Enhancements and Verification Tests for Portable Deflectometers

Transcription

1 Final Reprt Enhancements and Verificatin Tests fr Prtable Deflectmeters

2 Technical Reprt Dcumentatin Page 1. Reprt N Recipients Accessin N. MN/RC Title and Subtitle 5. Reprt Date ENHANCEMENTS AND VERIFICATION TESTS FOR PORTABLE DEFLECTOMETERS May Authr(s) 8. Perfrming Organizatin Reprt N. Olivier Hffmann, Bjan Guzina and Andrew Drescher 9. Perfrming Organizatin Name and Address 10. Prject/Task/Wrk Unit N. University f Minnesta Department f Civil Engineering 500 Pillsbury Drive S.E., Minneaplis, MN Cntract (C) r Grant (G) N. (c) (w) Spnsring Organizatin Name and Address 13. Type f Reprt and Perid Cvered Minnesta Department f Transprtatin 395 Jhn Ireland Bulevard Mail Stp 330 St. Paul, Minnesta Supplementary Ntes Abstract (Limit: 200 wrds) Final Reprt: Spnsring Agency Cde In this study, the accuracy f the stiffness estimate frm prtable deflectmeters is investigated, based upn the example f a particular device, PRIMA 100. The Beam Verificatin Tester (BVT) apparatus was develped at the University f Minnesta fr the Minnesta Department f Transprtatin t: (i) verify the perfrmance f the PRIMA device and (ii) t check the calibratin factrs f the sensrs f the PRIMA device. The bjective f such tests is t detect the ptential ccurrence f deteriratin f the sensr s accuracy. Assciated with the BVT apparatus, an enhanced setup fr the prtable device is examined. The incnsistency f the traditinal data interpretatin methd using peak values f lad and displacement time histries is pinted ut by cmparing the stiffness estimated frm the PRIMA device against the knwn stiffness f the beam. An alternative methd using Frequency Respnse Functins, spectral average, Single Degree f Freedm System analg, zer frequency estimates and curve fitting is prpsed t extract the static stiffness frm PRIMA measurements. Test results shw gd agreement between estimates based n the mdified analysis and true beam stiffness. Implementatin f bth the alternative data interpretatin methd and the enhanced device setup t quality assurance field measurements are prpsed. 17. Dcument Analysis/Descriptrs 18. Availability Statement Prtable deflectmeter In-situ elastic mdulus Accuracy f stiffness estimatin Interpretating stiffness estimates Dynamic analysis Spectral analysis Verificatin N restrictins. Dcument available frm: Natinal Technical Infrmatin Services, Springfield, Virginia Security Class (this reprt) 20. Security Class (this page) 21. N. f Pages 22. Price Unclassified Unclassified 101

3 ENHANCEMENTS AND VERIFICATION TESTS FOR PORTABLE DEFLECTOMETERS Final Reprt Prepared by Olivier Hffmann Bjan Guzina and Andrew Drescher University f Minnesta Department f Civil Engineering May 2003 Published by Minnesta Department f Transprtatin Office f Research Services 395 Jhn Ireland Bulevard Mail Stp 330 St. Paul, Minnesta The cntents f this reprt reflect the views f the authrs wh are respnsible fr the facts and accuracy f the data presented herein. The cntents d nt necessarily represent the view r plicy f the Minnesta Department f Transprtatin and/r the Center fr Transprtatin Studies. This reprt des nt cntain a standard r specified technique. The authrs and the Minnesta Department f Transprtatin and/r Center fr Transprtatin Studies d nt endrse prducts r manufacturers. Trade r manufacturer s names appear here herein slely because they are cnsidered essential t this reprt.

4 TABLE OF CONTENTS CHAPTER 1 INTRODUCTION Backgrund Research issues and gals Research apprach Reprt rganizatin 3 CHAPTER 2 - THEORETICAL BACKGROUND Sil mdulus Elaststatic mdel Review f the Single Degree f Freedm system Mtin transducers Frequency Respnse Functin (FRF) fr a SDOF system Furier transfrm Mbility functin M(ω) Dynamic stiffness functin K(ω) Practical cnsideratins Estimatin f FRF FRF fr a hmgeneus elastic half-space mdel Exact slutin fr the vertical cmpliance f a massless fting Vertical cmpliance f a massive fting 22 CHAPTER 3 - PFWD PRIMA 100 DEVICE Experimental setup Measurement principle Issues regarding the current PRIMA device Enhancement f PRIMA device CHAPTER 4 - BEAM VERIFICATION TESTER: TRUE STIFFNESS OF THE BEAM VS. PRIMA ESTIMATES Setup fr the BVT Chice f the setup Beam setup descriptin Static stiffness f the beam k s Theretical static stiffness f the beam Experimental Yung s mdulus f the steel Experimental static stiffness f the beam K true Estimated beam stiffness k est frm PRIMA vs. k s Pssible causes f the misfit between k est and k s PRIMA sensrs accuracy Theretical cmparisn f peak-based k est vs. k s. 40

5 CHAPTER 5 - ENHANCEMENT OF BEAM STIFFNESS ESTIMATION USING PRIMA DEVICE Cnsistent data interpretatin SDOF mdel fr the beam Frequency dmain analysis tls Experimental prcedure and results Data acquisitin setup Fitting prcess Estimated k est frm PRIMA vs. k s using the mbility functin Beam with additinal damping Auxiliary dampers External damping setup Experimental results PFWD BVT apparatus: summary and practical applicatins PFWD PRIMA Other prtable deflectmeter devices.. 57 CHAPTER 6 PRIMA DEVICE: ADDITIONAL CONSIDERATIONS AND RECOMMANDATIONS FOR FIELD USE Theretical cmparisn peak-based k est vs. k s fr field prfiles Hmgeneus half-space Layered half-space Prpsal fr PRIMA field-testing enhancements Backcalculatin analysis Hardware.. 63 CHAPTER 7 CONCLUSIONS. 65 References Appendix A1

6 LIST OF FIGURES page Fig Cmmn in-situ test cnfiguratin Fig Typical cyclic shear stress vs. cyclic shear strain curve (granular material).. 6 Fig Half-space laded by a circular fting Fig SDOF system - tp frce excitatin Fig SDOF system - base mtin excitatin Fig Plt f the magnificatin factr fr a SDOF system - base mtin excitatin Fig Ideal single input/single utput linear system Fig Plt f the mbility functin theretical example fr a SDOF system Fig Plt f the dynamic stiffness functin theretical example fr a SDOF system 17 Fig Relatinships amng FRFs Fig Theretical dimensinless cmpliance functins fr the vertical mde f vibratin f a massless rigid disk resting n an elastic half-pace Fig Steady-state frcing f the elastic half-space massive rigid fting Fig General illustratin f PRIMA 100 device Fig Detailed illustratin f PRIMA 100 device Fig Plt f PRIMA s sftware utput Fig Mdificatin f the setup f PRIMA device Fig Sketch f the verificatin setup Fig Verificatin setup Fig Idealized beam mdel layut Fig Fur-pint bending and static stiffness tests setup Fig PRIMA measurements n the 70 cm span beam, using a rubber hammer. 37 Fig Verificatin f PRIMA s gephne utput against an accelermeter Fig Peak methd applied t a theretical SDOF system Fig Influence f the pulse duratin n peak-based stiffness estimatin gdness - theretical SDOF system Fig BVT data acquisitin Fig SDOF analg fr the BVT setup Fig Typical measurement sequence and utput Fig Measured data and fitted SDOF curves- example f a 60 cm span beam.. 50 Fig Fitted SDOF real part f K(ω) - example f a 60 cm span beam Fig Simplified layut f the simply supprted beam with damping devices lcated n the cantilever parts

7 Fig Layut f a damping device lcated at the end f the verhanging part f the beam.. 53 Fig Testing setup fr the BVT with additinal damping Fig Cmparisn f velcity time histries undamped and damped beam setup Fig Example f time histry plts hmgeneus half-space Fig Example f plt k est /k s vs. T - hmgeneus half-space Fig Pssible field testing setup fr PRIMA device LIST OF TABLES Table 2.1. SDOF free vibratin slutins. 10 Table 2.2. Nmenclature fr FRF Table 4.1. Values f k s using direct measurement and beam thery.. 36 Table 4.2. Average k est frm PRIMA: riginal (with small height drps) and enhanced cnfiguratins vs. k s. 37 Table 5.1. Static stiffness estimates via fitting f the mbility functin undamped beam Table 5.2. Static stiffness estimates via fitting f the mbility functin beam with damping devices

8 LIST OF SYMBOLS The fllwing alphabetical list defines the main symbls assciated with the main parameters used thrugh the reprt. Symbl a a 1 β b c c s C vv C vv t ω d ε E E max E true f f peak F F 0 f f n FRF γ 2 G G max name fting radius distance between beam supprt and receptacle supprt tuning rati beam width damping cefficient shear wave velcity vertical cmpliance half-space/rigid fting nn-dimensinal vertical cmpliance half-space/rigid fting sampling perid angular frequency reslutin distance between receptacle supprts strain Yung s mdulus small strain r seismic elastic mdulus true Yung s mdulus f the beam frce time histry peak f the frce time histry Furier transfrm f f steady-state frce amplitude linear frequency undamped natural linear frequency Frequency Respnse Functin cherence functin shear mdulus maximum r small strain shear mdulus G ψψ (ω) ne-sided aut-spectral density functin f the signal ψ(t) G θ ψ (ω) ne-sided crss-spectral density functin between ψ(t) and θ(t) η shape factr (fr interface sil/fting stress distributin) h beam thickness i imaginary number I beam crss-sectinal mment f inertia I(ω) impedance functin k spring cefficient (stiffness) K(ω) dynamic stiffness

9 k est k static k s L m m eq M(ω) M M b ν p θ and Θ ρ σ S SDOF t t q T T T n ω ω ω j ω n ω d ξ u and U x x peak X X 0 x& X & & x& X & ψ and Ψ y z and Z estimated static stiffness static stiffness static elastic stiffness f a system beam (effective) span mass equivalent mass fr the SDOF system mbility functin mass rati bending mment Pissn s rati half the ttal lad applied t the beam linear system s arbitrary input time histry and its Furier transfrm sil s mass density stress cantilever (verhanging) part f the simply supprted beam Single Degree f Freedm System time discrete sampling time duratin f frce pulse f dimensinless perid SDOF system natural perid angular frequency dimensinless frequency discrete sampling frequency undamped natural angular frequency damped natural angular frequency damping rati steady-state grund displacement and its amplitude displacement time histry peak f the displacement time histry Furier transfrm f x steady-state displacement amplitude velcity time histry Furier transfrm f x& acceleratin time histry Furier transfrm f & x& linear system s arbitrary utput time histry and its Furier transfrm psitin alng the beam arbitrary time histry and its Furier transfrm

10 EXECUTIVE SUMMARY Nn-destructive testing is becming mre cmmnly used in field characterizatin f paved and unpaved subgrade prfiles. In particular, its applicatin t the quality assurance f newly cnstructed granular base (i.e. pavement fundatin) layers has becme mre widespread. Typically, prtable nndestructive devices are used fr an in-situ assessment f the elastic parameters (such as Yung's mdulus) f cmpacted sil layers n the basis f dynamic frce and displacement measurements. This study deals with the reliability f stiffness estimates stemming frm such prtable tls. T this end, the perfrmance f a particular device, Prtable Falling Weight Deflectmeter (PFWD) PRIMA 100, is examined and verified against the newly develped verificatin tester, the PFWD Beam Verificatin Tester (BVT). The BVT apparatus has been develped at the University f Minnesta fr the Minnesta Department f Transprtatin (Mn/DOT). This apparatus is intended fr verifying the perfrmance f PRIMA 100 devices. Using the BVT assciated with a spectral analysis methd, tests als can be cnducted t check whether the calibratin factrs f the PRIMA device sensrs, as given by the manufacturer, are accurate. The bjective f the verificatin tests is t detect the ptential ccurrence f deteriratin f the sensr s accuracy during the life f the field-testing device. The data interpretatin methd that is traditinally used in PFWD testing uses peak values f the dynamic frce and displacement recrds in lieu f their static cunterparts. By cmparing the PRIMA 100 device t the BVT, it is shwn that this peak values methd fails t prduce crrect estimates f the static beam stiffness, with an errr ften exceeding 100 percent. Systematic errr due t the peak methd, bserved experimentally fr the beam, can be expected in the field situatins as well. T eliminate this majr surce f systematic errr assciated with dynamic fieldtesting f sil stiffness, an alternative data interpretatin methd, based n spectral analysis, cncept f frequency respnse functin, and single-degree-f-freedm mechanical analg is emplyed t extract the true static stiffness f the supprt (i.e. beam) frm PRIMA 100 measurements. The results shw a gd agreement between the true static stiffness f the beam and its estimates determined using the mdified data interpretatin methd. Using an apprpriate mdel fr the sil, the same methd is prpsed fr quality assurance applicatins. On the basis f labratry experiments, a simplified, yet imprved versin f the cmmercial PFWD device als is prpsed that imprves the dynamic signature f the impact frce and minimizes the detrimental effects f nn-linear sil behavir in cmmn field situatins.

11 CHAPTER 1 - INTRODUCTION 1.1 Backgrund Althugh empirical methds are still widely used in pavement design, mechanisticempirical design prcedures are beginning t be implemented. Such methds are based upn the determinatin f the fundamental mechanical prperties f pavement layers. The material prperties typically used t characterize base and subgrade layers within the framewrk f mechanistic methds are the resilient mdulus and Yung s mdulus. The frmer can be measured via labratry tests (see the LTPP prtcl P46 [1]), whereas the latter mdulus, whse field measurement is the fcus f this study, is typically estimated using nndestructive testing (NDT) techniques. The field techniques addressed in this reprt are limited t dynamic NDT methds. Other field methds that als are cmmnly emplyed t estimate Yung s mdulus include static NDT methds (bearing plate tests) and destructive methds (penetratin tests). These methds d nt fall within the scpe f this study and will nt be discussed. In NDT f pavement prfiles, the prperties f sils, aggregate base and ht-mix asphalt (HMA) layers are cmmnly estimated frm field measurements using backcalculatin methds. The estimated prperty is ften the stiffness f the NDT device s lading plate-pavement system. Due t their shrt test duratin, the mst prmising devices are based upn a dynamic lading using vibratry r impulse surces. Such devices prvide a fast and accurate tl that has imprtant applicatins in cnstructin quality assurance and pavement deteriratin assessment. A gd example wuld be the Falling Weight Deflectmeter (FWD) devices. Indeed, the backcalculatin f the elastic prperties f pavement layers using FWD measurements is a wellrecgnized prcedure in pavement assessment. The reader can find mre abut this tpic in Lyttn [2]. Amng varius testing devices fr nn-destructive in-situ assessment f sil prperties, prtable deflectmeter-type devices have recently becme the fcus f increasing interest. These devices, being prtable, are easier t use and require nly ne r tw peratrs. These prtable devices are based n the dynamic frce and deflectin measurements. They can be classified int three main categries: 1) vibratry devices, such as the Humbldt GEOGAUGE, 2) impact lad devices based upn a transient type f lading, such as the Prtable Falling Weight Deflectmeters (PFWDs) PRIMA 100 and LOADMAN, and 3) devices based n the Spectral Analysis f Surface Waves (SASW), such as the Prtable Seismic Pavement Analyzer (PSPA), and its revisins called Dirt SPA (D-SPA) and Olsn SASW-2G. In cntrast t SASW tls, vibratry and impact lad devices are bth based n the s-called deflectmeter type analysis that will be the subject f this investigatin. Amng the latter, the mst cmmnly used devices are the PFWDs and the GEOGAUGE. 1

12 1.2 Research issues and gals This reprt addresses issues assciated with prtable NDT devices used by the Minnesta Department f Transprtatin (Mn/DOT), namely GEOGAUGE, PRIMA 100, and LOADMAN devices. The main cncern is t assess whether r nt the fting-n-sil stiffness (and sil mdulus) estimates btained frm these devices are reliable. These devices all fall int the categry f near-field testing devices, in the sense that the deflectin measurements are made in the vicinity f the applied lad. All f the abve devices measure the stiffness f the fting-n-a-subgrade system and prvide user with an estimatin f the equivalent hmgeneus elastic mdulus f the granular base and subgrade materials. GEOGAUGE recrds the data in the frequency dmain, whereas impact devices recrd the time histry f dynamic measurements. PRIMA 100 measures bth lad and displacement time histries; GEOGAUGE recrds the dynamic stiffness transfer functin, and LOADMAN acquires acceleratin t estimate peak deflectin and uses either an estimate f the lad (riginal versin) r a measured lad (LOADMAN 2) t calculate the elastic mdulus. It is well-knwn that the mdulus estimates btained in the field are strngly dependant upn the applied stress and therefre upn the particular NDT device being used. Fr example, ne can refer t cmparative studies in Chen & al [3], Siekmeier and al. [4], Mc Kane [5] and Van Gurp and al. [6]. The incnsistency f stiffness estimates stemming frm varius prtable devices is due t tw main reasns: 1) the stress dependency f the sil and therefre the dependency upn the testing device 2) the crrectness f stiffness estimates (i.e. the analysis) assciated with a given device. The dependency upn the device renders cmparisn between mdulus estimates btained frm several devices difficult. The treatment f the first issue is ut f this reprt s scpe. One way t vercme such disparity in the results is t investigate experimental crrelatins between devices r empirical relatinships frm labratry results. Cncerning the secnd issue, a ratinal apprach needs t be develped t check and validate the stiffness estimates stemming frm a given prtable device, which is the fcus f this study. The in-situ mechanical prperties f sils and granular materials are generally unknwn, especially wing t the stress dependency f the base and subgrade materials. T determine whether r nt the stiffness estimates btained frm a given field-testing device are reliable, verificatin against a well-characterized fundatin (r supprt) mdel is indispensable. The main issue f this prject is therefre t design a labratry setup t verify n a rutine basis the stiffness estimates resulting frm prtable deflectmeter-type devices. A secndary issue is t investigate enhancing such devices. One particular prtable device, PRIMA 100, has been chsen as a fcus fr this prject. Amng the devices available fr this study, PRIMA device was the best example pssible t set up the perfrmance verificatin prcedure, since bth measurements f lad and displacement can be checked. PRIMA testing is based upn measurement f the velcity due t impact lad applied by a falling weight nt a plate resting n the surface f the tested layer. The primary cncerns regarding verificatin f the perfrmance f PRIMA device are 1) accuracy f the frce and velcity sensrs embedded in PRIMA, and 2) scrutiny f the 2

13 internal backcalculatin prcedure that prduces stiffness estimates frm peak values. The Beam Verificatin Tester (BVT) apparatus has been develped at the University f Minnesta fr the Minnesta Department f Transprtatin (Mn/DOT). Using the BVT assciated with a spectral analysis methd, tests can be cnducted t evaluate the perfrmance f PRIMA stiffness estimatin. Als, the BVT can be utilized t check whether the calibratin factrs f the sensrs f PRIMA devices, as given by the manufacturer, are accurate r nt, the bjective being the detectin f a ptential ccurrence f deteriratin f the sensr s accuracy during the life f the devices. 1.3 Research apprach The interpretatin f FWD and PFWD measurements is in practice based n a simplified wavefrm analysis, which fcuses n the peak values f the frce and displacement time histries. This peak methd can be seen as an attempt t btain static prperties frm tests that are dynamic in nature. A majr drawback f this simplificatin is that the time ffsets between peak values and the inertia effects are disregarded. The use f the traditinal methds based n this peak methd t prvide an input t elaststatic backcalculatin is knwn t lead t errneus estimates f pavement layers prperties. Using the entire wavefrms as input t a dynamic backcalculatin methd wuld be assciated with a high cmputatinal cst, which is currently nt desirable fr rutine practice. As shwn recently by Guzina and Osburn [7], the elaststatic backcalculatins perfrm well when the peak methd is advantageusly replaced by a simple analysis invlving the cncept f Frequency Respnse Functin (FRF). Utilizatin f FRF-based methd in quality assessment has nt been implemented widely t date, althugh it has been revisited by several researchers. An example f FRF applied t pavement integrity evaluatin is given in the 1994 paper by Lepert and al. [8]. Stlle [9] used a FRF assciated with a Single Degree f Freedm System (SDOF) analg t estimate an equivalent hmgeneus Yung s mdulus. Briaud [10] applied such methd t the estimatin f the static sil-under-a-spread-fting stiffness using an impact test. The interpretatin f PRIMA 100 measurements is based upn a peak values methd assciated with an elaststatic calculatin algrithm. As a result, the verificatin f the perfrmance f the device invlves nt nly verificatin f the accuracy f the sensrs, but als mdificatin f the data interpretatin methd. An alternative methdlgy based upn a new data acquisitin prcedure and a determinatin f the static stiffness via FRF and spectral average cncepts is ne f the key cntributins t this study. 1.4 Reprt rganizatin The remainder f this reprt is rganized in six chapters. In Chapter 2, we will first examine the theretical elaststatic framewrk embedded in the classical deflectmeter- 3

14 type data interpretatin and review useful ntins such as SDOF systems, FRFs, and spectral analysis. In Chapter 3, we present the PRIMA device, explre hw it perates, and hw the measurements are interpreted in cmmn practice. Chapter 4 describes the prpsed verificatin setup and prcedure that will be used t verify PRIMA stiffness estimates. The verificatin assembly, s-called Beam Verificatin Tester (BVT), based n a simply supprted steel beam, will be intrduced. Its mechanical stiffness will be examined theretically and experimentally using a static calibratin test. Results f verificatin tests n the BVT apparatus will be discussed. The cnsistency f the stiffness estimates btained frm PRIMA device as intended by the manufacturer is investigated with a fcus n the data interpretatin methd. In Chapter 5, an alternative calculatin apprach is intrduced and used t verify the perfrmance the device. This apprach invlves bth an enhancement f the device and a mdificatin f the data acquisitin and interpretatin methd. The prpsed data interpretatin methd is based upn the full wavefrm analysis perfrmed via measurement f an apprpriated FRF, such as the mbility M(ω), frm which the true static stiffness can be extracted. The BVT apparatus will be prpsed as a tl t perfrm rutine verificatin tests f the perfrmance f PRIMA devices. Guidelines fr extensin f the BVT t ther prtable deflectmeter devices will be presented. Finally, recmmendatins fr the enhanced field perfrmance f the PRIMA device and cnclusins will be presented in Chapters 6 and 7, respectively. 4

15 CHAPTER 2 - THEORETICAL BACKGROUND As stated in Chapter 1, the Yung s mdulus f base and subgrade layers is the parameter targeted by NDT methds. Typically, this mdulus is inferred frm field measurements f the device s-circular-fting-n-tp-f-sil stiffness. Hwever, there is a need t clarify what is a sil Yung s mdulus. Indeed, the Yung s mdulus f a sil is nt unique and several definitins exist. The mdels used in practice t interpret field measurements generally assume linear elasticity and thus small defrmatins. The mdels develped t verify the field devices in this study als need t cmply with these requirements. We will first clarify the mdulus f interest. We will further examine the elastic frward mdel underlying the estimatin f Yung s mdulus frm frce and deflectin measurements, as intended by the manufacturers. We als will intrduce sme theretical cnsideratins cncerning the Single Degree f Freedm (SDOF) mdel and the Frequency Respnse Functins (FRFs), anticipating their need in the subsequent chapters. Finally, the slutin fr the cmplex cmpliance f an elastic half-space mdel is intrduced. 2.1 Sil mdulus Prtable deflectmeter devices are usually assciated with data interpretatin assuming a sil mdel that is semi-infinite, hmgeneus, and elastic. In many cases, hwever, the field situatin fr pavement fundatin testing can be mre accurately described as a unifrm layer n tp f a half space, as sketched in Fig In this case, the mdulus estimated using the device is a cmbinatin f the mduli f the base and subgrade layers. The resulting measured mdulus is referred t as the equivalent hmgeneus mdulus. Frce f Fig Cmmn in-situ test cnfiguratin 5

16 In the backcalculatin f pavement prfiles, the assumed frward mdels are cmmnly based n linear elasticity. Hwever, the materials we are dealing with are highly nn-linear. Indeed, sil prperties are knwn t be highly dependent upn pressure and strain level, amng ther parameters (see Hardin and Drnevich 1972 [11]). Previus experimental wrk by Pak and Guzina [12] als illustrated the stiffness dependence f a sil-fundatin system upn bth the size f the fundatin and the static pressure applied t the sil, n the basis f scaled centrifugal tests. In the literature many definitins f sil mduli are utilized: secant mdulus, tangent mdulus, maximum mdulus, resilient mdulus. Let us examine the nnlinear cnstitutive stress-strain relatin fr cyclic shear test depicted in Fig This plt shws that the shear mdulus G(γ) is strain dependent: it decreases as the strain level increases. τ cyclic G max G(γ) γ cyclic Fig Typical cyclic shear stress vs. cyclic shear strain curve (granular material) Therefre, the measured equivalent hmgeneus elastic mdulus will depend upn the particular cnditins assciated with hw each testing device applies the lad. As a result, Yung s mdulus estimates frm different devices r different testing setups inevitably will differ, and ne has t be very careful when cmparing different devices. Fig. 2.2 als illustrates the fact that fr small strains, G(γ) becmes cnstant. In this case, the shear mdulus is referred t as the maximum shear mdulus G max. The same cncepts apply t Yung s mdulus E, and E max is knwn as the seismic mdulus r small strain mdulus. E max is als the slpe f the initial unlading/relading hysteresis lp fr small defrmatins. As a matter f fact, it is accepted that at lw strain level, (f the rder f 10-4 % fr the cyclic shear strain), the sil exhibits an apprximately elastic behavir (e.g. Kramer [13]; Richart et al. [14]; Seed et al. [15]). In ther wrds, such strain level is nt large enugh t induce significant nn-linear stress-strain behavir in the sil. Typically, E max is estimated in the field using lw strain level nndestructive testing (NDT) techniques. Fr pavement cnstructin quality assurance and deteriratin mnitring purpses, the use f tests invlving small defrmatins wuld bring mre cnsistency in the sense that nly ne value f elastic mdulus (namely E max r G max ) wuld be 6

17 estimated. This wuld enable a direct cmparisn between varius deflectmeter devices. Such mdulus estimates als culd be directly cmpared t gephysics investigatin methds. Furthermre, fr a given dminant wavelength f the applied dynamic frce, the use f testing methds transmitting a smaller energy level t the sil is assciated with a smaller depth f penetratin. This is appreciable in quality assurance f pavement cnstructin where the mdulus f the tested surface layer (e.g. base) needs t be checked nly: The estimated equivalent hmgeneus mdulus wuld then crrespnd t the seismic mdulus f the targeted layer. 2.2 Elaststatic mdel Fr the sake f simplicity, it is generally assumed in prtable NDT methds that the tested pavement fundatin is elastic, istrpic, hmgeneus and unbunded. The backcalculatin prblem is further simplified by cnsidering the static lad-displacement relatinship. As illustrated in Fig. 2.3, the tested sil is mdeled as an elastic semi-infinite half-space cntinuum, characterized by tw elastic cnstants, Pissn s rati ν and shear mdulus G. Yung s mdulus E and shear mdulus G are related by the well-knwn relatinship frm elasticity thery (fr example, in Sklnikff [16]) E = 2 G (1 + ν) (1) Fig Half-space laded by a circular fting As mentined earlier, the devices under study are deflectin-based. The measured quantities are lad and displacement. Frm static measurements f lad and displacement ne can cmpute the static stiffness, defined fr any linear system as 7

18 f k = (2) x where x is the displacement f the surface due t the applied frce f, measured at the center f the fting. As a result, the stiffness k is a parameter f fundamental interest in the peratin f the prtable deflectmeter devices. During testing with PRIMA device, the lad is applied t the sil supprt via a circular base plate f radius a (see Fig. 2.3). The resulting displacement is measured under the lad, at the center f the plate. The Bussinesq equatin, classical elastic slutin f the prblem f a half-space subjected t a surface pint lad, can be used t derive the relatinship between displacement and stresses fr the case f a rigid r flexible fundatin sitting n the halfspace. Such relatinships can be fund, fr example, in Craig [17]. In the particular case where the displacement under the center f circular fting f radius a resting n the surface f the elastic half-space is cnsidered, the shear mdulus G and Pissn s rati ν are related t the static stiffness k accrding t k( 1 ν ) G = (3) η a where η is a shape factr depending upn the stress distributin at the interface sil/fting. If the stress is unifrmly distributed under the fting, which crrespnds t a flexible fting, then η = π. At the extreme ppsite, fr the case rigid fting is mdeled, the stress distributin is n lnger unifrm and η = 4. T cnclude, the elaststatic analysis prvides a clsed-frm static slutin frm which, fr a knwn r assumed value f the Pissn s rati f the sil, the sil Yung s mdulus can be cmputed frm measurements f the stiffness. 2.3 Review f the Single Degree f Freedm system The sectin s purpse is t briefly review the main results f the mtin f a Single Degree Of Freedm (SDOF) system. Fr mre details, the reader shuld cnsult the specialized literature. Reference examples are a bk f Meirvitch [18] fr the fundamental aspects and a bk f Inman [19] fr mre detailed engineering applicatins. A SDOF system is defined as a system that cntains nly ne significant rigid mass. The cnstitutive respnse f a SDOF system can be described in term f the mechanical Kelvin-Vigt analg. As depicted in Fig. 2.4, the SDOF system is cmprised f a mass m, a massless dashpt with damping cefficient c, and a massless spring f cnstant k. The spring represents the elastic part (stiffness) f the system whereas the dashpt represents the viscus damping. In this example the SDOF system is attached at its base t a rigid supprt and subjected t a tp excitatin time varying frce f = f(t). 8

19 f(t) mass m x(t) spring cefficient k dashpt cefficient c rigid supprt Fig SDOF system - tp frce excitatin With reference t Fig. 2.4, the SDOF system respnds with a displacement x = x(t) induced by the applied frce f = f(t). The crrespnding equatin f mtin is m && x + c x& + k x = f (4) The ntatin x& represents the velcity dx/dt and & x& stands fr the acceleratin d 2 x/dt 2. Using the definitin f the undamped natural frequency ω n and the damping rati ξ, i.e. ω = k n (5) m c and ξ = (6) 2mωn the equatin f mtin can be rewritten as 2 f && x + 2ξ ω x& (7) n + ωn x = m The slutin t this rdinary differential equatin (ODE) can be derived via the time dmain analysis r via the frequency dmain analysis. The slutin f the ODE can be written as the sum f a general r hmgeneus slutin crrespnding t the case f = 0, and f a particular slutin that takes int accunt the applied frce. The ttal mtin f the SDOF is crrespndingly decmpsed int a free vibratin part (hmgeneus slutin) and a frced vibratin part (particular slutin). Fr the free λ t vibratin case, enfrcing a slutin f the type x (t) = Ae, it can be shwn that λ1 t λ2 t x = A1 e + A2 e where A 1 and A 2 are cnstants t be determined frm the initial cnditins and where λ 1 and λ 2 are the rts f the characteristic equatin (8) 2 2 λ + 2ξ ωn λ + ωn = 0 (9) 9

20 Depending upn the value f the damping cefficient ξ, several cases can be distinguished: the undamped case (ξ = 0), the verdamped case (ξ > 1), the critically damped case (ξ = 1), and the underdamped case (ξ < 1). The rts f the characteristic equatin are synthesized in Table 2.1. rts f the characteristic equatin λ 1,2 undamped case verdamped case critically damped case ξ ω ± ω ξ 2 1 ω ± ω n i n n n underdamped case ξ ω ± iω n n 2 1 ξ slutin x(t) Asin( ω nt + φ A) B e 1 λ1 t + B 2 e λ2 t e ω t n ( C + 1 C2 t ) t e ωn ξ D sin( ω t + φ ) D d assciated definitins A = A 1 A2 D = D 1 D2 φ A A = atan 1 A φ D D = atan 1 D 2 ω ω d = n 2 1 ξ Table 2.1. SDOF free vibratin slutins The cnstants A 1, A 2, B 1, B 2, C 1, C 2, D 1 and D 2 must be determined frm the initial cnditins. They represent the amplitude f vibratin. The parameter ω d is called the damped natural frequency. Bth φ A and φ D are referred t as the phase angle. The free vibratin characteristics f the SDOF system depend nly upn the prperties f the SDOF and the initial cnditins f mtin. Cncerning the frced vibratin, the mst general case is the ne f an arbitrary time varying applied frce. The particular slutin can be recvered using the Duhamel r cnvlutin integral. Fr example, in the case f an underdamped SDOF system subjected t a frce f applied frm t = 0 t t = T, the particular slutin can be written in terms f Duhamel integral as t 1 ξ ω n ( t τ ) x( t) = e ω t τ τ dτ (10) mω sin( d ( ))f( ) d 0 The Duhamel integral can be determined in a clsed-frm manner nly fr a few cases where f is simple enugh and knwn analytically. In general, integral (10) has t be evaluated numerically. An alternative and yet very cmmn way t evaluate the system mtin is t perfrm a direct numerical integratin f the equatin f mtin. Amng the varius existing methds, ne can refer t the linear acceleratin methd in Wilsn and Clugh [20]. This step-by-step methd is easy t implement and gives accurate results. A particular case f frced vibratin is the steady-state case, in which the transient part is disregarded. In this case the analysis is simplified. Assuming a harmnic applied frce, fr example a sine frce described by f = F0 sin( ω t) (11) 10

21 ne can shw that the mtin is given by where X 0 is defined as X 0 = m F 0 x = X sin( ω t 0 φ F0 = k ( ω n ω ) + (2ξ ωωn) (1 β ) + (2ξ β ) ) 1 (12) (13) and φ is φ 2ξ ωω n 2ξ β atan = atan 2 2 ωn ω 1 β = 2 (14) In these equatins, ω is the driving frequency, f 0 / k crrespnds t the static displacement and β is the tuning rati defined as x s tatic ω β = (15) ω n A useful quantity is the amplificatin r magnificatin factr f the SDOF mdel, defined as X 0 1 = x (16) s tatic (1 β ) + (2ξ β ) Nte that all the equatins are written using the circular frequency ω, given in terms f the linear frequency f by 2.4 Mtin transducers ω = 2 π f (17) In the existing prtable deflectmeter devices, measurement f dynamic displacement is perfrmed using mtin transducers perating withut a fixed frame f reference, namely gephnes (velcity transducers) r accelermeters (acceleratin sensrs). The assciated backcalculatin prcess als utilizes these transducers, and it is imprtant t intrduce sme f the basic cncepts used fr the measurement f displacement. Gephnes are sensrs whse utput is prprtinal t velcity, whereas the utput f accelermeters is prprtinal t acceleratin. The displacement time histry can be btained by integratin f the velcity time histry r by duble integratin f the acceleratin time histry. Regular gephnes and accelermeters are bth referred t as seismic transducers, designed using SDOF system thery. Als cmmn are the piezelectric transducers, fr which acceleratin is sensed. Hwever, the utput f piezelectric sensrs als can be prprtinal t velcity by integratin f the acceleratin. Gephnes are sensrs designed with a lw natural frequency (typically f the rder f 3 t 20 Hz) whereas accelermeters are designed with very high natural frequency (typically several khz). 11

22 The perating frequency range f gephnes is typically lcated abve their natural frequency, whereas accelermeters are theretically accurate fr frequencies lwer than their natural frequencies. Pr nise/signal ratis, hwever, reduce the reliability measurements frm accelermeters at very lw frequencies. As a result, the utput f bth types f sensrs at very lw frequencies (f the rder f 3-15 Hz) is disregarded in practice. T illustrate the peratin f cmmn seismic transducers, the theretical respnse f a gephne, fr which the utput vltage is prprtinal t the velcity f the mass, is examined. The utput vltage f an SDOF-type velcity transducer is generated by a cil f mass m mving thrugh a magnetic field. Mrever, the utput vltage is directly prprtinal t the relative velcity between the cil and the magnet (Faraday s law). Cnsider a SDOF system subjected t a steady-state sine displacement u = u(t) at its base (see Fig. 2.5) where u =U sin( ) (18) 0 ω t x(t) k c magnet mass m cil surface in mtin (e.g. grund) u(t) Fig SDOF system - base mtin excitatin With regards t Fig. 2.5, the magnet is made part f the transducer s frame, which experiences the same mtin as the attached surface. In such scenaris, the utput vltage is, therefre, directly prprtinal t the relative velcity between the cil and the surface in mtin. Similarly t the case presented in the previus sectin, it can be shwn that the mtin x = x(t) f the transducer relative t its supprt is and that the magnificatin factr is X 0 U 0 x = X 0 sin( ω t φ) X& = U & 0 0 = ( 1 β ) + (2ξ β ) Based n equatin (20), it can be readily prved that fr β >> 1, i.e. fr frequencies higher than the natural frequency f the gephne, the magnificatin factr tends twards a unit value. That is, the gephne experiences the same mtin as its supprt fr β >> 1. 2 β (19) (20) 12

23 The cmplete plt f the magnificatin factr versus tuning rati is presented in Fig. 2.6 fr three different values f damping rati ξ. Frm Fig. 2.6 we bserve that fr β > 2 t 3, the magnificatin factr f a gephne can be cnsidered as cnstant and equal t 1. On the cntrary, fr β < 2 t 3, the magnificatin factr, and therefre the gephne utput, is highly nn-linear. With reference t Fig. 2.6, the amplificatin factr is cnstant n a larger frequency range fr a damping rati clse t 70% than fr smaller damping ratis. Indeed, 70% f damping is very ften the value chsen in the design f seismic sensrs when the gal is t btain a maximum peratinal range fr the amplitude f the measured signal. 4 4 amplificatin factr Magnificatin factr rapprt( β ) rapprt20( β ) rapprt70( β ) 3 2 ξ = 20% ξ = 0% useful range 1 0 ξ = 70% β Fig Plt f the magnificatin factr fr a SDOF system - base mtin excitatin The main cnsequence resulting frm the featured magnificatin factr is that the utput f an SDOF-type gephne shuld be disregarded at very lw frequencies. T prvide fr engineering applicatins, let us cnsider the cmmn case f a gephne with a natural frequency f the rder f 5 Hz. As a guideline, the utput f such a transducer shuld nt be taken int accunt fr frequencies belw Hz. 2.5 Frequency Respnse Functin (FRF) fr a SDOF system 5 Let us cnsider the ideal linear system depicted in Fig The system is subjected t an input θ(t) and respnds with an utput y(t). Let Θ(ω) and Y(ω) be the Furier transfrms f θ(t) and z(t). Frequency respnse functins FRF = FRF(ω) f the linear system are defined in frequency dmain as the rati between utput Y(ω) and input Θ(ω) Ψ( ω) FRF( ω) = (21) Θ( ω) 13

24 input θ (t) Θ(ω) Linear System utput y(t) Y(ω) Fig Ideal single input/single utput linear system FRFs cnstitute a particular case f the s-called transfer functins, which are defined in the Laplace dmain rather than in Furier dmain. Several types f frequency respnse functins can be defined, depending upn the quantities f interest taken as system input and utput. The mst cmmnly used type f FRF describes the relatin between a frce input and the system s kinematic utput, which can be displacement, velcity, r acceleratin. This type f FRF will be the nly ne discussed here. Table 2.2 presents a general nmenclature based n the utput s type. FRF definitin Displaceme nt Frce Velcity Frce Accelerati n Frce Frce Displacement Frce Velcity Frce Acceleratin FRF name Cmpliance Mbility Accelerance Dynamic Impedance Apparent adpted in stiffness mass the study 1 C ( ω) = K( ω) 1 M ( ω) = I( ω) 1 A ( ω) = AP( ω) F( ω) K ( ω) = X ( ω) F( ω) I( ω) = X& ( ω) F( ω) AP( ω) = X&& ( ω) Other names Dynamic - Inertance Impedance Mechanical Effective encuntered cmpliance impedance mass in the Admittance literature Receptance Table 2.2. Nmenclature fr FRF The tw FRFs used in this study, the mbility and the dynamic stiffness, will be presented in mre details in the cming sectins Furier transfrm Let us cnsider a cntinuus tempral signal z(t) and recall the definitin f its frward Furier transfrm Z(ω) iω t Z( ω) = z( t) e dt - (22) where i is the imaginary number defined byi = 1. The inverse transfrmatin can be written as 1 i ω e ω t z( t) = Z dω (23) π ( ) 2-14

25 By cnventin, in this reprt, the capital letters represent the Furier transfrm f the quantities in the crrespnding lwer case letters. The abve equatins are valid fr cntinuus signals f infinite duratin. Measurements, hwever, are made using digitized recrds f finite duratin, and, therefre, these equatins cannt be applied. In this case, it is cnvenient t use the Discrete Furier Transfrm (DFT). If we cnsider z(t), tempral signal f finite duratin T z, and z(t k ), discrete recrd issued frm z(t) sampled at t q = q t (with q = 0, 1, 2, N z -1), where t is the sampling perid, the DFT pair is Z( ω ) = t z( t q j ω ) = 2π N z 1 q= 0 N z 1 j = 0 z( t q ) e Z( ω ) e j iω j tq iω j tq (24) (25) where N z = T z / t is the length f the recrds (number f pints) and ω j = j ω = π j / T z. The frequency reslutin is given by ω = 2 π / T z. The DFT is usually implemented n cmputers using Fast Furier Transfrm (FFT) algrithms, which prvide significant cmputatinal time savings and render DFT calculatins efficient n persnal cmputers Mbility functin M(ω) Applying the frward Furier transfrm, equatin (22), t bth sides f the equatin f mtin, and using X && = iω X& = -ω ² X (26) where X & = X & ( ω) and X = X ( ω), ne btains 2 X& ω n iω + 2 ξ ω + i ω This equatin can be rearranged t yield X& ( ω) M ( ω) = F( ω) n = The mbility functin M = M(ω) is therefre given by The mbility functin f a SDOF system is cmplex valued and can be decmpsed int real and imaginary parts. It can be readily shwn that F m (27) (28) (29) Im{ M ( ω)} iω M = k (1- β ²) + (2 iξ β ) 1 2 ξ β ω Re{ M ( ω)} = k 2 (1- β ²) + (2 ξ β ) n = 2 k ωn (1- β ²) 2 3 ω ω β + (2 ξ β ) 2 (30) (31) 15

26 10 x 10-6 cmplex mbility M(ω) [m/s/n] 5 real part 0 imaginary part β = ω / ω n Fig Plt f the mbility functin theretical example fr a SDOF system Fig. 2.8 presents the mbility functin, fr a tuning rati β ranging frm 0 t 5. The rder f magnitude fr the amplitudes reflects nly the particular values chsen fr 3 the prperties f the SDOF system represented (m = kg, ξ = 20% and k = 4 MN/m) Dynamic stiffness functin K(ω) The dynamic stiffness functin is defined as K = K(ω) F( ω) K ( ω) = X ( ω) Fllwing the same pattern than fr the mbility functin, it can be shwn frm that 2 [( 1- β ²) (2 iξ β ) ] K = k + Fr a SDOF system, real and imaginary parts are Re{ K ( ω)} = k (1- β ²) Im{ K( ω)} = 2kξ β Using equatin (5), equatin (6) can be rewritten as k c = 2 mξ ωn = 2ξ k m = 2 ξ ω Plugging equatin (36) int equatin (35) yields Im{ K ( ω)} = cω n (32) (33) (34) (35) (36) (37) 16

27 Examinatin f equatin (34) implies that, fr a SDOF system, the real part f the dynamic stiffness (i) is a parabla and (ii) des nt depend n the damping. Similarly, equatin (37) crrespnds t a line f slpe c. In Fig. 2.9, the dynamic stiffness functin is presented fr β ranging frm 0 t 5. The same cmment n the meaning f the amplitude values than fr Fig. 2.8 applies. 8 x cmplex dynamic stiffness K(ω) [N/m] imaginary part -2-4 real part β = ω / ω n Fig Plt f the dynamic stiffness functin theretical example fr a SDOF system Practical cnsideratins Relatinships between FRFs can be cnstructed n the basis f integratin and differentiatin in the frequency dmain. Recalling equatin (26), integratin in time dmain crrespnds t a divisin in frequency dmain by (i ω), whereas a differentiatin in time dmain is equivalent t a multiplicatin by (i ω) in frequency dmain. Similarly, duble integrating in time dmain crrespnds t divisin by (-ω 2 ) in frequency dmain and differentiating twice t multiplying by (-ω 2 ), respectively. On that basis, ne can btain the relatinships presented in Fig If the data acquisitin invlves nly ne type f mtin transducer, nly ne type f FRF can be directly measured. Hwever, advantage can be taken f the relatinships in Fig t btain all remaining FRFs. Fr example, using a gephne, ne can measure directly M(ω) r I(ω) and btain by simple multiplicatin r divisin the ther respnse functins. The FRF are cmmnly used in prblems invlving structural vibratins, such as mdal testing and system identificatin. Experimental FRFs are used t extract the prperties f the equivalent SDOF r MDOF (Multiple Degree f Freedm) analgs f the system under testing, such as natural frequency, mass, damping, and stiffness. 17

28 integratin: integratin: Accelerance A(ω) divisin by iω Mbility M(ω) divisin by iω Cmpliance C(ω) duble integratin: divisin by ω 2 Apparent mass AP(ω) differentiatin: multiplicatin by iω duble differentiatin: multiplicatin by ω 2 Impedance I(ω) differentiatin: multiplicatin by iω Dynamic stiffness K(ω) Fig Relatinships amng FRFs Estimatin f FRF Let us cnsider the single input/single utput, cnstant parameter linear system f Fig Measurement f the FRF = FRF (ω) defined by Ψ( ω) FRF( ω) = (21) Θ( ω) can be cnducted using 1) measurements f bth input and utput signals, and 2) a direct divisin f the signals in the frequency dmain. Data can be acquired either directly in frequency dmain, either in time dmain. In the case f time histries data, a FFT algrithm is emplyed t btain the crrespnding frequency histries. This methd is nt very rbust t nise perturbatins and experimental variability. As shwn in Bendat and Piersl [21], spectral average is a tl indispensable t minimize the effect f randm nise and measurement errrs n the FRF estimates. It is shwn in [21], fr recrds frm statinary randm prcesses with zer mean value, that the s-called input/utput crssspectrum relatin is Gθ ψ ( ω) FRF( ω) = (38) Gθ θ ( ω) where G θθ is the ne-sided aut-spectral density functin and G θψ is the ne-sided crssspectral density functin. Switching ntatins t use frequencies in Hertz leads t Gθ ψ ( f ) FRF( f ) = (39) Gθ θ ( f ) These relatins apply fr ideal situatins where n extraneus nise and n time varying r nnlinear characteristics are present. In practice, when dealing with 18

29 N T (n = 1, 2, N T ) digital recrds f length N ψ each, the averaged DFT f θ and ψ are given respectively by Nψ 1 2π i tq f j [ Θ( f )] = t [ θ ( t )] e, j = 0,1,2... N 1 (40) j n q= 0 Nψ 1 2π i tq f j [ Ψ( f (41) j )] n = t [ ψ ( tq)] n e, j = 0,1,2... N 1 q= 0 ω j j where f j = = and tq = q t. With equatins (40) and (41), the ne-sided spectral 2π Nψ t density functins are given by N 1 T * G θ ψ ( f j ) = [ Θ( f j ) ] n[ Ψ( f j )] n (42) N T n= 1 N 1 T * G θ θ ( f j ) = [ Θ( f j ) ] n[ Θ( f j )] n (43) N T n= 1 where * dentes the cmplex cnjugatin. T reduce the effects f side leakage during the FFT evaluatin, the acquisitin prcess can use varius types f time windw functin Λ(t), such as bxcar, Hanning r triangular windws. When using a windw, the DFT f the n th measurement (n = 1, 2, N T ) f a tempral recrd z(t) f length N z becmes [ Z( ω )] j n = t N z 1 q= 0 Λ( t The spectral average apprach als prvides the user with a tl t estimate the 2 quality f the measurements via the cherence functin γ = γ 2 ( ω) 2 G ( ) 2 θ ψ ω γ ( ω) = (45) Gθθ ( ω)gψψ ( ω) It can be shwn that 2 0 γ ( ω) 1 (46) The case γ 2 = 1 crrespnds t a perfect cnstant parameters linear system assciated with zer nise in the measurements. The case γ 2 = 0 is encuntered when there is n crrelatin at all between input and utput. All intermediate cases can be due t the presence f extraneus nise in the measurements, f nn-linearity in the system, r t the cntributin f additinal inputs. Spectral average techniques, assrted with windwing ptins are implemented in mdern data acquisitin systems such as the SigLab spectrum analyzer used fr this study. A descriptin f the methds used t determine FRFs with SigLab can be fund in the manufacturer s technical dcumentatin [22]. q q n )[ z( t q )] n e iω j tq (44) 19

30 2.6 FRF fr a hmgeneus elastic half-space mdel On the ne hand, assuming that the sil-fundatin system depicted in Fig. 2.3 behaves as a linear system, ne can measure experimentally any FRF listed in Table 2.2 using the tls presented in previus sectins. On the ther hand, theretical frmulatins f these FRFs describing dynamic Sil-Fundatin Interactin (SFI) fr an elastic half-space are als available Exact slutin fr the vertical cmpliance f a massless fting The theretical slutin fr the vertical dynamic interfacial cmpliance C vv fr the case f a massless rigid disk resting n the surface f a semi-infinite hmgeneus and linearly elastic cntinuum is presented in Luc and Westman [23] and Pak and Gbert [24]. This slutin, develped fr the case f a vertical steady-state frce, can be written as C ( ω) = C (0)[ C ( ω)] (47) vv The dimensinless vertical cmpliance functin C (ω vv ) is essentially independent f Pissn s rati ν. The static vertical cmpliance C vv (0) f a circular rigid fting f radius a is given by 1 ν C vv (0) = (48) 4G a where G and ν represent the equivalent hmgeneus shear mdulus and Pissn s rati f the half-space. The dimensinless vertical cmpliance C vv is a cmplex functin that can be represented as the sum f the dimensinless frequency functins F 1 and F 2 C vv ( ω ) = F1 ( ω ) + i F2 ( ω ) (49) where F 1 and F 2 are respectively the real and imaginary parts f C vv and where the dimensinless frequencyω defined as ω a ω = (50) c s In equatin (50), c s is the velcity f the shear wave prpagating in the cntinuum with mass density ρ and is defined as G c (51) s = ρ A plt f Cvv versus ω, cnstructed using the values f F 1 and F 2 tabulated in the paper by Pak and Guzina [12], is presented in Fig vv vv 20

31 C vv C ( ω ) = C vv ( ) (0) vv ω F 1 F1 -F2 0.6 F ω Fig Theretical dimensinless cmpliance functins fr the vertical mde f vibratin f a massless rigid disk resting n an elastic half-space Fig Steady-state frcing f the elastic half-space - massive rigid fting 21

32 2.6.2 Vertical cmpliance f a massive fting The slutin presented in the previus sectin relates t the case f a rigid massless fting resting n the tp f an elastic half-space and subjected t a vertical steady-state frce. Hwever, the mass f the fting needs t be taken int accunt in rder t describe cmpletely the sil-fundatin interactin. When the massive fting is iω t subjected t a steady-state frce f( t) = F0 e, as depicted in Fig. 2.12, it can be shwn that the steady-state displacement X 0 is given by F0 X 0 = 1 (52) 2 mω C ( ω) where m is the mass f the fting, ω the driving frequency, C vv the massless cmpliance functin and (m ω 2 ) the fting s transfer functin. Therefre, the displacement can be cmputed, fr a given driving frequency and a given fundatin s mass, n the basis f equatins (50) and (52), by using the curves presented in Fig Als, the vertical m cmpliance fr the massive fundatin, C, is readily btained frm equatin (52) as vv vv C m vv X ( ω) = F = 1 2 mω C ( ω) vv (53) 22

33 CHAPTER 3 - PFWD PRIMA 100 DEVICE The PRIMA 100 is a prtable FWD device. The device used in this study was purchased frm Carl Br Paving Cnsultants, a Danish cmpany. During the study, Carl Br Pavement Cnsultants, and Kers Technlgy, Denmark, prvided technical assistance regarding the hardware and sftware. In the peratin f this device, the elastic (equivalent hmgeneus) Yung s mdulus f an unbunded pavement fundatin is estimated frm the measurement f the surface deflectin due t transient (impact) lading applied t the fundatin thrugh a circular lading plate. The device is cmmercialized with a sftware prgram fr data acquisitin and interpretatin frm a laptp cmputer. The sftware, develped fr a Micrsft Windws envirnment, enables user t chse the test setup, and visualize and save the test results. Displayed results include time histries and peak values f lad and displacement, as well as an estimated value f the Yung s mdulus. Maximum applied stress and lad pulse duratin als are displayed. 3.1 Experimental setup rubber buffers falling weight lading plate husing Fig General illustratin f PRIMA 100 device 23

34 The experimental PRIMA setup is shwn in Fig The device itself is cmpsed f three main parts: 1) sensrs with the assciated electrnics, 2) husing prtecting the sensrs, and 3) falling weight (sliding hammer). The device incrprates tw sensrs: a lad cell and a central gephne, shwn in Fig The lad cell is a frce transducer that measures static and dynamic cmpressive frces, with a nminal range 0-2 kn. Its measuring bdy is a steel spring with 8 strain gages attached. The gephne is a velcity transducer, i.e. a sensr with utput prprtinal t the velcity. Tw additinal gephnes als can be added t btain the deflectin away frm the lading pint, but nly the versin with ne central gephne will be studied here. This gephne is spring-munted inside the base part f the husing, s that it can measure the mtin at the center f the lading area. Bth sensrs are cnnected t an electrnic bx cmprising the data acquisitin system that stres the data befre sending them t the prtable cmputer. lad cell cap f the lad cell rubber buffers sliding hammer gephne electrnic bx gephne sliding hammer additinal part t btain a 20 cm diameter lading plate 10 cm diameter lading plate base f the gephne Fig Detailed illustratin f PRIMA 100 device The steel husing can be decmpsed int a base that huses the sensrs and a mvable upper part, the lad cell cap, which supprts fur rubber buffers and a central mvable 24

35 guidance rd. The rd cnsists f tw rds screwed tgether. It supprts a mvable hanger fr the falling mass and ends with a rubber handle. The lad cell cap is a mvable part that seats directly nt the lad cell lcated at the upper part f the base. The cntact area between the lad cell and cap resembles a pint cntact and is the unique pint f cntact between them. This design respnds t the necessity t btain a centered and integrally transmitted frce. The intermediate part f the base, under the lad cell, cnsists f a hllw cylinder that huses the central gephne. The lad is transmitted frm the lad cell via the walls f the cylinder. The base s bttm part, where the lad is finally transmitted t the tested supprt, is a 10 cm diameter circular plate. It can be seen frm Fig. 3.2 that the device is designed such that the gephne is in direct cntact with the sil. The gephne is munted n a spring t ensure a gd cntact. Optinal additinal rings can be attached nt the lading plate t btain either a 20 cm either a 30 cm diameter lading surface, t accmmdate different types f sil and cntact pressures. Fr a given lad applied, the use f larger plates prduces smaller stresses and surface deflectins. The falling weight is a nminal 10 kg sliding hammer that can be released frm variable heights nt the set f rubber buffers n tp f the husing. The maximum drp height is abut 0.85 m. The manufacturer als supplies tw additinal masses f 5 kg each. The duratin f the recrded frce and velcity signals can be chsen by the user in the range frm 10 ms t 120 ms. With such shrt time windws, nly the frce and velcity time histries crrespnding t the first impact are captured by the data acquisitin system. Further impacts f the falling weight (due t buncing) after rebund are nt taken int accunt. The lad pulse shape resembles a half-sine frm. Using varius drp heights and masses enables the user t btain different lading characteristics. Accrding t the manufacturer, the lad pulse duratin can apprximately range frm 15 ms t 20 ms, and the maximum frce that the device can experience is 15 kn. The crrespnding maximum stress underneath the bearing plate is abut 210 kpa fr the 30 cm diameter plate and abut 1.9 MPa fr the 10 cm diameter plate. The manufacturer recmmends chsing plate diameter, mass f the sliding hammer and drp heights such that the measured deflectin des nt exceed 2 mm. Als, they limit the use f the 30 cm and 20 cm plates t a calculated Yung s mdulus belw 125 MPa and 170 MPa, respectively. 3.2 Measurement principle The sftware prvided with PRIMA device uses the time histries f bth the frce signal btained frm the lad cell and the velcity signal btained frm the gephne. The displacement time histry is btained by integrating the velcity recrd. The user des nt have access t the primary velcity recrd, nly t the displacement recrd. After the test is cmpleted, the displacement and lad time histries are displayed n the cmputer screen. The results can be saved in specific files, with extensin.pkv and.crv. The 25

36 files prduced by PRIMA s sftware can be pened afterwards using a text editr, a spreadsheet prgram, r anther tl such as Matlab. Fig. 3.3 shws a plt cnstructed using Micrsft Excel frm the sftware s utput. The plt crrespnds t a test n the labratry s cncrete flr, but its characteristics are cmparable t nes that wuld be btained frm field tests. The displacement time histry reflects the cnventin that dwnwards displacements are negative. The utput als gives the maximum (in abslute values) deflectin x peak and lad f peak, in this case 43 µm and 6.16 kn respectively, as well as the lad pulse duratin, 14.8 ms. The key characteristics featured n Fig. 3.3 are: The peak value f the deflectin signal lags the peak value f the frce signal, and the general trend is that the deflectin signal amplitude decays after the peak f the first scillatin. Indeed, the expected behavir fr the deflectin time histry wuld be t die ff rapidly due t the effects f radiatin damping in the supprt. The time lag between the peak values is due t the effects f inertia. The amplitude decay is due t damping (energy dissipatin). In details, the deflectin signal in Fig. 3.3 shws after the end f the lad pulse scillatins that d nt decay twards a zer value, and that incrprate sme ringing. This can be explained by several factrs: a nn-perfect cntact between the device and the supprt, a nn-cnstant stress distributin acrss the cntact area (due t eccentricity) applied by the peratr, and the numerical integratin f (measured) velcity t btain the deflectin. These effects are particularly significant in this case where the tested system is very stiff. t lag 10 f peak Frce (kn) - Displacement (µm) displacement frce x peak time (ms) Fig Plt f PRIMA s sftware utput 26

37 In accrdance with the cmmn practice in FWD testing [2], an estimatin f the stiffness fting-ver-a-supprt-system is based upn the peak values f the lad and displacement time histries, i.e. f peak k est = (54) xpeak As a result, the quantity directly estimated frm the measurements is the stiffness f the system fting-n-a-supprt. With the tested versin f PRIMA 100, the estimated Yung s mdulus is cmputed in the manufacturer s sftware using the elastic half-space slutin, equatin (3), fr the case f a flexible circular fundatin, and under the assumptin that ν = ν 2 fpeak E est = 2 (55) π a xpeak Bth the manufacturer and a few experimental verificatins cnfirmed that equatin (55) was indeed the ne being used. Applying equatins (54) and (55), t the particular case illustrated in Fig. 3.3, fr example, yields K est = 143 MN/m and E est = 1.6 GPa. 3.3 Issues regarding the current PRIMA device During FWD data interpretatin, it is cnvenient t assume a unifrm stress distributin under the lad. T simulate this cnditin, sme full-scale FWD devices are equipped with a rubber membrane added between the lading plate and the sil. Even thugh PRIMA device des nt use a rubber membrane, this simplifying assumptin is embedded int the data interpretatin framewrk f the PRIMA device. Hwever, accrding t the manufacturer, a new release f the sftware is nw available. In this new versin, mre cntrl is given t the user. Fr example, the value f Pissn s rati in equatin (55) can be changed, and the influence f the cntact pressure distributin can be taken int accunt. The difference in estimates f Yung s mdulus resulting frm fully flexible cmpared t fully rigid cntact is f the rder f 20 percent. Anther cnvenient simplificatin is t assume that the lading area is a disk. Hwever, examinatin f the lading plate in Fig.3.3 shws that this in nt the case. The lading plate itself crrespnds t an annulus f uter diameter 10 cm and inner diameter 4 cm. Hwever, the center gephne (3 cm diameter) als cntributes t the lading area. The exact lading area is therefre a cmpsite area, apprximated in the featured data interpretatin as a disk. Als, PRIMA device is designed such that the center gephne measures the grund deflectin. This implies that grund and fting are assumed t experience the same mtin. The analysis presented in this reprt is based n these tw assumptins (circular fting and equality f displacements). Rigrus examinatin f the influence f these assumptins n the perfrmance f PRIMA device n the estimatin f the (elastic equivalent hmgeneus) Yung s mdulus falls ut f the scpe f this study, which fcuses n stiffness estimatin. With reference t Fig. 3.3, the recrded time histries can reflect measurements that d nt depend nly upn the sil s characteristics. The quality f the measurements 27

38 depends upn the peratr (i.e. hw the device is manipulated) as well as upn the seating cnditins f the device. Given the type f testing, where dynamic lads are applied, btaining reliable measurements at times greater than the lad pulse perid is difficult, especially n stiff supprts. Als, the displacement is cmputed via numerical integratin f the velcity, and numerical integratin is generally assciated with increasing errrs with increasing cmputatin time (that is, the cmputed displacement at the end f the time histry may be errneus). The advantage f estimating the stiffness frm the peak values measurements is that the peak values are less likely affected by these factrs. The respnse f the PRIMA gephne is similar t the curve presented in Fig. 2.6 fr 70 percent f damping. Fr such sensrs, the respnse is nt linear (and thus des nt reflect the true grund mtin) at very lw frequencies. Accrding t the calibratin guidelines frm Carl Br Pavement Cnsultants (CBPC) [25], the respnse f the gephne equipping PRIMA device is nt linear in the range 0 7 Hz. T partially cmpensate fr this nn-linearity and btain a flat respnse ver the range Hz, a digital filter is added t mdify the signal frm the gephne. The filter has been calibrated and implemented in PRIMA by the manufacturer. 3.4 Enhancement f PRIMA device PRIMA 100 is a PFWD designed fllwing the principles f standard full-scale FWD devices. It is cmmnly believed that, in assessing the characteristics f a pavement structure, using a dynamic impulse that simulates a mving wheel lad gives reliable results frm FWD measurements. The device is designed t prduce typical stress levels, as they wuld exist beneath paved radways. Fr this reasn the recrd duratin f FWD is typically in the range 15 t 60 ms, and the recrd duratin f PRIMA 100 des nt exceed 120 ms. Als, the maximum lad f 15 kn t 20 kn develped during impact simulates real traffic situatins n paved radways. This apprach, hwever, des nt take int accunt the traffic cnditins during cnstructin, which are ften mre detrimental than the lads taken int accunt fr the lng-term design. It is imprtant t pint ut that in the quality assurance f pavement cnstructin, the targeted prperty is the small strain Yung s elastic mdulus r E max. In this case, the cnditins f the test must prduce small defrmatins. Under small strains, the true value f the tested material elastic mdulus is unique (E max ) and shuld nt depend upn the characteristics f the applied lad (e.g. the pulse shape). Based n this argument, simulating traffic cnditins is nt relevant. As mentined in Sectin 2.1, it wuld be advantageus fr quality assurance purpses t use a device that prduces small strains in the sil mass. It will be shwn in Chapters 5 and 6 that the prpsed alternative determinatin f the static fting-n-supprt stiffness k s using FRF techniques des nt depend upn the applied frce, as lng as the behavir f the tested material remains apprximately linear. An enhanced experimental setup, as shwn in Fig. 3.4, can be prpsed fr FRF measurement purpses. As illustrated in Fig. 3.2, the lad cell s cap can be remved. This 28

39 allws fr tests where the lad can be applied directly t the lad cell, using, fr example, a rubber hammer instead f the sliding hammer. By using this alternative technique, the applied lad is lw enugh t ensure small defrmatins that can be used t characterize accurately the seismic mdulus E max beneath a paved rad. Enhanced (mdified) cnfiguratin Original cnfiguratin Fig Mdificatin f the setup f PRIMA device This enhanced setup, which limits the amplitude f vibratins due t applied impact frce, is als very advantageus during the verificatin tests cnducted n a simply supprted steel beam (reduced ringing ). Fr field-testing, this setup culd replace the riginal setup t reduce the weight f the device (mre prtable), prvide a small defrmatin test (crrespnding t Yung s mdulus), and ffer an even mre easy-tuse apparatus (less awkward and fewer parts). The repeatability f the tests wuld be imprved and the data interpretatin, based upn E max, greatly simplified. 29

40 CHAPTER 4 - BEAM VERIFICATION TESTER: TRUE STIFFNESS OF THE BEAM VS. PRIMA ESTIMATES The Beam Verificatin Tester (BVT) fr PFWD utilizes a simply supprted steel beam. The true static stiffness k s f the beam is measured using tw independent static calibratin tests. With such infrmatin, the perfrmance f the PRIMA device is examined using the BVT, and the stiffness estimate k est frm PRIMA is cmpared t k s. The reasns f the bserved pr crrelatin between k est and k s are investigated, and an alternative data interpretatin methd is prpsed that eliminates the bserved discrepancy. 4.1 Setup fr the BVT Chice f the setup In rder t examine the perfrmance f PRIMA device, PFWD tests need t be cnducted n a supprting structure with knwn stiffness. Als, it is advantageus t perfrm these verificatin tests n a supprting structure whse stiffness can be adjusted ver a range f values. Furthermre, the structure has t ffer the ability t be easily mdeled nt nly statically but als dynamically. Finally, the structure shuld be instrumented t allw fr direct measurements f supprt deflectin and lad applied t the structure. The use f a simply supprted steel beam matches thse bjectives. The idea is t use an instrumented simply supprted beam as a supprt with knwn stiffness k s fr the PRIMA device, and then t cmpare PRIMA utput t k s. It is understd that sils and steel beams have little in cmmn. Hwever, wing t its primary rle as a tl fr measuring the fundatin (i.e. supprt) stiffness, the PFWD device als shuld be suitable fr determining the fting-n-tp-f-a-supprting-beam stiffness. T apprximate in-situ cnditins, characteristics f the beam are chsen t match the realistic range f in-situ fting-n-base and fting-n-subgrade stiffnesses. Fr cmparisn, the static stiffness f the PRIMA s fting (10 cm t 30 cm diameter) resting n typical cnstructin sils ranges frm 6 MN/m t 24 MN/m Beam setup descriptin During the verificatin test, the PRIMA device stands centrally n a straight hmgeneus beam with a slid rectangular crss-sectin as illustrated in Fig. 4.1 and 4.2. As mentined earlier, the verificatin setup uses nly the bttm prtin f the PRIMA device tgether with a rubber hammer as a means t apply the impact lad. In additin t the issues discussed in Chapter 3, this setup ffers several practical 30

41 advantages. Owing t the lw amplitude f vibratin after impact, this mdified PFWD cnfiguratin is preferred ver the manufacturer s setup t btain quality measurements and t guarantee repeatability. These tw latter requirements are nt necessarily satisfied when the beam is impacted by the falling mass, because f uncntrlled ringing f the entire apparatus. 1, 2: uter supprts 3, 4: inner supprts base f PRIMA device lad cell adjustable clamps receptacle verificatin beam supprt beam fundatin beam Fig Sketch f the verificatin setup receptacle adjustable clamps lad cell base f PRIMA device verificatin beam supprt beam fundatin beam Fig Verificatin setup Fig. 4.1 and 4.2 shw the BVT setup fr verificatin testing. The BVT assembly cmprises (i) a cm (4 ) wide and 1.59 cm (5/8 ) deep O-1 tl type steel beam sitting n tw adjustable supprts (dented 1 and 2 in Fig.4.1), (ii) a supprt beam 31

42 attached t the fundatin beam, and (iii) a receptacle fr PRIMA device. The supprts fr the verificatin beam (1 and 2 in Fig. 4.1) and the supprts f the receptacle (3 and 4 in Fig.4.1) are made f a 2.54 cm (1 ) diameter hardened steel rd. The verificatin setup resembles a fur-pint bending test, with beam effective span L (in-between the uter supprts 1 and 2) and a s-called lading span d (in-between the inner supprts 3 and 4). There are tw main advantages t use such cnfiguratin, as ppsed t using a single supprt at mid-span r t sitting the device directly n the beam: The tested device is stable n the supprts, and any cntact between the lading plate edges and the beam during bending is avided. As shwn in Fig. 4.1 and 4.2, the PRIMA device is rigidly clamped t the BVT receptacle using mvable clamps munted n the receptacle. The clamping frce f these clamps is rughly adjustable via a screw system. The same type f clamp als is used t insure cnstant cntact between the verificatin beam and its supprts during impact testing. In the latter case, the clamping frce is set t a mderate value t reduce the clamps influence n beam bending. Five pairs f slts are machined n the tp f the supprt beam t receive the mvable supprts fr the tested beam (uter supprts 1 and 2). This allws fr verificatin tests using five different spans ranging frm 0.3 m t 0.7 m with an interval f 0.10 m. The supprt beam is rigidly clamped t a heavy S type steel fundatin beam. Its stiffness has been fund t be high enugh, at least fr the lnger beam spans tested, fr the supprt t be cnsidered rigid. Als, measurements frm accelermeters lcated n the fundatin beam shwed that the amplitude f its mtin during testing is negligible when cmpared t the receptacle mtin. The use f the fundatin beam was advantageus fr preliminary testing. Fr definitive use in Mn/DOT facilities, we recmmend that the supprt beam be attached t a heavy fundatin r cast int cncrete. The advantages wuld be t btain a heavier supprt with mre damping, which results in increasing stability and stiffness f the supprt and decreasing ringing in the supprt. The receptacle is a shrt cm (4 ) wide and 1.59 cm (5/8 ) deep steel beam. Its span when placed n the supprts (i.e. lading span) is d = 0.1 m. The receptacle is cnsidered rigid when cmpared t the beam. It has been verified that the receptacle stiffness exceeds the tested beam stiffness by a factr f ver 100. Cnsequently, the receptacle experiences the same rigid bdy mtin as the PFWD device attached t it. 4.2 Static stiffness f the beam Theretical static stiffness f the beam The theretical mdel describing the actual verificatin setup is develped using the engineering beam thery r Euler-Bernuilli beam mdel. The assumptins r apprximatins embedded in this mdel are that: 1) the beam is made f an istrpic linear elastic material underging small strains; 2) planar crss-sectins f the beam remain planar during defrmatin, which implies that the length f the beam needs t be significantly greater than its width, and 3) the beam is thin. Further, the verificatin beam 32

43 will be cnsidered t have a cnstant rectangular crss-sectin and t be simply supprted, even thugh the beam is slightly clamped n its supprt. Finally, all the cntacts between clamps, supprts, beam and receptacle will be apprximated as pintcntacts that prvide n mment restraint. The BVT setup presented in Fig. 4.1 and 4.2 can be idealized fr the purpse f the theretical analysis as sketched in Fig. 4.3, fr a beam f width b, thickness h and effective span L. In the freging static analysis, the verhanging part S f the beam is disregarded. The lad f applied at mid-span by the PFWD device sitting n the receptacle is split int tw equal frces p lcated at +/- (d / 2 ) apart frm the mid-span, d being the distance separating the supprts f the receptacle (inner supprts 3 and 4 in Fig.4.3). h b verificatin beam crss-sectin p p rigid receptacle verificatin beam 0 y x h a 1 b 1 d f = 2 p verificatin beam elevatin rigid supprt a 2 b 2 verhang cantilever S L Fig Idealized verificatin beam layut 1, 2: uter supprts 3, 4: inner supprts The classical slutin fr the static deflectin f a simply supprted beam under a single pint lad p applied at y = a 1 can be fund in many publicatins, fr example in Lardner and Archer [26]. Using this slutin and the principle f superpsitin, ne can cnstruct the slutin fr the displacement functin f the simply supprted beam subjected t tw pint lads as depicted in Fig Fr this lading cnfiguratin, it can be readily shwn that the stiffness k s evaluated under either f the receptacle s supprts (i.e. supprt 3 r 4) is given by f f 12 E I k s = = = (56) 2 3 x( a1) x( a1 + d) 3a1 L 4 a1 where x(a 1 ) is the deflectin at y = a 1 and, fr the rectangular crss-sectin, the crsssectinal mment f inertia I f the beam is given by I = b h 12 3 (57) 33

44 4.2.2 Experimental Yung s mdulus f the steel An accurate value fr the Yung s mdulus f the steel E true is necessary because it can be used t determine the beam true stiffness k s. The fur-pints bending beam test is a well-recgnized methd f determinatin f Yung s mdulus, using measurements f lad and strain. In this methd, ne r several strain gages are munted, in-between the innermst lads, n the surface f the beam specimen, t measure the strain experienced by the specimen subjected t a knwn applied frce. Frm the engineering beam thery [26], the maximum tensile stress σ at any crss-sectin y n the surface f the beam f Fig. 4.3 can be cmputed frm h σ ( y ) = M b( y) (58) 2 I Frm slid mechanics [26], the bending mment M b in a fur-pint cnfiguratin is cnstant between the innermst lads (pure bending) and linearly prprtinal t y therwise. Given the symmetry f the prblem, the analysis can be restricted t the left half-beam fr which M b( y) = p y fr 0 y a1 f L M b = p a1 = a1 fr a1 y (59) 2 2 Cmbining equatins (57), (58) and (59) yields the expressin fr the maximum tensile stress (ccurring at the beam s bttm fiber) in terms f the applied lad 3f a σ = 1, a1 y a1 + d 2 b h 2 (60) Assuming a linear relatinship between stress σ and strain ε, and als a uniaxial state f stress (narrw beam apprximatin), Hke s law can be used σ ( y) = E ε( y) (61) Equatin (61) is the basis f the experimental determinatin f the steel s Yung s mdulus E = E true using the fur-pint bending test, E true being the slpe f the curve σ vs. ε. Using a strain gage lcated at mid-span n the beam enables t take advantage f equatin (60), in which the measured strain des nt depend n the exact lcatin f the gage. The rientatin f the gage, hwever, is still a ptential surce f errr n the strain measurement. The gage must be aligned in the axial directin f the beam, which is the majr principal directin f the stress state. Experiments have been cnducted n the beam using a MTS lad frame lcated in the Rck Mechanics Labratry f the civil engineering department. Fig. 4.4 shws the testing setup. The lad was recrded using the internal lad cell f the testing machine; the displacement f the receptacle was measured using a Linear Variable Differential Transfrmer (LVDT), and the strain was read ut f the Vishay P-3500 strain indicatr. A Micr-Measurements 350 Ω strain gage was munted at mid-span n the tp surface f the beam. 34

45 MTS lad cell LVDT receptacle beam under testing supprt & fundatin beams strain gage Fig Fur-pint bending and static stiffness tests setup T estimate the influence f the length f the lading span d (distance between innermst supprts 3 and 4) when cmpared t the length f the supprted beam L (i.e. effective span, distance between utermst supprts 1 and 2), several beam cnfiguratins were used. Sme tests have been perfrmed with spans L = 0.6 m and L = 0.7 m assciated with a lading span d = 0.10 m, and thers with L = cm (32 ) assciated with lading spans d = cm (12 ) and d = cm (16 ). On the basis f equatins (60) and (61), the measured frce and strain yields an experimental value f E = E true fr each f the tested cnfiguratins. The strain measurements were crrected frm the gage transverse effects using the standard prcedure (e.g. Dally and Riley [27]; manufacturer technical dcumentatin [28]). Averaging the results btained frm the varius cnfiguratins leads t the experimental value that will be used later. This value is E true = ± 0. 7 GPa. The values f k s cmputed frm E true and equatin (56), labeled as beam thery, are presented in Table Experimental static stiffness f the beam The beam s static stiffness k s is the reference fr the PFWD device perfrmance verificatin. Therefre, rather than calculating the static stiffness frm the beam thery using an estimated value f the Yung s mdulus, it is highly preferable t directly measure the stiffness f the beam fr different spans. As an alternative t the beam thery apprach utlined in Sectin 4.2.2, experimental values f the beam static stiffness k s als were directly measured using a 35

46 series f lad-deflectin tests perfrmed with the MTS lading frame. As bserved in Fig. 4.4, the entire beam assembly, i.e. the tested beam itself munted n its fundatin beam, has been emplyed fr this calibratin. The frce was recrded frm the lad cell f the MTS machine, and the displacement f the receptacle n the beam was evaluated using a Linear Variable Differential Transfrmer (LVDT). The supprt f the LVDT was secured n the fundatin beam s tp, s that nly the deflectin f the verificatin beam tested was measured. Fr each test, the stiffness is determined by fitting a straight line, in the least-square sense, t the frce vs. displacement experimental data. Table 4.1 summarizes the results f the static tests. Beam span [m] k s [MN/m] Direct measurement Beam thery Table 4.1. Values f k s using direct measurement and beam thery Table 4.1 shws that there is a gd agreement between the theretical values f static stiffness derived frm beam thery and frm direct measurements. The difference between averages des nt exceed ± 1. 7 % fr the smallest spans (0.3 and 0.4 m) and ± 0. 8% fr the lngest spans. Therefre, the static stiffness directly measured is designated as k s. 4.3 Estimated beam stiffness k est frm PRIMA vs. k s Perfrmance f the PRIMA device is verified by placing it n the BVT with knwn stiffness, k s. In these tests, PRIMA s stiffness estimates k est are cmputed fllwing equatin (54), i.e. as the rati f the measured peak frce and the peak displacement, as intended by the manufacturer. Several types f tests were cnducted, using either the base f PRIMA device and a rubber hammer, r the entire device with the sliding hammer. Fr each test, peak frce, peak displacement, k est and duratin f the frce pulse were read directly frm PRIMA sftware display. k est was then cmpared t the knwn beam stiffness k s (see Table 4.1) assciated with each tested beam span. Table 4.2 summarizes the average results f these tests. The tests have been cnducted with the riginal PRIMA setup as well as with the enhanced setup (i.e. PRIMA base + rubber hammer), because the riginal setup had t be tested t establish a cmparisn. Nevertheless, the riginal falling weight is drpped nly frm small heights (apprximately 10 cm) t limit the beam s vibratin level and the beam ringing, and t avid beam rebund n its supprts. Als, the falling mass is carefully manually held during the rebund fllwing the impact, t avid further impacts. These precautins are necessary fr meaningful measurements. It is imprtant t nte that such precautins cannt be realized if the LOADMAN device is used, because 36

47 the falling mass is internal t the device. As a result, the BVT cannt be used as a tester fr this device withut further mdificatins. Beam span [m] k s [MN/m] k est [MN/m] Original Enhanced Original k est / k s Enhanced Table 4.2. Average k est frm PRIMA s: riginal (with small height drps) and enhanced cnfiguratins vs. k s The length f the frce pulse depends upn the mass f the falling weight (r hammer), the buffer characteristics, and the stiffness f the supprt. When using the riginal lading cnfiguratin, the measured perid f the lad pulse is abut 15 ms, with peak values reaching apprximately 1 kn. Using the enhanced cnfiguratin, the measured perid f the lad pulse ranges apprximately between 4 and 10 ms, with peak values being cnsistently belw 1 kn. Typical frce and deflectin recrds frm PRIMA s sftware fr the case when PRIMA is placed n the BVT are presented in Fig Frce (N) - Displacement (micrmeter) 400 frce measured by Prima displacement measured by Prima Time (ms) Fig PRIMA measurements n the 70 cm span beam, using a rubber hammer When cmparing the recrd in Fig.4.5 with the ne n Fig. 3.3, the displacement signal assciated with the beam deflectin des nt die ff at the end f the 60 ms acquisitin windw. This is due t the fact that the beam has very little damping, while 37

48 sils have much higher damping characteristics, which wuld result in a displacement time histry with smaller duratin. Frm Table 4.2, ne can bserve a severe discrepancy between the true BVT stiffness, k s, and its estimates frm the PRIMA device, k est. The tw sets f measurements invlving alternative PRIMA setups (riginal and mdified) give different results, but neither shws a gd agreement with the true values. The general trend shwn in Table 4.2 is a very great underestimatin f the stiffness fr shrt spans, and a significant verestimatin f the stiffness fr lnger spans. Further, the rati k est / k s ranges between 0.72 and 3.21, even fr the lnger spans, which crrespnds t an errr n the estimatin f k s ranging frm 30 percent t 220 percent. 4.4 Pssible causes f the misfit between k est and k s As shwn in the previus sectin, the estimates frm PRIMA device, k est, d nt match the true value f the beam stiffness k s. Tw pssible reasns culd cntribute t this misfit: a majr prblem with PRIMA s sensrs, r the fact that the methd f stiffness calculatin (peak methd), based n equatin (54) is nt apprpriate. T distinguish between these tw pssible causes, it is instructive t examine first the sensr accuracy. T examine the secnd pssibility, we will test the peak methd using a theretical Single Degree f Freedm (SDOF) mdel PRIMA sensr accuracy a) Lad cell The lad cell direct utput has been calibrated under static lading prvided by dead lads. This calibratin prved a gd agreement between the measured lad cell sensitivity and the calibratin factr indicated by the lad cell manufacturer (which is 0.1 mv/kn per unit input vltage). b) Gephne The beam was instrumented using accelermeters and a gephne placed directly n the receptacle. PRIMA s gephne utput has been verified against the utput f the accelermeters (cmparisn f integrated/differentiated signals) and f the gephne (direct cmparisn) during impact testing. The added sensrs used fr cmparisn were munted either n the tp surface f the receptacle (gephne) r n the bttm surface f the verificatin beam, under the exact lcatin f the receptacle s supprts (accelermeters). Fig. 4.6 presents an example f the cmparisn between the velcity recrd frm PRIMA (as recvered by numerical differentiatin f the displacement signal stred by PRIMA s sftware), and the velcity btained by numerical integratin f the acceleratin measured by an accelermeter. Cmparisns in time dmain such as the ne presented in Fig. 4.6 shw that the accuracy f the gephne s utput is satisfactry. Nte that such cmparisns nt nly 38

49 check the gephne accuracy itself but als the numerical errr assciated with the integratin frm velcity t displacement perfrmed by PRIMA Velcity frm external accelermeter Velcity frm PRIMA Frce frm PRIMA ] m y [ /s it Velcity [m/s] V elc N ] [k F 0 F rce -0.1 Frce [kn] Time [s] Fig Verificatin f PRIMA s gephne utput against an accelermeter The accelermeters used fr this study are high sensitivity PCB Pieztrnic ICP accelermeters. The gephne used is an IMI Sensrs (a PCB Pieztrnic divisin) industrial ICP accelermeter. This sensr is actually an accelermeter with a velcity utput (integratin perfrmed within the sensr). The cnditiner/pwer supply fr the accelermeters and the gephne is a PCB Signal Cnditiner. The analg/digital (A/D) cnverter and spectrum analyzer, used fr data acquisitin, cmprises tw DSP SigLab units f tw channels each. The cnditiners are used t bth pwer the sensrs and send the signals frm the sensrs t the SigLab analyzers. The SigLab units cllect the signals frm the cnditiners and are directly cnnected t a laptp where the data can be visualized, stred and pst-prcessed. Based n the abve results btained fr bth PRIMA gephne and its lad cell, the accuracy f PRIMA s sensrs is satisfactry and cannt explain the bserved misfit between k est and k s in Table

50 4.4.2 Theretical cmparisn f peak-based k est vs. k s Since the accuracy f PRIMA sensrs has been checked, the bserved stiffness predictin errr in Table 4.2 must be attributed t the peak-based data interpretatin methd used fr PRIMA s stiffness estimatin. T explain the discrepancies between PRIMA s estimates and true values f the static stiffness f the beam, we can examine the peak methd using a simple theretical mdel, with knwn k s, fr which we can cmpute k est using the peak methd. A simple Single Degree f Freedm (SDOF) system is chsen fr this purpse. As will be seen later, such system can be chsen as an analg fr the vibrating beam. With such a system, the features bserved in the field and with beam measurements can be reprduced: the mass m intrduces inertial effects (time lag between frce and displacement time histries), the spring cnstant k is the elastic stiffness (k = k s ) and the viscus damping cefficient c is respnsible fr the vibratin decay. With reference t the SDOF system presented in Sectin 2.3, the displacement time histry f a SDOF system due t external lading is shwn via the Duhamel integral, equatin (10), t depend upn the entire lading histry. On this basis, different lad pulse shapes and duratins will prduce different time histries. Unfrtunately, the peak-based methd takes int accunt nly ne pint f the entire lad pulse and therefre cannt prduce an accurate predictin f the SDOF stiffness in general. T illustrate the systematic errr in stiffness estimatin that arises frm the use f the peak methd, the effect f varying lad pulse duratins T n the estimated stiffness f a SDOF system is examined. The respnse, in terms f displacement, f a SDOF system subjected t a half-sine frce pulse applied at its tp can be cmputed either via the linear acceleratin methd (Wilsn and Clugh [20]), r using an analytical slutin derived frm the Duhamel integral. The true stiffness f the SDOF system (i.e. the spring cnstant, k s ) is an input t the displacement cmputatin, alng with the damping rati ξ and the (undamped) natural perid f vibratin T n f the SDOF system. Let us recall that T n = 1/f n where f n is the (undamped) natural frequency f the SDOF system, which itself depends n bth k s and the mass f the SDOF system, as shwn in equatins (5) and (17). In this exercise, the peak values are extracted frm the (prescribed) lad and (cmputed) displacement time histries as shwn in Fig The plt presented in Fig. 4.7 was cnstructed frm a by prescribed half-sine frce (sine perid 2.T) by means f the linear acceleratin methd. The estimated static stiffness k est is then btained frm equatin (54) and can be cmpared t its true value k s. When repeating this prcedure fr varius duratins f the lad pulse, ne can cnstruct the cmparisn presented in Fig This plt shws, fr different amunts f damping ξ, the effect f the lad pulse duratin T n the gdness f the peak-based estimatin f the SDOF system stiffness k s. The best agreement between peak methd estimates and true stiffness value is given by the hrizntal line k est / k s = 1.The diagram in Fig. 4.8 shws that using the peak methd t estimate the SDOF stiffness, k s, results in a significant systematic errr. This errr can be either an verestimatin r an 40

51 underestimatin, depending partly upn the damping rati f the SDOF and mainly upn the impact duratin relative t the natural perid T n f the SDOF. Example f a SDOF system with f n = 50 Hz, m = 15 kg and ξ = 20% 15 peak frce, f peak 15 Displacement [mm] Prescribed frce T Cmputed displacement Frce [kn] peak displacement, x peak Time [ms] Fig Peak methd applied t a theretical SDOF system Simulatin fr a 60 cm span beam using a SDOF system with f n = 50 Hz and m = 15 kg 2 ξ = 70% k est /k s ξ = 1% ξ = 20% Timpact / Tn PRIMA mdified cnfiguratin T = 4-6 ms PRIMA riginal cnfiguratin T = ms T/T n Fig Influence f the pulse duratin n peak-based stiffness estimatin gdness - theretical SDOF system 41

52 Fr engineering applicatins, apprximate ratis T/T n range fr PRIMA tests n a particular beam are indicated in Fig The stiffness f the SDOF system has been chsen t apprximate the characteristics f the verificatin beam fr a 60 cm span, which stiffness may be cmparable the in-situ stiffness fr a sft sil (k s = 1.71 MN/m): SDOF parameters that resemble this particular span are f n = 50 Hz, r T n = 20 ms. In this case the experimental results shwn in Table 4.2 and the trend bserved in Fig. 4.8 (fr light damping) lead t the same cnclusin that the true stiffness f the 60 cm span beam, apprximated by a SDOF system, is verestimated by the enhanced setup and underestimated by the riginal cnfiguratin f PRIMA device. The abve theretical analysis simulates an impact test perfrmed n a SDOF system. It demnstrates the strng influence f the lading time histry (pulse duratin) n the results f the peak methd. It als shws that significant systematic errrs are assciated with the peak methd when applied t a SDOF system. Based upn the analgy between the SDOF system and simply supprted vibrating beam, the same trend can be expected when applying the peak methd t the case f the Beam Verificatin Tester (BVT). 42

53 CHAPTER 5 - ENHANCEMENT OF BEAM STIFFNESS ESTIMATION USING PRIMA DEVICE The PFWD Beam Verificatin Tester (BVT), has shwn that the stiffness estimates k est frm PRIMA device d nt match the true stiffness k s f the beam. Upn verifying the accuracy f PRIMA s sensrs, it was cncluded that the mismatch was primarily due t the peak methd used t estimate the supprt stiffness frm dynamic frce and deflectin measurements. It was demnstrated theretically using the SDOF analg that the peak methd leads t significant systematic errrs in the determinatin f the supprt stiffness. This chapter examines a cnsistent methd t determine the static stiffness f the supprt frm dynamic measurements, based upn a spectral analysis f the recrded data. The key pint underlying this methd is that the static stiffness can be extracted frm dynamic measurements t estimate the stiffness at zer frequency, as shwn fr example in Guzina and Osburn [7] and Briaud and Lepert [10]. Therefre, the methd prpsed here fcuses n the lw frequency range measurements. The physical setup f PRIMA device used n BVT apparatus crrespnds t the enhanced setup as described in Chapters 3 and Cnsistent data interpretatin In the peak methd, the static stiffness is incrrectly estimated frm dynamic frce and displacement time histries as the rati f peak frce ver peak displacement. Indeed, the dynamic peak values d nt ccur at the sane time and d nt crrespnd t the static values. Strictly speaking, the static frce and displacement crrespnd respectively t the values f frce and displacement at zer frequency in the frequency spectra cmputed frm time histries. Therefre, the crrect methd t estimate the true static stiffness f the beam is t use the rati f frce ver displacement at zer (r near-zer) frequency. In this study, the frequency spectra are btained frm respective time measurements using Fast Furier Transfrm algrithms. Recalling the definitins f the Furier Transfrm presented in Sectin 2.5.1, the main practical requirement t perfrm a crrect transfrmatin int frequency dmain is that entire transient signals must be sampled. Recrding the entire time histries is therefre a key step. Due t PRIMA s hardware limitatins (maximum recrd duratin f 120 ms), hwever, it was nt pssible t btain the entire time histries fr the beam tests (duratin f mtin up t a few secnds) directly frm PRIMA device. As a result, an external data acquisitin system is used. Furthermre, rather than using the manufacturer s sftware t extract the measurements, an in-huse prgram is develped t recrd the time histries and interpret the data. 43

54 In additin t the frce and velcity sensrs, the data acquisitin system is cmpsed f the cnditiner/amplifier fr an independent mnitring f PRIMA s lad cell, a cnditiner/pwer supply fr the accelermeters and the gephne, the DSP SigLab spectrum analyzer, and a prtable cmputer. All these elements have been already described in Sectin 4.4.1, except the signal cnditiner/ amplifier used fr PRIMA s lad cell. This signal cnditiner is a Vishay strain gage cnditiner mdel 2120 cupled with a mdel 2110A pwer supply. The data acquisitin layut used in this study fr verificatin testing is summarized in Fig cmputer Spectrum analyzer (2 units) unit 1 unit 2 PRIMA gephne PRIMA lad cell Strain gage cnditiner / amplifier External gephne Fig BVT data acquisitin Accelermeter cnditiner / pwer supply The prpsed methd, based upn the measurement f the zer-frequency cmpnents f the frce and displacement recrds, is well-understd, hwever in practice a main limitatin arises due t the sensrs characteristics. As examined in Sectins 2.4 and 3.3, in the lw frequency range (typically 0 t 10/20 Hz), the sensr (gephne/accelermeter) utput des nt reflect the measured mtin quantity. As a result, the value at zer frequency cannt be estimated directly frm the measurements. T vercme this prblem, ne must extraplate experimental data in the lw frequency range t btain the value at zer frequency. It is well-knwn, hwever, that data extraplatin can be a dangerus prcess if reasnable cntrls are nt applied. The best way t ensure a crrect determinatin f the zer frequency value is t use a theretical mdel as a guide fr the extraplatin. The remainder f this chapter presents the theretical mdel used t extraplate the data, as well as the frequency dmain analysis tls used t estimate the beam s static stiffness. 44

55 5.1.1 SDOF mdel fr the beam The theretical mdel chsen t guide the dynamic stiffness data extraplatin twards zer frequency is the SDOF system. We will assume fr nw that the beam mtin accurately can be apprximated by a SDOF system, at least fr the lw frequency range. The use f this particular mdel is mtivated by its simplicity. Als, SDOF analgs are cmmnly used in structural vibratin prblems, such as determinatin f beams fundamental frequency. Nte that n rigrus theretical justificatin will be presented here. Hwever, the relevance f this assumptin is based upn previus analytical wrk, such as the analg SDOF system f a simply supprted beam carrying a mass at mid-span and underging free vibratins in Stkey [29] and in Kármán [30], and will be justified a psteriri by the results btained. As a result, an alternative data interpretatin methd is prpsed based upn the determinatin f the stiffness (i.e the spring cnstant) k f an equivalent SDOF system. In this methd, the static stiffness k s = k is recvered frm the time histry f bth lad and displacement by fitting, in the lw frequency range, a measured FRF t the crrespnding FRF f the SDOF system. During testing n the BVT, an impact shck prduces a transient excitatin f the beam cntaining varius frequencies. It was fund, hwever, that the BVT reacts t an impact lad (applied t the receptacle) primarily thrugh the fundamental mde f vibratin. As a result, the SDOF simulatin f the beam respnse pertains nly t the lwer vibratin frequencies lcated arund the fundamental resnant frequency. Fr higher frequencies, the SDOF system des nt mdel the beam behavir well. Althugh the size f the useful frequency windw culd be adapted t each span f the tested beam, it is chsen in this study t match the range Hz. Fig. 5.2 illustrates the analgy tested beam / SDOF system. Fllwing this utline, the equivalent mass m eq crrespnds t the ttal mass cmprised between the beam and the pint f measurement f the applied frce. In ther wrds, m eq cmprises the mass f the PRIMA device (enhanced setup) and the mass f receptacle, plus the mass f receptacle supprts and clamping devices. Furthermre, the mass f the sensr(s) lcated n the receptacle and a (unknwn) part f the mass f the cable assemblies als shuld be included. 45

56 impact frce f(t) impact frce f(t) m eq x(t) x(t) k c BVT setup: fur-pint beam test SDOF analg Fig SDOF analg fr the BVT setup Frequency dmain analysis tls In the ensuing analysis, the entire time histry f the lad and displacement signals is used as an input t the SDOF apprximatin f the beam respnse. During testing with the BVT, the beam respnse is linear, and s is the respnse f the SDOF equivalent system. This allws us t use the FRF fr linear systems as defined in Sectin 2.5. Let us recall the definitins f the dynamic stiffness K(ω) and mbility M(ω) f a linear system, equatins (32) and (28): F( ω ) K ( ω ) = (32) X ( ω ) X& ( ω) M ( ω) = (28) F( ω) where F(ω), X(ω) and X & (ω ) are respectively the cntact frce, the displacement and the velcity at the PRIMA-BVT interface, in frequency dmain. The advantage f using this methd is that FRFs characterize the entire system, are unique (linear system), and are independent f the applied frce. Accrding t the apprach f Sectin 2.5.5, which is built int the SigLab data acquisitin system, the measured (average) FRF f chice is btained directly frm the spectral density functins estimates. Let us recall at this pint that FRFs are cmplex-valued functins, cmprised f a real and an imaginary part. Fr this prject the mbility functin M(ω), bth real and imaginary parts, is used t fit the experimental data. The mbility functin is used because it crrespnds directly t the quantities (velcity and frce) being measured. The average M(ω) functin can be btained either directly frm PRIMA sensrs (gephne 46

57 and lad cell) either frm external sensrs munted n the calibratin beam (accelermeter / gephne). The tests presented in this reprt are realized using the lad cell frm PRIMA and a gephne munted n the receptacle. Based upn the analytical slutin fr the mbility functin M(ω) f a SDOF, presented in Sectin 2.5, an in-huse Matlab cde is used t fit the experimental mbility data t the theretical M(ω) curve f a SDOF system and t btain the ptimal set f its fundamental prperties, namely static stiffness k s, damping rati ξ and equivalent mass m eq. At this pint the static stiffness f the SDOF, and thus that f the BVT, already has been estimated frm the fitting prcess. Hwever, it als can be directly read frm the real part f the dynamic stiffness curve f the fitted SDOF system, at zer frequency: k s = Re{K(ω = 0)} (62) Nte that the SDOF dynamic stiffness functin K(ω) can be cnstructed either frm the fitted SDOF parameters, r by inverting the fitted M(ω) functin t get the impedance functin I(ω) and then multiplying I(ω) by (iω). The cherence functin γ 2 (ω), als intrduced in Sectin 2.5, is a gd indicatr f the quality f the measurements and f the linearity f the system. A cherence significantly less than unity indicates presence f nise in the measurements, nnlinearity between input and utput, r bth. The cherence is therefre used t define the usable frequency range that can be emplyed fr the fitting prcess and as a criterin t accept r reject a series f measurements. Typically, all the measurements fr which the value γ 2 (ω) is significantly belw the unit value (less than 0.95 fr this study) will be disregarded in the analysis. 5.2 Experimental prcedure and results Data acquisitin setup It has been shwn previusly that an independent data acquisitin system is needed t capture the entire lad and displacement time histries relevant t the BVT testing, which are characterized by an extended duratin relative t field measurements. Therefre, the utput plugs frm bth PRIMA s sensrs, lad cell and gephne, have been directly cnnected t an external data acquisitin equipment described earlier. Ding s, hwever, ne cannt access the sensrs calibratin parameters that are embedded in the PRIMA s sftware. In this study, vltage utput and crrespnding velcity and frce have been related using the values given by the sensr manufacturers. As mentined earlier, an independent check using dead lad calibratin fr the lad cell and direct cmparisn fr the gephne demnstrated that the sensrs were accurate. Fig. 4.1, 4.2 and 5.1 shw the general experimental setup f the BVT. In the verificatin prcedure, each test is cmpsed f a series f ten measurements f the frce frm PRIMA s lad cell and the induced mtin velcity frm PRIMA s gephne. The 47

58 data are sent directly t the SigLab analyzer where the mbility M(ω) is estimated using an FFT algrithm as a spectral average f the 10 measurements, tgether with the cherence functin γ 2 (ω), as depicted in Fig Nte that the measurements presented in Fig. 5.3 crrespnd t the cnfiguratin f lng spans fr the beam (50 cm t 70 cm) nly. Examinatin f the cherence functin in Fig. 5.3, pltted in the range 0 t 150 Hz, indicates that the useful range can be taken as 10 t 150 Hz. Nte that a lwer limit f 20 Hz wuld be even mre cnservative, but experience shws that this precautin is nt necessary. The cherence degrades at very lw frequency because the gephne s utput is nt linear: Recalling the gephne descriptin given in Sectin 2.4, n accurate measurements can be expected at very lw frequencies (belw 10/20 Hz). Fr frequencies higher than 150/200 Hz, measurements are assciated with an imprtant decrease f the cherence functin, which indicates that the respnse includes nnlinear effects, nise and stray vibratins. Therefre, the analysis is limited t frequencies up t 150 Hz. f(t) [kn] M(ω) real part x 10 [m/s/kn] -3 M(ω) imaginary part x 10 [m/s/kn] x&(t)[m/s] x y Lad cell t [ms] FFT Gephne t [s] f [Hz] X( ω) G ( ω xy ) M( ω) = & = F( ω ) G ( ω ) Fig Typical measurement sequence and utput Let us pint ut that, depending upn which channel is chsen as reference (i.e. input f the ideal linear system), the FRF resulting frm velcity and frce measurements can be either mbility M(ω) r impedance I(ω). Als, a very useful feature implemented in the SigLab analyzer sftware enables t integrate/differentiate directly the measured FFTs. The advantage is that K(ω) culd be directly given as an utput f the SigLab 48 xx 2 G ( ω) 2 xy γ ( ω ) = G ( ω ) G ( ω ) xx yy che re nce f [Hz] f [Hz]

59 analyzer frm measurements f I(ω) and direct spectral differentiatin (n additinal data manipulatin). Fr this prject, hwever, the use f M(ω) is preferred Fitting prcess Once the experimental M(ω) is recrded, an in-huse Matlab cde fits the measurements t the theretical SDOF mbility functin t estimate the system prperties, k, ξ and m eq. The prgram ptinally can be called directly frm the SigLab analyzer sftware after the test, s that the results are displayed n the same screen as the acquisitin prgram. The fitting prcess fllws an ptimizatin methd in which an bjective functin is being minimized. Fr this prject, a built-in Matlab functin, based upn an uncnstrained nnlinear ptimizatin methd, is used. This methd is a direct search methd. It des nt use numerical r analytic gradients. The built-in Matlab functin finds the minimum f a scalar real-valued functin f several variables, starting with an initial estimate. The minimum fund is the lcal minimum f the bjective functin t minimize, in the vicinity f the initial estimate prvided t the ptimizatin prcess. The initial estimate can be a scalar, a vectr, r a matrix. Cnsistent with current practice, the functin minimized is the square f the difference between measurements and trial fitted curve. Bth the real and imaginary parts f M(ω) are used fr curve fitting. The analytical expressin used is given by equatin (29). The initial parameters used t initiate the fitting prcess crrespnd t bth autmatic estimatin f the fundamental frequency f the beam and initial guess (damping rati and mass, and als frequency windw width). The natural frequency f the SDOF analg is cmputed as being the fundamental frequency autmatically estimated frm the spectral data, using the peak in the real part f M(ω). Nte that the fundamental frequency als culd be estimated frm the FFT f the velcity time histry. The initial values fr the SDOF damping rati, equivalent mass and stiffness crrespnd t values cmprised in the expected range. Mre precisely, the selected values are 10 kg fr the SDOF equivalent mass and 1 percent damping rati. An initial value fr the stiffness f the SDOF analg is cmputed frm the equivalent mass and the natural frequency estimate using equatin (5). As mentined earlier, the apprpriate frequency range is Hz. Hwever, t take int accunt the frequent drp in cherence functin bserved at the vicinity f the fundamental frequency (see in Fig. 5.1 at apprximately 50 Hz), it is decided t exclude the pint crrespnding t the natural frequency, as well as tw additinal pints n each side f the frequency spectrum. Using the freging scheme, the chice f an initial guesses fr the mass and the damping rati d nt influence the fitting prcess, and a stable cnvergence f the minimizatin functin is reached Estimated k est frm PRIMA vs. k s using the mbility functin Verificatin tests n the enhanced PRIMA cnfiguratin and the techniques presented in the previus sectin were cmpleted fr spans f the BVT between 0.3 m and 0.7 m. Fig. 5.3 shws a typical measured mbility functin M(ω). Fig. 5.4 shws the plts crrespnding t the same data tgether with the crrespnding fitted SDOF curves. 49

60 It can be seen frm Fig. 5.4 that the SDOF mdel matches the experimental data well fr lnger beam spans. Frm this fitting prcess, the parameters f the equivalent SDOF system are effectively estimated. In particular, the stiffness k f the equivalent system, which crrespnds t the static stiffness f the beam, is estimated. 2.5 x x 10-3 Real {M(ω)} m/s/kn Im {M(ω)} m/s/kn f [Hz] f [Hz] measured average pints fitted SDOF curve Fig Measured data and fitted SDOF curves - example f a 60 cm span beam K(ω = 0) = 1.70 kn/m Re {K(ω)} [MN/m] Measured average pints Fitted SDOF curve f [Hz] Fig Fitted SDOF real part f K(ω) example f a 60 cm span beam As mentined earlier, using the values f the SDOF system resulting frm the fitting prcess, it is pssible t cnstruct the curve crrespnding t the real part f the dynamic stiffness K(ω) f the equivalent SDOF system. Fig. 5.5 presents such a plt, 50

61 crrespnding t the beam span f 60 cm. Frm this curve the change f the stiffness with frequency is clear and can be used t estimate the static stiffness, at f = 0. The value f the static stiffness k est = 1.70 MN/m btained by this fitting prcess cmpares well with the true value which is fr this 0.6 m span equal t 1.71 MN/m. Table 5.1 shws the results assciated with the fit f the experimental mbility data with the theretical M(ω) functin f a SDOF system fr all beam spans. These results shw that the prpsed methd is able t estimate the true static stiffness f the beam k s within a few percent fr the lnger spans, namely L = 0.5, 0.6 and 0.7 m. Cnversely, the results regarding the shrter spans (L = 0.3 and 0.4 m) shw a pr agreement with the true value. Beam span [m] k s [MN/m] k est [MN/m] k est / k s Table 5.1. Static stiffness estimates via fitting f the mbility functin undamped beam The mismatch assciated with the shrter spans can be assciated with the particular gemetry f the beam, with prduces a very stiff beam with lng cantilevers. Fr lw spans, the cantilever parts f the beam, n each side f the supprts, are lng cmpared t the beam span. The presence f these lng cantilever parts intrduces additinal significant masses away frm the receptacle and therefre additinal degrees f freedm, s that the SDOF analg is n lnger apprpriate. The range f spans apprpriate fr verificatin tests is therefre 0.5 m t 0.7 m, fr which the errr ranges frm 2 percent t 5 percent Beam with additinal damping During testing with PRIMA, the impact generates stress waves radiating away frm the area f impact (surce) in the tested supprt, which was the beam fr this study and is the sil in field-testing situatins. Wuld the supprt be cnstituted f a perfectly elastic material, the ttal energy imparted t the supprt and carried by the wave wuld be cnserved withut lss. Hwever, the beam and sils d nt behave purely elastically, energy is dissipated, and the waves amplitude is attenuated (damped). Frm Fig. 5.2, ne may nte that the damping f the BVT assembly, which results frm material damping and energy dissipatin at the mechanical cntacts, is significantly lwer than the bserved in-situ damping (see Fig. 3.3). In the case f sils, damping is generally very high. The attenuatin f the stress wave amplitude results frm tw damping prcess, material damping (cnversin f elastic energy int heat) and radiatin damping (e.g. Kramer [13]). Radiatin damping, 51

62 als referred t as gemetrical damping, is related t the reductin f the specific energy (elastic energy per unit vlume) as the wave travels away frm the surce due t spreading f the energy ver a greater vlume f material. The ttal damping in the beam, experimentally estimated in this study via the damping rati f the equivalent SDOF system, des nt exceed a few percent. As a result, the duratin f mtin during testing n the BVT is significantly lnger than that bserved in the field. It is imprtant t stress here that the need t prvide additinal damping t dissipate energy des nt affect the validity f the stiffness estimatin methd. The static stiffness f the beam r its SDOF analg des nt depend upn the amunt f damping. Indeed, fr the dynamic stiffness f the equivalent SDOF system with reference t equatins (34) and (37), the real part is independent f the damping and the imaginary part is zer at zer frequency, which indicates that the static stiffness is independent f the amunt f damping. Furthermre, as it will be seen in the fllwing, tests shwed that the additin f external damping n the beam des nt affect the stiffness estimates. Hwever, anticipating a pssible use f the BVT device as a tl fr rutine verificatin f PRIMA-type devices, we have t expect bjectins frm the field-testing practitiners wh might argue that the duratin f the mtin in BVT testing des nt resemble the duratin f the mtin experienced in the field. T address this issue, it was decided t decrease the mtin duratin by adding sme external damping t the beam Auxiliary dampers Let us examine hw damping can be added t the BVT using external devices. Damping devices such as auxiliary mass absrbers ften are used (e.g. Reed [31]) t dissipate the vibratin energy t reduce excessive vibratins in a structure. The simplest frm f auxiliary damper is a SDOF system attached t the structure where additinal damping is sught. In that case it is suppsed that the amplitude f mtin f the structure t be damped, the s-called primary structure, is unaffected by the presence f the auxiliary system, and that all the energy dissipatin takes place in the damping element f the auxiliary system. Expressins t cmpute the energy dissipated by viscus damping per cycle f vibratin in the case f a SDOF system submitted t steady-state vibratins, such as the ne depicted in Fig. 2.5, can be fund in the literature (e.g. Kramer [13]; Reed [31]). It can be shwn that fr such a case, efficient energy dissipatin is assciated with an auxiliary system with a large mass, a small damping rati and a stiffness tuned t β = 1. In ther wrds, efficient energy dissipatin will be btained fr a driving frequency clse t the natural frequency f the auxiliary damper. Fr example, in the case f a beam with a span f 60 cm (measured fundamental frequency f abut 50 Hz), the tuned value f the spring cnstant wuld be apprximately 0.2 MN/m. As a result, a sft spring will be required fr ptimal energy dissipatin External damping setup T add sme external damping t the beam withut changing its static stiffness, it was decided t attach auxiliary damping devices n the beam s cantilever prtins. 52

63 Befre presenting the damping devices used fr this study, let us first shw that the static stiffness f the beam is nt affected by their presence. Let us cnsider the additinal masses crrespnding t tw identical damping devices psitined n the cantilever prtins f the beam, as shwn in Fig m Auxiliary damper Fig Simplified layut f the simply supprted beam with damping devices lcated n the cantilever parts Befre impacting the base f PRIMA represented by the mass m, the beam initial defrmatin is due t the mass m and als t the mass assciated with the devices. This defrmatin state f the beam is the riginal state r equilibrium state. During testing, an additinal defrmatin ccurs due t the impact. Hwever, the beam behaves as a linear elastic system and the defrmatin measured during testing crrespnds nly t the perturbatin f the equilibrium state. In ther wrds, the static stiffness f the linear elastic beam is nt affected by the initial defrmatin state. As a result, adding sme mass des nt theretically affect the static stiffness f the beam. Hwever, the stiffness estimatin is based upn the SDOF analg fr the beam, s that the added mass shuld be small enugh t avid the intrductin f supplementary degrees f freedm in the system. added mass beam crss-sectin rubber layer aluminum plate Fig Layut f a damping device lcated at the end f the verhanging part f the beam 53

64 Let us nw describe the design f the damping devices. Based upn the guidelines fr auxiliary damping systems presented in the previus sectin, damping devices with high vibratin attenuatin are assciated with a small damping rati, a large mass and a sft spring. Hwever, a large mass wuld intrduce an additinal degree f freedm t the system. Therefre, a small mass needs t be used. As a result, the damping devices designed fr the BVT are cnstituted f a small mass seating n a rubber layer and lsely cnnected t the cantilever prtins f the beam, as shwn in mre details in Figs. 5.7 and 5.8. The lse cnnectin is equivalent t the use f a sft spring and intrduces a phase delay (respnsible fr the damping) between primary system and added mass mtins. The rubber layers used are cm wide, 8.5 cm lng and 0.63 cm deep. They are supprted by aluminum plates f same width and length. The assembly rubber layers plus aluminum plates is attached t the beam using an adjustable aluminum frame, n the tp f which a mass is attached. Fig Testing setup fr the BVT with additinal damping Experimental results Tests n the BVT shwed that the efficiency f the auxiliary damping devices highly depends n the precise tuning f the devices. They als shwed that the beam with smaller spans did nt behave as a SDOF system due t the intrductin f relatively imprtant additinal masses. As a result, the verificatin tests were cnducted n the lnger spans (0.6 m and 0.7 m) nly. Fr this study, a mass f 2.3 kg (5 lb) was used fr each damping device. We did nt attempt t estimate the exact tuning values, but fcused n the attenuatin effects. The stiffness f the cnnectins was manually tuned, by adjusting the frce in the nuts, in rder t btain the ptimum vibratin attenuatin. Typical plts f the measured velcity are presented in Fig. 5.9, t shw the reductin f the duratin f the velcity time histry due t the additin f the damping devices. 54

65 x& [m/s] undamped beam x& [m/s] t [s] beam with damping devices t [s] Fig Cmparisn f velcity time histries undamped and damped beam setup The damped beam stiffness was estimated using the same tls and prcedures that were used fr the case f the undamped beam. Table 5.2 presents the results f the beam stiffness estimatin fr the BVT with additinal damping, and shws a gd match between estimated values k est and beam stiffness k s. Cmparisn with Table 5.1 indicates that the featured damping devices d nt have any significant effect n the accuracy f the results. 55

66 Beam span [m] k s [MN/m] k est [MN/m] k est / k s Table 5.2. Static stiffness estimates via fitting f the mbility functin beam with damping devices 5.3 BVT apparatus: summary and practical applicatins Several departments f transprtatin thrughut the wrld are preparing t use PFWD devices fr quality assurance purpses. In this field, it is essential t use well-calibrated and rigrusly validated devices. By design, the BVT can be used t effectively check the crrectness and accuracy f stiffness estimates frm PFWD devices PFWD PRIMA This sectin shwed that the true stiffness f the beam accurately can be estimated using PRIMA device, prvided that the mdified device (i.e. enhanced physical cnfiguratin f the device) is utilized fr testing, and that a cnsistent (spectral-based) data interpretatin methd is used, rather than the peak methd embedded in the riginal PRIMA data interpretatin scheme. Under these cnditins, therefre, the BVT can be used t check and validate the perfrmance f PFWD devices such as PRIMA, using a tw-step prcedure: 1) the calibratin factrs fr the frce and mtin sensrs, as used by the PFWD device data acquisitin system, must be checked using the independent data acquisitin system f the BVT, 2) the stiffness estimates can be directly verified against the true stiffness f the beam, using spans between 0.5 m and 0.7 m. A match between k est and k s within a few percents will indicate that the PFWD perfrmance is satisfactry. In the case f a mismatch between k est and k s, a recalibratin f the sensrs will be necessary. Nte that in the case f PRIMA device, bth the gephne and the lad cell can be checked separately t determine the cause f the mismatch. As seen in Sectin 4.4.1, the lad cell calibratin factr easily can be verified under static lading using dead lads. Als, the utput f the gephne can be calibrated during dynamic testing by cmparisn with the utput f an independent BVT gephne, either in time dmain, as presented in Sectin 4.4.1, r using frequency spectra. A user s manual has been develped and is included as an appendix, t prvide a supprt fr rutine testing using the BVT. This dcument describes a cmplete testing 56

67 prcedure t verify the perfrmance f PRIMA devices, and als presents sme guidelines and testing prcedures t verify the accuracy f the individual sensrs embedded in PRIMA Other prtable deflectmeter devices In this prject, the PFWD verificatin effrt was fcused n the PRIMA device because it ffers direct measurement f bth frce and velcity. Anther advantage f PRIMA is that the device can be mdified s that the impact lad is lw enugh t allw fr accurate measurements n the BVT. Other deflectmeter devices generally have specific characteristics in terms f (i) excitatin type, such as impact (LOADMAN) r steady-state vibratry frcing (GEOGAUGE); (ii) frce and mtin measurement methds; and (iii) frcing magnitude and physical setup. As a result, adaptatins f the BVT device t particular deflectmeter tl must be dne n a case-by-case basis. Hwever, a general restrictin applies. The BVT can be adapted t check the stiffness estimates f ther PFWD devices as lng as they can perate with an impact energy cmparable t the ne used with the enhanced PRIMA cnfiguratin. Therefre, as mentined in Sectin 4.3, the LOADMAN device, where the height f free fall f the weight is nt adjustable, cannt be used n the BVT. When aiming t check the perfrmance f a given deflectmeter device, ne must distinguish between the estimated stiffness k est frm the device and the sensr accuracy. Indeed, the displayed k est incrprates a data interpretatin scheme specific t each device that might mask the effect f sensr calibratin. Sensr accuracy can be checked nly if the device allws fr a direct independent check f each individual sensr. If nt, a direct cmparisn k est vs. k s n the BVT can nly check the verall device perfrmance withut distinctin between the effects f sensr calibratin and thse f embedded data interpretatin. 57

68 CHAPTER 6 - PRIMA DEVICE: ADDITIONAL CONSIDERATIONS AND RECOMMENDATIONS FOR FIELD USE It was shwn in this study, using the Beam Verificatin Tester (BVT), (i) that the peak methd ften used in practice fr static stiffness estimatin can lead t significant systematic errrs and (ii) that reliable estimates f the static stiffness frm PRIMA device measurements can be btained nly if the nn-truncated lad and velcity time histries can be extracted frm the device and used as input t a cnsistent data interpretatin based upn FRF zer-frequency estimates f dynamic stiffness and spectral average. We nw will investigate the pssibility f implementing the cncepts develped in the previus sectins during field testing. It will be demnstrated that the peak methd als can lead t systematic errrs in the case f hmgeneus and layered half-space. As a result, enhancements f the PRIMA analysis, hardware and sftware develped in cnjunctin with BVT testing need t be extended t field applicatins. 6.1 Theretical cmparisn peak-based k est vs. k s fr field prfiles Hmgeneus half-space In Sectin the peak methd was applied t the theretical SDOF system. The results demnstrated that the rati k est /k s was strngly and nn-linearly dependent n the duratin f the frce pulse, T, relative t the natural perid T n f the SDOF system. Fllwing the same lgic, we will examine the relatinship between k est /k s and T in the case f fieldtesting using PRIMA device as an example, where the fundatin (i.e. base n tp f subgrade) is mdeled as a hmgeneus elastic half-space. The steady-state laddisplacement relatinship fr a rigid massive disk resting n the tp f a hmgeneus half-space, as depicted in Fig. 2.12, is given by equatin (52) F0 X 0 = 1 (52) 2 mω C ( ω) vv where m is the mass f the fting, F 0 the maximum frce amplitude, X 0 the maximum displacement amplitude, ω the driving frequency, and C vv the massless vertical cmpliance functin. In the case f testing with PRIMA, the frce pulse is nt steady state but can be apprximated as a half-sine functin. Decmpsing the transient halfsine frce pulse int a series f harmnic functins, equatin (52) can be rewritten as 58

69 F( ω) X ( ω) = (63) 1 2 mω Cvv( ω) where F(ω) is the Furier transfrm f the applied frce f(t), X(ω) the Furier transfrm f the displacement x(t), and m the mass f the fting. Let us recall frm equatin (3) that the theretical static stiffness k s f a hmgeneus half-space under the actin f a rigid, frictinless circular punch f radius a is given by 4G a k s = 1 ν Let us nw examine the prcedure used t cmpute the field rati k est /k s resulting frm a knwn, i.e. prescribed, f(t). The transient frce time histry is cnstructed frm a knwn duratin T and knwn maximum amplitude f peak. Applying a FFT t f(t) t get F(ω), and using the theretical curves fr C vv (ω) presented in Sectin (Pak and Guzina [12]), ne can cmpute X(ω) using equatin (63). The peak displacement x peak is calculated as the maximum amplitude f the displacement x(t) btained frm the inverse FFT f X(ω). The stiffness k est, estimated using the peak methd, is cmputed frm equatin (54) as k est f = x peak peak (64) (54) 0 Deflectin x (t) [µm] x(t) f(t) Frce f f (t) [kn] [κν] Time [s] Fig Example f time histry plts - hmgeneus half-space 59

70 Fr given sil prperties G and ν and given fting radius a, k est /k s is cmputed frm equatins (54) and (64). Nte that G can be calculated frm the given shear wave velcity c s and sil s mass density ρ using equatin (51). Fig. 6.1 shws an example f deflectin time histry cmputed fr a prescribed frce with duratin T = 15 ms in the case f an elastic half-space with c s = 100 m/s, ν = 0.30 and ρ =1800 kg/m 3. If the prcedure is repeated fr varius values f frce duratin T, a plt f k est /k s vs. T can be cnstructed. Nw, rather than using directly T, the dimensinless quantity T, defined as cs T = T (65) a is preferred t present the results. Fig. 6.2 presents an example f the k est /k s vs. T diagram, cnstructed assuming Pissn s rati ν = 0.30, and mass density ρ = 1800 kg/m 3. k k est s 1.4 pssible range fr PRIMA using 5 cm radius fting 1.2 massless fting fting a = 5 cm, m = 7.2 kg cs T = T a Fig Example f plt k est /k s vs. T - hmgeneus half-space Fig. 6.2 presents nly tw curves. The first ne crrespnds t a massless fting and cnstitutes a theretical limit as the mass decreases fr a given fting rati. One may nte that this curve is independent f a, c s and ρ. The secnd curve simulates the cnditin assciated with a test using PRIMA device with the smallest lading plate (10 cm diameter). In this case the actual mass f the PRIMA fting (lading plate plus husing) is m = 7.2 kg. This cnfiguratin f PRIMA is the ne that yields the greatest mismatch between k est and k s. Fr larger plates, and using the crrespnding true fting mass, the curves tend tward the massless case curve. Cmputatins shwed that, fr a 60

71 given plate diameter, the results fr k est /k s vs. T were independent f the shear wave speed c s. Indeed, ne can intrduce an additinal parameter, the mass rati M, defined as 3 ρ a M = (66) m In diagrams such as Fig. 6.2, M can be used as a unique variable frm which all the curves k est /k s vs. T can be cnstructed. Pssible field cnditins are described with the fllwing parameters: (i) fr the half-space, c s ranges frm 50 m/s t 300 m/s, Pissn s rati and mass density are assumed cnstants and (ii) fr PRIMA device, the 10 cm diameter plate is prescribed and T ranges between 15 ms and 20 ms. Using these values, the range fr T cvering mst pssible in-situ cnditins is reprted n Fig. 6.2 t describe the effect f the peak methd n the results fr the half-space mdel fr realistic cnditins. It can be seen frm Fig. 6.2 that the peak methd als can lead t systematic errrs when applied t a hmgeneus half-space mdel. Fr the chsen example (10 cm fting diameter), the use f the peak methd tends t verestimate the stiffness assciated with very sft sils (lw T ) and t underestimate the stiffness crrespnding t sft t stiff sils (mderate values f T ). The rati k est /k s tends twards unity nly fr very high values f T. A similar trend, yet less accentuated, is bserved fr ther fting diameters (i.e 20 cm and 30 cm). One may recgnize that this trend is similar t the ne bserved in Fig. 4.8 in the case f the SDOF system (diagram k est /k s vs. T/T n ). In bth cases there is n ne-t-ne relatin between the errr k est /k s and the supprt prperties. As a result, the peak methd cannt be used t prduce an accurate index parameter. Therefre, fr field-testing, using stiffness estimates frm the peak methd as index values culd yield inaccurate interpretatins Layered half-space The half-space mdel rarely accurately describes in-situ prfiles. Hwever, the nncnsistency f the peak methd shwn in the case f hmgeneus half-space can be generalized t ther situatins. Indeed, it is knwn that the use f a static backcalculatin frm FWD measurements assciated with the peak methd can yield significant errrs, especially in the case f a shallw stiff layer, as shwn in Resset and Sha [32]. It is further shwn in Guzina and Osburn [7] that using a cnsistent data interpretatin methd, based upn zer frequency cmpnents, rather than the peak methd, imprves the cnsistency f the static backcalculatin. As a crude generalizatin, ne culd see that the accuracy f the peak methd depends n the particular but unknwn in-situ prfile being investigated during rutine testing. This fact has imprtant implicatins since it means that 1) the peak methd shuld nt be used fr estimatin f the value f fting-n-sil stiffness withut additinal infrmatin n the in-situ prfile, and 2) this methd shuld nt be used even fr relative cmparisns cncerning unknwns sil prfiles. 61

72 6.2 Prpsal fr PRIMA field-testing enhancements Based upn the cnclusins related t the data interpretatin methd, and the physical setup mdificatins f the PRIMA device discussed earlier, the use f PFWD PRIMA device culd be enhanced fr field-testing purpses assciated with the cnventinal static backcalculatin prcedure. This sectin briefly explres the pssibility f such enhancements. Develpment f an entire design fr an enhanced methd using PRIMA device wuld require a specific study. Hwever, in summary, the prpsed methd wuld invlve: 1. using the cncepts f FFT, FRF, spectral average, and zer-frequency estimates, 2. defining an adequate theretical FRF (e.g. fting-n-a-layered-half-space mdel), 3. utilizing the enhanced PRIMA setup, 4. adapting apprpriate data acquisitin system and assciated sftware t measure experimental FRF functins using the entire frce and velcity time histries frm PRIMA, 5. develping prgrams fr fitting the measured FFT with the chsen theretical FFT and extracting the static stiffness Backcalculatin analysis The PFWD backcalculatin analysis, as perfrmed in practice, is based upn the use f the static stiffness as input t an elaststatic mdel t estimate the sil s Yung mdulus E. The key pint is that the estimatin f the static stiffness frm the measurements shuld nt be based upn peak frce and displacement values but rather shuld use a cnsistent apprach utilizing FFT, FRF, zer frequency cmpnents f dynamic stiffness and spectral averaging. In the case f the BVT, a SDOF system was used as theretical mdel fr the beam t extract the static cmpnents frm the measured data. In the case f field measurements, the cnsistent methd presented fr the BVT can als be applied, prvided that an apprpriate half-space mdel is chsen t guide the extraplatin f measured data twards zer frequency. Fr example, whenever an in-situ prfile can be apprximated as a hmgeneus elastic half-space, the FRF C vv (ω) presented in Sectin 2.6 can be used as reference mdel fr fitting the experimental C(ω). Such methd successfully was used by Pak and Guzina [12] in an experimental wrk using scaled mdels n a getechnical centrifuge. This wuld cnstitute the basis f an imprved data interpretatin, using the same techniques as the labratry beam verificatin. In ther cases, such as the layered elastic half-space, the analysis wuld have t be perfrmed using apprpriate FRFs. Fr all cases, the measured and theretical FRFs shuld be used in the fitting prcess t extract the true static fting-n- supprt stiffness. Nte that the chice f the FRF t use is nt limited t the cmpliance C(ω); ther FRFs, such as the dynamic stiffness K(ω) can be 62

73 utilized. Als, the cherence functin can be used as an index f the usable frequency range. Nte that the suggested analysis is based upn the frce signal as measured by the lad cell. Hwever, due t inertial frce f the deflectmeter device, the true frce transmitted t the sil is different frm the frce measured by the lad cell. This issue des nt affect the estimatin f the static stiffness k s = K(ω = 0) theretically (zer acceleratin and thus zer inertial frce) but wuld affect the selected extraplatin prcedure used t estimate the static stiffness frm the measured dynamic stiffness. Indeed, the fting mass is embedded in the mdels chsen t guide the extraplatin f the measured data twards zer frequency (the utput f the mtin sensrs cannt be used at very lw frequency). Fr illustratin purpses, let us examine hw the effects f the fting mass were taken int accunt in this study. Cncerning BVT testing, PRIMA mass was incrprated in the equivalent mass f the SDOF analg, which was estimated as a fitted parameter. In the theretical example presented in Fig. 6.2, the prescribed frce was applied n the tp f PRIMA device, and the mass f the latter was taken int accunt in equatin (63) Hardware a) PRIMA setup With regards t the cmments raised fr the physical setup f PRIMA device in Sectin 3.4, it seems preferable, fr quality assurance purpses, t use the enhanced cnfiguratin f the PRIMA device. As a remainder, the s-called enhanced cnfiguratin is cmpsed f the base f PRIMA nly, and the lad is applied using a rubber hammer. Using such a dynamic lading, nt nly the energy imparted t the sil is minimized, but als the dminant frequency f the frce spectrum is increased, that is, a smaller frce wavelength is prduced. Such cnfiguratin therefre minimizes bth the applied stress and the depth f penetratin, resulting in small-strain estimates f the in-situ stiffness, which can be used t btain an estimate fr the in-situ equivalent hmgeneus seismic elastic mdulus. Dealing with such small-strain mdulus, as discussed in Sectin 2.1, wuld be advantageus fr bth cnstructin quality assurance and deteriratin assessment applicatins. The shallwer depth f penetratin assciated with the enhanced setup is appreciable in quality assurance f pavement cnstructin where the mdulus f the tested layer (e.g. base) needs t be checked nly. b) Data acquisitin system T perfrm a cnsistent analysis f the measured frce and deflectin data, nntruncated time histries need t be recrded. Furthermre, in PRIMA data acquisitin system, the velcity autmatically is integrated t yield the deflectin time histry. The integratin prcess, hwever, is generally bserved in practice t yield increasing inaccuracies (distrted displacement time histries) as the time recrd increases. Therefre, it is recmmended that the velcity utput f the PRIMA gephne be recrded. 63

74 The equipment used fr data acquisitin in the labratry is nt practical fr fieldtesting. The cnsequence is that new pieces f hardware, mre adapted t a daily utilizatin in the field, have t be fund t perfrm data acquisitin and FFT, t allw fr spectral average and FRF estimate. The experimental setup fr field-testing culd resemble the ne sketched in Fig PRIMA device Rubber hammer Measurement with PRIMA s sensrs lad cell pwer + amplifier data acquisitin and A/D cnversin hardware fitted prperties tested surface cmputer data acquisitin sftware data interpretatin and fitting prgram wireless transmissin f digital time histry data t PC Fig Pssible field-testing setup fr PRIMA device The exact definitin f the necessary hardware wuld require a specific study. Hwever, general ideas can be presented. Fr example, the use f an A/D cnverter and FFT card that can be directly inserted in a laptp wuld be advantageus. An external pwer surce/amplifier fr the lad cell wuld als be needed. T ensure a small frce cmpatible with the linear system assumptin embedded in the data analysis, the data acquisitin system needs t include an autmatic limitatin f the frce level. Fr example, the system culd recrd the data nly fr frce levels belw a predefined threshld value. Tgether with the field-adjusted data acquisitin system, suitable pieces f sftware r prgrams wuld need t be develped t mnitr data interpretatin, spectral averaging, and FRF estimatin. 64

75 CHAPTER 7 - CONCLUSIONS The study s main purpse was t design and develp a labratry tl fr verificatin f the perfrmance f PFWD devices used fr quality assurance purpses in pavement cnstructin. This study als aimed t examine pssibilities t enhance the perfrmance (physical setup and data interpretatin) f existing PFWD devices. A particular PFWD device, PRIMA 100, was chsen as example and an assciated verificatin testing setup was prpsed. The verificatin test fr the PRIMA device is based upn the Beam Verificatin Tester (BVT) apparatus, which is a simply supprted beam assembly. The BVT apparatus has been develped at the University f Minnesta fr the Minnesta Department f Transprtatin (Mn/DOT). This apparatus is intended fr verifying the perfrmance f PRIMA 100 devices. Using the BVT assciated with the spectral analysis presented in the reprt, tests als can be cnducted t check whether the calibratin factrs f the sensrs f PRIMA devices, as given by the manufacturer, are accurate. The bjective f the verificatin tests is t detect the ptential ccurrence f deteriratin f the sensr s accuracy during the life f the field-testing device. Fr the featured PRIMA device, the calibratin f the internal frce and velcity sensrs prved t be satisfactry. Hwever, it was demnstrated that the data interpretatin methd embedded in the device, based upn peak values, intrduces significant systematic errrs, in terms f the estimated supprt stiffness, bth in the case f BVT and field testing. T btain cnsistent results frm the device during verificatin testing n the BVT, an alternative data interpretatin scheme was used. Based upn an enhanced device setup and spectral analysis f dynamic signals, it was shwn that the static stiffness f the beam mdeled as a SDOF system culd be accurately recvered experimentally, fr apprpriate span length, f bth undamped and damped beams. As a result, the BVT apparatus is prpsed as a rutine tl t check the perfrmance f PRIMA devices. Als, extensin f the BVT use t ther devices is discussed. Finally, a basis fr the enhancement f PRIMA device in field-testing situatins is prpsed. Enhancements are based upn the same cnsistent data interpretatin methd than the nes used fr the BVT. The key cnclusins and recmmendatins stemming frm this research are: 1. PFWD devices such as PRIMA deflectmeter cnstitute an effective and reliable tl fr lad and deflectin measurements. It appears, hwever, that the traditinal data interpretatin scheme, s-called peak methd, used t estimate the static fting-n-sil stiffness frm these measurements, is capable f prducing systematic errrs and therefre needs t be replaced. 2. In this study, the riginal data interpretatin, based upn peak values, was replaced by a cnsistent dynamic methd in rder t estimate the static fting-n-supprt stiffness using the reactin supprt ffered by the BVT apparatus. The mdified data interpretatin is based upn spectral analysis 65

76 and relies n a SDOF analg as theretical guide. In additin t this first mdificatin, the physical setup f the device itself is als replaced with a simplified, yet enhanced, setup that enables testing with small impact energy. The BVT apparatus prves t be a ptentially useful tl fr rutine verificatin f the perfrmance f PRIMA 100 devices. 3. Furthermre, similar enhancements f PRIMA device fr field-testing situatins can be investigated. In that case, the spectral-based data interpretatin shuld incrprate an adequate theretical mdel (such as the fting-n-a-layered-half-space mdel, and replace the SDOF analg used fr the beam). 4. Fr quality assurance purpses, field in which stiffness estimatin fcuses n the tp layer f pavement prfiles, the enhanced physical setup presented in this study is recmmended. In additin t the advantage f ffering a shallwer depth f investigatin (cmpared t that assciated with the riginal falling weight setup), the use f the enhanced setup aims tward the estimatin f the seismic mdulus, cmparable t that stemming frm gephysics investigatin methds. This research fcused n a labratry assessment f the perfrmance f PRIMA 100 device. Applicatins t field-testing fr this device, as well as extensin t ther prtable deflectmeter devices culd nly be tuched n briefly. Specific cmplementary studies t address these issues wuld be necessary t ptimize the use f deflectmeter devices. Further investigatins culd encmpass the fllwing prpsitins: 1. The BVT apparatus culd be adapted n a case-by-case basis t ther devices fr perfrmance assessment. 2. Further studies are necessary t develp the data analysis techniques, as well as the necessary hardware and sftware adaptatins, required fr the cnsistent and enhanced field peratin f PRIMA 100 and similar PFWD tls. 3. The traditinal data interpretatin (i.e. the peak-based methd) can prduce systematic errrs. Nevertheless, it presents the advantage t be simple and rbust. A field-testing study shuld be cnducted in rder t examine the perfrmance f the prpsed enhancements in field cnditins. 4. There is a need t investigate the relatins between seismic mdulus, btained frm field-testing, and resilient mdulus, used in pavement design. On-ging labratry research at the University f Minnesta aims t examine crrelatins between resilient mdulus and small strain mdulus. 66

77 REFERENCES [1] Resilient Mdulus f Unbund Granular Base/Subbase Materials and Subgrade Sils, Lng-Term Pavement Perfrmance (LTPP), Prtcl P46, U.S. Department f Transprtatin, Federal Highway Administratin, Virginia, [2] Lyttn, Rbert L., Backcalculatin f Layer Mduli State f the Art, Nndestructive Testing f Pavements and Backcalculatin f Mduli, ASTM STP 1026, A.J. Bush III and G. Y. Baladi, Eds., American Sciety fr Testing and Materials, Philadelphia, pp. 7-38, [3] Chen, Dar-Ha, Bilyeu, Jhn, and He Rng, Cmparisn f Resilient Mdulus between Field and Labratry Testing: A Case Study, Prceedings f the 78 th Annual Meeting f Transprtatin Research Bard Annual Meeting, 25 pp., January 10 th -14 th, Washingtn, D.C, [4] Siekmeier, Jhn A., Yung, D., and Beberg, D., 1999, Cmparisn f the Dynamic Cne Penetrmeter with Other Tests During Subgrade and Granular Base Characterizatin in Minnesta, Nndestructive Testing f Pavements and Backcalculatin f Mduli: Third Vlume, ASTM STP 1375, S. D. Tayabji and E. O. Lukanen, Eds., American Sciety fr Testing and Materials, West, Cnshhcken, Pennsylvania., [5] McKane, Ryan, In Situ Field Testing f Mechanical Prperties, University f Minnesta, 48 th Annual Getechnical Engineering Cnference Yung Engineer Paper Cmpetitin, Feb. 18, [6] Van Gurp, Christ, Grenendijk, Jacb, and Beuving, Egbert, Experience with Varius Types f Fundatin Tests, Fifth Internatinal Sympsium n Unbund Aggregates in Rads (UNBAR5), 21 st -23 rd June 2000, Nttingham, United Kingdm, [7] Guzina Bjan B. and Osburn, Rbert H., An Effective Tl fr Enhancing the Elaststatic Pavement Diagnsis, Transprtatin Research Recrd 1806, pp , [8] Lepert, Ph., Simnin J.M., and Meignen, D., An Alternative Apprach t Bearing Capacity: Impulse Dynamic Investigatin, Prceedings f the Furth Internatinal cnference n the Bearing Capacity f Rads and Airfields, Vl. 1, July 17 th 21 st Minneaplis, MN, [9] Stlle, Dieter F.E. and Peiravian Farideddin, Falling Weight Deflectmeter Data Interpretatin Using Dynamic Impedance, Canadian Jurnal f Civil Engineering, Vl. 23, Issue 1,

78 [10] Briaud, Jean-Luis and Lepert, Philippe, WAK Test t Find Spread Fting Stiffness, Jurnal f Getechnical Engineering, ASCE, Vl. 116 (3), p , [11] Hardin, Bbby O. and Drnevich, Vincent P., Shear Mdulus and Damping in Sils: Design Equatins and Curves, Jurnal f Sil Mechanics and Fundatin Divisin, ASCE, Vl. 98 (SM7), p , [12] Pak, Rnald Y.S. and Guzina, Bjan B., Dynamic Characterizatin f Vertically Laded Fundatins n Granular Sils, Jurnal f Getechnical Engineering, ASCE, Vl. 121 (3), p , [13] Kramer, Steven L., Getechnical Earthquake Engineering, Prentice-Hall Inc, New Jersey, [14] Richart, F. E., Jr., Hall, J. R., Jr., and Wds, R. D., Vibratin f Sils and Fundatins, Prentice-Hall, Inc, New Jersey, [15] Seed, Bltn H., Wng, Rbert T., Idriss, I. M. and Tkimatsu, K., Mduli and Damping Factrs fr Dynamic Analyses f Chesinless Sils, Jurnal f Getechnical Engineering, ASCE, Vl. 112 (11), pp , [16] Sklnikff, Ivan S., Mathematical thery f elasticity, Secnd Editin, McGraw-Hill Bk Cmpany, Inc, Yrk, Pennsylvania, [17] Craig, R. F., Sil Mechanics, Sixth Editin, E & FN Spn, an imprint f Chapman & Hall, Lndn, United Kingdm, [18] Meirvitch, Lenard, Fundamentals f Vibratins, Third Editin, McGraw-Hill Cmpanies, Inc., New Yrk, [19] Inman, Daniel J., Engineering Vibratin, Secnd Editin, Cyril M. Harris Editr, Prentice Hall, Inc., Upper Saddle River, New Jersey 07458, [20] Wilsn, Edward L. and Clugh, Ray W., Dynamic Respnse by Step-by-Step Matrix Analysis, Sympsium n the Use f Cmputers in Civil Engineering, Labratrie Nacinal de Engenharia Civil, Lisbn - Prtugal, 1-5 Octber, 1962, pp [21] Bendat, Julius S. and Piersl, Allan G., Randm data analysis and measurement prcedures, Third Editin, Wiley series in prbability and statistics, Jhn Wiley & sns, Inc., [22] Estimating Transfer Functins with SigLab, Applicatin Nte, DSP Technlgy Inc., available at [23] Luc, Juan E. and Westmann, Russell A., Dynamic respnse f circular ftings, Jurnal f the Engineering Mechanics Divisin, ASCE, Vl. 97, pp ,

79 [24] Pak, Rnald Y.S. and Gbert, Alain T., Frced Vertical Vibratin f Rigid Discs with Arbitrary Embedment, Jurnal f the Engineering Mechanics Divisin, ASCE, Vl. 117 (11), pp ,1991. [25] Clemen, Rene, Guidelines fr Calibratin f Falling Weight Deflectmeters, Carl Br Pavement Cnsultants, Denmark, available at [26] Lardner, Thmas J. and Archer, Rbert R., Mechanics f Slids, an Intrductin, Third Editin, Cyril M. Harris Editr, McGraw Hill, Inc., New Yrk, [27] Dally, James W. and Riley, William F., Experimental Stress Analysis, Third Editin, Cyril M. Harris Editr, McGraw-Hill, [28] Errrs Due t Misalignment f Strain Gages, Technical Nte, TN-511, Measurements Grup Inc., Raleigh, NC, als available at _grup, [29] Stkey, William F., Vibratin f System having Distributed Mass and Elasticity, Shck and Vibratin Handbk, Third Editin, Cyril M. Harris Editr, McGraw-Hill Bk Cmpany, pp , [30] Kármán, Thedre vn and Bit, Maurice A., Mathematical Methds in Engineering, First Editin, McGraw Hill Bk Cmpany, Inc., New Yrk, [31] Reed, Everett F., Dynamic Vibratin Absrbers and Auxiliary Mass Dampers, Shck and Vibratin Handbk, Third Editin, Cyril M. Harris Editr, McGraw-Hill Bk Cmpany, pp , [32] Resset, Jse M. and Sha, K-Yung, Dynamic Interpretatin f Dynaflect and Falling Weight Deflectmeter Tests, Transprtatin Research Recrd 1022, pp. 7-15,

80

81 Appendix A: User s Manual fr the Beam Verificatin Tester (BVT) - Technical infrmatin and experimental prcedure fr PRIMA device verificatin testing

82 Appendix A: User s Manual fr the Beam Verificatin Tester (BVT) - Technical infrmatin and experimental prcedure fr PRIMA device verificatin testing A - 1

83 This user s manual is an appendix f the reprt Enhancements and Verificatin Tests fr Prtable Deflectmeters and shuld be used nly in cnjunctin with this reprt. The Beam Verificatin Tester (BVT) apparatus has been develped at the University f Minnesta fr the Minnesta Department f Transprtatin (Mn/DOT). This apparatus is intended fr verifying the perfrmance f Prtable Falling Weight Deflectmeter (PFWD) PRIMA 100 devices. Using the BVT assciated with the spectral analysis presented in the reprt, tests can als be cnducted t check whether the calibratin factrs f the sensrs f PRIMA devices, as given by the manufacturer, are accurate r nt. The bjective f the verificatin tests is t detect the ptential ccurrence f deteriratin f the sensr s accuracy during the life f the field-testing device. This user s manual presents the apparatus, as delivered t Mn/DOT, the necessary cmplementary equipment references, and the prcedure t fllw fr rutine testing f PRIMA devices. A 1 - DESCRIPTION OF THE BVT APPARATUS ELEMENTS A 1.1 Listing f the elements delivered The PRIMA deflectmeter verificatin tester apparatus is delivered t Mn/DOT with: ne PRIMA 100 device, with a unique central gephne, tw additinal 20 cm diameter plates and tw cable assemblies (this device is the ne being used fr the study), ne manufacturer installatin sftware (PRIMA 100 sftware 2001) n CD, ne verificatin steel beam (dimensins: 4 x 5/8 x apprximately 93 cm), ne supprt steel beam, machined t receive the supprts f the verificatin steel beam at several spans, tw supprts fr the verificatin beam (each cmprting a rectangular steel munting, a hardened steel rd, and tw adjustable clamps), ne receptacle fr PRIMA device with its tw supprts (each cmprting a hardened steel rd, and tw adjustable clamps) and tw adjustable clamps n the tp surface t secure the deflectmeter device; the receptacle is equipped with fur screw t center PRIMA device, a remvable annulus t amelirate the seating f PRIMA device n the receptacle, tw remvable damping device assemblies, ne external gephne fixed n the tp f the receptacle with its cable assembly, perating manual and calibratin certificate. The gephne munted n the BVT is an IMI Sensrs (a PCB Pieztrnic div.) industrial ICP accelermeter mdel VO622A01, were VO stands fr Velcity Output. The sensitivity f the sensr delivered with the BVT, as given by the manufacturer, is 3854 mv/m/s at 100 Hz, all necessary screws t assemble the varius elements and rubber buffer pieces t amelirate the quality f the cntact between clamps and steel structures, A - 2

84 cable assemblies necessary t cnnect sensrs and data acquisitin system (2 special cable assemblies plus tw BNC cable assemblies), ne rubber hammer with additinal rubber tip, ne sftware package cmprising: the prgrams necessary t perfrm the cnsistent data interpretatin presented in the reprt and btain an estimate f the stiffness f the beam, and t cmpare the utput frm PRIMA gephne, using PRIMA sftware, and the utput f the external IMI gephne, independently acquired, the reference SigLab files fr the beam stiffness estimatin applicatin and fr the acquisitin f the external IMI gephne time histry alne, sme examples f measured data files. The prgrams are written in Matlab, and the reference files are SigLab.vna files. lad cell base f PRIMA device adjustable clamps receptacle verificatin beam supprt beam fundatin beam Fig. A.1.1. Sketch f the verificatin setup receptacle adjustable clamps lad cell base f PRIMA device verificatin beam supprt beam fundatin beam Fig. A.1.2. Verificatin setup A - 3

85 remvable damping device Fig. A.1.3. Testing setup fr the BVT with damping devices Fr the sake f clarity, the reader will find displayed n the previus page and abve sme figures (Fig A.1.1, Fig A.1.2 and A.1.3) brrwed frm the bdy f the reprt. It will be easier fr the reader t identify the elements detailed in this users guide. A 1.2 Additinal elements needed A Fundatin f the BVT The BVT is delivered with its machined supprt beam, but withut the fundatin beam used fr the evaluatin testing (cmparable t a W12x35 type steel beam, 39 lng, 6.5 wide and 12.5 deep). The supprt beam needs t be fixed nt a rigid fundatin. We recmmend fixing the apparatus either t a heavy beam (steel r better, cncrete) as shwn in Fig A.1.1 and A.1.2, either directly t the testing facilities cncrete slab. A gd cupling between the BVT apparatus and its fundatin is necessary. A Data acquisitin hardware Mst parts f the acquisitin system are nt delivered with the BVT and will therefre need t be btained separately. These parts were nt purchased as part f the Mn/DOT cntract. The necessary hardware t be used with the BVT t acquire the data frm PRIMA device is as fllws: a laptp cmputer (desktp pssible with special adaptatins) under Windws system, a secnd cmputer (ptinal, as explained next page) an analg/digital (A/D) cnverter and spectrum analyzer: tw-channels DSPT SigLab unit mdel This mdel is a 50 khz bandwidth analyzer. Other SigLab mdels, featuring a 20 khz bandwidth, culd als be suitable (Mdel fr 2 channels and Mdel fr 4 channels), an Adaptec SCSI card t cnnect the SigLab analyzer t the laptp, A - 4