then the analytical impedance of the measured response because of thevalues would be obtained from (F/V)*(L*/L). analytical assumption of plane strain response.

What is critical, however, is that the inclusionof a hysteretic backfill soil model alters the

To isolate the effects of soil hysteretic nonlin- shape of the response impedance diagram byearity upon impedance versus forcing function supressing the resonance mode at 180 hertz.frequency relations, results are presented fortwo assumed backfill soil models. The first soil For the dual in-phase vibration test, using a 500model is an elastic-plastic CAP model and the lb (2224 N) force at drive points F2 and F3, the

second is a viscoelastic CAP model. effect of backfill soil hysteresis is illustrated inFigure (7). For the case of elastic-plastic

For the single vibration test, using a 500 lb backfill soil, a distinct trough in the impedance(2224 N) force at drive point F1 (crown of curve is obtained. This occurs at 140 hertz and

arch), the effect of backfill soil hysteresis is represents structural resonance. The use of aillustrated in figure (4). For the case of elas- viscoelastic-plastic backfill soil model does nottic-plastic backfill soil model, two distinct supress the resonance occuring at 140 hertz.

troughs in the impedance curve are obtained. The comparisons of both analytical results withThese occur at 100 and 180 hertz and represent the experimental data for the dual in-phasefrequencies where structural resonance occurs. vibrator test are shown in Figures (8) and (9).However, the use of a viscoelastic-plastic The experimental response of the gage A3 isbackfill model supresses the resonance occur- shown. Note that the magnitude of the com-ing at 180 hertz and smooths out the impedance puted impedance curve differs from the magni-trough which occurs at 100 hertz. Thus the use tude of the measured response because of theof a viscoelasticity backfill CAP model reduces plane strain limitation. As seen in Figure (7),the tendency of the structural arch to resonate the inclusion of the hysteretic soil model did notwithin the frequency band considered. supress the calculated resonance mode at 140

The comparisons of both analytical results with hertz. This resonance mode was not observed in

the experimental data for the single vibration the experimental records.

test are shown in figures 5 and 6. The exper- For the dual out-of-phase vibrator test, using aimental response of the two gages, CL6 and 500 lb (2224 N) force at drive points F2 and F3,CL8 are shown. The response of the gages CL6 the effect of backfill soil hysteresis is illustratedand CL8 illustrate the change in impedance in Figure (10). For the case of elastic-plasticalong the length of the arch's crown. If the arch backfill soil, a distinct trough in the impedanceresponded in plane strain modes, these two curve is obtained. This occurs at 110 hertz andgages should have identical responses. This is represents a frequency where structural reso-not observed as gage CL8 has a greater nance occurs. However, the use of a viscoelas-

9

tic-plastic backfill soil model suppresses the For the single vibrator test and the dualresonance occuring at 110 hertz. Thus the use out-of-phase vibrator test, the inclusion ofof a viscoelastic backfill CAP model reduces viscoelasticity into the backfill soil modelthe tendency of the structural arch to resonate served to supress resonant modes. The supres-within the frequency band considered. sion of these resonance modes agreed with the

experimental data which showed that for theThe comparisons of both analytical results with arch structure, resonance was supressed in thethe experimental data for the dual out-of-phase covered tests as compared to the uncoveredvibrator test are shown in Figures (11) and (12). tests.The experimental response of gage A3 isshown. For the elastic backfill soil model, thecomputed resonance at 110 hertz is not ob- Forthe dual in-phase vibratortest, the inclusionserved in the experiment gage record. For the of viscoelasticity into the backfill soil modelviscoelastic (hysteretic) backfill soil model, the did not serve to supress the resonant modes.computed response supresses the resonance at This supression was observed however in the110 hertz. The inclusion of a hysteretic backfill experimental gage records. Possibly, the reasonsoil model alters the shape of the response for this difference is that the plane strainimpedance diagram by supressing the reso- assumptions were not at all valid for thisnance mode at 110 hertz. in-phase test series.

10

SECTION 4

CONCLUSIONS AND RECOMMENDATIONS

4.1 CONCLUSIONS. observed in the covered arch experimen-tal results.

1. Resonant frequencies for the coveredarch with a single vibrator force at drive 5. Impedance testing excites nonlinear soilpoint F I are analytically supressed when response typified by hysteretic energya viscoelastic-plastic soil backfill CAP losses and radiation and friction energymodel is used. dissipation.

2. Resonant frequencies for the covered 6. The applicability of impedance testing toarch with a dual out-of-phase vibrator define buried arch response is question-force at drive points F2 and F3 areanalytically supressed when a viscoelas-tic-plastic soil backfill CAP model is 4.2 RECOMMENDATIONS.used.

3. Resonant frequencies for the covered Additional experimental and analytical studiesarch with a dual in-phase vibrator test at are needed to expand the data base of burieddrive points F2 and F3 are not analytical- structural response. Soil parameters definingly supressed when a viscoelastic-plastic hysteretic energy losses need to be moresoil backfill CAP model is used. accurately defined to assist in establishing

viscoelastc-plastic soil constitutive models to4. The inclusion of a viscoelastic-plastic simulate the hysteretic energy losses which

soil model is critical in helping to occur during the forced vibration tests of buriedexplain the resonant mode supression structures.

11

SECTION 5

REFERENCES

1. Crowson, Roger D. , "ESSEX-Diamond Ore Research Program: Vibration test and analysesof ESSEX V Model Structures". Weapons Effects Laboratory, U.S. Army EngineerWaterways Experiment Station, WES TR N-78-2, August 1978.

2. Isenberg, J. , Levine, H.S. and Pang, S.H., "Numerical Simulation of Forced Vibration Testson a Buried Arch". Weidlinger Associates Contract No. DNA 001-76-C-0110, Report No.DNA 428 IF, Defense Nuclear Agency, 1977.

3. Merkle, D.H. and Merkle, L.D., "High Frequency Analysis of Circular Arches," AppliedResearch Associates Inc. Contract No. F29601-85-C-0029, Report No. AFWL-TR-86-22,Air Force Weapons Lab, 1986.

4. Wojcik, G. and Isenberg, J., "Effects of Radiation Damping on Vibration of a Shallow-BuriedRectangular Structure," Weidlinger Associates Contract No. DNA 001-77-C-0233, ReportNo. DNA 4600F, Defense Nuclear Agency, 1978.

5. Daddazio, R.P., Ettouney, M. and Sandier, I., "Nonlinear Dynamic Slope Stability Analysis,"Journal of Geotrechnical Engineering, ASCE Vol. 113, No. 4, pp. 285-298, (1987).

6. DiMaggio, F.L. and Sandier, I.S., "Material Model for Granular Soiils," Journal of the Engi-neering Mechanics Division, ASCE, Vol. 97, No. EM3, pp. 935-949, (1971).

7. Sandier, I.S., DiMaggio, F.L. and Balardi, G.Y., "Generalized Cap Model for GeologicalMaterials, "Journal of the Geotechnical Division, ASCE, Vol. 102, No. GT7, pp. 683-699(1976).

8. Sandier, I.S., "The Cap Model for Static and Dynamic Problems," Site Characterization - U.S.Symposium on Rock Mechanics, W.S. Brown, ed., University of Utah, Salt Lake City, Utah,pp.1A2-1 - 1A2-11, (1976).

9. Sandier, I.S. and Daddazio, R.P., "Analysis of Earthquake Induced Slope Failures" Phase ITechnical Report, NSF July 1988.

10. Sandier, I.S. and Rubin, D., "An Algorithm and a Modular Subroutine for the Cap Model,"International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 3. pp.173-186, (1979).

12

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24

NOTATION

CA, CB CC, CD, CW, FCUT, TCUT CAP fit Parameters

eF elastic strain deviator

4 trace of strain tensor

FMAX maximum harmonically applied force

Gf shear modulus under fast loading

G, shear modulus under slow loading

J trace of stress tensor

K bulk modulus

S stress deviator

VMAX maximum velocity

(0 relaxation frequency

25

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