Dynamic Analysis of Falling-Weight Deflectometer Data · DYNAMIC ANALYSIS OF FALLING-WEIGHT...

TTI: 2/3-18-88-1175-1

[EXAS [RANSPORTATION lNSTITUTE

DYNAMIC ANALYSIS OF

FALLINC3-WEIGHT DEFLECTOMETER DATA

RESEARCH REPORT 1175-1

COOPERATIVE RESEARCH PROGRAM

TEXAS TRANSPORTATION INSTITUTE

THE TEXAS A&M UNIVERSITY SYSTEM

COLLEGE STATION, TEXAS

STATE DEPARTMENT OF HIOHWAYS

AND PUBLIC TRANSPORTATION

in cooperation with the

U.S. Department of Transportation Federal Highway Administration

TECHNICAL REPORT STANDARD TITLE PAGE

1. R .. port No. 2. Govt"rnment Acce1i 11on No. 3. Recipient's Catalog No.

TX-89/1175-1 4. Title and Subtitl• S. Reporr Oare

September 1988 Dynamic Analysis of Falling-Weight Deflectometer Data 6. Performing Organ• rot1on Code

Research Report 1175-1 7. Aut"or' sl 8. Performing Organ• ration Report No.

Allen H. Magnuson

9. Performrng Orgoni rattan Nome and Address 10. Work Unit No.

Texas Transportation Institute The Texas A&M University System 11. Contract or Grant No.

College Station, Texas 77843-3135 Studv No. 2;3:.18-88-1175 13. Type ol Report and Period Covered

12. Spansaring Agency Name and Add•ess September 1987 Texas State Department of Highways and Public Interim - September 1988 I Transportation; Transportation Planning Division P .0. Box 5051 14. Sponsoring Agency Code

I Austin, Texas 78763 IS Supplementary Nares

Research performed in cooperation with and for the U.S. Department of Transportation, Federal Highway Administration

-16. Abstroct

Falling-Weight Deflectometer data from three in-service pavement sections has been used to analyze dynamic response characteristics. The analyzed data will be used later in the project along with computer predictions of pavement response to back-calculate engineering properties of pavements. The pavemeets had thicknesses of 1.5 in., 9 in., and 12 in. , and two load levels were tested. The full pulse data is pre-SPnted for the dropweight force and all seven surface displacement sensors. The ana -vsis was performed in the frequency domain using frequency response functions computed from the pulse data. The frequency response functions were computed by dividing the Fast Fourier Transform (FFT) of the displacements by the FFT of the force. Magnitude and phase angles of the frequency response functions are presented. An undesirable oscillation was present in the frequency response functions. This was attributed to a discontinuity in the displacement pulses resulting from zero-packing in the FFT Program. The pulses have a non-zero "tail" value at the end of the 60 msec sample period which causes the discontinuity. The pulse 11 taiii' may be due to drift, permanent deformation of the pavement, or premature truncation before the transient response has died out. A linear correction to the pulses that eliminated the discontinuity was applied to one test case. This eliminated the undesirable oscillation, giving the responses a more regular behavior similar to predicted theoretical responses. This correction should facilitate comparison studies of computed versus measured FWD responses.

17. Key Words 18. Distribution Stat...,ent

Non-Destructive Testing of Pavements, Non-Destructive Evaluation of Pavements, Falling-Weight Deflectometer, Dynamic Analysis of Pavements, Asphaltic Concrete Pavements

No restrictions. This document is available to the public through the National Technical Information Service 5285 Port Royal Road Snrinafield- Vircinia 22161

19. Sacurity Clauil. (of thia raport) 20. Security Clanil. (ol this page) 21· No. of Pages 22. Price

Unclassified Unclassified 90 Form DOT F 1700.7 11-u>

DYNAMIC ANALYSIS OF

FALLING-WEIGHT DEFLECTOMETER DATA

Allen H. Magnuson

Research Report 1175-1

Research Study No. 2/3-18-88-1175

Sponsored by

Texas State Department of Highways and Public Transportation

Maintenance and Operations Division

and

In Cooperation With The U.S. Department of Transportation

and Federal Highway Administration

Texas Transportation Institute Texas A&M University

College Station, Texas 77843

September 1988

METRIC (SI*) CONVERSION FACTORS

APPROXIMATE CONVERSIONS TO SI UNITS Symbol When You Know Multiply By To Find

In ft yd ml

ln1

ft1 ydl ml1

ac

oz lb T

fl oz gal ft' yd'

Inches feet yards miles

square Inches square feet square yards square miles acres

LENGTH

2.54 0.3048 0.914 1.61

AREA

645.2 0.0929 0.836 2.59 0.395

mllllmetres metres metres kilometres

millimetres squared metres squared metres squared kilometres squared hectares

MASS (weight)

ounces 28.35 pounds 0.454 short tons (2000 lb) 0.907

fluld ounces gallons cubic feet cubic yards

VOLUME

29.57 3.785 0.0328 0.0765

grams kilograms megagrams

mlllllltres lltres metres cubed metres cubed

NOTE: Volumes greater than 1000 L shall be shown In m'.

TEMPERATURE (exact)

°F Fahrenheit 5/9 (after Celsius temperature subtracting 32) temperature

• SI Is the symbol for the International System of Measurements

Symbol

mm m m km

mm1

ma ma km2

ha

g kg Mg

.. -

.. -

"'

..

... ... "4 ... ... ...

=

E ...

APPROXIMATE CONVERSIONS TO SI UNITS Symbol WhenYouKnow Multiply By To Find

mm m m km

mm2

ma km2

ha

g kg Mg

ml L

ml Ill metres metres metres kilometres

LENGTH

0.039 3.28 1.09 0.621

AREA

millimetres squared 0.0016 metres squared 10.764 kilometres squared 0.39 hectores (10 000 m2) 2.53

Inches feet yards miles

square Inches square feet square miles acres

MASS (weight)

grams 0.0353 kilograms 2.205 megagrams (1 000 kg) 1.103

millilitres litres metres cubed metres cubed

VOLUME

0.034 0.264 35.315 1.308

ounces pounds short tons

fluid ounces gallons cubic feet cubic yards

TEMPERATURE (exact)

°C Celsius 9/5 (then Fahrenheit temperature temperature add 32)

Of Of 32 98.6 212

-~ •• I~ I I .1~, I·.~. l . 1~. I • 1~. I .2?°J -.O '-io ' o 2o 1 T4o ' 60 ' 8o ' 100 ~ ~ ~

These factors conform to the requirement of FHWA Order 5190.1A.

Symbol

In ft yd ml

ln2

ft2 ml2

ac

oz lb T

fl oz gal ft3 yd'

ABSTRACT

Falling-Weight Deflectometer data from three in-service pavement sections has been used to analyze dynamic response characteristics. The analyzed data will be used later in the project along with computer predictions of pavement response to back-calculate engineering properties of pavements. The pavements had thicknesses of 1.5 in., 9 in., and 12 in., and two load levels were tested. The full pulse data is presented for the dropweight force and all seven surface displacement sensors. The analysis was performed in the frequency domain using frequency response functions computed from the pulse data. The frequency response functions were computed by dividing the Fast Fourier Transform (FFT) of the displacements by the FFT of the force. Magnitude and phase angles of the frequency response functions are presented. An undesirable oscillation was present in the frequency response functions. This was attributed to a discontinuity in the displacement pulses resulting from zero-packing in the FFT program. The pulses have a non-zero "tail" value at the end of the 60 msec sample period which causes the discontinuity. The pulse "tail" may be due to drift, permanent deformation of the pavement, or premature truncation before the transient response has died out. A linear correction to the pulses that eliminated the discontinuity was applied to one test case. This eliminated the undesirable oscillation, giving the responses a more regular behavior similar to predicted theoretical responses. This correction should facilitate comparison studies of computed versus measured FWD responses.

iv

DISCLAIMER

The contents of this report reflect the views of the author who is responsible for the opinions, findings, and conclusions herein. The contents do not necessarily reflect the official views or policies of the Federal Highway Administration or the Texas State Department of Highways and Public Transportation. This report does not constitute a standard, specification, or regulation.

v

ACKNOWLEDGEMENTS

The author wishes to express thanks to Mr. R. Briggs of the Texas State Department of Highways and Public Transportation for supporting this research project, for his guidance and encouragement and for useful suggestions on modifications of the draft copy of the report.

The author also wishes to acknowledge the contributions of other members of the TTI staff: Dr. R.L. Lytton for helpful suggestions and for encouragement and support; Mr. F. Germann for acquiring the Dynatest FWD data, Mr. P. Chan for developing the FFT and frequency-response program listed in Appendix A and Mr. V.S. Torpunuri for performing the data analysis computations and preparing the data plots.

vi

Chapter

I

II III

IV v

TABLE OF CONTENTS

INTRODUCTION . • . . . . . . . . . . . . . . . . . . . DYNAMIC ANALYSIS OF PAVEMENTS ........... .

TIME PULSE DATA FROM THE FALLING-WEIGHT DEFLECTOMETER PAVEMENT SECTION FREQUENCY RESPONSE FUNCTIONS SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS REFERENCES . . . . . . . . . . . . . . APPENDIX A - DATA FOR FM ROAD 79 . . . . . . APPENDIX B - DATA FOR STATE HIGHWAY 82 . . .

APPENDIX C - LISTING OF DATA ANALYSIS PROGRAM FOR FFT AND FREQUENCY-RESPONSE FUNCTIONS ......... .

vii

Page

1 8

20

29

54

57 58

68

78

Figure

1

2

3a

3b

3c

4

5

6

7

8

9

10

11

12

13

14a

14b

15

LI ST OF FI GU RES

Location of Monitored Highway Sites Selected for Dynamic Analysis Using Falling-Weight Deflectormeter Data .....

Schematic Diagram of the Dynatest Falling-Weight Deflectometer . . . . . . . . . . . . . . ...

Input-Output Relation in Time Domain.

Phasor Diagram of Input and Output ...

Sketch Showing Lag Angle Introduced by Internal (Material) Damping ........... .

Sketch of Dynamic Testing of Pavements ...... .

In-Phase Dimensionless Normal Displacement for r = O and Uniform Pressure Distribution versus Dimensionless Frequency for Poisson's Ratio of 0, 1/4, 1/3, and 1/2 (Sung, 1954) ..................... .

Out-of-Phase Dimensionless Normal Displacement for r = 0 and Uniform Pressure Distribution versus Dimensionless Frequency for Poisson's Ratio of 0, 1/4, 1/3, and 1/2 (Sung, 1954) ....................... .

Dimensionless Magnitude of Normal Displacement for r = 0 and Uniform Pressure Distribution for Poisson's Ratio of 1/3: Comparison of Theory and Computed Values ...... .

Phase Angle of Normal Displacement for r = 0 and Uniform Pressure Distribution for Poisson's Ratio of 1/3: Comparison of Theory and Computed Values ...

Load versus Time, Lowest Load (SH 19 and 24).

Load versus Time, Highest Load (SH 19 and 24)

Deflections versus Time, Lowest Load (SH 19 and 24) ..

Deflections versus Time, Highest Load (SH 19 and 24) ..

Possible Extrapolated Behavior of Pulse ....•...

Truncated Pulse with Zero-Packing Showing Discontinuity

Linear Correction to Pulse to Eliminate Tail Discontinuity.

Corrected Deflection versus Time, Lowest Load (SH 19 and 24) . . . . . . . . . . . . . . .

viii

Page

2

4

9

9

9

13

15

16

17

18

22

23

24

25

27

30

30

31

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

Real Part of Fast-Fourier Transform of Force Pulse, Lowest Load (SH 19 and 24) ................... .

Imaginary Part of Fast-Fourier Transform of Force, Pulse Lowest Load (SH 19 and 24) ............... .

Real Part of Fast-Fourier Transform of Deflection Sensor 1, Lowest Load (SH 19 and 24) ..•....•......

Imaginary Part of Fast-Fourier Transform of Deflection Sensor 1, Lowest Load (SH 19 and 24) ........ .

Magnitude of Frequency Response Function versus Frequency, Lowest Load, 0-300 Hz (SH 19 and 24) .......•...

Magnitude of Frequency Response Function versus Frequency, Lowest Load, 0-125 Hz (SH 19 and 24) ............ .

Magnitude of Frequency Response Function versus Frequency, Lowest Load N = 2048 (higher resolution), 0-125 Hz (SH 19 and 24) ....•..................

Magnitude of Frequency Response Function versus Frequency, Lowest Load N = 512 (lower resolution), 0-115 Hz (SH 19 and 24) ...................... .

Magnitude of Frequency Response Function versus Frequency, Lowest Load, Corrected Pulse with No Discontinuity, 0-125 Hz (SH 19 and 24) ................. .

Magnitude of Frequency Response Function versus Frequency, Highest Load, 0-300 Hz (SH 19 and 24) .......... .

Phase Angle versus Frequency, Lowest Load, 0-125 Hz (SH 19 and 24) . . • . • . . . . . . . . . . . . . .

Phase Angle versus Frequency, Lowest Load, 0-300 Hz (SH 19 and 24) ....•...............

Phase Angle versus Frequency, Lowest Load, 0-125 Hz, N = 2048 (higher resolution), (SH 19 and 24) .....

Phase Angle versus Frequency, Lowest Load, 0-115 Hz, N = 512 {lower resolution), (SH 19 and 24) .....

Phase Angle versus Frequency, Lowest Load, 0-125 Hz Corrected Pulse with No Discontinuity (SH 19 and 24)

Phase Angle versus Frequency, Lowest Load, 0-125 Hz Using Corrected Pulse with Branch Jumps Eliminated (SH 19 and 24) • • • • . . . . . . . . . . . . . . .

Phase Angle versus Frequency, Highest Load, 0-300 Hz (SH 19 and 24) • • . . . . . . . . . . . . . . .

ix

32

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35

37

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39

40

41

43

45

46

47

48

50

51

52

Table

I

2

3

4

LIST OF TABLES

Pavement Sections Selected for Dynamic Analysis Using the Falling-Weight Deflectometer ..... .

Description of Analyzed Pavement Sections ..

Temperature and Moisture Content Data ...

Tail Ratios for Three Pavements and Two Load Levels

x

Page

3

6

21

44

Background

CHAPTER I

INTRODUCTION

This study is part of the Texas State Department of Highways and Public Transportation Project 2/3-18-88-1175 entitled "Development of Dynamic Analysis techniques for Falling-Weight Deflectometer Data." The report has been written in preparation for Task 2 of this project: "Measurement and Verification of Deflection and Surface Wave Data."

Task I of the Project 1175 called for selection of pavement sections for nondestructive test measurements. Thirty highway sections were selected under this task. The sections are from Districts 1, 8, 11, and 21 and the TTI pavement Test Facility. The sections selected make maximum use of data from related studies. Most sites are monitored under Projects 1123, 1159, 473, and NCHRP 10-27. The locations of the test sites are shown in Figure l. The characteristics of the in-service highway sections are shown in Table 1 where the pavement type and subgrade conditions, as well as special cases, are indicated. Special cases include rock ledge subgrade and cement-stabilized base sections.

Dynamic data from all these sites will be acquired using the Falling-Weight Deflectometer system which consists of the dropweight apparatus and a series of geophones mounted on a trailer as shown in Figure 2. The trailer is towed by a van containing electronic apparatus (computer and associated peripheral devices) for data acquisition, remote operation of the test and data analysis and display. During testing, the dropweights are raised by a hydraulic piston and released remotely by the operator in the van. At a given test site, the geophones are raised and lowered hydraulically. The test operation, including data acquisition and analysis, is performed remotely by the operator in the van. Because the whole operation is automated, great quantities of data can be acquired in a short time. Normally, only the peak values of the dropweight force and deflection are measured and recorded. The peak values are assumed to represent a static deflection basin. The peak values are used for the backcalculation analysis which assumes the pavement is a layered linear elastic (static) solid. However, the full-time history of the force and deflection pulses can be recorded if desired for dynamic analysis. This report will show how the full-time history can be analyzed and interpreted for dynamic analysis purposes.

l

N

LOCATION OF TEST SITES

District 8

Figure 1. Location of Monitored Highway Sites Selected for Dynamic Analysis Using Falling-Weight Deflectometer Data

Subgrade Conditions

wet ~ ·dry IO ,.....

w u

wet "O c:

dry IO Vl

Shallow Bedrock

Cement Stabilized Base

TABLE 1.

Pavement Sections Selected for Dynamic Ana1ysis Using the Falljng-Weight Deflectometer.

Pavement Type

Surf ace Medium (2 11 -S 11 )Thick Thick (6 11 or more) Treated (Thin) A.C. on Granular Base A.C. on Granular Base 8.6, 2J. .2 21.4 8.5, 11.1, 21.5 1. 3, 11. 3, 11.4 1.2, 1.5, 21.6 1.1, 1.4

21.1 8.2 21. 3

1.6, 8.1, 11.5 (8.3) 11. 2, (8.4)

8.3 8.4

11.9 11.8 11. 7

GEOPHONES ~;::::::::...,......:::=::~~~liiil;;;;o;---__.¢:::::.~~_,c....-,::::.~~~:::.......-=::::-.: TIME

/EFLECTIONK? •

------THE FALLING WEIGHT DEFLECTOMETER

Figure 2. Schematic Diagram of the Dynatest Falling-Weight Defl ectometer

The FWD nondestructive test data to be collected for this project includes the complete load impulse and geophone response signals, making available for the first time masses of pavement data that can, if analyzed properly, give useful information on the dynamic response of pavements. This research builds on the extensive past and ongoing work at TTI on pavement technology, design, research, testing, and evaluation, (Lytton, Roberts, and Stoffels, 1986; Lytton, 1988).

The overall approach or strategy of this research project is to extend or build on existing methods of extracting pavement properties from FWD data. Current practice is to use elastic (static) models to extract information on the properties of pavements, e.g, back-calculating elastic moduli for each layer (Uzan, Lytton, and Germann, 1988). Full use is not being made of the data which contains information on the dynamic properties of the pavement layers. Viscoelastic behavior such as creep and relaxation is exhibited in the pavement layers. To extract this information, new techniques must be developed that use analytical models of the dynamic properties. This is the ultimate objective of Project 1175. The work of Magnuson (1988) is the first step in developing a fast, efficient predictive computer program for pavement dynamics including viscoelastic effects. Successful completion of the project should result in new methods for:

1. More accurate determination of surface layer moduli, 2. Determination of layer thicknesses by data analysis rather than by

labor-intensive core sampling and laboratory analysis, 3. Development of an improved structural index, 4. Prediction of remaining life of pavements, and 5. Possible in situ measurement of the rutting properties of pavements,

using the FWD as a pavement evaluation tool.

Monitored sections from District 1 were chosen for a pilot study of FWD dynamic analysis. (See Table 1). The characteristics of the pavement section selected for the pilot study are given in Table 2. This study is based primarily on State Highways 19 and 24 having a 9-inch thick asphaltic concrete pavement. For comparison purposes, data is also presented for a thin (1.5 inch) pavement from Farm-to-Market road 79 and a thick (12-inch) pavement from State Highway 82.

5

Table 2. Description of Analyzed Pavement Sections (District 1)

Pavement Section County Section Route

1.1 Lamar Thick State 19 & 24

, .4 Lamar Thick State 82

, .6 Lamar Thin FM 79

location and/or

Mi le Post

Paris, Texas to Sulphur River

East of Loop 271

0.7 miles East of Pine Creek

Surf ace Base

911 AC 711 Crushed Stone

1211 AC 22 11 Sandy S/G

1.511 AC 6" Flex

Subbase Subgrade Groundwater

10" Lime- Clay No

Treated Clay

Clay No

611 Lime-treated Silty No

Material Sand

Objectives

The major objectives of this study were to introduce dynamic pavement analysis methods to the highway engineering community and to prepare the way for the upcoming large-scale data acquisition and analysis effort. Other related objectives are to:

I. Present proposed standardized procedures for reducing and presenting data for use in subsequent comparison studies of computed versus measured FWD responses. In the comparison studies, key engineering properties related to pavement life and performance will be extracted by an iterative back-calculation procedure.

2. Develop a computerized system for data analysis and plotting. 3. Demonstrate how the dynamic response data looks and how to interpret

it. 4. Point out any problems in analyzing data and present ways to resolve

them.

7

CHAPTER II

DYNAMIC ANALYSIS OF PAVEMENTS

The purpose of this chapter is to introduce some "linear systems" concepts that are helpful in understanding pavement dynamics. The system concept involves a transformation of an input into an output. The system can be described or characterized by how it transforms the input into the output; that is, the system can be treated as an unknown "black box": where only the input and output are known.

The system description can be done in the time domain or in the frequency domain. The most direct approach is to use the time domain. However, the frequency domain characterization provides more insight into specific phenomena occurring within the system; therefore, it is a better "diagnostic" or analysis tool. The system concept will be described briefly here as it applies to pavement dynamics in both the time and frequency domains. The system approach is described in many books, for example Bendat and Piersol (1971), Bendat and Piersol (1980) and Lee (1960).

The FWD provides both input and output data to characterize the pavement section as a system. The input (excitation or forcing function) is the force, and the output or responses are each of the surface deflections, as shown in Figure 3a. The output for a linear system, z(t), is related to the input, F(t), by the impulse response function, h(t), which is used in the convolution (or superposition) integral as follows:

m

z(t) = J h(r} F (t - r} dr (1)

where r = Dummy time variable associated with the "shift" of the input F.

The FWD force and deflection time pulse data can, in principle, be used to compute the impulse response function, h(t), through a computational deconvolution procedure. This will characterize or describe the input-output relationships for the given system. Note that the response from each displacement sensor will give a different impulse response function, or a different "system."

Generally, when analyzing a relatively unknown system (such as the dynamic

8

f( t)

·I Z( t)

h(t) • Input output

system

Figure 3a. Input-Output Relation in Time Domain.

z(w) F<w>

Figure 3b. Phaser Diagram of Input and Output.

PAVEMENT MATERIAL

TIME

Figure 3c. Sketch Showing Lag Angle Introduced by Internal (Material) Damping.

9

response of a highway section), it is preferable to work in the so-called frequency domain. This is because more physical insight can be obtained on the system, even though the method appears to be less direct than working in the time domain. Another advantage of working in the frequency domain, is that the theoretical predicted responses are more readily computed in the frequency domain. Therefore, comparisons between theory and experiment can be made more readily in the frequency domain and differences can be "diagnosed" better. For example, specific phenomena such as natural frequencies, normal modes or surface waves, can be detected and damping effects can be identified directly.

To work in the frequency domain, it is necessary to perform a transformation on the time pulse data. This is done using a Fourier integral transform (Lee, 1960). If the time pulse signal is designated as f(t), its Fourier integral transform is denoted F(w), defined as follows:

CIO

F(w) • J f(t)e-M dt -a)

where: i2 = -1, and w = is the radian frequency.

The inverse transform recovers the time pulse from the frequency domain transform F(w) as follows:

CIO

f(t) = },, J F(w)eM dw -a)

(2)

(3)

Equation (3) can be considered to be a synthesis or addition of oscillatory responses with F(w) acting as the weighting function used to synthesize a given pulse f(t). The exponential term eiwt is the kernel of this transform and it can be expanded as follows:

e:tiwt = cos wt ± i sin wt

This means the transform F(w) is a complex number having a real and imaginary part.

(4)

In practice, the Fourier Transform (2) is performed on experimental data using a computer program with a Fast Fourier Transform (FFT) algorithm (Davis and

10

Rabinowitz, 1975). The FFT treats the pulse as a periodic (repeated or cyclic) function with a period T which varies with the number of sample points and the sampling interval. It is customary to use zero-packing to extend the pulse to a convenient point depending on the desired resolution of the transformed function. This will be discussed in more detail in Chapter IV.

The frequency domain counterpart to the convolution integral input-output relation in Equation (I) is written as:

where: Z(w) = F(w) =

Z(w) = H(w) F(w)

Fourier-transformed displacement (output), Fourier-transformed force (input), and

H(w) = frequency-response function.

(5)

This relation (5) is simpler than the convolution integral relation, which is part of the reason for working in the frequency domain. The frequency response function is a complex number having a magnitude and phase angle associated with it, both defined as follows:

where:

(HI =

tPa =

real part, imaginary part,

(H2 + H2 )% =magnitude, and R I

tan- 1 (H1 /HR) = phase angle. The significance of the response function may be illustrated by taking an

oscillatory cosine forcing function:

where: Re indicates taking the real part. From Equations (5) and (6), the time domain response z(t) is:

(6)

(7)

This result indicates why H(w) is referred to as a frequency response function: for a unit cosine forcing function (F0 = 1), the magnitude of the response of the system is the magnitude of the frequency response function IHI,

11

and the phase shift or phase angle response is the phase angle (¢8 ) of the frequency response function.

This can be interpreted graphically with a phasor diagram shown in Figure 3b. The oscillatory force F(w) and the response z(w) are drawn as rotating vectors in the complex plane, both rotating counterclockwise at an angular velocity, w. The response leads the force by the phase angle ¢H while the magnitude of the response is given as:

That is, the output to an oscillatory input for a given frequency is characterized by a change in magnitude and a phase shift relative to the input, as indicated in Equation (7). The phase shift for the pavement is caused by inertia or wave effects and internal damping, as shown in Figure 3c.

The pavement system is to be characterized dynamically by the frequency response function. This is done by solving for H(w) in Equation (5):

H(w) = Z(w)/F(w) (8)

That is, Equation (8) indicates that the frequency response is obtained by dividing the FFT of the response at each sensor location by the FFT of the force pulse. This has been done for pavement sections and the results are shown in Chapter IV.

To this point, the discussion has treated the "system," the highway section as an unknown "black box." The response of the system can also be predicted theoretically (at least to the level of simplifying approximations made in the formulation of the theory). To illustrate what the frequency response functions should look like for a highway section, the theory of Reissner (1936), as modified by Arnold, Bycroft, and Warburton (1955) and Sung (1953), can be used. In this theory,the highway is modeled as a uniform elastic halfspace with a circular uniform oscillatory pressure distribution acting on the surface {Figure 4). The surface displacement at the center of the pressure distribution was computed. This simulates the sensor 1 data on the FWD apparatus after the pulse data is transformed using an FFT.

Reissner introduced a dimensionless representation for the response as follows:

12

Oscillatory Force Surface Wave

f(l) n-l i1 ~- ...... ~,._...-,....------ r ·777117;077l/11 77 ~

llllllllllll

77777777

Figure 4. Sketch of Dynamic Testing of Pavements.

13

u (9) z

where: uz = surface vertical displacement {dimensional), p = force magnitude, G = halfspace rigidity modulus, a = radius of pressure distribution,

f 1 = real part of the dimensionless response, and

f 2 = imaginary part of the dimensionless response. indicates that the response is proportional to the force and

inversely proportional to the rigidity and the disk radius. The dimensionless functions f 1 and f2 are in turn functions of Poisson's ratio {µ) and the dimensionless frequency (a

0) defined as:

Equation (9)

wa = t:

T

where: the shear wave speed (Cr) is given as

That is,

Cr (G/p)% and p mass density (~=pg).

f 1 f 1 ( ao' µ)

f 2 = f 2 ( ao ' µ)

These functions, f1 and fz, are plotted against a0 for various values of Poisson's ratio in Figures 5 and 6 taken from Magnuson (1988).

(10)

It is more convenient and easier to interpret the results if the dimensionless functions are expressed in magnitude and phase form as follows:

( 11)

where: I f I = < f ~ + f ~ ) ~ , ¢£ = tan- 1 {fz/f1 ).

The magnitude and phase for a Poisson's ratio of 1/3 are shown in Figures 7 and 8, respectively, taken from Magnuson (1988). Elastic results are from Sung's

14

~

01

0.35

nu - 0 nu - 1/4

0.30 nu - 1/3 T"-

"I- nu - 1/2 .. .....

c: 0.25 Q)

c: 0

----- ..... - - - - - - -c. E 0 0.20

-------- - - - - - - - - - - - - ---u ---- ....... Q) .......

UJ c .c a.. c

0.15

....... ....... ---------- .... ---- ---------. --------........---------------..._ ------------

....... ....... .......

....... .......

0.10 ---... ---... --0.0 0.2 0.4 0.6 0.8 1.0 1.2

Dimensionless Frequency

Figure 5. In-Phase Dimensionless normal Displacement for r = O and Uniform Pressure Distribution versus Dimensionless Freque~cy for Poisson's Ratio of 0, 1/4, 1/3, and 1/2 (Sung, 1954).

---1.4

N '+-

.. ~

c: Q)

c: 0 a. E 0 u Q) en 0

..c: a... '+-0

~

:J 0

0.25

nu -nu -nu -0.20 nu -

0.15

0.10

0.05

0.0 0.2

0

1/4

1/3

1/2

---------

0.4 0.6 0.8 1.0 1.2 1.4

Dimensionless Frequency

Figure 6. Out-of-Phase Dimensionless Normal Displacement for r = 0 and Uniform Pressure Distribution Versus Dimensionless Frequency for Poisson's Ratio of 0, 1/4, 1/3, and 1/2 (Sung, 1954).

__, ..........

Q)

"'O :J

-+-c Ol 0 ~

(/) (/) (l)

c 0 (/)

c (l)

E 0

0.20 _r------------------1.....a.-...:::-..-L:..::::;--=--==-=-===-:::..::-~-=-===--==-=--=--~::..

0.15 -

0.10-

0.05-Elastic Theory

1/ 4 3 Damping

1 /2 % Damping

0.00-4-~~~~1~~~~~1~~~~.-1~~~.1~~~~.,~~~--,.-~~~,1~---

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Dimensionless Frequency, a_zero

Figure 7. Dimensionless Magnitude of Normal Displacement for r = 0 and Uniform Pressure Distribution for Poisson's Ratio of 1/3: Comparison of Theory and Computed Values.

__. 00

50

O> 40 Q)

-0

.. O> 0 30 _J

Q) en 0

..c 20 a_

10

0.0

Elastic Theory

1/4 % Damping

1 /2 % Damping

0.2 0.4

~ ,,..::-.

,.....:, -u.

0.6

.; ........... ...........

.... / ,,, 'l"

............ .............

0.8

,, .......... ,, / ... .,,.,,

.............. ,, ........

........... ,, ...........

,, ........ ...........

,, .......... ,,.,,.,, ,,,,,.....

,, ........... ...........

........... ... ............

,./ ,,,,,,.....

1.0 1.2 1.4

Dimensionless Frequency, a_zero

Figure 8. Phase Angle of Normal Displacement for r = 0 and Uniform Pressure Distribution for Poisson's Ratio of 1/3: Comparison of Theory and Computed Values.

(1953) computations while the damping = % percent, and damping = % percent results were computed as part of this project by Magnuson (1988). These plots indicate that the magnitude response falls off gradually as frequency increases and the phase angle is nearly linear in frequency. Damping is seen to decrease the amplitude and increase the phase angle. In Chapter IV, Figures 7 and 8 will be shown to resemble the frequency response functions computed from FWD data. However, the FWD data includes effects of layering, inhomogeneities, etc., which are not accounted for in the Reissner-Sung theory, so a quantitative comparison cannot be made.

19

CHAPTER III

TIME PULSE DATA FROM THE FALLING-WEIGHT DEFLECTOMETER

Time-pulse data for SH 19 and 24 are presented and discussed in this Chapter. Appendices A and B contain similar data on FM 79 and SH 82, respectively. Characteristics of these pavement sections are given in Table 2. Test time, date, temperature and moisture content are given in Table 3. The load or force pulses are shown in Figures 9 and 10. The data is sampled every 0.2 msec and the sampling time is 60 msec. The lowest load is shown in Figure 9 and the highest load is shown in Figure 10. {Data was acquired for four load levels.) Note the double peak in the pulse. The second peak is higher than the first for the highest load. The load is given in units of kiloPascals {kPa).

Deflections are shown in Figures 11 and 12 for the lowest load and highest load, respectively. The deflections are plotted for all seven seismometer sensors. The sensors are all spaced at varying distances from the center of the dropweight as follows:

Sensor 1 2 3 4 5 6 7

Distance (ft.) 0 1 2 3 4 5 6

Note from Figures 11 and 12 how the deflections decrease as the distance increases. The pulse peaks occur at later times as the distance increases. This indicates wavelike behavior where the response is delayed the farther the distance from the dropweight.

Normal practice for static analysis is to take the peaks of each of the deflection pulses as the deflection basin. The static force is taken as the peak of the corresponding force pulse. This practice involves some approximations because the peaks do not occur at the same time and the shapes of the pulses are distorted. The time delay and the pulse distortion indicate dynamic effects are present. This will be confirmed in the frequency domain analysis in the next chapter.

20

Table 3. Temperature and Moisture Content Data

Highwa~

SH 19 & 24 FM 79 SH 82

Test Date 3/30/88 3/29/88 3/30/88 Test Time 9:30 a.m. 2:00 p.m. 10:00 a.m.

Thermocouple Depth (in.) TCl 9 2 6 TC2 16 6 12 TC3 13 14 15

Temperature (•F) Surf ace 65 58 67 TCl 62 63 60 TC2 66 68 62 TC3 72 72 64

Moisture Content (Bars) Base Saturated Saturated -0.167 Subbase Saturated Saturated Saturated

21

N N

PEAK LOAD 364 kPa

SH 1924:PT 0

- LOAD 300

n3 0... ~ 200

c <( 0 ....J

100

-100-r-~~~~r-~~~--,-~~~~-r-~~~--,..~~~~-.--~~~--I

0 10 20 30

TIME(MILLI SEC.)

Figure 9. Load versus Time, Lowest Load (SH 19 and 24).

40 50 60

PEAK LOAD 11 63 kPa

1200

SH 1924: PT 0 1000 I - LOAD

I 800 I

ltJ I Cl. ~

I 600

I

0 <(

N 0 w _J I

400

200

0-+---L-~~~~~~~~~~~--=============================

-200~1~~~~~,~~~~~1~~~~~,~~~~~,~~~~~,~~~----i 0 10 20 30 40 50 60

TIME(MILLISEC.)

Figure 10. Load Versus Time, Highest Load (SH 19 and 24).

Vl c: 0 ~ u

•r-E ....._..

z 0 ~ 0 I.LI _J LL. I.LI 0

PEAK LOAD 364 kPa

80

60

40

20

0 10 20 30 40

TIME(MILLI SEC.)

Figure 11. Deflections versus Time, Lowest Load (SH 19 and 24).

SH 1924:PT 0

SENSOR 1 SENSOR 2 SENSOR 3 SENSOR 4 SENSOR 5 SENSOR 6 SENSOR 7

50 60

N U1

en c: 0 Su

•r-E -z 0 -l-o LLI _. LL w 0

PEAK LOAD 11 63 kPa

300-T------------~---------------~~--~--~---------------------

250

200

150

100

50

I I

/ I

I

,. -.... ' '\

\

I , ...... - .. , ,

I , I

I I

\ \

\ \

I I ~-- \ \ -~ .. ~ \ I I / ... \ \

I , ' I ; ' ' \'

I , -- ,,,

I I I ,,,, ""'",'\~ I ' ,-' ""-'' \ I I I / ----- '-''

/ ,-' -..... '\' ~ I I , ,- ' .. '"'

I I / , ---- ... ·_....: '\": I I I / ,,' ,.,,'" ................. 'l

I I I I / ,,'/ I I / ,' , I I , ,/

I I / ,, /

'/ '1 ,, /'

SH 1924:PT 0


'J_, I, / ,,' ,.,' o ....... __ _....._~~-"7"''-~~~~~~~~~~~~~~~~~--~__,..~,,.._---e4

0 10 20 30 40 50 60

TIME{MllLI SEC.)

Figure 12. Deflections versus Time, Highest Load (SH 19 and 24).

Another significant feature is the truncation of the pulses at the end of the 60 msec sample time. Ideally, the pulses should "die out" or go to zero at the end of the sample period. Instead, the deflection pulses usually cross the time axis and become negative before they are truncated. What causes the truncated "tails" of the pulses is not known at this time, however one or more of the following factors are probable causes:

I. "Drift" caused by digital integration and digital filtering of the seismometer.

2. Permanent deformation caused by plastic flow of the asphaltic concrete pavement or compaction of the granular material in the base or subbase.

3. Premature truncation of the pulses before the transient response has died out. The pulses may have a damped oscillatory behavior as shown in Figure 13.

Recommendations for resolving the pulse truncation problem are given in Chapter V. The pulse truncation causes problems when the time-domain data is transformed into the frequency domain by using a Fast Fourier Transform (FFT) algorithm. Why the problem occurs and how the problem can be alleviated is discussed in the next Chapter.

Pulse data on a thin pavement (FM 79) is given in Appendix A (Figures Al to A4) and thick pavement data is given in Appendix B (Figures Bl to B4). Comparisons of force pulses in Figures 9, 10, Al, A2, Bl, and 82 show that the first peak is higher than the second for the lowest load. For the highest load, the second peak is higher than the first peak for the thicker pavements (SH 19 and 24 and SH 82). For the thin pavement (FM 79), the first peak is higher for the highest load. This indicates that the force pulse may be dynamically interacting with the pavement depending on pavement stiffness.

The pavement deflection pulses for FM 79 and SH 82 are shown in Figures A3, A4, and B3 and B4, respectively. The behavior of the pulses at truncation is summarized in Table 4 where the ratios of the pulse "tail" at truncation to pulse peak is shown for the force and all the deflection sensors. SH 82 pulse data behaved similarly to the SH 19 and 24 data discussed above whereby the relative size of the tails is higher for the lower loads. The thin pavement data (FM 79), Figures A3 and A4, showed a different pattern. The lowest load tails were moderate in size for all sensors while the first two sensors for the highest load

26

c: 0 ...... .µ u QJ ,..... ~

N QJ ....... Cl

0

Pulse

Assumed Extrapolated Behavior of Pulse

I --- - - -- ---,,,,,.,,, --,,,

Figure 13. Possible Extrapolated Behavior of Pulse.

time

behaved differently. The sensor 1 pulse drops tangent to the horizontal axis and then rises again. Sensor two crosses the time axis becoming negative, then rises again and crosses a second time. The other sensors (sensors 3, 4, 5, 6, and 7) have very sma 11 "tails."

28

CHAPTER IV

PAVEMENT SECTION FREQUENCY RESPONSE FUNCTIONS

The time domain data in Chapter III was transformed into the frequency domain, and the results are presented here. A Fast Fourier Transform (FFT) was performed on the force pulse and on each displacement sensor for each test case. The FFT is discussed in Davis and Rabinowitz (1975). The FFT numerically performs the Fourier Transform in Equation (2) on digitized data.

The FFT's are computed in the program listed in Appendix C. The FFT works on a digitized signal which is a series of discrete values sampled at fixed intervals of time. The sampling rate of the Dynatest FWD is 0.2 msec, giving 300 samples for the 60 msec pulse. The FFT sample size must be a power of two. The normal sample size is 210 = 1024. To obtain this sample size, the FWD data is supplemented by a series of zeroes. This process is called zero-packing. When the data pulse has a truncated tail, which is usually the case for FWD data, the full pulse has a discontinuity as shown in Figure 14a. It will be shown shortly that the pulse discontinuity produces an undesirable effect on the FFT of the pulse. It will also be shown that the effect can be alleviated (if not totally eliminated) by adjusting the pulse with a linear correction, as shown in Figure 14b. The corrected deflection pulse data eliminating the tail discontinuity is shown in Figure 15.

The FFT of the force pulse for the lowest load (SH 19 and 24) is shown in Figures 16 and 17 for the real and imaginary components, respectively. The FFT algorithm sets the FFT frequency range as the reciprocal of the sampling rate, or 1/0.0002 = 5000 Hz. The normal frequency resolution ~f is given as:

1 ~f = (0.0002 x 1024) = 4·8828 Hz

Note in Figures 16 and 17 that most of the energy of the force pulse is in the 0-250 Hz range. The real and imaginary components of the sensor 1 deflection for the lowest load (SH 19 and 24) are given in Figures 18 and 19.

The frequency response functions are obtained by dividing the FFT of the deflections [Z{w)] by the FFT of the force [F(w)]. Since the FFT's are complex numbers, the frequency response function is also a complex number. If the frequency response function is denoted H(w), one has:

29

c: 0 ...... +' u Q)

...... 4-Q)

C1

1

Zero -Packed Portion of Pulse

____ ......__! --- - - -1023

299 Sample No.

Figure 14a. Truncated Pulse With Zero-Packing Showing Discontinuity.

c::: 0 ...... +' u Q)

...... 4-Q)

c

Uncorrected Pulse

/Corrected Value

l ts

60 msec Linear Correction

Figure 14b. Linear Correction to Pulse to Eliminate Tail Discontinuity.

30

w ......

c.n c: 0 s... u

•r-E

z 0 1-u LU _J LL LU 0

PEAK LOAD 364 kPa

DATA ADJUSTED FROM 50 100-r-~~----~--~~~~~~~:-=-~~~~~~--~-----------------.,

80

60

40

20

0 5 10 15

J

J J

20

J

v \. \. J ., J '\.

25 30 35

TIME(MILLI SEC.)

40

SH 1924:PT 0


45 50 55 60

Figure 15. Corrected Deflection versus Time, Lowest Load (SH 19 and 24).

PEAK LOAD 364 kPa

400000 (REAL)

SH 1924:R=O 300000 -

- FORCE FFT

200000 -N :c .......... ttj a.. 100000 ~ ...._...

w J-

u LL Ct: 0 LL 0

w LL N

-

I ~ L A 'i v~ vv ~ 1 \

I -100000 -1

I -200000 -

-300000 -

-400000 I ' I I 0 1000 2000 3000 4000 5000

FREQUENCY(HZ.)

Figure 16. Real Part of Fast-Fourier Transform of For.ce Pulse, Lowest Load (SH 19 and 24).

'""" N ::i:: ........... tt:S

o_ ~

w f- u LL. a:: LL.

w 0 w u_

PEAK LOAD 364 kPa

400000-,-~~~~~~~~(~IM_A_G_l_N_A_RY~)~~~~~~~~~~-

SH 1924:R=O 300000-

-- FORCE FFT

200000-

100000-

0 IA.

r ~

-100000-

-200000 -

-300000-

-400000;---------~.-l----~---,-l----------~I~--------~,----------~

0 1000 2000 3000 4000 5000

FREQUENCY(HZ.)

Figure 17. Imaginary Part of Fast-Fourier Transform of Force, Pulse Lowest Load (SH 19 and 24).

w .a::-

N :c ............ Vl c 0 s... ·u .... E

...........

I-L1.. L1..

z 0 1--4

I-u LLJ _J

L1.. LLJ Cl

PEAK LOAD 364 kPa

(REAL) 0.4-r-----~~~~~~--~~~;__~~~~--~~~~----~----~-----

SH 1924:R=O 0.3-

-- DEFLECTION FFT

0.2-

0.1 -

0.0 A. - .. v .. y~

-0.1 -

-0.2-

-0.3-

-0.4--+-----~--~~,.---~~~----.l~~~--~--.l~~~~~---..l----~~--~

0 1000 2000 3000 4000 5000

FREQUENCY(HZ.)

Figure 18. Real Part of Fast-Fourier Transform of Deflection Sensor 1, Lowest Load (SH 19 and 24).

w U1

V> c: 0 s.. u

•r-E ....._

I-LL. LL.

z 0 ....... I-L) LLJ _J LL. LLJ Cl

PEAK LOAD 364 kPa

(IMAGINARY) 0.4.,--~~~----~~~~~~~~~~~~--~~~~--~------

SH 1924:R=O 0.3-

-- DEFLECTION FFT

0.2-

0.1 -

L J J 0.0 ~r ~y

-0.1 -

-0.2-

-0.3-

-0.4-r-----~----.,--~~-----,-,----~--~-r-,----------...1----------_j 0 1 000 2000 3000 4000 5000

FREQUENCY(HZ.)

Figure 19. Imaginary Part of Fast-Fourier Transform of Deflection Sensor 1, Lowest Load (SH 19 and 24).

H(w} = Z(w} = F'"{W}l =

where the magnitude f HI and the phase ¢8 are given by:

IHI = (H2 + H2 )% R I

tPa = arctan ( ~ )

The magnitude IHI has dimensions of in./lb.

(12}

Figures 20 and 21 show magnitudes of the frequency response functions for the lowest load (SH 19 and 24) for frequency ranges of 0-300 Hz and 0-125 Hz, respectively. For a given sensor, one expects the magnitude response to decrease with frequency similar to the theoretical result in Figure 7. One sees from Figure 20 that the noise apparently takes over at about 120 to 150 Hz, where the magnitude starts to increase. The noise becomes dominant when the FFT of the force pulse becomes small because of the division indicated in Equation 12. The dominance of numerical "noise" starts at about 250 Hz, as can be seen from Figures 16 and 17 where the FFT of the force is seen to be small. Figure 21 shows the magnitude response over the relatively noise-free range. One sees from Figure 21 that (despite the marked oscillation), on the average, the response decreases with distance as well as with frequency. Later, the frequency oscillation will be shown to be associated with the truncation of the pulse and the resulting discontinuity.

A study of the effect of the responses to the resolution of the FFT was made by taking a sample size of 211 = 2048 (higher resolution) and 29 = 512 (lower resolution}. The results are shown in Figures 22 and 23 for higher and lower resolutions, respectively. Figure 22 shows improved resolution relative to the •normal" FFT in Figure 21. Figure 23 shows an unacceptable deterioration of resolution, as the frequency step size was about 10 Hz, which was too crude to characterize the oscillations which had a "period" of about 20 Hz. The same oscillation was present regardless of the resolution.

Figure 24 shows the responses for the adjusted pulses with no discontinuity, using the correction scheme shown in Figure 14b. The oscillations present in the previous four plots have all but disappeared, giving a smooth, well-behaved,

36

. ..a ........... . c: .,....

w Cl ::> I-z (.!) <( w ...... ~

.40E-05

SH1924:PT

SENSOR 1 SENSOR 2

.30E-05 SENSOR 3

SENSOR 4

SENSOR 5 SENSOR 6

.20E-05 SENSOR 7

.1 OE-05

0 25 50

PEAK LOAD 364 kPa

N=1024

0

r ·1 I ~ ' ~· I f II

't ', '~ , , t I I I 1\ I

I I I I t,

I \

75 100 1 25 150 175 200 225 250 275

FREQUENCY(HZ.)

Figure 20. Magnitude of Frequency Response Function versus Frequency, Lowest Load, 0-300 Hz (19 and 24).

w co

PEAK LOAD 364 kPa

. 1 oE-QS-,-~~~~~~~~~~N~1_0_2_4 _____ ~--~~~~~~~~

.. BOE-06 .c ,_. .......... . c

•r-

~ .60E-06 :::> ..... z (.!)

~ .40E-06

.20E-06

--- -'

- - - ... - -

' \

' \ \

' ' ' ' '

' , __ -, ----... --- -........, ' " ' --- ' ' --- -.....\ ,, .... -- ........ ' .... -- ' ' ' - - "' __ ,

SH1924 • •

SENSOR SENSOR SENSOR SENSOR SENSOR SENSOR SENSOR

.,...,.- ', /,. - I /' \ I

PT 0

1

2

3

4

5

6 7

.,,,,--.., ... ------,. ~ \ /- /\~ I \

___ ,...,,,.... '\.' I / I ' I .- ~ -.?"\..~

__ ... - ------------:~ "----" / '\ "'' I /.,......,, '~.... ,' ,--: ""- ,, ,~ / / _;.~~' ', ;'/ / ~ ,'v I/ /-""'"'\~ .... , ' ,,,---

, ............ I / ' / "'~' '/ I '' -' / °'I::~ /' /"' -...'/'•./ _,,,.,.- \_ ..,Y, I \ '"'' / r "' ,7-- 91<"', /... ~--~--' --~.:-- v / ... ' ,,. ' _... \ 7"..,,, __ ,, .OOE+OO-+---~-----r~--~~___,.----~~-----~-'-/---------~-'-----~------'---~-.... ...::::...:-~-~

0 20 40 60 80 100 120

FREQUENCY(HZ.)

Figure 21. Magnitude of Frequency Response Function versus Frequency, Lowest Load, 0-125 Hz (SH 19 and 24).

PEAK LOAD 364 kPa

.10E-05----------N_=_2_0_4_8 ___________ _

. .a

~ .60E-06 :::> 1--z (!)

~ .40E-06

.20E-06

.OOE+OO 0

SH 1924: PT 0 - SENSOR 1 - - SENSOR 2 - - .. SENSOR 3 -- SENSOR 4 --- -',',,\ - - SENSOR 5 ---- SENSOR 6 --- SENSOR 7

-- ------- '\\ ~ (" ' ,~ \ ', ' - \ -

\ ,_ ..... "" ,. ' _.,,..-- ... , ' \ I \ \ / \ ~

....--"""'' ', ,,-, \ / ..... , \ I \ .,,..-- '\ ' , ' " , ' ' I ,-, - .....

_...;.- '- \ ' ; -\ I _, 6'

------...... , ' ' - "" ''t\.. I I \ '\: - .,,,; ' .,..- , ~ \ / / .. r~'' I ... ~ \ I \ '

--:::..._..:-.:._______ ',~ ', ,/ I / ,, \\ I / .,...;~... I ,,....:;~ , - ... '1' -- - ... ..._ '~ '- / I '.\ \ \ I ,' "", I ' , "'~~ , '

'~\ ... // ..--'~~"<,I ;;,, 'l\!! // / ...... '\,', ,' ,_....-;~

20

~ I , \~ '"' I , ,,,, - .- '\ I I ' ~> / , ,. ' ...... , / "\,"' .. i.' .. \/~.:."I I \,:- ', / , I ' ... _, ~ / \.. ~- , -- y , ..

40

... \ ' , "' ' , .,,,,.- _,/ , ' / .... ... \ ~.. ..... .... ... ....- ... r 60

FREQUENCY(HZ.)

80 100 120

Figure 22. Magnitude-of Frequency Response Function versus Frequency, Lowest Load N = 2048 (higher resolution), 0-125 Hz (SH 19 and 24).

. ..0 r--.......... . c: .,....

w Cl ::> t-z (!)

..s::- <( 0

~

PEAK LOAD 364 kPa

.10E-05-r-------~-----------~-N_=_5 __ 12~~-----~------------------

.BOE-06

.SOE-061

.40E-06

.20E-06

SH1924: PT 0 I I

SENSOR 1 ii

~ SENSOR 2

- - - .... .... '\ SENSOR 3 I ..... ,,, \ SENSOR 4 II

SENSOR 5

, SENSOR 6 I

- - - - - - - ' ''' ·\ SENSOR 7 I ',,_ ', I ~ I -------, \ - - - ',,\ / ,/', \ / . - ""' I

... ,, '' '\ , / / ' ' v .,,,. .................... ~. ~Ji - ' .,. ," .,,,. "" -- -- '\ ' .,." ' / ,, " ' _,,____. ~ ' ',' \ I ', '-

-------- "" /... \ I ' - ... ' - - - I ,--- ', ' ,F "'\ "" I ' .- ,.. .- ' j

-- '' ' / '' '-....' I ' - - - - ' _-_-:::::. ----------... '~ '\.,,' -- ::'< """ ...-----~ ,,~, ' , ' ... '~ _,, ...... , ... ~~ ,,,,;" ' ./7 ' ' , '-,'~ _.,,,-. , I .._ , ' ' , .,,--~ , .,._______ ...... ~'} ,, /,. "' /- / .,, " -..'\. ~ .,

... ' ' .,, , '.....,, ,,r / .,"' ' , ... > , I ........ ., ' , , ... _~ / ' ' ~ ..... - ' ,/ ~-- -~- / ......

... .,,,,' ' ' , ,, ... _ ~ ... ...' ' ..... ... .,,t:.-~

' / ------ ' ;;'-..... OOE-1-00 ,.. v ............. 1 . . -+-------.,..-~-r--~~,--~~,~~-,.--~--1~--!.~1~---,-----1~~-,----;__1~:.::J

0 10 20 30 40 50 60 70 80 90 100 , 10

FREQUENCY(HZ.)

Figure 23. Magnitude of Frequency Response Function versus Frequency, Lowest Load N = 512 (lower resolution), 0-115 Hz (SH 19 and 24).

. ..c ....-.......... . c: .,...

· PEAK LOAD 364 kPa

.1 oE-os ....,.--______ D_A_T_A_A_DJ_u_s_T_E_D_FR_o_M_s_o _______ _

.BOE-06 -' ' ' ' '

SH1924:PT 0

SENSOR 1 SENSOR 2

SENSOR 3 SENSOR 4

~ .60E-06 :::>

- ... ' ' ' ' ' '

SENSOR SENSOR

5

6 1-z SENSOR 7

' ... ... ... ...

C> ... -..............

' ... , ~ .40E-06 ...

... ... ...

' ' ' '

.20E-06

... --........... ..., .... ' '---------.-.--.--.-... '

........... ' ', ---- ' ... , ... ----......... ' ' ', ---- ... ..._..... .................. " ' ' -... ...... ... ... -... _ ......... ' .......... ... - - -... ... _ .................... __ ............................... ~ .................

-. ....... - - .... .....

,_. - - - .... ,_, ... .. _ .. ____ _

...... ...... ---

......... ...... ..... -.... ... .... ....._ --~ ..... ------___.. --------..... .............. ....._____ ----~ .__________ .. -..... ----------....... ._--. - - ... -----..:::==---=--==--=:-::-._~ --- ---- -~ - - ~

--------==~-:~=~-:~:-=--:-~--~= .OOE+OO-t-~---~~r-~~-----,..~~---~-r----------~,.-~~---------~~---__..J

0 20 40 60 80 100 120

FREQUENCY(HZ.)

Figure 24. Magnitude of Frequency Response Function versus Frequency, Lowest Load, Corrected Pulse with No Discontinuity, 0-125 Hz (SH 19 and 24).

almost monotonic decrease in response with frequency and distance. This response is more consistent with computed responses from models based on elasto-dynamic or viscoelastic theory (Figure 7 for example).

The response magnitude for the highest load (SH 19 and 24) is shown in Figure 25 for a frequency range of 0-300 Hz. Comparison of Figure 25 with Figure 20 (lowest load) shows that noise effects (as manifested by an increase of response magnitude with frequency) are less pronounced for the highest load case, where noise effects are apparent only at the highest frequencies (250-300 Hz). The highest load response also exhibits much less oscillation than the lowest load. The noise effects are smaller for higher loads, probably because the noise is independent of load. This reduces the effect of division by a small number (containing random noise), as indicated in Equation 12.

The correlation between the discontinuity caused by the truncated pulse tail and the oscillations can be seen more readily by referring to Table 4, which shows the ratio of the pulse "tail" discontinuity to the pulse peak for the force pulse and all the displacement sensors. High values of the "tail ratio" in Table 4 consistently indicate severe oscillations in the magnitude and phase of the frequency response functions. The "tail ratio" is consistently higher for the lowest load cases, apparently because the pulse tail is relatively independent of load. Therefore, low load cases will have relatively large "tails."

Nonlinear effects in the magnitude responses can be determined by a frequency-by-frequency comparison of Figures 24 and 25. If the pavement response system is linear, the responses in Figure 24 and 25 for a given frequency and sensor (distance) should be equal, since the frequency response function is defined on a per unit load basis.

Phase angle data for the lowest level (SH 19 and 24) is shown in Figures 26-31. The uncorrected phase angle data in Figures 26-29 is difficult to interpret because of "jumps" or discontinuities caused by the arctangent function, which is restricted to the range between -180° and +180°. The jumps occur when the phase angle gets more negative than -180°, causing the arctangent to go to the next "branch" starting at +180°. In Figure 26, the "jumps" start at about 35 Hz, with the outermost sensors jumping first. The first three sensors show no jumps at all because the phase angle never gets more negative than -180°. The "jumps" are so numerous in Figure 27 (which shows the extended 0-300 Hz frequency range) that it would be impractical to try to correct the data. The improved resolution seen in Figure 28, with sample size of 2048, helps in interpreting the

42

PEAK LOAD 11 63 kPa

.10E-05-r------------·--~-----~~~----------~--~--------------

.. BOE-06 ..0

. c: ·.-

~ .60E-06 ::J 1--z CJ

~ .40E-06

.20E-06

SH 1924: PT 0

- SENSOR 1

- - · SENSOR 2 I --- SENSOR 3 I --- SENSOR 4 I

-- SENSOR 5 I ... ----· SENSOR 6 I

---· SENSOR 7 i \ \ \ ! ' ' I

-\., '~ I \ \ \

--.. \" , , -/"\. I \\\ ' . 'v I _.-... \ ' \ ,, ' -'

~"'\ ' ~ ' i ,\ ' _, I ... ,..-, \ \ ' \ ' ~ I ,, ...._ - , I ,, \ ' \, - ' , ' ' ..... ,,,,.....

"'- ', ·'- ' ,'' - - ' '-../"~ /· 1 ' _...., I I~ ,, - ...... - - h fl

,, v.'' ' .. ~_ '~ _,,-" ' , ... ' ' ..._ ~.,,...·

,~,,-~\/ M' '-"\~' ,. , ., , , , _ - - - - ..... _ _ ~ y; - \ , / - 'r ~ ,/~'' ' - , ... - - / '· _ "'- '''J}·.

~, \:~'I JA.'/' ~: . ..../~~~_...;~~~~--~~~._.,.,--- ... ;(1 j .OOE+OO-t-----r----..,..-~~-'---1 -r-~---r~v-~----r---~~--Tc.:-__ -~_~~,--~~~~~~~...:;:....~----.....~~~-~~-----~~--'-J___Ji

I i I I I I 0 25 50 75 100 125 150 175 200 225 250 275

FREQUENCY(HZ.)

Figure 25. Magnitude of Frequency Response Function versus Frequency, Highest Load, 0-300 Hz (SH 19 - 24).

Table 4. Tail Ratio for Three Pavements and Two Load Levels.

load Force Sensor Highway Level load

1 2 3 4 5 6 7

Lowest 0.0027 -0 .104 -0.0864 -0 .172 -0.238 -0.3125 -0.4167 -0.25 SH 19 & 24

Highest -0.0094 -0.0218 -0.0084 -0.017 -0.0652 -0.103 -0 .117 -0.1617

lowest -0.016 -0 .190 -0 .138 -0.174 -0 .194 -0.258 -0.35 -0.375 FM 79

Highest -0.037 -0.055 -0.010 -0.032 -0.065 -0.083 -0.121 -0 .172 .,i::.. .,i::..

lowest -0.032 -0.051 -0.064 -0 .156 -0 .192 -0 .192 -0 .175 -0.161 SH 82

Highest -0.009 +0.048 0 .161 0.00787 -0.0078 -0.027 -0. 0511 -0.091

~ (J'I

SH 1924:PT 0

- SENSOR 1 - - SENSOR 2

~

Ul LtJ

100 - - • SENSOR 3

-- SENSOR 4 LtJ 0:: c:>

- - SENSOR 5

SENSOR 6 w Cl

--- SENSOR 7 ..._,, w _J

c:> z <(

LtJ Ul <( :r: a.. -100

0 20

PEAK LOAD 364 kPa

........ I ... I 1 , I I

' ' I

I I

I I I I I I

I I

I

40

N=1024

j\

I ' 1, ~ ; \1~ I \ f I ' '

I ~\ / r ,~, I '\ I \ \i

I\' ; I \, / ; t \' I '\ ; i \ I 1 ~, ; i \

I \

I I

60

FREQUENCY(HZ.)

80 100

Figure 26. Phase Angle versus Frequency, Lowest Load, 0-125 Hz (SH 19 - 24).

120

,....... Vl LLJ LLJ ~ (!) LLJ 0 ..._,, LLJ ...J (!) z < w Vl < :r: a..

PEAK LOAD 364 kPa

N=1024 200-----~~~~~~------~--~~----~-----

il ,, ,, 100

-100 11 I

~·· ,.

l '; ,, f

o 'i'6 '6a 1'6 , ac , ri'6 , <::>a , 1'6 'l-aa 'iri'=> ri'6° '1-1'6

FREQUENCY(HZ.)

SH 1924:PT

- SENSOR 1

- - SENSOR 2

- - - SENSOR 3

-- SENSOR 4

- - SENSOR 5

---- SENSOR 6

--- SENSOR 7

Figure 27. Phase Angle versus Frequency, lowest Load, 0-300.Hz (SH 19 and 24).

0

,,,....,. (I) w I.LI 0:: (!) LaJ c ~ UJ ..J (!) z <

N=2048 PEAK LOAD 364 kPa

200,----------------------------~-----------------------------

t-. • ! 'f\1 f \ f, ~ ~',,f\ {\,. . f, I •\\'~ ! \'i\ I I\

__ SENSOR • I 1 r~, A 11 r1l:~,~ 1• ~l\ r-r.

.,. I I l \ I ' I ·.~ I '\' '1 \ ! I I - • SENSOR 5 ! I ! I 1' ~ \ J \ ! ~ \~ I I I \ \ I '\

'

---- SENSOR 6 I \,!I I I ·, \ ; ,! I ! 1

\ ~ 11 ! \~~\\ j I ! --- SEtlSOR 7 I I \ I I I \ \ I I I ~ i ~ ,· \ l\ I I : \\ \\

1

,1 I I L~, I :. o~~---------~-------+----__;'r--+~1-+-{-~-'~..;+''-+'~·--..J-f.--~'-\4'4'~-1--L-L..---'-'-i~·-~---...L-U

' • ' ',r, I \ ,~, I 'II I\ II I '\

,,,~~~ I ' I I I I 1

\'' 1 I I ' \\ \ ' ', ........ s I ' I I ' ' I I ( : \ ', . '

' ',, ............... ,....._~~ 11

• ~ ' t I .. ,,.- , \1 ' ,,, ... '~ ,,. \ / '1

100

SH 1924: PT 0 i'-./t I I

I t I I

I l I

t I

'

- SENSOR 1 - - SENSOR 2 - - • SENSOR 3

', ',"- ''<' .... - - - 1- - -:. i ..... ' I ~,- ~~ I I I I , .,,,. ,-\.\...._ l t l ' ',' "-. ' ' ... r' , ... '1, , ...., '"'f II \ I I I ·r ... / I . \ \ I / - ~I ~ . ,, ',,, .... - . ...--... ,, : . r ,~ , ...... ,, , 'U L ,,, , ,

,, -~ ,'-,...J~/ '""I I ·~ ' I I "l ' d ,\ ' i .!. ', ', ' 11 I)' i, ~ '1-, ~I'' I I ,' ~~I I 1' ,/ \ '~ f !

\ \ ..... I ...... '1 \ \ ' ~J t ' . '\' I I '<',, i / I i '1~\1 i \I \\I'~,' '.\l '1- ~ ,' \~\i,,

LLJ

~ -100~ '' '' I I I I 't\ ' I " ~ ~ ! \ ~ \.J , \~\i,~ i ' ',.. / \ i \f \ 1' 1 \\ l\J' \ \ l\ I r

'J ,_., ' I f \ I '" ~ \I - ' I ~ J

-200-r----------,-,----------r,----------T,-----------,-----------,----------,--1-20 40 60 80 1 00 , 20 0

FREQUENCY(HZ.)

Figure 28. Phase Angle versus Frequency, Lowest Load, 0-125 Hz, N = 2048 (higher resolution), (SH 19 and 24).

PEAK LOAD 364 kPa

N=512 200-------------1 SH 1924:PT 0

ft l ~ - SENSOR 1 j \ l1 I'\. 1/\

I l 9. f ,'1 \ \ ' - - SENSOR 2 I \ /,~,~-------- \ \ 1/ \\

- - .. SENSOR 3 i ' 1:; \ I ~\ ti' '~\~\ - - SENSOR 4 1 ' ~' ' \_\ J ' ' - .. SENSOR 5 I \ I:! \ I \\ f \

I \ \ ~~ I ,', ~ ---· SENSOR 6 ! , c'! , I \'\ "j------ SENSOR 7 ! \ tf! \\ J \/ I \\~

I \ f I \ , ,,

100

I ' I q I \ I \ t,. 0 ---, ~, ----·------!,~---,\-- ; ' J J ' t

'\,:,''"':..":_~~~ ' \ ,,,, 'G 7'- \ i1 ;,,

-100

0

' ~"' ~ I I,~ ' ·1 1 • ~ ,,~ ' ', ... , ~ ' I t . \ \ ' '&' ....._ '-...; ~ I \ It \ I \ ' ',, .. , .. -_ ........ t i·.,..-~, if ~ ..... 'l'

' ' .... ' ... .. - - - - - ~ - .:L. / _ - ':\'ll ........._ .... '~\\

10

' '' ' ' " I 1 f,1 I - - - -'\i:.._ ""-- _/ .,,,.. '- - r\ ', ',' "-, -r-----"'-41 W -::::-----0,,, /\\', j'" -\l ' '" ..... ' \ ,.. "\' - , \ ...... \\

', ',, ' 1--------T-V.! I '.. ,, ,, I\' , I '" ' ' ' "" I \ fl\ ,, ' I \l Ii , ... \

' ', "" /' ., ' ,, I \ ' , , \ ' If , , ... ... \ ', ,, ' / \/ !\ I \ ' , I \' 'h,

20

, r,, ~ ,'' / , , , , , I • " ,, ' , \ \ I \ I \\ I

30

',, ,/ I ; \ I ' I \~, ' , '· \I .. I \1 " ~ ' \,

40 50 60 70 80

FREQUENCY(HZ.)

90 100 110

Figure 29. Phase Angle versus Frequency, Lowest Load, 0-115 Hz, N = 512 (lower resolution), (SH 19 and 24).

discontinuities, but interpretation is still difficult. The poor resolution in Figure 29 with sample size of S12, makes interpretation impossible. The phase data in Figure 30 shows the effect of applying the linear correction to the pulse to eliminate the tail discontinuity, as indicated in Figure 14b. Figure 30 shows a single jump for each of the last three sensors.

The jumps in the last three sensors were eliminated by subtracting 360° from the phase angles immediately after the jump. The results are seen in Figure 31. The phase angle data is seen to behave regularly, with the phase angle for each sensor increasing with frequency and the phase angles increasing with distance. The phase angles are nearly linear with frequency for the first 20-30 degrees. Some irregularity is present in the phase data for the higher frequencies and the outer sensors. This is probably due to noise in the sensor or force signal. A comparison of the uncorrected phase data in Figures 26 and 28 to the corrected data in Figure 31 shows the dramatic improvement made possible by the correction.

The phase angle data for the highest load (SH 19 and 24) is shown in Figure 32. Except for the first three sensors, the phase angle data has many discontinuities due to branch jumps and is therefore difficult to interpret. The oscillations are not as pronounced as they were for the lowest load, again because the tail discontinuity was moderate due to the high load level (Table 4).

Frequency response data on a thin pavement (FM 79) is given in Appendix A (Figures AS to A9), and thick pavement data (SH 82) is given in Appendix B (Figures BS to 89). No frequency resolution studies or pulse corrections were made on the data in Appendices A and B. The data is presented for comparison purposes to illustrate the changes in pavement thickness. The most interesting comparison is the correlation of the pulse tail ratio (Table 3) to the oscillations. The frequency response functions for SH 82 behaved similarly to SH 19 and 24, whereby the oscillations are worse for the lowest load. The thin pavement {FM 79) behaved differently; the moderate "tails" in the lowest load pulses resulted in moderate oscillations in the frequency response functions in Figures AS and A6. The highest load data in Figure A7 showed severe oscillations by sensor 2 which had the worst tail. Sensors 3 through 7 had very little oscillation because the "tails" were small {Table 4). The phase angle data for the FM 79 highest load (Figure A9) had the least number of discontinuities, or jumps from -180° to +180°, of any of the uncorrected data in the frequency range of O to 12S Hz because the oscillations were small for the outer sensors. All of the other phase data for SH 82 and FM 79 is very difficult to interpret because

49

01 0

200

100

0

-100

SH1924:PT 0

- SENSOR 1

- - SENSOR 2

- - - SENSOR 3

-- SENSOR 4

- ·SENSOR 5

---- SENSOR 6

--- SENSOR 7

PEAK LOAD 364 kPa

DATA ADJUSTED FROM 50

.. , :', ,----------- ........... __ I" ........._ I .......... I ... ___ ... ....... '

' ... I ....... I -- ---'

't-- ,,,,, .... ---... ', I ' - ... ..., : ............... ............ I I I

I I

I I

I I

' I

I I

I '....._ I ... _ '1------------I --: ... , I ,,,,.,.. ... - .......... I ... , I ... , I ... , I I ' I ... I : ~-~

: I I I

~... I ~ ...... ~~''';:-.;;_ I I I ,, ...... ... - ... - -:..~---+--"""----, ... _....... ................ ... ... - - -1.

,,,........... ............ ... ... .... - - -+f - - I - -

',' ........ .......... "" "" - I - - - - - - - - - - - - - - - - - - -', -...., ... ...__ T - - - I ... , ', ............. ... ........... I T - - - - - ...

' ' ............. - I ..

'

', ,....__ I - - - - - ... ' ... ... ' I ........... "' - - - ,... .... ' ... .............. .............. ..._ -

' ', f' I .... ......_ -- ... ""' ""' ' ' '° I ------- --- ... __ ,_ - -' ', I ............. ---', ',J ' ..........__

' ,....... ~ ,,,,.,, -- I -----... , I ........... I ' / ' --- ... '...J .... ..._ - - - -

-200-+--------------------------------------------------------------~__.J

0 20 40 60 80 100 120

FREQUENCY(HZ.)

Figure 30. Phase Angle versus Frequency, Lowest Load, 0-125 Hz Corrected Pulse with No Discontinuity (SH 19 and 24).

U'1 --'

~

(/) w w c::: (.!) w 0 ......_,, w --' (.!)

z 4:

w (/1

4: I a...

PEAK LOAD 364 kPa

DATA ADJUSTED FROM 50 0--.c"~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~----t

-100

~ ... ~ ' .._.....;::

~~"-..':::'-~ .... "'-"' .. : ~-~---~~~~~-----' ' ' '"-- .... - -- --. - - - - ---, , .............. -...... - - -

' ', ..... ...........__ ... - - - - - - - - - - - - -' ' .......... -......_ - - - - - - - - - - -' ' ........ -....... ' ', ........ --....... ' ...... ............ ---'.... .......... ............ ', ... , .......... ' '... ............ ... _____________ __

.... , ......... .......... ............... __ .... , ............ ............... / - ""' ---.. I

',~ ........ ""' - - .,,,/ ' -- .................. ''...... ... ......... ~ ... - -- '

~ ' ----- " ................. .... ___ .. .............. --

... , ---- ',, - -'... ' ' ·',

' ...

- - - - -- - - -- - - -

-200

SH 1924:PT 0 ... , ... - ___ ,

..... ..... _______ _ -- SENSOR 1

- - SENSOR 2 -300 - - • SENSOR 3

-- SENSOR 4

'""' _..,...,,,. .......

' r ' ... ', I '

' I ', ' I ' , __ _; '

' - • SENSOR 5 ---· SENSOR 6

--- SENSOR 7

0 20 40 60 80 100 120

FREQUENCY(HZ.)

F1gure 31. Phase Angle versus Frequency, Lowest Load, 0-125 Hz Using Corrected Pulse with Branch Jumps Eliminated (SH 19 and 24).

(J1 N

~ (/) w LtJ a:: (!) w 0 '-" w _J (!) z <(

La.I (/) -ct: :::c a..

PEAK LOAD 11 63 kPa

200-r---~--~------~--~~---------------·--

I l

tJ1

-200 I I ( I I I ' I l I I

•j

o 'i".J <:>o 1 ~ 'co 'ci'=> '.;§::, (1 <:> 'ioo ri,.'i <:> ri.. ~o 'J..1 <:>

FREQUENCY(HZ.)

SH 1924:

- SENSOR 1

- - SENSOR 2 -· - .. SENSOR 3

-- SENSOR 4 - .. SENSOR 5

---- SENSOR 6

--- SENSOR 7

Figure 32. Phase Angle versus Frequency, Highest Load, 0-300 Hz {SH 19 and 24).

PT 0

of the phase discontinuity problem. Whether this behavior is typical of thin pavements will not be known until more thin pavement data is analyzed.

53

CHAPTER V

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

Falling-Weight Deflectometer (FWD) full-pulse data from three in-service highway sections have been analyzed to obtain dynamic response information. The three sections analyzed have thin (1.5"), medium (9") and thick (12" asphaltic concrete pavements. Data for two load levels for each pavement have also been analyzed. The time series data for the force pulse and the seven seismometer surface deflection pulses at r = 1, 2, 3, 4, 5, and 6 ft. are presented. The force and deflection pulse data were Fourier transformed using a Fast Fourier Transform (FFT) algorithm and pavement frequency response functions were computed. The frequency response functions are the FFT deflections divided by the FFT force. They are preseOnted in the form of separate magnitude and phase angle plots versus frequency for each displacement. These plots have an oscillatory component superimposed on them that is not predicted by the elasto-dynamic or viscoelastic theory. The oscillation makes the phase angle data for the outer sensors extremely difficult to interpret because it produces many "jumps" or discontinuities between -180° and +180°. This will make the phase angle data reduction very expensive and time-consuming. In general, the severity of the oscillations is higher for the low load levels. The oscillation is attributed to the truncation of the pulses at 60 msec. This results in a discontinuity in the pulse because of zero-packing of the data in the FFT algorithm. The discontinuity occurs because the pulses do not decay out to zero at the end of the fixed sample time. The pulse "tails" may be due to one or more of the following factors:

I. Drift from digital integration and filtering of the raw seismometer data,

2. Permanent deformation of the pavement, or 3. Premature truncation before the transient response has decayed out.

The relationship between the "tail" and the load level is shown by taking ratios of the "tails" to the peaking of the respective pulses. The results are shown in tabular form. A definite correlation between the "tail ratio" and the severity of oscillatory behavior of the frequency response functions is evident

54

by visually comparing the frequency plots to the "tail" ratios for both load levels.

To verify if the discontinuity in the displacement pulses was causing the oscillations, the pulses were adjusted by introducing a linear (in time) correction such that the discontinuity at 60 msec was eliminated. For the SH 19 and 24 lowest load data, the frequency response functions were recalculated using the corrected pulses. The result was that the superimposed oscillations were virtually eliminated for both magnitude and phase responses, giving the data a much smoother, regular variation with frequency. This greatly simplified interpretation of the phase angle data which without the correction has many discontinuities (from -180° to +180°) due to branch interchanges of the arctangent function. Making this correction to the data will enable investigators to make more accurate comparisons of computed responses to measured FWD responses. The improvement will make it economical and practical to interpret phase angle data for the full frequency range of 0-125 Hz.

Reconnendations

I. For use in back-calculation studies to be performed later in the project, FWD full-pulse data should be presented in the form of frequency-response functions as defined in this report. The results should be presented in magnitude and phase-angle format. This format for presenting the data is similar to the format used in the computer programs developed for this project. The phase angle data (as shown, for example, in Figure 31) is particularly sensitive to damping effects, which are directly related to rutting and cracking properties of pavements.

2. As an interim measure to eliminate oscillations in frequency response functions, the FWD full-pulse data should be corrected to eliminate the "tail" discontinuity by using a linear correction in time as described in this report.

3. An investigation of the cause or causes of the pulse "tail" should be conducted and means of improving measurements for dynamic data analysis should be developed to reduce or eliminate the "tail." This investigation requires: a. Dynatest FWD software and/or hardware modification so that a 100

msec or 120 msec pulse sampling period can be used as an option.

55

b. Time-domain synthesis of responses should be computed from a linear layered viscoelastic media computer program using actual force pulse data as input (for comparison with extended time pulses).

c. The manufacturer of the Oynatest units should investigate possible causes of drift and recommend ways to reduce or correct for drift.

d. The possibility of pavement permanent deformation as a contributing cause of the pulse "tail" should be investigated. This will require analysis of responses using nonlinear stress-strain relationships locally, using finite element analysis.

56

REFERENCES

Arnold, R.N., G.N. Bycroft, and G.B. Warburton. "Forced Vibrations of a Body on an Infinite Elastic Solid." Journal of Applied Mechanics, Transportation ASME Vol. 77, 1955, pp. 391-401.

Bendat, J.S. and A.G. Piersol. Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York, 1971.

Bendat, J.S. and A.G. Piersol. Engineering Applications of Correlation and Spectral Analysis. Wiley-Interscience, New York, 1980.

Davis, P.J. and P. Rabinowitz. Methods of Numerical Integration. Academic Press, New York, 1975.

Lee, Y.W. Statistical Theorv of Communication. John Wiley and Sons, Inc., New York, 1960.

Lytton, R.L., F.L. Roberts, and S.M. Stoffels. "Determination of Asphaltic Concrete Pavement Structural Properties by Nondestructive Testing." Final Report, Phase I, NCHRP Project 10-27, July 1986.

Lytton, R.L. "Backcalculation of Pavement Layer Properties." Presented at the First Symposium on Nondestructive Testing of Pavements and Backcalculation of Moduli. ASTM, Baltimore, Maryland, 1988.

Magnuson, A.H. "Computer Analysis of Falling-Weight Deflectometer Data, Part I: Vertical Displacement Computations on the Surface of a Uniform (One-Layer) Halfspace Due to an Oscillating Surface Pressure Distribution." Texas Transportation Institute Research Report 1215-1, October 1988.

Reissner, E. "Stationare, Axialsymmetrische Durch Eine Schuttelnde Masseerregte Schwingugen Eines Homogenen Elastischen Halbraumes." lngenieur-Archives, Vol. 8, No. 6, 1936, pp. 381-396.

Sung, T.Y. "Vibration in Semi-Infinite Solids Due to Periodic Surface Loadings." Symposium on Dynamic Testing of Soils, ASTM, STP 156, 1953, pp. 35-68.

Uzan, J., R.L. Lytton, and F.P. Germann. "General Procedure for Backcalculating Layer Moduli." First Symposium on NOT of Pavements and Backcalculation of Moduli, ASTM, 1988.

57

APPENDIX A

DATA FOR FM ROAD 79

Figure:

Al Force pulse, lowest load A2 Force pulse, highest load A3 Deflection pulses, lowest load A4 Deflection pulses, highest load AS Magnitude of frequency response function, 0-300 Hz, lowest load A6 Magnitude of frequency response function, 0-250 Hz, lowest load A7 Magnitude of frequency response function, 0-300 Hz, highest load AS Phase angle of response, 0-300 Hz, lowest load A9 Phase angle of response, 0-300 Hz, highest load

58

SH 79:PT 0

- LOAD 300

ttj a... ~

200

c.n Cl \.0 <{

0 _J

100

0 10

Figure A-1. Force pulse, lowest load.

PEAK LOAD 375 kPa

20 30

TIME(MILLI SEC.)

40 50 60

1200

1000

tO a.. 800 ~

Cl <{ 600 0 _J

400

200

PEAK LOAD 1 222 kPa

SH 79:PT 0

- LOAD

0-r---~~~~~~~~~~~--==============================!

0 10 20

Figure A-2. Force pulse, highest load.

30

TIME(MILLI SEC.)

40 50 60

m

(/)

c 0 Su .,... E

z 0 ~ (.) L&J ...J LL. LaJ c

PEAK LOAD 375 kPa

400-T---------~----~--~--~~------~~~~~-----------------

300

200

100

--_, ' ,. ' ,,, - ' I \

I \ I \

I \ \

I ---...... \ I , ,- ... ,, \

.,, ... I , ----- ... \ , ,,- ... '

I ,,. ....... '~ , ' .,,""' - ,--- - ~ ......... I / _.,- ---------' " .,.,. ---- ------' ,/ .,,, ___ .,, _ _,_,,,,,

SH 79: PT 0

SENSOR 1 SENSOR 2

SENSOR 3 SENSOR 4 SENSOR 5 SENSOR 6 SENSOR 7

-' .,,,. ,- ..... -- ---"" O-:-----------------....... -----...c:":==--"""-~~--~~-----~~-....::-~~~c::'"!:":~~

-100....----------------------------------~~---------~------t

0 10 20 30

TIME{MILLI SEC.)

40 50 60

Figure A-3. Deflection pulses, lowest level.

0 10 20 30 40 50 60

TIME(MILLI SEC.)

Figure A-4. Deflection pulses, highest load.

°' w

-. ..c r--............ . c: .,...

w c :::> f-z (!)

< ::E

PEAK LOAD 375 kPa

.70E-05-r-~------~---·~--~--~--~

.SOE-05

.SOE-05

I .40E-051

I i

.30E-05

.20E-05

.1 OE-05

SH79:PT 0

SENSOR 1 SENSOR 2 SENSOR 3

--- SENSOR 4 - - SENSOR 5 ----· SENSOR 6

---· SEf\!SOR 7

25 50 I

75 100 125 150 175 200 225 250 275

FREQUENCY{HZ.)

Figure A-5. Magnitude of frequency response function, 0-300 Hz, lowest load.

PEAK LOAD 375 kPa

.30E-05-t--------------~~----------------------------------------

.25E-05 .

..c

c: .,... .20E-05 w 0 ::» !:: .1 SE-05 z (!)

< ~

.1 OE-05

\ \

-- '\ - ' \ \ \ ' ,\

' ' \ I \ v \ ,, /\

SH 79:PT 0


--- SENSOR 4

- - SENSOR 5

----· SENSOR 6

---· SENSOR 7

... - ... , ..... , I \ / -\ , ' ' \ I \ .. /~ I\

, \ ' / ' / ' \,._...I" \ ~ ![\} -1 ' ' / \ .; J \ /) ' (I ... -~ \ \ / ' ' I I\ -, I

,/ '\, , .. - , /,,..,, , ~, _. , \"f ',/, ·x1.r , .50E-06· --- -..... \ \ - , ' , \ ~ ~ \ I I (V... I 11'"-}j \ I/, \

_,,,,, '\ \ I \ I I I .!' \ ... -, /I ---------- ... ~\... /"~ I ~\ , ' '' , '\ I ... ~ I \ '../.. I ~I 'f.' ~ ,,,_. _..., i --------... ~, /'"\, ;' '' I .,--, I I ,.....\... I r-~ 'r J. I / ',(' y

I ~ ..,-\ / ~- I .A_, I ~,..-~~, ~ ......_.1 _-- ':-;:- ....._ >f. _ _,, /' ' I\ ' , ~ --"- ,/ / ' , / '."\ ~ ~ .. , ' , ,_., __ ,,. .. ,' ~-«-~ ... --~ ...... -- ~~ ... - :-'L---------' .OOE+OO - I I - '-;, .... -- --- ...- --

0 25 50 75 100 125 150 175 200 225

FREQUENCY(HZ.)

Figure A-6. Magnitude of frequency response function, 0-250 Hz, lowest load.

°' U"1

. .D r--......... . c: ·.-

LU c ::::> t--z (.!) <( :?

PEAK LOAD 1 222 kPa

.30E-Q5-,-·----------------~~~---~--~ ·------------------~------

.25E-05 \

SH 79: PT 0 i

l ~\ -~~1 ,,

I \ -- SENSOR 2

\ - - • SENSOR 3 I

I \\ \ -- SE."'JSOR 4 1

.20E-05 - /\ ! - • SENSOR 5 I

\,\,, ---· SENSOR 6 1'

--- SENSOR 7 \ I

\ I I

.1 SE-05 I \ \ ,, ,..i.,\ (· I \ \ ~.J 1\\ \J

... '" /' '\ I"\ !' ! j , , \ \ ' \ \ I 1 /~ ~ \ . ""' ;'\ I ! .10E-Q5 ,' ', \/ \\

1 I I I·/\ \_/\,./'\_./"'"'\,/"\.,/\/ v V i

r /,......._, \ ' ·' V I I \ I\ / \ ii

I \ \ ¥ ,1 \ I \ \ \ I \ ;'

.SOE-06 _( / - ,, \ - ' ' f I I I ,' \ /\ I I \ /\ ,' \ I 11 I v , ... --- ... , \ .... ' I ... ' \I \ I \ I \ f \ I \ I \ I \ I \I !

.. /' ,,-.... ,,~, '" r--- ' - - ... 1 '1 ' / ' / ' / \ / \ / '1 I .,. ,,, ~'°'' ... ...._, ... ' ... -... ... \ ~.1 \ \ \ I I - - - ·1· ~ .... :::--.- ,,,,,- ~ - ... _... - - -. - - -/- - - ~ - - .., - - _\ - - -- _. ... -_. ... ~ ....... -:..__ ___ ,:-- -..'""-- ' ,. ' ,, ~-" ... ---------..::::-~------L-----~-----.:..::::=!:'::--~~ OOE+OO --

1 .... _... ....:..:- ~~~~-~c~~~~-:..:::1

• I I I l I l I ' I I 0 25 50 75 100 125 150 175 200 225 250 275

FREQUENCY(HZ.)

Figure A-7. Magnitude of frequency response function, 0-300 Hz, highest load.

PEAK LOAD 375 kPa

200

tt \

• ,\h '

l .. II

~

,, (/) iOO l I

, , \ 11

w ~: I

w \ I

0:: ·'- l I

(.!) i . ~: w f, . c ' . ~

' f

LaJ I I

0 t I

..J ~ Ol C) ,u

,. °'

1 . z ' h.- . l ,, • <( i\~ I· UJ

'' ' SH 79:PT 0 \t f (/) " :

<( \' ' - SENSOR 1

:I: -100 r a.. \ . - - SENSOR 2 - t t •

. j - - • SENSOR 3 t :i -- SENSOR 4

,1 \.

'i. 1 - - SENSO~ 5

' i ---· SENSOR 6 ' . I

--- SENSOR 7 -200 ' I

' l l I

0 '},~ ~o 1<:> ,oo ".~~ ,~o ,1~ '},oO '},ri<:> 'J,.<:>0 v" FREQUENCY(HZ.)

Figure A-8. Phase angle of response, 0-300 Hz, lowest load.

~

en LLI w ~ (!) w c '-' w ..J (.') z <(

w en <( :c a..

PEAK LOAD 1222 kPa

200-----------------------~---------------------------------

100

•,, ~'" t '\ : -' -" -- i J\ ~ ; I ~ r1 ~ , 1\J , I " 1 .... r ... 1 ~ • , ~ , \I' ....,, \ ' I l I\ ' ' ... i - ... ~I l 11 I \ . •\' I ' J t 1' • \ I I \ I \ A J ' i1 -~ 4 L\ Jh \ \ I 1t •'\\ l ' I f ~ ' 1\1~ ~ f 1 I, I , •• I\ 11 \ i....1 I ' I\ """"I I \ 1\11 ti 1\11 ti #,l, •"\l-t

: ', l '\ ' 11 I ' I\• '1~111f1:11" I\ 1'"\ 11\"1 :1,\ \ ... ',{ \ ~-, /ij 1 \ 1~\~ i:ltl, ii\~111'" ,r,1~' I ~ I ... I '....i \ I I 1' I ~r l I , I I I \ i.\ \I

I I ' . ' I I! 1'' 1 I I 111 ·~· I ~I '11 \ 1 I,. 1

' 1 I l \, I\ \ I I ' '"''l I '. I~ .,, 1: ·~1 I I I 111 I : \ '\ ,

1 11 I ,'--ht\ ,:r ~:~:11\ ii 'I I

. 1

, \' , 'I • • • \ • • ... \, r . 1 ii • U ,. , lJ, h , ,

' ' ' "'""' '\ , 1 i ' ' , ~ r1~ ~ .... f ' ' ' 1 ~1 i~1 ' ' J \ I I \ I I I I ' I I • • I ,, '"'-. ! I ) l 1 I ~

1 ' ' I ~ \ ..._ I I l ,,_ ' t1 ~ll!lll_L~ .. I I ~ H \ Q !lit... I 1-1 \ 1 '~ \ ,.t '\ . r:r, jl i '~Tr . '\

!t,,, I : I ' ' .. '. /j ~ I '\ i I : ~ \'1: "1 ,·, • ,,:,~ h I! r. 11 \,,1-~N ' 1 \ 1' I I " A 1 I 1 '

1 1 ti \ : • 11 t "'i I :it\ 'I

''\\' I I I ' \ I I I \... l ! I ·~'1 1 1' '1't1 .,, It f,~;h, 1 \\ ,~, ' l\ ' \ ~ ' , ' ' !\1 ~ ~ ,, • ' , ' I'\ ( r ' ' ·1 ,\\\' {f . I I, '" \ ,... : I I I '-i ~~11: 1: i1"' 1: it I' ''.J ''~\,' 1 ~~ / ' ' r~ \ 1,'J • t 1 • 'i '..U 1.!. 1: , • 1 . ' 1~,11 ' l \\,~,:~I \ l ,~ I\ \ ! '\ '\_);~~t ~ 11 : !I \ j !"'i{ ~rt! \ 11 r\·tf \ \ \ \ I .... \ '}. I ' \ t I \ I -..... f>:-, , , J I • I "•1 I ' I • I '

-1,QQl \\\ 11\~I 1 la\\ ~ ,~ ''(' , )\. ~'I 'l' :ue \· ,1;\ \ ' I~ " \ ' l I I I I \ ,, • ' t I I I"'' f I 1; '\''" ' , J '\1 'l ' I', I~ I I }' '( I ,, .,, I ,, II l ~ ... 11 j!\I ' 'f. ' I - - l \ l I l I I ' ,, I \ ._I 'i I " I I I I II • \~ :1 \

I ' 1\1 1' J ,' It \, '1 I\ !' '1 \ I'/ 1 \ \U' 1 if n \ \;II~ f\' \ l \ I \ \ ' I • ' \ \ I ' I I I ' \ I\ . ' ' ' • ,, ... I \I ' ~ , , I \ / ' \ I '\I \ I \ I ... I I " ,,

''i~ , _ _,.4 , , -- : ,; • ,~ ,, • ~ ': ' '1 \J \: \u v . , , , , I -200_._ ______________________ .y-______ ~--------__,----

c rz,,~ ~o 1'6 ~sJ~ , ri,'::> , ~o ,1~ 'i0o ri,'l-'6 '],,':Jo i\~

FREQUENCY(HZ.)

Figure A-9.· Phase angle of response, 0-300 Hz, highest load.

SH 79:PT 0

- SENSOR 1 - - SENSOR 2 - - • SENSOR 3

-- SENSOR 4

- ·SENSOR 5

---· SENSOR 6

--- SENSOR 7

APPENDIX B

DATA FOR STATE HIGHWAY 82

Figure:

Bl Force pulse, lowest load 82 Force pulse, highest load 83 Deflection pulses, lowest load 84 Deflection pulses, highest load BS Magnitude of frequency response function, 0-300 Hz, lowest load 86 Magnitude of frequency response function, 0-250 Hz, lowest load 87 Magnitude of frequency response function, 0-300 Hz, highest load 88 Phase angle of response, 0-300 Hz, lowest load 89 Phase angle of response, 0-125 Hz, highest load

68

°' '°

PEAK LOAD 365 kPa

SH 82:PT 0

- LOAD 300

co a..

200 ~

0 <( 0 _J

100

-100-+-~~~~~~~~--~~~~---~~~---~~~~~~~~---4

0 10 20

Figure B-1. Force pulse, lowest load.

30

TIME(MILLI SEC.)

40 50 60

-....J 0

1000

800

ltl 0... ~

600 0 <( 0 ....J

400

200

0 10

PEAK LOAD 1098 kPa

20 30

TIME(MILLI SEC.)

figure B-2. Force pulse, highest load.

SH 82:PT 0

- LOAD

40 50 60

PEAK LOAD 365 kPa

ao

SH 82: PT 0

- SENSOR 1 en 60 -SENSOR 2 c: 0 s... SENSOR 3 u .,.... E SENSOR 4

II -"' \ \ z 401 "'

.. SENSOR 5 0 I \

1"' .. SENSOR 6 i== "" .,-' .. f\~'\.. I\

.......... (.) _r .. ~ - SENSOR 7 w - \ __.

-1-v-1.. \,, _. LL

20~ r -v\ \.~

LaJ \ \ \.

0

-20-+-~--~----r--~--~--~,~~~~---,~-------~-,~-------~~,-----------t 0 10 20 30 40 50 60

TIME(MILLI SEC.)

Figure B-3. Deflection pulses, lowest load.

....... N

V> c: 0 s... u .,... E

z 0 i== (J w ....I LL. w c

0 10

PEAK LOAD 1102 kPa

20 30

TIME(MILLI SEC.)

Figure B-4. Deflection pulses, highest load.

SH

40

82: PT 0

SENSOR 1 SENSOR 2 SENSOR 3 SENSOR 4


50 60

....... w

. .a r--........... . c: .....

w 0 :::l f--z (!) -<( :::?

PEAK LOAD 365 kPa

.BOE-05 ~END I I

.70E-os

1

r - 02 n I

1 --. 03 r ;

.60E-05 - - - 04 11\ I '~\ ,.

l --- 05 :~.~! i

.50E-05 -I~ - - 06 i~~\ I ----· 07 ir 'I~ .,

·u "\ /&' ~: !

.40E-05 ---· DB :~Ai\ I I tk1\l \i

.30E-05 4 /~ "~ J f \ JI;, ,~~,

.20E-05 I i /.~ t117 I I 11\~..,11 1

_j ·11 '~-' I .rnE-os-1 ~. rdf:'\// I

-_-..:=..: ... "- --.::---. /.:::;;~;:s.;a_.,~};/'.,.c-i-;' I ~--~~-:::----~~ ~~-~-~~-------::.... ~~...;.~-- .._~ I I

OOE.J.00 ---- -· ..,,,.....:.__. - ..... i ~-,.~ ... , -• ' I r- 1 --r ·-,---, --,-----

0 25 50 75 100 125 150 175 200 225 250 275

FREQUENCY(HZ.)

Figure B-5. Magnitude of frequency response function, 0-300 Hz, lowest load.

....... ..i=::.

. ..c r--.......... . c:

•r-.........

L&J 0 ::> t:: z C> -<( ~

.14E-05T:

I SH 82:PT 0 · 12E-OS l - SENSOR 1

- - · SENSOR 2

. i OE-05 J SENSOR 3 l SENSOR 4

.BOE-06 I SENSOR 5

SENSOR 6

---· SENSOR 7

PEAK LOAD 365 kPa

I

100 125 150 175 200

FREQUENCY(HZ.)

Figure B-6. Magnitude of frequency response function, 0-250 Hz, lowest load.

225

......, 01

. .0 .--............ . t: .,....

LLI c ::> t:: z (!)

< ~

.BOE-061

.70E-0St \

.SOE-06 \\ \ \

' ' .SOE-OS , '~

PEAK LOAD 11 02 kPa

--~------

I ', \\ ' \ \

1 \ •• \

-... \ \ ""' \

SH 82: PT 0

SENSOR 1

SENSOR 2 SENSOR 3 SENSOR 4 SENSOR 5 SENSOR 6 SENSOR 7 .40E-OS 1· ' \ \

. \ ' l \ \ \ \ ,...... .30E-061- ~ , , , ,...__./'\ I \ ;

: \ \ \ \ "" \ ,1\ :

i ' ' '- ..... , . I \ , ___ .,.'\ \ \... \ - A\ , !

.20E-06 \ \ '\' , ... ~ ,. i:'\ A r-; ..... -"' ' .. ... '- ;, '\f v I i ... \:\ "\ i" '. ,/~:, , , \ _. \./ ;\ . .\ /""- /'\ /~/\ I \ I A

', - - "\' ,, \ I "' .. , ,"'- , ... , ..... ~ _. ..... \.../ 'v v \ \ I y ,1 '' 1 OE-06 ' ~~ ~~ ''/ <\\ l A\' "'' , - , .. - - '_.I - - - , v {f.'l 1.1' • - _. \\ v I \ 1, '1 '/ ' , \ -. !'X:,' ' ,- " .,. -- '- - .... / "-..£~.J;

\ I II \• .. IV 1 ... ..l ,/ '... ... ' , ' - - ' /.,. .... \A'~ \v:/ 'l \~~,' / C.,.~1 )< I,., '~i. /1";l":" 'A ... ,,~' '/'.a-.-'~-:,~l/'~./,f'I V \, \fl\/ "- 'J<,/ ~# ~~·,:~,///~..{ .,.1- I e.' \/\/\;"'- ,, "'':

OOE 00 t/ -~ ~·... "'\::¥ , J · + ._.....-----,- I I f I l -1 I I

0 25 50 75 100 125 150 175 200 225 250 275

FREQUENCY(HZ.)

Figure B-7. Magnitude of frequency response function, 0-300 Hz, highest load.

""' VJ w LL.I 0:: (!) w c .__.,,,,,, UJ ....I t:> z <(

PEAK LOAD 365 kPa

200,..-----~--------~----~--~----~---------·-----

FREQUENCY(HZ.)

Figure B-8. Phase angle of response, 0-300 Hz, lowest load.

SH 82:PT O

- SENSCR1 - - SENSOR2 - - - SENSOR:S -- SENSOR4

- ·SENSORS

---· SENSORS --- SENSOR7

~ (/) LU w Q! C) w Cl ..._,, La.J ..J (!) z < LU (/) <( ::r: Cl.

PEAK LOAD 11 02 kPa

200 ' ---· . ------------·· ---,

I ! SH 82: PT 0 •' ... -. f, ~ f

I ! I ~ ;', ~ \ t ~ '~\ ,,-- \ i - SENSO~ 1 I I •, I '1 ~\ '~\ " \ \ l - - SENSOR 2 l \ :.~ ~\ \ ~ \ 1~ j: \f' \ .

J I I :•, D \ I I \ 1il I: \ \ !

1 OO . - - • SENSOR 3 I \ ~1~ l \ ~ l \{ •~ / 1 11 ' \ \ ! ' SENSOR ~ r 5 1 ' ' 1 1 1

\ ' · I -- ., I ~ :, .~ \ ~ ,,,, f: I \ ~ \ l

l - · SENSOR 5 1• 1 ~u i~ l' \ ' 1• I : \ ' ' I

SENSOR 6 I 11 ' ' I l I I I \ ' \ I

---- • ,· 1 t; l~ n 1 •\\ , • ,~. l I SENSOR 7 I J I '' I .. \ I I I ,, I I I --- ' 1 n ~ ' l' • 't \/ ' f ,, i \ I l I ~ 11 ' t I } II\ \ I f I I . ' ~ ~ ~ t' J_ ~ : I , \ ~

0 ~ 1 I rl--4----,\!-- - ~-f-t----\r-:

I ._, ! I ~ l ' •\ I ~I : J ~! l '~- I I ; ' ~ " . I " I I ,, I '' ......... - ' R ii " I I '\ I '\ I

I '''" '..'-'~- I 1 I ! I \1 I I \ l '' ' ' ... '-:::::----- I ------- '_.,...._ F. I ,• 'l r f 't I I ,,,,' ...... ---·::..-~ - ;...--;:-.._..r , ....... , ~-....... "~ I _, I /"'' .1~11

' ' '-. ' ... ... ~ - - I - :.a -.II - ' v I I ,.- fl'lf ' ' '- ............. _ ......... -. I - -I"" - I - -, II I' '_,_,,,· 11~

l ',',, '-, -------... -; - - .... .,,--...L ... 1·- --tr... r· - ~ - ... t -~, - 1 r--'4 - - - 1 '· ', -- "). __ ,,... '"' / ' .. ... , \ ' ... ,..., , .. ,u;

- , 00 -ii ', '----... '.. 1 / -" ' ., ',, ,....._,, ' t - - ·~·' '-t... , # ·~ ' ... ' ~ / I \ • I • .. I -\ \. ... .. 11

I '.... , , ... 1 v ,, ~, \ J t\ \ l ', , r -1- \ I '1 ~ ", I ~~ ......... ~-,, f', "< \\ l ../ \\f\ " I \\.,!. __ / ~: \ ~, /- '1'

L. '-1 '.,' \ \1 \I \ I \ii ~: \ 'v'

\) ,, \ / \: . 'I I I I

-200 -: ·--,- l .J

0 20 40 60 80 , 00 120

FREQUENCY(HZ.)

Figure B-9. Phase angle of response, 0-125 Hz, highest load.

APPENDIX C

LISTING OF DATA ANALYSIS PROGRAM

FOR FFT AND FREQUENCY-RESPONSE FUNCTIONS

78

C THIS PROGRAMME IS TO COMPUTE FAST FOURIER TRANSFORMS AND C TRANSFER FUNCTIONS OF TJIE RAW FWD DATA c c~~~~~~~~~~~-.-~~~~~~~~~~~~~~~~~~~~~~~~~~~-

c * DEFLECTPHASE.FOR COMPLEX TEMPX1(1024),TEMPX2(1024),TEMPX3(1024) REAL MAX,NTAN,MAG(l024,7),PHASE(1024,7),SENSOR(0:1023,7) REAL PI,R,T,TT DIMENSION ALOAD(0:1023},

* XIM(l024),X(0:1023),A(0:1023),TEMP(0:1023},XRE(l024}, * B(0:1023),TEST(0:1023)

INTEGER ILOAD(0:1023),IENSOR(0:1023,7) OPEN(UNIT=5,FILE='S1Hl924.DAT',STATUS='OLD') OPEN(UNIT=6,FILE='LLM1924.DAT',STATUS='NEW') OPEN(UNIT=7,FILE='LLP1924.DAT',STATUS='NEW'} OPEN(UNIT=8,FILE='LLFT1924.DAT',STATUS='NEW'}

C OPEN(UNIT=9,FILE='Dl924.DAT',STATUS='NEW') C OPEN(UNIT=10,FILE='Sl924.DAT',STATUS='NEW') c-------------------------------------------------------------------------------c AR! is to convert micron to 'mu' i.e. milliseconds i.e. 1/1000 c AK2 is to convert KPa to Psi c-------------------------------------------------------------------------

c c c Clll

c

N = 1024 PI=3.1415926535891924 MAX= -1. AKl = .0001/2.54 AK2 = 109.73/6.894757 XMAX 1023. XMIN 0. YMIN 0. YMAX 1. r=O. t=O tt=O WRITE(6,*)'SH1924.DAT:point=O:LOWEST LOAD:N=l024' WRITE(7,*)'SH1924.DAT:point=O:LOWEST LOAD:N=l024' WRITE(8,*)'DATA FOR SH1924.DAT :point=O :LOWEST LOAD:N=l024' WRITE(9,*}'SH1924.DAT:point=O:LOWEST LOAD:N=1024' WRITE(10,*)'SH1924.DAT:point=O:LOWEST LOAD:N=1024' WRITE(6,lll) FORMAT(2X,'FREQUENCY',10X,'FFTLOAD',10X,'FFTDEFLECTION') DO 220 I=0,299,2

READ(5,*)ILOAD(I},IENSOR(I,1),IENSOR(I,2),IENSOR(I,3), * IENSOR(I,4),IENSOR(I,5),IENSOR(I,6),IENSOR(I,7),ILOAD(I+l}, * IENSOR(I+l,l),IENSOR(I+l,2},IENSOR(I+l,3),IENSOR(I+l,4), * IENSOR(I+l,5),IENSOR(I+l,6),IENSOR(I+l,7)

220 CONTINUE c------------------------------------------------------------------------c THE FOLLOWING STEP IS TO WRITE ALL THE DATA IN A SYSTEMATIC FASHION C IN A FILE c--------------------------------------------------------------------------c DO 221 I=0,299 c write(10,113)ILOAD(I),IENSOR(I,1),IENSOR(I,2),IENSOR(I,3), C * IENSOR(I,4),IENSOR(I,5),IENSOR(I,6),IENSOR(I,7) Cll3 FORMAT(5X,8(I6,3X)) C221 CONTINUE C ALOAD(299)=FLOAT(ILOAD(19,)) C DO 114 J=l,7 C114 SENSOR(299,J)=FLOAT(IENSOR(299,J)) C WRITE(*,*)(ALOAD(299),SENSOR(299,J),J=l,7)

c--------------------------------------------------------------------c c~~~~A-D-J~U~S~T~I~N~G=--T~A~I~L~o=r-=T~H~E,_...,,,P~U~L~S~E=--~~~~~~~~~~~~~~~~~~~~-

c~~~-.,,,D~0~2~2~3.---::I~=~0..--, 2~9,.....,....9~~~~~~~~~~~~~~~~~~~~~~~~~~~~

79

DO 222 K•l,7 C T•(I-50) C TT•(299-50) C R•T/TT

ALOAD(I)•FLOAT(ILOAD(I)) SENSOR(I,K)•FLOAT(IENSOR(I,K))

C ALOAO(I)•(ALOAO(I)-(ALOAD(299)•R)) C SENSOR(I,k)•(SENSOR(I,k}-(SENSOR(299,k}*R)) 222 CONTINUE c write(9,lll)ALOAD(I},SENSOR(I,1),SENSOR(I,2),SENSOR(I,3), C * SENSOR(I,4),SENSOR(I,5},SENSOR(I,6),SENSOR(I,7) Clll FORMAT(SX,8(F6.0,3X)) 223 CONTINUE

c--------------------------------------~----------------------------------------c ZERO PACKING:

c -------------c the following loop is to put zeroes for load & sensor values after 300 c i.e. after sixty milliseconds value,since we have data only until c sixty m.sec and we have defined n value to be larger than 300 c ------~o~a~2~aMor-=1:----t!3~0~0~,~N~-~1------------------------------------------------~

DO 230 J•l,7 ALOAO(I) • 0. SENSOR(I,J)• 0.

230 CONTINUE 240 CONTINUE

C ____ ....,.D~0_...5~0~0..__,,I~•-O~,~N~--1--------------------------------------------------------T EM P (I)• ALO AD (I)* AK 2

TEMPX2(I+l)•CMPLX(TEMP(I),0.) 500 CONTINUE

c CALL FFT(TEMPX2,10)

DO 110 J•l, 7 DO 120 I•O,N-1

TEMP(I)•SENSOR(I,J)*AKl TEMPXl(I+l)•CMPLX(TEMP(I),0.}

120 CONTINUE CALL FFT(TEMPXl,10)

c-----------------------------------------------------------------------c THE NEXT STEP IS TO WRITE TRANSFER FUNCTIONS OF LOAD ANO C DISPLACEMENT FOR RAW DATA

c--------~--..---=--.-......-,or-.--------------------------------------------------~ DO 160 I•0,1023

160

c

c

X(I+l)•I/.0002/1023 CONTINUE

do i•l,1024 write(8,•)x(i),tempx2(i),tempxl(i) end do

DO 140 I•0,1023 IF (tempxl(i}.EQ.0) THEN IF (tempx2(i).EQ. 0} THEN GO TO 140 end if end if TEMPX3(l)•TEMPXl(I)/TEMPX2(I)

c~----------~~---"":"'f-=:-:-:":"i'""::'::~~..,-;--r-r----~~--~--~----~--------~ MAG(I,J)•CABS(TEMPXJ(I))

c

Dl•AIMAG(TEMPX3(I)) D2•REAL(TEMPX3(I)) MAY BE THIS:-- WRITE(7,*)0l,D2 IF (Dl.EQ.0) THEN IF (02 .EQ. 0) THEN GO TO 140 ENO IF END IF

80

PHASE(I,J)•ATAN2(Dl,D2) PHASE(I,J)•(l80.0*PHASE(I,J))/PI

140 CONTINUE 110 CONTINUE c C~~- THE MAX VALUE 'N' FOR I DEPENDS ON C (5000/1024)•(100/N) C E.G.FOR A FR OF 100, N•(l00*1024/5000)

c------~--.l("'lll""-=---__,~r------------------------------------------------------~ DO 122 I•l,1024 X(I+l)•I/.0002/1023

122 CONTINUE DO I•l,62

C FOR MAGNITUDES WRITE(6,900)X(I),(MAG(I,J),J•l,7)

900 FORMAT(F10.4,2X,7(El0.4,1X)) C NOW FOR THE PHASE ANGLES

WRITE(7,920) X(I),( PHASE(I,J),J•l,7) 920 FORMAT(Fl0.4,2X, 7(F8.2, lX))

END DO STOP

ENO

c ______ -:-:-=-=~=-=~:o-:=-=-=:"7'":':~~----------------------------------------s u e Ro u TINE FFT(X,M) COMPLEX X(l024),U,W,T N•2**M PI•3.l415926535891924 DO 20 L•l,M LE•2**(M+l-L) LEl•LE/2 U•(l.0,0.0) W•CMPLX(COS(PI/FLOAT(LEl)),-SIN(PI/FLOAT(LEl))) 00 20 J•l,LEl DO 10 I•J,N,LE IP•I+LEl T•X(I)+X(IP) X( IP )•(X( I )-X( IP)) *U

10 X(I)•T 20 U•U*W

NV2•N/2 NMl•N-1 J=l DO 30 I•l,NMl IF(I.GE.J) GO TO 25 T•X(J) X(J)•X(I) X(I)•T

2 5 K•.NV2 26 IF(K.GE.J) GO TO 30

J•J-K I<•K/2. GO TO 26

30 J•J+K RETURN END

81

Dynamic Analysis of Falling-Weight Deflectometer Data · DYNAMIC ANALYSIS OF FALLING-WEIGHT...

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