TECHNICAL REPORT S-76-3

AN ITERATIVE LAYERED ELASTIC COMPUTER PROGRAM FOR RATIONAL PAVEMENT DESIGN

by

Yu T. Chou

Soils and Pavements Laboratory U. S. Army Engineer Waterways Experiment Station P. O. Box 631, Vicksburg, Miss. 3 9 180

February 1976 Final Report

Approved For Public Release; Distribution Unlimited

TA Prepared for Office, Chief of Engineers, U. S. Army, and 7 Federal Aviation Administration, Washington, D. C. .W34t S-76-3 Under Inter-Agency Agreement DOT FA73WAI-377 1976 '•& ..

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4. T IT L E (and Subtitle) 9- 5. TYPE OF REPORT & PERIOD COVERED M ^ i t e r a t i v e l a y e r e d e l a s t i c c o m p u t e r p r o g r a m f o r Final Report RATIONAL PAVEMENT DESI® '-- j 6. PERFORMING ORG. REPORT NUMBER

Jr A U T H O R ^ 8. CONTRACT OR GRANT NUMBERfs) *Yu T. Chou DOT FA73WAI-377 Ò

9. PERFO^IING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS U. S. Army Engineer Waterways Experiment Station Soils and Pavements Laboratory P. 0. Box 631, Vicksburg, Miss. 39180

11. CONTROLLING OFFICE NAME AND ADDRESS Tj T r e p o r t D A T E Office, Chief of Engineers, U. S. Army, and ^“February 1976^ Federal Aviation Administration 13. NUMBER OF PAGES Washington, D. C. 67 14. MONITORING AGENCY NAME & ADDRESSfif different from Controlling Office) 15. SECURITY CLASS, (of thia report)

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18. SUPPLEMENTARY NOTES Designated Report No. FAA-RD-75-226 by Federal Aviation Administration, Washington, D. C.

19. KEY WORDS (Continue on reverae aide if necessary and identify by block number) Computer programs Finite element method Layered systems Pavement design

2Qv ABSTRACT (Continue am reverae a£c&a ft neceetamry and identify by block number) This study was conducted to develop a simple and easily operated computer program for the rational design of pavements. The program must yield results approximating those computed by the more sophisticated nonlinear finite element program. A linearly layered elastic computer program originally developed by the Chevron Oil Company was expanded and modified to include an iterative procedure for rational pavement design. Pavement response to multiple loads can be predicted using the nonlinear moduli of pavement materials. The new (Continued)

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20. ABSTRACT (Continued). program is designated CHEVIT. The program uses the iterative procedures to determine the nonlinear moduli of pavement materials and then computes the stresses and strains in the pavement under a single wheel load by the Burmister linear layered elastic computational method. The principle of superposition is used to account for multiple wheel loads. For completeness, the CHEVIT program also accounts for linear problems and single loads. Results computed by the iterative layered elastic program were compared with those computed by the nonlinear finite element program. Modifications and improvements to the iterative layered program were made accordingly. Comparison of computed and measured stresses and displacements are presented in Appendix A. Flow charts and an input guide are presented in Appendix B. Example problems and illustra­ tion of the use of the input guide are also presented.

Unclassified SECURITY CLASSIFICATION OF THIS PAGEflWien Data Entered) PREFACE

The study described herein was jointly sponsored by the Federal Aviation Administration as a part of Inter-Agency Agreement FA73WAI-377, "New Pavement Design Methodology," and by the Office, Chief of En­ gineers, U. S. Army, as a part of the Military Construction RDTE Proj­ ect UA762719AT01+, "Pavements, Soils, and Foundations," and Military Engineering RDTE Project ^Al6ll02B52E, "Research in Military Engineering and Construction." The study was conducted by the U. S. Army Engineer Waterways Experiment Station (WES), Soils and Pavements Laboratory. Dr. Yu T. Chou, under the general supervision of Messrs. James P. Sale, Richard G. Ahlvin, R. L. Hutchinson, and H. H. Ulery, Jr., was in charge of the study and is the author of this report. The iterative procedure de­ scribed herein was developed by Dr. Walter R. Barker, who also assisted in the preparation of the computer program. COL G. H. Hilt, CE, was Director of the WES during the conduct of this study and the preparation of this report. Mr. F. R. Brown was Technical Director.

1/2

CONTENTS Page PREFACE ...... 1 METRIC CONVERSION FACTORS ...... 5 INTRODUCTION ...... 7 BACKGROUND ...... 7 PURPOSE ...... 9 S C O P E ...... 9 ITERATIVE LAYERED ELASTIC COMPUTER PROGRAM (CHEVIT) ...... 10 THEORETICAL BACKGROUND ...... 10 ITERATIVE TECHNIQUE ...... 13 MULTIPLE WHEELS ...... 16 NONLINEAR FINITE ELEMENT COMPUTER PROGRAM ...... 21 COMPARISONS OF COMPUTED RESULTS, ITERATIVE LAYERED ELASTIC AND NONLINEAR FINITE ELEMENT PROGRAMS ...... 22 DISCUSSION OF THE ITERATIVE LAYERED ELASTIC PROGRAM ...... 27 CONCLUSIONS AND RECOMMENDATION ...... 28 APPENDIX A: COMPARISONS OF COMPUTED AND MEASURED STRESSES AND DISPLACEMENTS...... 29 APPENDIX B: CHEVIT PROGRAM...... 39 EXAMPLE...... 39 FLOW CHARTS AND INPUT GUIDES ...... 39 APPENDIX C: NOTATION ...... 67 R E F E R E N C E S ...... 68

TABLES

1 Comparison of Computed Results, Nonlinear Finite Element and Iterative Layered Elastic (CHEVIT) Programs ...... 23

FIGURES

1 Nonlinear layered elastic pavement system ...... 11 2 Bilinear representation of the relationship between resilient modulus and deviator stress for fine-grained soil ...... 13 3 Coordinate system of one main gear of C - 5 A ...... 18 L Comparisons of computations by CHEVIT program and nonlinear finite element program ...... 25

3 FIGURES (CONTINUED) Page A1 Variations of modulus with different stress-strain relations . 30 A2 Comparison of the computed and measured deflections along the load axis of a 30-kip single-wheel load ...... 32 A3 Comparison of measured and computed vertical stresses along the load axis of a 30-kip single-wheel l o a d ...... 3^ Ab Comparisons of measured and computed deflections along the load axis of two single-wheel loads ...... 35 A5 Comparisons of measured and computed vertical stresses along the load axis of two single-wheel l o a d s ...... 36 A6 Comparisons of measured and computed deflection basins along the offset axes of a 30-kip single-wheel l o a d ...... 37 B1 Pavements illustrating the use of CHEVIT program input guide . Ho B2 Grid pattern for computational points for a Boeing 7^-7 twin-tandem gear assembly...... kl B3 Input guide for CHEVIT p r o g r a m ...... b2 B^ Input guide for CHEVIT, linear analysis...... 1+3 B5 Input guide for CHEVIT, nonlinear analysis ...... » ...... 51 B6 FORTRAN listing for CHEVIT, linear analysis ...... 62 B7 FORTRAN listing for CHEVIT, nonlinear analysis ...... 6b

U METRIC CONVERSION FACTORS

n Approximate Conversions to Metric Measures 0200004801013002010202020202020011009048000202020000024853480002000132020100000000010002020200020002020001020100000001000902001004M Approximate Conversions to Metric Measures «4 M Symbol When You Know Multiply by To Find Symbol Symbol When You Know Multiply by To Find Symbol 5 LENGTH a ______LER6TH mm millimeters 0.04 inches in e> cm centimeters 0.4 tndMs in m meters 3 3 «set ft in inchas *2.5 centimeters cm mmZ m m meters 1.1 yaids fé ft feet 30 centimeters cm •4 —Z km kilometers 0.6 m iles m yd yards 0.9 meters m —: mi mi las kilometers km t* —£ AREA AREA —I 40 #*• cm2 square centimeters 0.16 square inchee in2 -E in2 square inchas 6.5 square centimeters cm2 m2 square meters 1.2 square yards yd2 ft2 square feet 0.09 square meters m2 —£ km2 square kilometers 0.4 square mi lee mi2 yd2 square yards 0.8 square meters ni2 - ♦ ha hectares (10.000 m2) 2.5 acres mi2 square miles 2.6 square kilometers km2 — r hectares acres 0.4 ha —E « ON 9» — MASS (wtiflbt) MASS (wtiifct) — = M 0 grams 0.035 r ------T s c oz ounces 28 grams 8 - £ kg kilograms 22. ptwdt ft lb pounds 0.45 kilograms kg _ t tonnes (1000 kg) 1.1 short toes short tons 0.9 tonnes t (2000 lb) * - £ e VOLUME - r VOLUME e» fluid ouaces tsp teaspoons 5 milliliters ml ml m illiliters 0.03 f i ec « Tbsp tablespoons 15 milliliters ml 1 lite rs 2.1 pints m fl oz fluid ounces 30 milliliters ml w —E 1 liters 1.08 quarts 9» c caps 0.24 liters 1 _ 1 liters 0.28 gallons m i pt pints 0.47 liters 1 m3 cubic meters 35 cubic feet ft3 quarts 0.95 liters 1 —r m3 cubic meters 1.3 cubic yards vrf2 qt «a gal gallons 3.8 liters 1 *™ Z r ft3 cubic feet 0.03 cubic meters m3 M •n yd3 cubic yards 0.76 cubic meters m3 —E TEMPERATURE (•net1 — z TEMPERATURE («iteti ° c C e lsiu s 9/5 (then Fahrenheit •f — r temperature add 32) temperature ’ f Fahrenheit 5/9 (after Celsius °c temperature subtracting temperature — •F C4 °F 32 98.« a t -SO 0 I 40 80 I 180 l« 0 200 I t i i i It i i i 1 • o o * 1 in. = 2.54 (exactly). For other exact conversions and more de­ 1 1 i* 1 I 1 * i 1 7 4 ■*T— I p Y «e - 4 0 -2 0 0 20 |40 .“ tailed tables, see NBC Misc. Publ. 286, Units of Weights and Mea­ % i •c 37 sures, Price $2.25, SD Catalog No. C13.10:286.

INTRODUCTION

BACKGROUND

A rational method for the structural design of pavements requires knowledge of the stresses and strains within the layers comprising the pavement structure. It is generally accepted that the fatigue cracking of asphalt pavements is caused by repeated applications of excessive tensile strains at the bottom of the asphalt-bound layer, and that the rutting of pavement surfaces is due mostly to excessive compressive stresses or strains in the subgrade soils. The adequacy of the design may be evaluated by comparing the computed stresses and strains at critical points in a pavement system with the allowable values obtained from laboratory tests on the component materials. The use of elastic theory for determining stresses and displace­ ments in a homogeneous linear elastic medium was first developed by Timoshenko and Goodier'*' and later extended to layered systems by 2 3 Burmister and Mehta and Veletos for multiple layers. Since the advent of high-speed electronic computers, the finite element technique has gained wide recognition in pavement structure analysis. Both linear and nonlinear stress-strain relations of pavement materials have been used. Boussinesq's homogeneous and Burmister's layered elastic solu­ tions have been widely used in pavement analysis because of their simplicity. The basic assumptions behind the theory relating to the nature of pavement problems, however, are always subject to questions and criticisms. For instance, Sowers and Vesic^ and Vesic^’^ questioned the validity of applying Burmister's layered theories to flexible pave­ ments. Their experimental results showed that normal stresses tended to exceed those computed by the layered theory. The recorded results usually were in the range of, or higher than, the homogeneous analysis of Boussinesq. Test data obtained at the U. S. Army Engineer Waterways y Experiment Station (WES) support Sowers' and Vesic's conclusion. Burmister's layered theories assume that the materials in each 6 layer are homogeneous, isotropic, and linearly elastic. Vesic con­ cluded that anisotropy of the paving materials and deformation moduli

7 variation in compression and tension were the main factors contributing to the observed differences between the measured and computed values. Linearity implies the applicability of the superposition principle. The material constants must be invariant relative to the state of the stresses. Since the stress-strain relation is linear, a doubling of the load doubles the deformation. Even though the superposition principle provides the backbone for the mathematical theory of linear elasticity, it is this principle itself that so drastically limits the application of the theory to the field of pavement design. Generally, most paving materials do not exhibit linear stress-strain behavior and their elastic moduli vary with the state of the stresses. The finite element method incorporated with nonlinear material characterization is generally considered the most promising method for solving pavement problems. This method has the potential to overcome most of the inherent difficulties of Burmister's linear layered elastic theory. A study was conducted to compare the stresses and displacements computed by the nonlinear finite element program with the measurements from one of the full-scale test pavements constructed at the WES. Pre­ dicted and actual measurements were very similar, and the results are presented in Appendix A. More information on the comparisons can be found in Reference 8. Although the nonlinear finite element program is considered to be the best current program for pavement design and analysis, the use of the program requires a large computer and long computer time. Under­ standing the input form also requires basic knowledge of the finite element method. Apparently the nonlinear finite element computer pro­ gram can be best used for pavement research and analysis. For routine pavement design work, however, engineers need a workable program -which is as simple in use as the linear layered elastic program but which yields results as good as those from the available nonlinear finite g element program.* A program developed by the Chevron Oil Company, which is based on Burmister's linear layered elastic solution, was used

* The available nonlinear finite element program is explained on page 19.

8 to develop the initial design procedure for flexible airport pavements in a study jointly sponsored by Office, Chief of Engineers (OCE), and Federal Aviation Administration (FAA).^ The iterative layered elastic program developed in this study and described herein will be used in future revision of the procedure by FAA.

PURPOSE

The purpose of the study reported below was to develop a workable computer program for the rational design of pavements. The program should incorporate the stress-dependent moduli of pavement materials, and be easy and economical to operate.

SCOPE o The existing Chevron program, which was developed from the Burmister linear layered elastic solution, was expanded and modified to incorporate the stress-dependent moduli of pavement materials through the use of iterative procedures. For completeness, the program also ac­ counts for linear problems. Comparisons were made between the computed results of the nonlinear finite element program and the iterative lay­ ered elastic program (the program developed in this study). Locations of offset distances were determined from these comparisons for averaging the stress intensities to determine the stress-dependent moduli, and the accuracy of the iterative layered elastic program was evaluated and determined. A superposition subroutine was added to the program to ac­ commodate multiple wheel loads. The developed program is called CHEVIT. The limitations of the CHEVIT program are discussed below. The flow chart and input guidance are included in Appendix B with example prob­ lems and illustrations of the use of the input guide.

9 ITERATIVE LAYERED ELASTIC COMPUTER PROGRAM (CHEVIT)

THEORETICAL BACKGROUND

As stated previously, the iterative layered elastic program (CHEVIT) is a modified version of the Chevron program,^ with the inclu- sion of iterative procedures to determine the stress-dependent moduli of pavement materials and a superposition subroutine for multiple-wheel loads. The program is similar in principle to the Chev5L^ developed for dual wheels. Also, it will be recalled that the Chevron program was 2 based on the Burmister linear elastic theory. The basic assumptions behind Burmister1s theory in relation to pavement structures are: a. Single-wheel loads are vertically and uniformly distributed to the pavement over a circular area. b. Pavement materials in each layer are homogeneous, isotropic, and linearly elastic. A constant modulus and Poisson’s ratio represent the material property for the whole layer. £. There is continuous contact between each interface of the layered pavement system.

Since the moduli of resilient deformation of pavement materials are stress dependent, and the stresses and strains are determined by the modulus values, a compatible solution requires successive approx­ imations. Because the stresses in the pavement induced by the moving loads may vary appreciably both vertically and horizontally, the resil­ ient mod-ulus within the pavement may vary considerably from place to place. Modulus variation in the vertical direction can be approximated by subdividing the pavement into many horizontal layers, each layer hav­ ing a constant modulus throughout its thickness. Since variations along a horizontal plane cannot be considered in linear elastic layered theory, it was necessary to determine a single modulus value to represent the entire layer. The value is based on an effective state of stress deter­ mined by averaging the stresses at several points within the layer. Figure 1 shows a layered elastic pavement system. Each layer of

10 R

P

AC ( M r ) , . V, CONSTANT (MR) i AND V,

(M r >2* V 2

BASE (Mr )3, V3

(m r ) 4 . v *

(M r ) 5 , V 5 <

SUBBASE (M r )6 , V6 (

(MR)7, l*; 1>

* ^ * ^ ,

SUBGRADE (MR)n.1t V„.y

W r ) ^ . K n

Figure 1. Nonlinear layered elastic pavement systein

11 granular materials is divided into layers normally 6 to 10 in.* in thick­ ness. When subgrade soil is considered to be nonlinear, the thicknesses of divided layers may be increased at greater depths. For simplicity, however, the lower portion of subgrade may be assumed to be linear be­ cause stress intensities are very small in that region. Since it is known that stress varies greatly with depth in the base course layer, and stresses at midpoint of the layer do not truly represent the average stress conditions of the entire layer along the depth, three or four points along the depth should be used to determine the average stress (see Figure l). Since the stress does not vary much with depth in the subbase and subgrade soils as compared with that in the base, stresses at midpoint may sufficiently represent the average stress conditions of the entire layer (see Figure l). The computational procedure used for analysis was as follows: a,. The initial modulus of each nonlinear layer was estimated and the stresses at preselected depths along the load axis and at offset points in each layer were computed. For layers in which computations were made at more than one depth, the stresses at middepth of each layer at each offset were ap­ proximated by averaging the stresses computed at all the pre­ selected depths. The stresses at middepth along the entire horizontal plane of each layer were then approximated by aver­ aging the stresses at the preselected offsets. The resilient moduli M^** were computed by the following equations:

For unbound granular materials:

K2 (1) ^ = Kie

For fine-grained soils:

, for CTd < K ± (2a) “e - K 2 + (KI - V K3

, for ct, > K (2b) «E ' K2 + (°d -

* A table of metric conversion factors is presented on page 5- ** For convenience, symbols are listed and defined in the Notation (Appendix C).

12 where Mp = resilient modulus K^, Kg, Kg, and K^ = constants 0 = a + cr + a = bulk stress x y z a, = deviator stress d

b. The moduli corresponding to the stresses determined in Step a were computed from Equation 1 or 2 for each nonlinear layer. c_. The moduli obtained in Step b were compared with those esti­ mated in Step a. If they were different by an amount greater than a specified tolerance, the whole procedure was repeated until the computed moduli of two consecutive iterations were within the specified tolerance. A two percent criterion for closure was used in this study. d. The stresses, strains, and displacements in the pavement at any location were computed using the last computed moduli.

A bilinear representation of the functions is shown in Figure 2.

Figure 2. Bilinear representation of the relationship between resilient modulus and deviator stress for fine-grained soil

a:

DEVIATOR STRESS crj

ITERATIVE TECHNIQUE

When using Burmister's layered elastic solution to solve pavement problems, tensile stresses usually are developed near the bottom of the

13 of some layers, consequently, becomes either very small or negative (a negative stress is a stress causing extension), which results in small or undefined values when Equation 1 is used. The situation is worse when strong, well-graded, dense base materials are placed directly on weak subgrade soils. Since the modulus of a negative bulk stress is not defined, a small positive modulus (or minimum modulus) is assigned to the layer when the computed bulk stress becomes negative in the CHEVIT program. Under this situation, the convergence of solution be­ comes rather difficult because of "overrelaxationT:, this is explained below. In the first analysis, large tensile stresses were developed in the bottom layers and minimum moduli (modulus that may be less than the supporting subgrade modulus) were thus assigned to these layers. Reason­ able moduli were then obtained in the upper layers. In the second iter­ ation, positive moduli (much greater than the subgrade modulus) were obtained in the bottom layers but minimum moduli were generated in the upper layers; the poorly supported underlying layers have excessively low moduli generated from the previous iteration. The results of the third iteration were very similar to those of the first one; i.e., mini­ mum moduli were obtained in the bottom layers while large moduli were obtained in the upper layers. The results of the fourth iteration were then very similar to those of the second one; i.e., minimum moduli were obtained in the upper layers and large moduli in the bottom layers. This oscillation continued with slow or no convergence of solution. 12 Dehlen also reported the absence of convergence after six iterations working with a homogeneous sand. Using the underrelaxation technique, Dehlen modified the iterative solution so that instead of letting (new) moduli in the (*■ + 1) iteration equal com- puted which were computed from the stresses of the i iteration, the change was damped as follows: N,+ (3)

Ik where ^ = new used in the iteration

fM^X/ = new used in the i*th iteration

= c o m P u'*:'e<^ ^ in the iteration

In the development of the CHEVIT computer program, Equation 3 was modified as ' . ftX * m (U) 1 + f (U+1 where f is a constant determining the rate of relaxation. Equations 3 and U are identical when f = 1 . The value of f is evaluated by the procedure described below. a. In the first iteration, f is always equal to unity; i.e., new is the average of computed and initial . If tensile bulk stress developed in the layer, instead of reducing the drastically from its initial value to the minimum values, nearly one-half of this value is used (as determined by Equation U). b. In the second iteration, if the difference between the com­ puted and initial follows the same direction as that of the first iteration, the new f will be twice the old f to speed up the relaxation. If the difference follows the reverse direction, the new f will be one-half of the old f to slow down the relaxation. c_. The same principle applies to the remaining iterations until the convergence criterion is met.

The following numerical examples illustrate the procedures: Example 1: Iteration 1. Initial Mp = 20,000 psi, computed = 2,000 psi, f = 1 , and

new = (20,000 + 2,000)/2 = 11,000 psi

15 Iteration 2. Initial Mp = 11,000 psi, computed = 5,000 psi; hence f = 1 x 2 = 2 , and new Mj^ = [11,000 + (2 * 5,000)]/(l + 2) = 7»000 psi

Iteration 3- Initial = 7000 psi, computed = 1+000 psi, hence f = 2 * 2 = L , and [7000 + (k x k000)]/(l + b) = b600 psi Example 2: Iteration 1. Initial = 20,000 psi, computed = 2,000 psi, f = 1 , and new = (20,000 + 2,000)/2 = 11,000 psi

Iteration 2. Initial Mp = 11,000 psi, computed M^ = 15>000 psi; hence f = 1/2 = 0.5 , and new Mj^ = [11,000 + (0.5 * 15,000)]/ (1 + 0.5) = 12,330 psi

Iteration 3. Initial M^ = 12,300 psi, computed M^ = 8,000 psi; hence f = 0.5/2 - 0.25 , and new Mr = [12,300 + (0.25 x 8,000)]/ (1 + 0.25) = 11,U00 psi Iteration 1+. Initial Mp = 11,1+00 psi, computed = 8,000 psi; hence f = 0.25 x 2 = 0.5 » and new Mg = [11,1+00 + (0.5 * 8,000)]/(l + 0.5) = 10,270 psi In Equation 1+, with variable f , convergence is much faster than in Equation 3 (f = l). In a problem in which convergence, could not be obtained with the original Chev5L program even after 20 iterations, with the same program modified by replacing Equation 3 with Equation U, convergence occurred on the fifth iteration. Averaging the stresses of several depths in one layer, when the stresses vary greatly along the depth, also assists in the convergence of solution.

MULTIPLE WHEELS

The Chevron computer program, from which the CHEVIT program is

16 derived, deals only with a single-wheel load. A superposition sub­ routine was added to the iterative layered elastic program to handle multiple-wheel loads. It should be noted that the multiple-wheel stresses are not used in determining the modulus of nonlinear layers. In the use of the superposition principle, the stresses, strains, and displacements were first computed at nine offset points: 0, 0.5, 1, 1.5, 2, 3, U, 8, and 12 radii. The values at desired offset distances for the multiple wheels were then interpolated from the known values of the nine points. The Gregory-Newton Interpolation Formula with three 13 points of equal intervals was used in this study, as explained in the following paragraphs. An interpolating parabola was expressed in a second-order poly­ nomial p(x) through the three points. It should be noted that equal intervals exist in the following sets of three points. 0, 0.5, and 1 radii 1, 1.5, and 2 radii 2, 3, and ^ radii U, 8, and 12 radii

Assuming the known values at the three points in each set are f(x^) , i equals 1, 2, and 3 , and the interval between two points is h . The polynomial has the form

A2f(x ) p(v) = f(xx ) + Af(x1 )y + ---g---n(p - l) (5)

where

Af(x1 ) = fia^ + h) - fCx^

A2f(xx ) = f(x + 2h) - + h) + f(xx )

The following numerical example illustrates the use of Equation 5.

17 PROBLEM

The coordinate system for a C-5A 12-wheel gear assembly is shown in Figure 3. Each wheel has an equivalent radius of 9*5^ in. The sur­ face deflections under one wheel load of C-5A computed hy the Chevron program at the nine offset points 0, 0.5» 1» 1.5» 2, 3» ^» 8, and 12 radii are 0.102, 0.1, 0.098, 0.092, 0.08, 0.05, O.Olt, 0.019, and 0.005 in., respectively. The surface deflection at the point x = 0 and y = 0 induced by the wheel load at location x = 3^ in. and y = 0 is needed.

18 SOLUTION

The computer would first determine that a point at a distance R - 3 ^ in. lies within the interval of offset points of 2, 3, and U radii. R is the radial distance and equal to Vx2 + y2 . The following values for deflections at offset points of 2, 3» and U radii are assigned to the computer. a. f(x1 ) = f(x = 19.1 ) = 0.08 in.

f(x2 ) = f(X;L + h) = f(x = 28.7) = 0.05 in.

f(x3 ) = f(xx + 2h) = f(x = 38.2) = O.Oli in.

b. The following values are computed. _ X " X1 3^ - 19.1 , u = — --- s^r— ” 1-58

Af(xx ) = f(x1 + h) - f(Xl) = -0.03

A2f(X;L) ,= f(x^ + 2h) - 2f(xx + h) + f(xx ) = 0.0U - 2 x 0.05

+ 0.08 = 0.02

c_. The surface deflection p(x = 3*0 can he computed from Equa­ tion 5*

A2f(xn ) p(x = 3 M = f(xx ) + Af(x1 )y + ---^--- wiv1 ~ !) = 0.08 - 0.03(1.58) + 0.01 x 1.58(0.58) = 0.0lt2 in.

Deflections induced by other wheels are interpolated similarly, and the total displacement for the C-5A 12-wheel gear assembly (shown in Figure 3) is the linear summation of individual wheel load deflections. The principle of superposition is the method of obtaining the actual resultant effect by adding or combining independent partial effects. The method is applicable only if a linear relationship exists between the loads and the effect they produce. Since granular materi­ als axe highly nonlinear, the use of the superposition principle to

19 obtain solutions for multiple-wheel loads is theoretically unsound. Unfortunately, there is no computer program now available capable of ob­ taining solutions for multiple-wheel loads, which can be economically operated without the use of the superposition principle. Therefore, the results for the multiple-wheel loads computed by the CHEVIT are subject to the common criticism of the superposition principle.

20 NONLINEAR FINITE ELEMENT COMPUTER PROGRAM

The nonlinear finite element program used for comparison purposes was obtained from the University of California at Berkeley. The program is suitable only for analysis of axisymmetric solids. The pavement structure is first idealized as an assemblage of a finite number of dis­ crete structural elements interconnected at a finite number of joints or nodal points. The sizes of the elements are chosen to vary in accor­ dance with the anticipated stress gradients. The elements are actually complete rings in the horizontal direction, and the nodal points are circular lines in plan view. There is a boundary on which the nodal points are fixed and a vertical boundary on which the nodal points are constrained from moving radially. The load at the surface is circular and is assumed to be applied in steps so that the nonlinear stress-strain behavior and the modulus- stress dependency of the material can be included in the analysis. Therefore, the accuracy of the solution is a function of the number of load increments used, with greater accuracy being associated with smaller load increments. The accuracy can also be improved by increasing the number of elements into which the pavement structure is divided. In this study, ten load increments were generally used. Equations 1 and 2 were used in computing the resilient moduli of nonlinear materials and are the same as those employed in the CHEVIT program.

21 COMPARISONS OF COMPUTED RESULTS, ITERATIVE LAYERED ELASTIC AND NONLINEAR FINITE ELEMENT PROGRAMS

As stated above, the purpose of this study was to develop a com­ puter program that is simple and economical to use and will yield re­ sults approaching those of the nonlinear finite element program. The CHEVIT program can be simply and economically used, and can incorporate the stress-dependent moduli of pavement materials in the computations. To determine whether CHEVIT yields results close to those of the non­ linear finite element program, computations were made on many selected pavement structures to find the differences between results computed by the two programs. Modifications to improve the CHEVIT program were made acordingly. Three types of pavements were used in the computations: (a) a 3-in. asphaltic concrete surface and 21-in. high-quality, well-compacted crushed stone base, (b) a 9-in. asphaltic concrete surface and 15-in. gravelly sand subbase, and (c) a 3-in. asphaltic concrete surface, a 6-in. high-quality well-compacted crushed stone base, and gravelly sand subbase with thicknesses varying from 6 to 32 in. These were typical test sections constructed at the WES. For these pavements, the loads were varied from 15 to 75 kips and the subgrades were varied from 2 to 16 CBR to provide a wide range of conditions. A total of 39 pavements were used in the computations, as given in Table 1. For each pavement, both the nonlinear finite element program and the CHEVIT program were used to compute the stresses and strains in the pavements, and the results were compared. In the computations, a mod­ ulus of 100,000 psi was assigned to the asphaltic concrete. Equation 1 was used to characterize the nonlinear behavior of the granular mate­ rials with KL^ and equal to 8300 and 0.71* respectively, for base materials, and 2900 and 0.kj9 respectively, for subbase materials. The relation of M^ = 1500 CBR was used to determine the subgrade modulus values. Because a bottom boundary on which the nodal points are con­ strained from moving vertically was used in the nonlinear finite element

22 Table 1 Comparison of Computed Results, Nonlinear Finite Element and Iterative Layered Elastic (CHEVIT) Programs

______Programs______Nonlinear Finite Iterative Layered Element Elastic (CHEVIT) Thickness. in. Sub­ r Percent e x io -3 e x IO-3! Difference Load Sur­ Sub­ grade z z e Payement kips face Base base CBR W . in. in./in. W , in. in./in. W z

Asphaltic 15 3 21 0 0.039 0.78 0.01*0 0.77 +2.6 “1.3 concrete 30 21 0 2 surface, 3 0.107 1 .5U 0.130 2.1*0 +21.5 +25.8 crushed 30 3 21 0 k 0.072 1.22 O.O76 1.1*5 +3.0 + I 8 .9 stone 30 21 0 8 0.050 -6.0 base 3 0.91* 0.01*7 0.91 -3.2 30 3 21 0 12 O .O k l 0.76 0.036 0.70 -12.2 -7.9 30 3 21 0 16 0.036 0.65 0.031 0.58 -13.9 -10.8 50 3 21 0 k O.lll* 1 .71* 0.122 2.33 +7.0 +33.9 50 3 21 0 8 0.078 1.36 0.073 1.1*2 -6.1* +l*.l* 50 3 21 0 12 0.061* 1 . 1 k O.O56 1.07 -12.5 -6.1 75 3 21 0 k 0.163 3.31 0.175 3.35 +7 .1* +1.2 75 3 21 0 8 0.112 1.77 0.101 I .98 -9.8 +11.9

Asphaltic 30 9 0 15 2 0.129 2. U7 0.155 2.93 +20.2 +18.6 concrete 30 0 I .76 +6.2 surface, 9 15 k 0.097 0.103 1.97 +11.9 gravelly 30 9 0 15 8 0.071 1 .1 6 0.073 1.27 +2.8 +9.5 sand 30 0 12 O.06I O.O6O O .96 -1.6 subbase 9 15 O .89 +7.9 30 9 0 15 16 0.055 0.73 O.O5U 0.78 -1.8 +6.8 75 9 0 15 k 0.232 1*.1*2 0.21*1* U .69 +5.2 +6.1 75 9 0 15 8 0.167 2.76 O.I67 3.06 0.0 +10.9 75 9 0 15 12 0.1U3 2.15 0.138 2.3I* -3.5 +8.8

Asphaltic 30 3 6 6 2 0.193 6.00 0.200 5.1*0 +3.6 -10.0 concrete 30 6 6 k 3.90 0.121 -6.2 -13.1 surface, 3 0.129 3.39 crushed 30 3 6 6 8 O.O82 2.37 O.O7O 2.ll* -ll*. 3 -9.7 stone 30 3 6 6 16 1.1*9 O.O5I 1.31 -10.5 -12.1 base, 0.057 gravelly 50 3 6 6 k 0.203 6.10 O.I87 5.16 -7.9 -15.1* sand 6 6 8 0.128 3.67 0.117 3.28 -8.6 -10.6 subbase 50 3 50 3 6 6 12 0.102 2.82 0.091 2.50 -10.8 -11.3 50 3 6 6 16 0.088 2.32 O.O77 2.01* -12.5 -12.1 30 3 6 15 k 0.113 2.10 0.108 2.00 -3.5 -U.8 50 3 6 15 k 0.173 3.22 0.16Ì* 3.07 “5.2 “1*.7 1 30 3 6 2k 2 0.lk5 2.20 0.11*0 2.06 -3.1* ON 30 3 6 2k k 0.113 I.5I* 0.100 1.39 -11.5 -9.7 30 3 6 2k 8 O.O92 1.03 0.078 O .90 -U.3 -12.6 30 3 6 2k 12 0.083 0.79 O.O68 0.67 -18 .I “15.2 30 3 6 2k 16 0.079 O .65 0.061* 0.5l* -19 .O -16.9 60 3 6 2k 2 0.11*5 2.20 0.11*2 2.25 -2.1 +2.3

60 3 6 2k k o . r r r 1.80 0.175 2.50 -1.1 +38.9 60 3 6 2k 8 0.11*3 1.62 0.130 1.63 -9.1 +0.6 60 3 6 2k 12 0.127 I .26 0.112 1.21* -11.8 -1.6 60 3 6 32 k 0.177 I .80 Ö .167 1.80 -5.6 0.0

23 program, a cutoff was used in the CHEVIT program at a depth about 300 in. beneath the surface of the subgrade soil. The exact depth varies with the magnitude and contact area of the wheel load. A high modulus of 555*555 psi was arbitrarily assigned to the cutoff layer. Higher moduli would not change the computed results noticeably, but may create some problem in the computations because of the extremely small modulus ratio between the subgrade and the cutoff layer. In the use of the CHEVIT program, computations were first made with moduli determined by the stresses only along the load axis. It was found, however, that the computed values of stress and strain were much larger than those computed by the nonlinear finite element program. This can be logically explained because the stresses along the load axis are generally greater than those at offset points. Since the response of pavement structures does not depend solely upon the materials along the load axis but also upon the materials at offset distances, average modulus values determined from the points at the load axis and at offset points should be used in the computation. In so doing, the computed values were reduced as the number of offset points was increased. Com­ puted values of the asphaltic concrete surface deflection and vertical strain at the subgrade surface for the 39 pavements computed by the two different programs are compared in Table 1. In the CHEVIT program, the moduli in the granular layers were computed based on the average stress intensities at offset distances of 0, 2, k , and 8 radii. For purpose of illustration, the comparison procedures are shown in Figure U. Table 1 indicates that the differences between the values com­ puted by the two programs are rather small for the majority of pavements. The computation time required by the CHEVIT program in most cases was about 1/20 of that required by the nonlinear finite element program. In the percent difference column in Table 1, the plus sign indicates that the computed value by the CHEVIT program was greater than that com­ puted by the nonlinear finite element program; the minus sign indicates that the reverse was true. Concerning the comparisons, the following points should be noted: a. The and values used in both computer programs do

2k NONLINEAR NONLINEAR LAVEREO-EL ASTIO FINITE ELEMENT P P

. ' !

w v / — i* — AC M r = CONST A ^ — j BASE ^ 1 v J A — K------A — f — j — V— ----fj ------

J SUBBASE M r = K , 0 K« F

J r

1> # 2 f # « f # e r j 7 7 7 7 7 7 7 7 7 7 7 7 ------— 1r P - €v «V J r SUBGRADE M r = 1500 CBR > J — f

— 1 -----

Figure k. Comparisons by CHEVTT program and nonlinear* finite element program

25 not significantly affect the comparisons. In other words, the same conclusions would he derived if different and Kg values (within reasonable range) were used in the compu­ tations. The determination of and Kg values used in this study is explained in Reference lH. b. The nonlinear properties of subgrade soils were not con­ sidered in the computations for the pavements shown in Table 1 because the information on the K values (Equa­ tion 2) for different CBR subgrade soils is not available.

2 6 DISCUSSION OF THE ITERATIVE LAYERED ELASTIC PROGRAM

Incorporating nonlinear characteristics of materials in the CHEVIT program is a great step forward in the improvement of the linear layered elastic program. However, the CHEVIT program still possesses many pitfalls which make the program far from perfect. These are the assumptions of (a) isotropy of materials, (b) identical deformation modulus in tension and compression, (c) continuous contact along each interface, and (d) existence of tensile stresses in the granular layers. Since the CHEVIT program is basically a linear layered elastic program, the materials in each layer are homogeneous, isotropic, and linearly elastic during each iteration. Excessive amounts of tensile stresses are developed frequently at bottom layers of the granular ma­ terials. This is, of course, physically impossible, as granular mate­ rials cannot take large tensile stress. Therefore, the use of the CHEVIT program to evaluate the radial stresses in granular layers should be used cautiously. Despite this abnormality, however, the responses, such as stress, strains, and displacements, computed by the CHEVIT program were generally very close to those of the nonlinear finite element program, in which tensile stresses usually do not develop pro­ vided the moduli of granular materials are not excessively high. There­ fore, the CHEVIT program can be used for rational pavement designs.

27 CONCLUSIONS AND RECOMMENDATION

The iterative layered elastic program (CHEVIT), developed for rational pavement design, has the capability of incorporating the stress- dependent characteristics of pavement materials. Pavement responses computed by CHEVIT yield results comparable to those computed by the nonlinear finite element program; furthermore, the CHEVIT program is much easier and more economical to operate. It is recommended, there­ fore, that the CHEVIT program be adopted in the rational design of flex­ ible pavements.

28 APPENDIX A: COMPARISONS OF COMPUTED AND MEASURED STRESSES AND DISPLACEMENTS

In this portion of the study, the vertical stresses and displace­ ments measured in item 3 of the multiple-wheel heavy gear load test 7 section at the U. S. Army Engineer Waterways Experiment Station (WES) were compared with those computed by the nonlinear-finite element method incorporated with various nonlinear stress-strain relations of the pave­ ment materials. The multiple-wheel heavy gear load test section was composed of a 3-in. asphaltic concrete (AC) surface course, a 6-in. well-compacted limestone base, a 2U-in. sand and gravel subbase, and a 1+-CBR buckshot clay subgrade soil. Four-inch WES pressure cells and linear variable differential transducers (LVDT) were installed at various depths in the test section to measure stresses and displace­ ments under various loading conditions. Details of the instrumentation, construction, and testing of the test section are presented in Refer­ ence 7 9 Volume III. The meshes of the finite element grid are shown in Figure A-l. Since the thickness of the AC surface course was only 3 in., a constant modulus was considered to be appropriate and was used in the computa­ tions. Pavement temperature during measurement was about 90°F, and a modulus of 100,000 psi was chosen. Three different computations were made with different stress-strain relations for the granular materials and subgrade soil: a. Stress-strain relation No. 1. (l) Granular materials:

(A-l) " r - Kl°32

and v = 0.3 • For base materials, Kj = 13,126 and Kq = 0.55 ; for subbase materials, Ki = 7 »650 and Kg = 0.59 •

29 RADIAL DISTANCE, IN. 120 60 40 20 30 16 12 0 6 J . 8 4 2 ^ - > * - ^ - ^ - - - ^ - P - —^ —P NOTE: VALUES WITHIN EACH ELEMENT ARE MODULI CORRESPONDING TO CORRESPONDING MODULI ARE ELEMENT EACH WITHIN VALUES NOTE: i V w V L V P 8. 144 .0 88 2 / ) . ; , STRESS-STRAIN RELATIONS NO. 1, RESPECTIVELY. 2, 3,AND RELATIONS STRESS-STRAIN J L r f f f „ Ï „ v 152

\ ~ Figure A-l. Variations of modulus with different stress-strain relations stress-strain different with of modulus Variations A-l. Figure 7.85 7.85 7.85 7.85 7.85 7.85 7.85 7.85 1.52 7.88 7.86 7.86 L 1.52 1.52 L 1.52 1.52 1.52 1.52 1.52 « 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 52 52 ^ 7.85 . 90. 68 .0 0 9 1.0 j r L 47 7.57 7.56 7.57 7.57 7.85 7.70 7.64 7.61 7.59 7.58 1.48 1 1.48 1.47 1.48 1.47 1.47 1.47 1.47 1.48 1.47 1.48 1.50 1.52 1.49 1.48 1.48 1.48 1.47 1.48 1.49 1.48 1.48 . 47 ^ 7.57 1 - - - r « 6.97 6.97 7.0 7.0 7.0 1 7.02 7.37 7.21 7.10 7.04 1.40 1.40 7.78 1.40 1.38 1.38 1.38 1.40 1.38 1.40 1.46 1.39 1.45 1.38 1.51 1.51 L 1.40 1.40 Ml 1.41 L U U 140 . 8 3 43 41 * "

7.0 À 5 : r UBGR 0 . 30 3. 330 27. . 1! 1.0 2 .0 7 2 3.0 3 36.0 43.0 4.0 5 .0 6.24 6.22 6.25 6.30 6.80 6.34 6.26 6.54 6.40 7.11 7.72 1.27 1.31 1.30 1.27 1.30 1.31 1.29 1.40 1.32 1.28 1.31 1.51 L 1.37 1.27 1.28 1.51 1.35 1.31 1.32 1.34 ADE 42 J

— J B 1

6.39 5.36 5.34 5.38 121 5.40 6.91 1.15 5.48 7.68 5.57 1.20 5.94 5.69 1.14 1.20 1.20 1.20 1.21 1.15 1.16 1.15 1.50 1.37 1.51 1.39 1.32 1.26 1.23 1.17 1.18 1.29 1.21 1.22

^ 5 .« J 02 10

05 10 109 6.06 1 4 6 4.39 4.37 4.34 4.56 4.70 4.44 5.34 4.94 m 1 1.09 L 1.09 1.09 LQ 1.01 1.0 L 1.10 1.03 1.51 1.27 1.51 L 1.14 1.04 1.37 L 1.22 I I I 1.11 . 00 09 06 36 18 2

’* 4.49 À u 0.98 0.89 0.98 6.80 0.99 0.90 0.89 0.88 5.86 0.91 3.55 3.51 3.47 4.98 7.68 . 2 0.9 3.62 0.95 3.80 4.41 4.02 1.00 1.01 1.51 1.36 1.24 1.50 1.03 1.36 1.13 1.07 1.18 1.03 - - - D 6 U PH IN. EPTH, , 10.5 10.3 10.1 10.2 10.4 12.7 10.2 0.81 9.41 9.31 9.37 8.89 9.24 9.32 8.42 2.65 2.44 7.58 3.58 5.22 4.13 2.57 7.31 3.74 3.55 1.52 1.54 1.84 1.54 — 1.47 1.58 03

10.4 i --- l f J 6,26 11.9 11.9 11.7 11.1 11.7 10.9 10.7 1 6.62 6.06 5.30 4.82 9.22 0.98 9 4 6 6.18 1 4 6 8.45 9.31 8.92 7.99 0 4 7 7.03 3.02 7.54 6.16 4.37 — - — — — L 3

11.9 SUBI 3 r

A « E S i .82 - - -

13.0 10.9 114 13.7 13.9 14.1 14.0 /1 8 9 9.12 9.57 8.41 8.17 3 4 9 6.46 9.37 9.34 2.79 1 4 9 5.37 8.71 9 4 0 8.88 5.92 3.19 7.43 4.18 2.93 2.38 9 4 7 7.96 7.15 0.93 1.25 -

71 71

13.7 J >.0 >.0 r

« “ 16.1 14.2 14.9 134 16.4 19.3 14.6 12.4 12.2 17.2 10.1 18.1 11.4 10.4 6.21 6.21 6.11 8.20 9.01 3.95 7.66 5.26 4.90 7.03 7.30 1.49 ~ - - - — - - t .3 J 0 6. 0 1.5 .0 3 .0 6 .0 9 í f - ~ «—

22,2 43.5 47.5 30.5 24.5 33.0 20.4 20.3 26.9 23.2 32.1 36.5 39.3 20.1 12.3 184 17.7 12.8 17.4 8.64 2.95 8.59 7.41 5.89 2.88 5.55 - - — - — — “ -

34.1 BA J i SE ------67.9 64.8 23.7 60.3 67.9 67.6 57.4 34.4 4 2 3 50.1 52.3 24.3 24.7 24.7 11.5 11*5 12.8 17.1 8.43 4.33 4.48 4.39 2.68 4.50 3.84 7.49 - - — “ - - - H \ H u 2 i s o 1 c T1 r 1 Z st 3 1 n » r

s T

»

9.5246 IN. 9.5246 "

RADIUS TIRE (2) Subgrade soil:

J" Rf(l - sin 0) (a - a )"l2 Et = L1 2C cos 0 + 2a3 sin 0 J Ei ^A_2^

and v = 0.4 . b. Stress-strain relation No. 2.

(1 ) Granular materials:

(l bis) “ r = K3°

and v = 0.48 . For base materials, K 3 = 8300 and = 0 .7 1 ; lor subbase materials, K = 2900 and = 0.47 . 3 (2) Subgrade soil: Equation A-2, where v = 0 . h . c. Stress-strain relation No. 3-

(1 ) Granular materials: Same as in stress-strain relation No. 2, (2 ) Subgrade soil:

( a b i s ) « R = K2 + (Ki - °a)K3 > f°r °d < h 2

M r = K 2 . (oa - , for ^ (2a b i s )

and v = 0 .1* .

The three numbers shown in Figure A-l within each element are moduli computed by stress-strain relations Nos. 1, 2, and 3* respectively. A large value of Poisson's ratio was used in stress-strain relations Nos. 2 and 3. It is believed that a large value for Poisson's ratio represents field conditions better than the small value that is used in stress-strain relation No. 1. Both static and resilient moduli for sub­ grade soil were determined from laboratory tests on undisturbed samples.

The determination of values for to were described in Refer­ ence 13. Figure A-2 compares computed and measured deflections along the

31 Also, the use of static moduli for the subgrade soilsubgradestatic forinadequateof usethebe­ themoduli was Also, terials and subgrade soil. Equation A-l assumes that the elastic Equationassumesthatthe moduli A-l subgradesoil.and terials tom of the granular layers, the elastic moduli reduced drastically asdrasticallyelastic layers,reduced granularthe of themodulitom o 1 tecmue auswr uhhge hntemaue. Thiswas computed valuesthantheNo. 1, higher theweremuch measured. the load increment increased, as indicated in the computerinindicatedincrementincreased, asoutputs.load thethe anticipated because of the inadequate values used forinadequategranular ofused the anticipated valuesthebecause ma­ stress-strainrelation Using 30-kipsingle-wheelaload.loadaxisof resilient moduli are generally much greater than the staticgreaterones.generallyare thanthe resilient much moduli soilinsubgradeshouldofused the Resilient modulibe curve.of the of granular materials depend solelystressdependonof principlegranular the materialsminor opttos hc ol euti euto fdfetos sincedeflectionsinresultareductionofthecomputations, would which cause the measured deflections were elastic, i.e.,deflectionscauserebound portion the thewere measured D E P T H , IN cniigpesr) We esl stressesdevelopedat thebot­ tensilewere When (confining pressure). Figure A-2. Comparison of the computed and measured deflectionsand Comparisoncomputedof themeasured FigureA-2. along the load axis of a 30-kipsingle-wheelofloadloadaxisa along the The use of Equation 1 can greatly increase the elasticofcanincrease1greatlyEquationofthe Theuse moduli E C DEFLECTI I . IN , N IO T C E L F E D IC T S A L 32

granular materials. The major principle stresses a are generally in compression and are very much greater than the intermediate and minor principle stresses and a , which are in tension with load at the bottom of the layers. Hence, in most cases, the first stress invariant 0 , the sum of the principal stresses, is positive in magnitude. The computed deflections based on Equation A-l are shown in curve 2 of Fig­ ure A-2. The computed deflections, however, were still much greater than the measured. This was also anticipated because static moduli were used in the computations. Nevertheless, it is clearly illustrated that the use of the first stress invariant 0 is superior to the use of the minor principle stress in characterizing granular materials. When the static moduli of subgrade soil were replaced by the resilient moduli in the computations. The resulting deflections were plotted in curve 3 of Figure A-2. The deflections were greatly reduced and agreed very well with the measured deflections. It is interesting to note that the portion of curve 3 above the subgrade is parallel to curve 2. This is due to the material characterizations for materials above the subgrade soil being the same for these two computations. Figure A-3 compares measured and computed vertical stresses for a 30-kip single-wheel load. Curves 1 and 2 were computed with static moduli of the subgrade soil; therefore, the computed stresses at the surface of the subgrade soil were much too low. Since resilient mod­ uli of the subgrade soil were used (providing better subgrade support), the computed vertical stress at the subgrade surface increased and approached the measured value (curve 3). For comparison, the curve for Boussinesq's solution is also presented in Figure A-3. Figure A-U compares measured and computed deflections along the load axes of 15- and 50-kip single-wheel loads. The computations were made with stress-strain relation No. 3. It can be seen that the com­ puted values were quite close to the measured ones. Figure A-5 is simi­ lar to Figure A-U except that it shows the vertical stresses. The com­ putations here were also in good agreement with the measurements. Figure A—6 compares measured and computed deflection basins at various depths under a 30-kip single-wheel load. The computations were

33 VERTICAL STRESS, PSI

Figure A-3- Comparison of measured and computed vertical stresses along the load axis of a 30-kip single-wheel load CLASTIC OCCLCCTIOH. IN.

Figure A-k. Comparisons of measured and computed deflections along the load axis of two single­ wheel loads V E R T IC A L STRESS. RSI

Figure A-5. Comparisons of measured and computed vertical stresses along the load axis of two single-wheel loads

36 Figur single-wheel loadsingle-wheel 30-kipaaxesof offsetflectionalong the basins ELASTIC DEFLECTION. IN e A-6. Comparisons of measured and computedde­ and Comparisonsof measured A-6.e 37 made with stress-strain relation No. 3. Unfortunately, the agreement between the computations and the measurements along the offset axes was not as good as that along the load axis.

38 APPENDIX B: CHEVIT PROGRAM

EXAMPLE

To illustrate the use of the input guide for the CHEVIT program, the pavement structure shown in Figure B1 was used as an example to compute the stresses, strains, and displacements in the pavement. A single-wheel load of 30,000 lb with 105-psi contact pressure and a Boeing Jb7 twin-tandem load of 2^0,000 lb with 210-psi contact pressure were used in the computations. In the nonlinear case, the moduli shown in Figure B-lb were the initial values; the true values were determined by the computer program with the following equations:

For base materials: ML = 6000(a + a + a (l bis) k x y z

For subbase material: Mg = 2900(ax + ay + oz)°-kT (l bis)

For fine-grained soils; Mg = U500 + (5 - ad ) x 700

for a, < 5 (2a bis) a Mg = 1*500 + (ad - 5) x

( —1+00) for a. > 5 (2b bis) d where the constants were chosen arbitrarily. For single-wheel load, the computed values were at offset dis­ tances 0, 9.5^, and 19-08 in. and at depths 0, 3, 38, 50, 70, 100, 150, 200, 250, and 300 in. For multiple-wheel loads, the computed values were at depths 0 and 38 in.; the offset distances are shown in Fig­ ure B-2 by the points on the grid pattern. The coordinate system for thè Boeing 7bj twin-tandem is also shown in Figure B-2.

FLOW CHART AND INPUT GUIDES

The flow chart and input guides for linear and nonlinear anal­ yses are given in Figures B-3 through B-7•

39 SINGLE OR MULTIPLE WHEELS

(Mft)4 = 555,555 PSI l/4 = 0.4 h4 = ®

«. PAVEMENT WITH LINE AR MATERIAL PROPERTIES

SINGLE OR MULTIPLE WHEELS

(LINEAR) (Mr ), = 150,000 PSI v, =0.5 h, = 3 IN.

(Mr )2 = 20,000 PSI vz = 0.4 h2 = 7 IN. BASE (MR )f = 17.500PSI t/s = 0.4 hj = 7 IN.

(Mr )4 = 14,000PSI l/4 =0.4 h4 = 7 IN.

SU BBA SE (MR )5 = 11,500 PSI v% = 0.4 h5 = 7 IN.

(Mr )« = 9,000 PSI 1* = 0.4 h« = 7 IN.

NONLINEAR) LAYERS (Mr )7 = 5.000 PSI U f = 0.4 ► h7 = 1 0 IN .

(MR). ^ 6.000 PSI l% = 0.4 ho = 20 IN.

6,000 PSI l* = 0.4 > ho = 30 IN. (Mr )* = §

(Mr )io s 6.000 PSI y ,0=0.4 <' h ,o = 3 0 IN .

(Mr ),, = 6,000 PSI v \ , =0.4 hn =210 IN. (LINEARI w m r (Mr )|2 = 555,555 PSI is,2 = 0.4

S. PAVEMENT WITH NONLINEAR MATERIAL PROPERTIES Figure B-l. Pavements illustrating the use of CHEVIT program input guide 40 V

Figure B-2. Grid pattern for computational points for a Boeing jkj twin-tandem gear assembly

hi C A L L E X IT

Figure B-3. Input guide for CHEVIT program 42 GENERAL PURPOSE DATA FORM

p r o g r a m DATE

REQ U EST CO BY PREPARED BY C H E C K E O BY 1 I 2 S{ 4 S i* I7 • » 11*12 11 14 IS 1C 17 1« 1« 20 21 22 23 24 2S| 2f 27 2t 2» 30 11 12 1 1 ) 4 H M )7 N N « |m ♦*[«• «7 m SI S 2 S 1 SC SS SCIS7 SC SC CO Cc|c7 cc|c» iH7>Hw1w 77 n a No. 1 Option Card (A5 or A3) In ease of start and continuation of input» columns 1-5 should read:

S .T .A M i

In case of termination of input columns 1-3 should read:

END . . I .

ii i ! IQ! 11 21 22 21 24 2S SI S2 SI 14 SS SC S7 SC

No. Identification Card (20AU) Problem description (title) can be described in columns 1-80. - r — I— I__I L.—L— 1 J L_

_l__ I I— I— I— I— j--1— I— . , 1 . iTETLEiil) to TITLE &2Q1

■ i f f- >j 11* 12 11 14 IS 1C 17 iCj IS j 20 t. 11 12 11 14 IS 1C 17 41 42 41 44 4S 4C 47 4C 4C SI S3 SI S4 SS SC 57 SCiSC > 71 72 71 74 7S 7C 7717» 7» I

WES j u u 74 1233 • COITION OC SCR 42 WILL SE USCO UNTIL EXHAUSTED

Figure B-U. Input guide for CHEVIT, linear analysis (sheet 1 of 8) GENERAL PURPOSE DATA FORM

PROGRAM

REQ UEST CO BY PREPARED BY C H E C K E D BY « 2! s 4 sic 7 a » n?i2jis|i4 t» te 17 10 14 iso 2t 22 za 24 25 ac 27 2a 29 so si S2 ss se ss se 37 41 42 4S 44 49 47 44 49 90 SI 92 SS S4 SS SC 57 St! SO CO 6l|C2 CS C4 CS CS €7 CO SS 70 71S TlfrS 74j7S(7C 77 TOITS Ho, 3 Wheel Card (2F10.0) Columns 1-10 WOT * wheel load, lb 11-20 psi * tire contact pressure, pel 21-30 Zero 31-bO Zero

5xTi_

4S •P* 4 S I C 7 S • ho 11 12ill IS 14 H | S 2 So. b Layer Card (F10.0) Columns 1-10 XSS » number of layers

-J __ I___I__ I__ I__ L_ XjNtSj t » «

So. $ Layer Properties Card (8F10.0) Columns 1-10 £(l) * modulus of elasticity for layer 1 11-20 v(l) « Poisson's ratio for layer 1 21-30 E(2) * modulus of elasticity for layer 2 31-bO v(2) «Poisson's ratio for layer 2 bl-50 E(3) « modulus of elasticity for layer 3 51-60 v (3) * Poisson's ratio for layer 3 61-70 E(b) * modulus of elasticity for layer b (if needed) ______71-30 v(b) « Poisson's ratio for layer b___ E (l) i l l ± JBCg). ,v(,2l ill v(3) Z$A1 'JL S I 4 I s C 7 a I 1 0 '111 12 IS 14 IS 14(17 20 21 22 2S 24 ; 2S 2C 27 2C 29 IO S I S2 S3 S4 SS SC 37 41 42 43 44 4S S I S3 S3 54 SS SC S7 SC 41 S2 CSC4 CS 44 47 71 7 2 73 74 7S 7C 77 7t 79 WES . 7« 1233 COITION o r SEP 42 WILL OC USED UNTIL EXMAUSTEO

Figure B-U. (sheet 2 of 8) GENERAL PURPOSE DATA FORM

R CQU EST CO BY PREPARED BY C H E C K E D BY

« 21 3 j sj 9¡9 j 7 • t IO H II 1) U 15 19 17 io ai n ss a« 29 27 29 23 90 31 32 39 34 35 39 97 39 30 42 42 47 42 42 10 SI 52|53 54 22 94 27 2 0 ‘91 42 23 94 99 22 97 >«[?> 7« 7* t s rr|7i Ttfi I l i 1 If the number of layers Is greater than 1», continue to next data card using .same format,until the number of layers is satisfied*

_l I__1 I I_I—L_ Bo. 6 Layer Thickness Card (8F10.0) Columns 1-10 HH(l) ■ thickness of layer 1, in. 11-20 HH(2) * thickness of layer 2, in. 21-30 HH(3) ■ thickness of layer 3, in. 31-^0 HH(U) * thickness of layer U, in. Ul-50 HH(5) * thickness of layer 5, in. 51-60 HH(6) * thickness of layer 6, in. 61-70 HH(7) * thickness of layer 7, in. 71-80 KH(8) * thickness of layer 8, in. (if needed) -P* + Ui 1 a l l ' s sis 7 19 j 2 It ta 13 14 15 17 12 13 20 2! 22 23 24 25 25 27 22 22 32 31 32 33 34 32 32 37 S3 94 92 52 97

_J—I_I-- 1_I_L -I_I_I-- L_J— I_I_L_

_l_I_I_I_l_ _l_I_I_I_I_L. h h (6) HH(7) . . . W , i-t...... FJ*L _i I_I_I I_I— i— If the number of layers is greater than 8, continue to next data card using - same format until the number of layers is satisfied.

1 2 3 4 9 9 17 0 • 110 11 12*13 14 15 19 17 19 13 2D 21 22 23 24 25 22 27 22 23 30 31 32 33 34 35 32 37 30 33 53 54 55 59 57 21 92 93 24 95 99 97192(92 71 72 72 74 79 72 77 72 73 22

COITION OP SC* 42 WILL S C U5EO UNTIL EXHAUST CO

Figure B-U. (sheet 3 of 8) GENERAL PURPOSE DATA FORM

REQUESTED «Y PREPAREO BY CHECKED BY

t ij » « i 717 4 9 toIni 12ÌIS 14 14 14 17 «• It SO 21 21 29 2« » 24 2t 90 31 92 19 94 9S 94 in * » 99 « «1 42 49 44 4t 40 47 SI 52 S3 S4 SS 54 S7 sijsilssjss 44 47 TO J71 ^ w |t« ?» n 7» 7t|y -L *EE - 4 — Wo. 7 Offset Card (F10.0)

Columns 1-10 X1R * number of offsets computations made.

This card is needed only for single-wheel load. For multiple wheels» set XiR * 0 and branch out to Ho. 9.

.»■ a i I__i.J__ «__l _ -1 ---»---L_ -I I— I_ I__ I I__I__ L_ XiR A, .1 I__ 1— 1_ I_ l_

Wo. 8 Offset Distances Card (8F10.0). Wot applicable for multiple wheels

Columns 1-10 RR(1) distance to first offset» in. 11-20 RR(2) * distance to second offset» in. 21-30 RR(3) * distance to third offset» in. ON 31-1*0 RR(!*) * distance to fourth offset» in. 1*1-50 RR(5) * distance to fifth offset, in. 51-60 RR(6) « distance to sixth offset, in. 4 y «.|43 44 44 72 74 74 74 74 77 7« 79 61-70 RR(7) * distance to seventh offset, in. 71-80 RR(8) » distance to eighth offset, in. (if needed) » » 1 I X . J..J .1_1__L_ _l_I_L_I_l_ RK(2) * ( ») 8R(5) PRJ[6.) 8B(TJ ... p w ... _1_I__I_L.-J8 _1 t . FW ■

-J—I__ I------I__ I___I__ I------I___L _ i _ If the number of offsets is greater than 8, continue to next data card

using same format until the number of offsets is satisfied.______

,Rg<*i$)

■ f - H 1 2 9 44147(lilt! 11 12 13 14 14 14 17 14 19>20 21 22 23 24 24 24 27 24 91 32 33 94 34 34 97 94 99 40 >3|44 72 71 74 74 74 eleo

W E S JwtTl 1233 COITION OP SC* 42 « IL L 4 C USED UNTIL CXMAU4TEO

Figure B-U. (sheet 1+ of 8) GENERAL PURPOSE DATA FORM

P R O G * AM DATE

REQUESTED BY PREPARED SV C H E C K E O B Y P A G E OP » |t|» |*|s SJT « |s |wjiiji2 is|i4|it|i* it]I*|IS a | t i 2 2 ) 2 s| sa | ssj ssj 2? a ] a 24 j 2 1 22 22 24 2S 24 27 24 24 44 41 42 «9 44 4S 4e |47 44 44j44 41 4S|42|S4|44|»e |»7 4 4 |» 44 •’ “M-M-H-M" Ho. 9 Computational Depths Card (F10.0) .... 1 1 .. 1 Columns 1-10 XiZ * number of depths t<» computational po ......

f * * * *»»»«*»»»

i 1 1 1 1 t 1 1 1 « i t 1 1 1 & 1 1 titillili A A A -A AIA AA

MV* AW w vitua l»W A——BIAM VM V 4W IH U F W A U V » V « VWfAU.V J Columns 1-10 ZZ(l) * first eoaputati jonai u s a l j aepxn,I a m AW in. _ 11-20 ZZ(2) * second coaputat lonal depth, in. A I A . AIA Al 21-30 ZZ(3) » third eoaputati onal depth, in. 31-1*0 ZZ(k) m fourth coaputat lonal aeptn, m. ll-50 ZZ(5) * fifth eoaputati onal depth, in. 51-60 ZZ(6) « sixth eoaputati onal depth, in. 61-70 ZZ(7) * seventh compute viVIM i UC|M«U, iB .-1----1 * ' » 1 1 1 1 1 41j4 s|4 s|4 4 | 4s| 44 4t |4s| 4s| 4r | 44| 4e | s 7 E i | e 2 |e 9 E4 49|44|47 71 | t ^ 7 9 | 77 74 71-B0 ZZ(8) * eighth coaputational depth, in. (if needed) i**|*7|»|»|«p J-L 74 (74 74 74 44 1 HHH» -w- aaaaaaa ai .. . » tu . . . j , . . m s ) . . ^ ■ ■ ■ g (3). . . . ■ . ■ w ... . , . P i5 ) . . , . . . MQ . . . , ,. mn ...... mn .., u ssv nwocr ox ocpuu is greater uhu oO , continue______Ai______A- lo______nextA data card using saae format until the number of depths is satisfied. 1 1 • a « ...... < 1 1 * 1 » 1 * 1 i 1 1 1 1 1 1 1 1 i i i i i . i j 1 | | ZZ(XiZ) • 1 1 1 1 * ii i AA. A Alili.

Bote: In the aain program of GHEVIT, ZZ(i) has a dimension of 99. If ftflaiftiaift ZZ falls on the interface, two coaputatlonal points iuse generated, I ll 1 A A 1 A A one in each layer* Subroutine WHEELS is dimensioned for 15 points » 1 1 * j — i —i 1— »— only.. Therefore, if subroutine WHEELS is utilised, 1 the maximum « 1 * 1 « 1 1 1 1 1 * 1 1 1 1 < 1 1 1 1 AA A. AA A AA

number of ZZ*s is 15. However, i t should be remembei red that at each i- Interface depth. ZZ(l) represents 2 ZZ*s in storage. * 1 ft « 1 a 1 1 j « > j« : 7 • 1 • jio n j «a|«sj«a|is is

WES 1233 EDITION o r SC» *2 WILL DC USED UNTIL EXHAUSTED

Figure B-U. (sheet 5 of 8) GENERAL PURPOSE OA TA FORM

Figure B-U. (sheet 6 of 8) GENERAL PURPOSE DATA FORM

4S vO

Figure B-U. (sheet 7 of 8) GENERAL PURPOSE DATA FORM

Figure B-U. (sheet 8 of 8) GENERAL PURPOSE DATA FORM

Figure B-5. Input guide for CHEVTT, nonlinear analysis (sheet 1 of 11) GENERAL PURPOSE OATA FORM

U l to

WES 1233 EDITION o r SEP «2 «ILL BE USED UNTIL EXHAUSTED

Figure B-5. (sheet 2 of 11) GENERAL PURPOSE DATA FORM

DATE

b e q u e s t e d b y C H E C K E O B Y PACE

* *! 4 ! * j* 7 • • *jnjt» 17 *• ** *• 17 »>¡12 »j»« 2} 21 24 2S 2»|n 22 22 30 j 31 »2 21 14 M H 27 » 3»j 41 142 4 1 4 4 A 4 S 47 4S si »«[n 03 2¿j 23 «4 C || m |«7| I 7« 71173 7« 72172 7? 72 1 If the number of layers is greater than 1», continue to next data card using same format until the number of layers is satisfied.

»

6 Layer Thickness Card (8F10.0) Columns 1-10 HH(l) * thickness of layer 1» in. 11-20 HH(2) * thickness of layer 2, in. 21-30 HH(3) * thickness of layer 3» in. 31-bC HH(U) « thickness of layer U9 in. ui U> i»l-50 HH(5) * thickness o f layer 5* in. »1 12 «1 14 II H n • 2 « 2 « « 9% 99 I 71 72 74 7» 72 1 51-60 HH(6) * thickness of layer 6, in. 61-70 HH(7) • thickness of layer 7, in. 71-60 HH(8) « thickness of layer 8, in. (if needed)

_J__ I__ I__ I__ L_ -4—A- Mi.L .PP L MM M i l . HR(6¿ S ill

llll i t i i i i i i t i i. If the number of layers is greater then 8, continue to next data card using same format until the number of layers is satisfied.

!.. !.. I__ I__ I__ I__ L _J—I__I— I 1— 1 I__ I— L_ - I __ I__ I___I__|_ HH(XNS-l)

,11 12 12 14 12 12 17 I t 1» 20 21 22 22 24 22 22 27 22 22 30 31 22 22 24 22 22 27 41 42 42 44 42 42 47 42 21 22 23 24 22 22 27 22 22 70> 71 71 73 74 72 72 77 72 7» I

WES 1233 COITION OF SEN 22 WILL 22 U2C0 UNTIL EXHAUSTED

Figure B-5. (sheet 3 of 11) GENERAL PURPOSE DATA FORM

p r o g r a m

r e q u e s t e d b y PREPARED BY CH E C K E D BY I i t r 'i 1 ! »! 3j oj ojo | 7 I* j T io Yljta 1} fjl> [l«|t7jl»ho » it i 22| 23 24 2 S j2 « j2 7 20{2» 30 31 32 33 34 3f 36 37 3 0 1 30 40 41 46 47 40 40 «0 01 02 S3 04 90 06 07 SO 60 61 62| 63 j6 4 1 0 9 6 6 6 7 I 72170 74 70 7« 77 1

Ko. 7.1 Nonlinear Offset Card (F10.0)

Columns 1-10 XiR ■ number of offsets to use for computing reference stresses

to determine the average modulus of the nonlinear layers.

_ i — I 1 L - 1 X .

-i~i— i_ j_I— - I — I I I---- 1 J----1 1 L_

_J__I— I-----1-----L_

I t 1 ^ 4 —I___I___I---- I— I— lo. 7*2 Nonlinear Offset Distance Card (8F10.0)

Columns 1-10 RR(l) * distance to first reference stress, in. 11-20 RR(2) * distance to second reference stress, in. 21-30 RR(3) * distance to third reference stress, in. 31-1*0 RR(U) ■ distance*to fourth reference stress, in.

Ul-50 RR(5) * distance to fifth reference stress, in. 72 72 74 70 76 1 Ln 51-60 RR(6) * distance to sixth reference stress, in. - r 61-70 RR(7) * distance to seventh reference stress, in. 71-80 RR(8) * distance to eighth reference stress, in. (if needed)

.Mil m W

If the number of offsets is greater than 8, continue to next data card using same format until the number of offsets is satisfied.

-l.-i-.l__ 1 _J__ l_ _ i___I ___ j. j

_ J ___I___1___J__ L_ -1-11 1 1 I

-U-U 1 | 2 3 I 4 0 6r j t 7 fj 2 j 9 10| 11 j 12 13 14 lO j 16 17 1« 1»|2P 2 lj 22 23 24 20 26 27 26 j 2» 30 31 32 « | 3 4 | 3 0 36 41 42 43 44 40 46 47 91 02 03 04 99 06 07 6 6 '7 0 7 7 j 76 * i WES j u L " TM4 1233 EDITION OF SEN 62 WILL 6E USED UNTIL EXHAUSTED

Figure B-5. (sheet k of 11) GENERAL PURPOSE DATA FORM

REQUESTED BV PREPARED BY CHECKED BY OP EEC to It! 12 IS 14 is to 17 to 1» 30 21 22 22 24 2S 20 27 20 2» 10 It 22 22 24 3S 20 27 SO 41 42 42 44 40 40 47 40 40 10 01 S2 02 Sojos 00 07 00 Qo|oQ 0t 02 02 04 00100 07 1 74 70170EH 77]70 TtlOO

4 0*1 01 -1_I-I_I_I_I_L- _l— I-1_I_L

_j— i_i i i i— _J-1_I_I_I_L- - I—I__ I------1— X _ 1 _ J __ J__ L_ For problems involving only single-wheel loads, a blank card should be placed after No. 11 input. Thus ends the input into the main program. For problems Involving multiple wheels, the input continues.

No. 12 Multiple Wheels Card (F10.0) Ln Columns 1-10 XNV * number of wheels of multiple-wheel gear assembly. v£> ■+-4- -4- -4—4- « ! s j o j~7 o t j to! ti ; 12! t* to to to 17 ioj to 20 21 22 22 24 20 20 27 20 a » 24 SO 20 27 20 30 41 42 42 44 40 40 02 03 04 00 00 07

. m

No. 13 Coordinates Card (2F10.0) Columns 1-10 XVf(l) * X-coordlnate of first wheel, in. 11-20 YV(1) * Y-coordinate of first wheel, in*.

-J ___I-----1___I_4__

-J __ I___L _ I - X _ l___I___I___I__ I___ L_ ___I___I___I___L_ _J— I_L

■I I i ! ! M I ! -f-(- , , . -Í- ' i * I * i * ! * I * iT i * ! * i'°i ” i '*!'* 14 tOi 101 17 IO 10 20 21 22 23 24|2S 20 27 20 20 20 31 32 33 34 30 30 37 3». 30 41 42 43 44 40 40 47 01 OS S3 04 SO 00 57 SO 50100 0t| 02 03 04 00 { ► 07 00 <0 70 7« 72 73 74 70 70 77 7» 70 I

COITION OF SEP 02 WILL SC USED UNTIL EXHAUSTED Figure B-5. (sheet 9 of 11) GENERAL PURPOSE DATA FORM

r e q u e s t e d b y PREPARED BY CHECKEO BY PAGE

1 ¡ 2 j 3j 4! s 's |7 • [» |l»iu itljlsj« « |lS 1« IT jls jltj» 31 SojsS 34 35 IS 37 38 SsjsB 41 |42|43¡44|ss|«s|«) stlszlsa sJssissIsT ssjss so |s iis 2 j ssjss {ssjss ST 40 89 t o ]T1 | t ^ T 3 | t 4 | t s | t o | TT j TO j T9 SO '‘ • “ " F F - F - * ! “ * 1 * 1 » This format is repeated until the numbe r of wheels XNW is satisfi i aaaaaaaaa aiiiiaaaa

...... W ( 2 ) . . . . aaaaaaaaa a a a 1 a a » a a iitaiiaii

...... V0>. ... . aaaaaaaaa aaaaaaaaa a ■ a a a a a aa

* « i • i i i i * a a a a a a a a a aaaaaaaa a a a a a a a a a a

x u t r m ) r w ( x a w ) . . . A r *. . ■ i . ■ i i ■ i * i ■ > aaia a a a a a aaaaa aaaa a a a a 1 a a a a aaaaaaaaa

i a a a a a a a a ___a__a__a__a___a__a__a__a___a__aaaaaaaa a aaaaaaaaa

No. I k Grid Card (F10.0) 1___a__a__ a__ a__ I__ a a a a a a 1 1 a

Columns 1-10 XNGRID * number of arid generator cards to follow _ . « . . . aaaa aaaaa a a a a 1 a a a 1 a a a a a a a a a aaa aaaaa a

__1__1___1___1___1__1__1___1__1_____I___a__ii__a___i__a__a__a___a____i___i__i__a__1__a__1__a___a____a__1__1___a »__1___1__1___1_____1___1__1___1__1___1__1__a___1__ CT* , , , ,30)51} ID , , ¡ . O a a 1 1 1 If i 1 1 j ai 3j « 1 4 ■ 6 ! 7 |s j 9 jlo jl1 :1 2 j'3 14 IS IS IT is i s a> 2 l| 22¡2S¡24 2s |a s 27 29 29 » 31 32 33 34J 3S 38 37 39 39 40 41 4 2 4 3 4 4 4S «L 4 8 49 19 SI 5 2 j S3154 s s s s ST 90 S o lio S it S2ÌS3 SojsS SS ST s s js s 70 T1 74)75 7S 77 70¡79 SO 4 * 1 j iiiiiiiitj iiitiiiii __t__1___1___1__1___1__1___1__1__ aaaaaaaaa

! No. 15 Grid Generator Card (6F10.0) a a a a a a aaaaaaaaa aaaaaaaaa! aaaaaaaaa aaaaaaaaa uoiumns x-iu ax * initiai A-coorainaxe or gria 11-20 YI * initial Y-coordinate of grid _ 1 a a a a a . a 1 a t a a a a 1 21-30 SX * step sise in X-direction 31-^0 SY * step size in Y-direction. a a a a a a aaaaaaaaa aaaaaaaai aaaaaaaaa a a a a i a a a a Ul-50 XNUM * number of lines in X-direction . 51-60 YNUM « number of lines in Y-coordinate of grid a a a a a a aaaaaaaaa a a 1 a a a 1 a 1 a a a a a a_ a a a

___l__1__ii___a__a__ a__a__a__a___ aaaaaaaaa a 'a a a a 1 a a a a a a i a a a a a a a a a a a a a a

» « * » * « •• * i i * i i » * « • « i I a a a a a a a a a i ■ a a i a a aaaaaaaaa 1 a a a a a a a a X I YI SX SY x m é Y N U M t » • « » « * • • i i itiiritii 1 1 i i i a i 1 1 aaaaiiili 1 1 a a _ a a a a a a- a a a aa a la

aaaaaaaai 1 a a. a a a i ■ ■ (Repeat this card fpr each XNGRIO) 1 t t f f 1 a ill l j « | s S ] 7 ’ 0 I 9 jlO , 111 12 111 14 IS i s j IT IS 2lJ 22 23 24 25 as 27 28 » 30 31 32 33 34 35 as 37 38 39 40 41 4 2 4 3 44 45 48 O r i s i 49 90 91 s a js s 54 s s 54 ST 58 j 58 j SO sijssjssjss ss SSIST S s |s 9 * TO T1 72 73 74 Tsjssj TTj 7oj79jsO —L , i i . i i i l i .1 - 1 . ± j - 1 _ -, .i-L - WES 1233 c o it io n o f SCP S2 WILL OE USEO U N TIL EXHAUSTEO

Figure B-5* (sheet 10 of 11) GENERAL PURPOSE OATA FORM

REQUESTED BY PREPARED BY CHECKED BY OP t : 11, 14Î11 12jl3 14 IS 14 17 1« 1» ®(*1 22 23Ì24 2S 24Ì27 24 29 30 31 32 33 34 3S 34 37 3f 34 4t 4» SO SI S2S3 » 71172s j 73 74179174 77 7S 79 00 This completes the input for one problem. If another problem follows, return to Bo. 1 and use the START option. If no additional problems are desired» use the END option.

ON • ’ 9 I w i n 30 31 32 33 34 3S 34 37

n il 10 11 12 13 14 IS 14 17 14 19(20 21 22 23 j 2412S 34 27 31 32 33 34 3S 34 37 S4 SS S4 S7 94 39 40 41 42 43 44 4S 44 47 44 71 72 73 74 7S 74 77) 74 79

WES ” 7 « 1233 TlON e r SE»» 42 W IL SC USEO u n t i l e x h a u s t e d Figure B-5» (sheet 11 of 11) GENERAL PURPOSE DATA FORM

PROGRAM OATE

REQUESTED BY PREPAREO BY C H E C K E D B Y p a g e OP I ’ l l — i— 1 » i ^ %' 4| s i* '7 ! • !» I f 17 21 22 20 2c! 27 2C 20 30 31 32 33 34 30 30 37 30 30 40 41 42 43 44 40 40 47 40 40 00 01 02Ì03 04 00 00 07 SO 9» 00 Cl Ics'cslocioo 00 07 oojeo 70 71 toIto 74 70 00 ___ 1______,___ ;___ i_ i i___ !_ » 3 =

___1___l___i___1___i___I___i___1___1__ 1___1___1___1___1___1___1___1___1___

k i ,Sâi ,Q, g ii • i . w , h t i e . ; X t9 a <* 0 B.R 8 u . b ft r a i d . lB 1 lO j 0 0 Û lC Q Y J u n e T i C i B i D i . 1 ( : L i li ni e % r ) ■ t ! 3 , 0 l O, i 10 . . ___ 1___1___1 _l___1___1___1___1___ 1___ 1 1___ 1___ __ i J 1__Ji iL LIA

u , . i __i__ 1 A__ 1___1___1___ 1___1 - 1 i___ 1___1___1___ 1___1___1__ 1___1__ 1 ! 1__ 1___1 1___1___I___1 1 1 . . 1 , 5 OiOiOiOi•i , L j Q ^ 5 i • 1 ■ 7 .8 .0 , 0 0 • i* l 5 ______6 i O i 0 0 0 1* L 5 i 5 i 5 l5_i 5 i,5 ___1______O n i V i______i___t___ i • 3 ,. 1 1 1 1 I 1 • 3 , 5 , . , i 3 j 0 j .O iLi_J ______111111111 i i i i i i i i i •i it ti it i i i t i il i i i * ii,ii iii it ...... 31 • 1 1 1 1 1 1 1 ___i___i___i___i titillili

! ! 1___1___L__i___1___1___1___J ? » . S i i j1______i___ .L l 2 jLSJLP. 8 __ i1___ 11111111 1— • 1*11*111 Iltlllll 1 j 1 t 1 1 1 i 1 1 lj titillili

i t i i i i i i i i l i Q i . 1___1___1___1___1___I___1___1___ 1___1___1___1___1___1___1__j ___1___ 1 • 11111(11 ___i___i___i___i__i___i___i___i___i______i___i___i___i___i___i___i___i___i___i___i___i j___i___i___i__i___i___i___:___i__i___i___i___i__i__i___i___i___ . 1___1___l J___1 1 * L ÌUXJ i___i 3 i 8 t » i___i___ ,___i___i___ i___L ^«^1 * 1___1___ 1___1___1___U - l___ 7 0 X 0 lO ; i i 5 i O 1 *1 ___ 1 : 2 i 0 i O n t i j i 2 , 5 ' 3 . 0 , 0 , ___i___i______1___1___1___ 1 ___i___ T * j 2 1 Ji« |sj« j7 jcj» loj n. la'uj IS 1C 17 1C it JO 21 22 22|24 20 20 JO 32 33 34 30 34 37 30 30 40 41 42 43 44 40 40 47 40 40 so 01 02 00 00 07 SO loo CO|C1|02 csjccjo s 07 co! 00 70 72 7» 74 70 70 77 to It o ! oo h 3 3 1 » e ___1___ l” ___»___i___ i B . 1 , »a tn , X t ,c , a i f d _ , h . e . lX j ___i___ 1 1 • i ■ 1 1 1 1 t 1 1 1 1 1 1 1 1 1. 1 1 1 A . 1 J 1 1 1 iiiii«iii iiiiiiiiijtiiiiiiit

S i T i A j R i T i ___1___1___1___1 ■___ ,___1___«___I___1___1___,___« I______1___1___I___1___1___1___I___I___1___i___ 1___1___1___I___1___1___1___1___1______1___ 1___1___1___1 1___1___1___1______1___ t ___1___ 1___1___ 1___J___1___ 1___ !___ 1___ 1___ 1___ 1___ 1___ 1___1___ 1___1____!___ 1___1___ 1___1___ 1___1___ 1 1___ 1___

i _ 8 U b 9 1 ]i i iBiO tBi_ij_Q a ' .7-!*.' 7 T v , i n i T i a i n , d e ,m t u c f i R 6 Vj l i l C j V,1 Cll L J i U n . e . T B , p i i L I V E A.F.)

1 i 6 , 0 , 10 10 10 ■. i i i ! 2 . 1 , 0 . . I j Oi lQj ___il___ I___ 1___1______1______i___ i___ i i i i » » * « Ì i l i 1 ___1___1___1___I___L I___1___1___1___1___ 1___| l ___1___1___1___1___1___1______1___1___1___1___1___1___1___ 1___1______i___i___ i___i___ i___i__j ___i___ i______1___ 1__ 1___1___1___1___1___1___1___!___1___1___1__ 1___1__ 1___1___A 1

,0,0, 0, 0, , 0 , . , 5 . T 0 1 , 5 1 7 . 8 . 0 1, !\ o k •5 6 0 t ill 0 0 1« k ___i1___ .5 *5 ,5 l5 i ___1___ L _JL t k ___t___i__ ...... i 3 ^ ! 3 . 5 . . i i___i ___1__ 3 . o . o . . . __ i1___ 1 ___ i i i i i i i i i llllfllil ! 1 1 1 1 11111 î ...... ' j Oi y 1 » 1 1 L 1 1 • 1 1 1 1 1 « 1 i 1 i i i i i i i i i j >i i i i i i i i i titillili tl Iltlllll ¡1111 llllt . 2 |_M ___1___1___1___1___1___1___»___ _ I___1___I___1___1___ 1___1___1___1 i i « * i i i_ i l î i i i l t i i l i lliltlilft* titillili lllll llllj (111 11111 i J k c j l__ 1___ 1___ 1__ J ___ 1___ 1___ 1 3 , 8 , . , i j A___ ltiiiii il) i i it i i i i i • i i i i i i i i 1 1 1 1 1 1 i 1 1___1___I J___1__j___1___1___1___L_____1__1___1 1__ll___1___L 1 j U ! * i ___i___1___1___1___1___ 1___i i ___1___1 i 1___1___«___1___1___1______1___1___1___1___1 - 1___1___L 1___1___1___1__1___1___1__1 i __1___1___ • t t i i t t ■ i ___1___ 1___1___1___1___1___1___L 1

i v J 1 j f } j J : 0 , • f it i ~ l 3 J 4 j 5 j 0 j 7 ; 0 j 0 jlö ( 11:12ll3 ,4 , s 1C 23 !» 20 20 32 33 34 30 36 37 3B 30 40 41 42 43 44 40 40 47 40 40 SO 01 0 . 03 04 00 SO 07 SO; CO 01 02 03 c o le s ¡0 0 7 ooicoi 71 72 73 74 70 70 JiU 1 * 1 F “ 1 n i i!31 1" _Q_ —L _ 1 5 3 WES ;uf“ 1233 COITION OF SCP C2 Wl LL SC USCO UNTIL EXHAUSTED

Figure B-6. FORTRAN listing for CHEVTT, linear analysis (sheet 1 of 2) GENERAL PURPOSE DATA FORM

Figure B-6. (sheet 2 of 2) GENERAL PURPOSE OATA FORM

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Figure B-7• FORTRAN listing for CHRVIT, nonlinear analysis (sheet 1 of 3) GENERAL PURPOSE OATA FORM

Figure B-7. (sheet 2 of 3) GENERAL PURPOSE DATA FORM

Figure B-7* (sheet 3 of 3) APPENDIX C: NOTATION

f = constant determining rate of relaxation h = interval between two points k1, k2, k3, ki+ = constants resilient modulus Nr new Mp used in the i ^ iteration new used in the (i + l) iteration

p(x) = second order polynomial W = surface deflection = subgrade strain ez 0 bulk stress x - x^ y h o , = deviator stress d a + a + 0 = bulk stress x y z

67 REFERENCES

1 . Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw- Hill, 1951. 2. Burmister, D. M. , "The General Theory of Stresses and Displacements in Layered Soil Systems," Journal of Applied Physics, Yol 16, 19^5, pp 89-9^ 126-127» 296, and 302. 3. Mehta, M. R. and Veletos, A. S., "Stresses and Displacements in Layered Systems," Civil Engineering Studies, Structural Research Series No. 178, 1959» University of Illinois, Urhana, 111. k. Sowers, G. F. and Vesic, A. S., "Vertical Stresses in Subgrades Beneath Statically Loaded Flexible Pavements," Bulletin No. 3^-2, PP 90-123» 1962, Highway Research Board, National Academy of Sciences— National Research Council, Washington, D. C. 5. Vesic, A. S., "Theoretical Analysis of Structural Behavior of Road Test Flexible Pavements," Report 10, 1962, National Cooperative Highway Research Program. 6 . Vesic, A. S., "Validity of Layered Solid Theories for Flexible Pave­ ments," Proceedings, International Conference on Structural Design of Asphalt Pavements, 1962» pp 283-290. 7. Ahlvin, R. G. et al., "Multiple-Wheel Heavy Gear Load Pavement Tests," Technical Report S-71-17 (Vols I-IV), Nov 1971» U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss. 8. Chou, Y. T., "Evaluation of Nonlinear Resilient Moduli of Unbound Granular Materials from Accelerated Traffic Test Data," Nov 1975» prepared for Department of Transportation, Federal Aviation Adminis- stration, Washington, D. C., and Office, Chief of Engineers, U. S. Army, by U. S. Army Engineer Engineer Waterways Experiment Station, CE, Vicksburg, Miss. 9. Michelo, J., "Analysis of Stresses and Displacement in an n-Layered Elastic System Under a Load Uniformly Distributed on a Circular Area," International Report (unpublished), Sep 1963» California Research Corp., Woodland Hills, Calif. 10. Barker, W. R. and Brabston, W. N., "Development of a Structural De­ sign Procedure for Flexible Airport Pavements," Aug 1975» prepared for Department of Transportation, Federal Aviation Administration, Washington, D. C., by U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss. 11. Monismith, C. L. et al., "Asphalt Mixture Behavior in Repeated Flexure," Report No. TE 70-5» 1971» University of California, Berkeley, Calif. 12. Dehlen, G. L., The Effect of Nonlinear Material Response on the Behavior of Pavements Subjected to Traffic Loads, Ph. D. Disserta­ tion, University of California, Berkeley, Calif., 1969*

68 13. Southworth, R. W. and Deleeuw, S. L., Digital Computation and Numerical Methods, McGraw-Hill, 1965, PP 291-295» lU. Chou, Y. T . , Hutchinson, R. L., and Ulery, H. H . , Jr., "Design Method for Flexible Airfield Pavements," Transportation Research Record, No. 521, 197^.

69 In accordance with ER 70-2-3, paragraph 6c(l)(b), dated 1$ February 1973* & facsimile catalog card in Library of Congress format is reproduced below.

Chou, Yu Tang An iterative layered elastic computer program for rational pavement design, by Yu T. Chou. Vicksburg, U. S. Army Engineer Waterways Experiment Station, 1976. 69 p. illus. 27 cm. (U. S. Waterways Experiment Station. Technical report S-76-3) Prepared for Office, Chief of Engineers, U. S. Army, and Federal Aviation Administration, Washington, D. C., under Inter-Agency Agreement DOT FA73WAI-377. Report No. FAA-RD-75-226. References: p. 68-69.

1. Computer programs. 2. Finite element method. 3. Layered systems. 4. Pavement design. I. U. S. Army. Corps of Engineers. II. U. S. Federal Aviation Administration. (Series: U. S. Waterways Experiment Station, Vicksburg, Miss. Technical report S-76-3) TA7.W34 no.S-76-3