a . . . . . 1:. _: fin..1 .. 11.9.3.5: . ,2 .M i=3. . 1171:... 3 . : .a 3:». : ("3:1 yfiuu ..7)1 :2. r ‘ .Yv )i..; .C. . 151....xtxz a 1 .Oorv . Av ii .L‘T. .2. I 2..., Fr. . (is? . 9.1%.? .3 va; ; .Ahtfit . 9.134,. 111:2... . ; é I is 5.} 55.... VA. . 9!. ‘t_..‘ ”:1. 11.1. . ‘ Inn, . .- f1... THE8t8 Illlllllll lllllIlllllllllllllllllllllll 31293 01411 1110 This is to certify that the thesis entitled AN ANALYSIS OF THE LONGITUDINAL REINFORCEMENT IN A JOINTED REINFORCED CONCRETE PAVEMENT presented by Julie Marie Vandenbossche has been accepted towards fulfillment of the requirements for Masters degree in Civil Engineering Zfl Major professor Date 3/4/96 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution LIQRARY Michigan State 1 University PLACE N RETURN BOXto romanthh chodtoufllom your record. To AVOID FINES return on or baton dot. duo. DATE DUE DATE DUE DATE DUE MSU I: An Afflnnntivo Action/Equal Opportunity institution WM! AN ANALYSIS OF THE LONGITUDINAL REINFORCEMENT IN A J OINTED REINFORCED CONCRETE PAVEMENT by Julie Marie Vandenbossche A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1995 US Sp pr pr cf ”0' pr pr ABSTRACT AN ANALYSIS OF THE LONGITUDINAL REINFORCEMENT IN A J OINTED REINFORCED CONCRETE PAVEMENTS by Julie M. Vandenbossche The longitudinal steel in many jointed reinforced concrete pavements (JRCP) designed using current procedures have failed prematurely, resulting in excessive crack widths, spalling, faulting, loss of subgrade support, etc. A review of current procedures shows that only extensional tensile stresses are now considered in design. A new design procedure must be developed that will consider all sources of stress and will prevent premature failures. This study identified the sources of stress in the longitudinal steel of a IRCP, evaluated the effectiveness of current longitudinal steel design procedures and included an analysis of the stress state of the longitudinal steel that can be used to check the adequacy of any given reinforcing design. The analysis revealed that even pavements which contain a sufficiently high amount of longitudinal reinforcing, such that all stresses are within their allowable limits, are failing prematurely. One reason for this may be because neither current design procedures or the proposed mechanistic analysis consider fatigue. A mechanistic design procedure which aCCOUI load It desi g1 accounts for fatigue can be developed after the effects of multi—axial loading and varying load magnitudes are determined. Once this is done, a truly comprehensive mechanistic design procedure can be developed. DEDICATION This thesis is dedicated to my parents and my sister and brother, for always being there with support and encouragement; to my grandmother and grandfather for being an inspiration; and to my friends, for helping me keep things in perspective. The suc assistan The M1 Tramp. Deterit thesis i Comp]: f0r he thesis. prOVii ThCE nude] iv ACKNOWLEDGMENTS The successful completion of the research described herein required the input and assistance of many individuals the author gratefully acknowledges: The Michigan Department of Transportation and the Great Lakes Center for Truck Transportation Research for funding the research project ("Factors Affecting the Deterioration of Cracks in Jointed Reinforced Concrete Pavements") which inspired this thesis topic and provided her with both research experience and financial support while completing her degree. Dr. Thomas Wolff for being the author’s advisor and Dr. Thomas Wolff and Laura Taylor for helping with the administrative duties required for the defense and submittal of this thesis. Dr. Ronald Harichandran and Dr. Gilbert Baladi for the technical contributions they provided during the production of this thesis. The author wishes to extend a special thanks to her technical advisor, Dr. Mark Snyder, under whose direction this research was performed. The help he provided with the grammat apprecia annng: grammatical and technical editing and his constant support and encouragement was greatly appreciated. Most importantly, Dr. Snyder helped the author realize the importance in striving above academic mediocrity. LIST 03 LIST 0: KEY Ti CONVE .N .N .N .N N. L 1. '_/. L. 'J) t ) OUNTQ 3. cu WWWLAJ TABLE OF CONTENTS Bag; LIST OF TABLES ............................................................................................... v1 LIST OF FIGURES .............................................................................................. vii KEY TO SYMBOLS .' ........................................................................................... x CONVERSION FACTORS .................................................................................. X11 1. INTRODUCTION ......................................................................................... l 1.1 Problem Statement .................................................................................. 1 1.2 Objective .............................................................. 3 1.3 Scope ........................................................................................ 3 2. CURRENT DESIGN PROCEDURES ........................................................... 5 2.1 Traditional Reinforcement Designs ......................................................... 5 2.2 Uniform Temperature and Moisture Changes ......................................... 5 2.3 Subgrade Drag Design ........................................................................... 12 2.4 Modified Subgrade Drag Theory ............................................................ 14 2.5 Revisions to the Modified Subgrade Drag Theory ................................... 15 2.6 JRCP-S Computer Program .................................................................... 17 2.7 Comparing he Procedures ....................................................................... 22 2.8 Summary ................................................................................................ 23 3. CURRENT DESIGN PROCEDURES ............................................................ 26 3.1 Introduction ........................................................................................... 26 3.2 Stresses Due to Differential Expansion and Contration ........................... 26 3.3 Effects of Uniform and Differential Temperature and Moisture ............... 29 3.4 Traffic Loadings ..................................................................................... 31 3.4.1 Load Transfer Concepts ............................................................... 32 3.4.2 Group Action of the Reinforcing System ...................................... 43 3.4.3 Bearing Stress .............................................................................. 53 3.4.4 Shear Stress ................................................................................. 61 3.4.5 Bending Stress ............................................................................. 61 3.5 Curling and Warping ............................................................................ 66 3.5.1 Curling Due to Daily and Seasonal Environmental Changes ........ 67 3.5.2 Contruction-Related Warping ..................................................... 71 3.5.3 Warping Due to Seasonal Environmental Changes ........................ 72 3.6 Combined Stresses ................................................................................. 74 3.6.1 Combined Bending Stresses ........................................................ 74 3.6.2 Combined Axial Tension and Bending .......................................... 75 3.6.3 Combined Axial Tension and Shear .............................................. 76 CONCEPTS FOR THE MECHANISTIC DESIGN OF THE LONGITUDIANL STEEL IN A JRCP ......................................................... 78 4.1 Introduction ........................................................................................... 78 4.2 Mechanistic Analysis of Longitudinal Steel ............................................. 79 4.3 Summary of the Mechanistic Analysis ..................................................... 91 4.4 The Effects of Fatigue on the Longitudinal Steel ..................................... 92 CONCLUSIONS AND RECOMMENDATIONS ......................................... 96 5.1 Conclusions from an Assessment of the Cument Design Procedures for the Longitudinal Steel in JRCP .............................................................. 96 5.2 Conclusions from the Mechanistic Analysis of the Longitudianl Steel in JRCP .................................................................................................... 95 5.3 Recommendations for a New Design Procedure for the Longitudinal ' Steel in a JRCP ..................................................................................... 99 LIST OF REFERENCES .............................................................................. 103 1&9 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table. 2.6 Table 4.1 Table 4.2 Table 4.3 Table 4.4 LIST OF TABLES Page Friction factors for a variety of materials underlying the slab (8). 9 Reference pavement parameters. .................................................... 11 Frictional resistance factors for various types of base material (25).... 15 Variables considered in the development of the regression equations (26). .............................................................................................. 17 Results of the regression equation analysis for the reference pavement. ...................................................................................... 21 Longitudinal steel required in reference pavement based on various design procedures. ......................................................................... 22 Information required for mechanistic analysis of longitudinal steel in jointed concrete pavement. ......................................................... 80 The magnitude of compressive stress in the bonded steel of the reference pavement for various shrinkage coefficients. .................. 84 Percent load transfer obtained for the AGG value tried in each finite element analysis performed. ........................................................... 87 Results of the mechanistic analysis of the reference pavement. ....... 92 FigUre : Figure I Figure Figure Figure Figure figure Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 LIST OF FIGURES Reinforcement design chart for JRCP from the AASHTO Guide (1). ................................................................................................ Reinforcement required by the modified subgrade drag equation for pavements of various thicknesses. .................................................. Comparison of the effects of slab length on required longitudinal steel for various design procedures. ............................................... Comparison of the effects of slab thickness on required longitudinal steel for various design procedures. ............................................... Comparison of frictional resistance on required longitudinal steel for various design procedures. ....................................................... Visual definitions of deflection load transfer; joint/crack effectiveness and stress transfer. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Slab bending factor calculations for load transfer adjustment. ......... Deflection basin for the reference pavement as determined with JSLAB-92. ............................................................................. Approximate relationship between load transfer efficiency based on deflection and load transfer efficiency based on stress (1 1). .................................................................................... Influence of joint opening on effectiveness: (top) 9-in concrete slab, 6-in gravel subbase; (bottom) 7-in concrete slab, 6-in gravel subbase (7). .......................................................... Load transfer efficiency and joint opening relationship for pavement thicknesses 5 to 13 in (10). OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14 Figure 3.15 Figure 3.16 Figure 3.17 Figure 3.18 Effective wire calculation for the reference pavement. Adapted from (30). ........................................................................ Influence of mechanical load transfer device on joint effectiveness after 12 yrs. of service (31). ....................................... LTEd as a function of dimensionless joint stiffness (AGG/k!) (25). .................................................................................... Friberg analysis vs. JSLAB-92 for determining the effective length of the reference pavement. ................................................... Deflections and deflection load transfer efficiency determined for the reference pavement using JSLAB-92 and a 9,000-1b load applied at the transverse crack.. .............................................. Bending stress in each longitudinal wire of the reference pavement determined using JSLAB-92 and a 9,000—lb comer load applied at the transverse crack. ............................................... Timoshenko‘s equation applied to the reference pavement. Adapted from (30). ........................................................................ Deflections in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. ............................................................................................ Deflections, shear stresses and bending stresses in the critical wire of the reference pavement with a 9,000-lb corner load is applied at the transverse crack. ...................................................... Deflection and bearing stress in the critical wire of the reference pavement when a 9,000 -lb comer load is applied at the transverse crack. .................................................................. Shear stresses in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. ............................................................................................ Schematic of marks on surfaces bending fatigue fractures produced under loading conditions at high and low nominal stress (3). ....................................................................................... 46 48 51 51 52 56 Figure 3.19 Figure 3.20 Figure 3.21 Figure 3.22 Figure 3.23 Figure 3.24 Figure 3.25 Figure 3.26 Figure 3.27 Figure 3.28 Figure 4.1 xi Photo of a bending fatigue failure found in the longitudinal wire of a specimen tested during the referenced research project (6). .................................................................................... Bending stresses in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. ............................................................................ Bending andshear stresses in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. ...................................................................... Effects of curling and warping on steel bending at transverse cracks of a jointed concrete pavement. ........................................... Effects of curling and warping on the shape of a jointed concrete pavement. ........................................................................ Effects of various thermal gradients on the deflections of the reference pavement when no load is applied, as determined using JSLAB-92. ........................................................................... Effect of various thermal gradients on bending stress in the reference pavement, as determined using J SLAB-92. ..................... Example moisture gradient in pcc (6). ............................................ Bending stress induced in the reference pavement for various thermal gradients when a 9,000-lb comer load is applied at the transverse crack, as determined using JSLAB-92. ..................... a.) Shear load applied at the crack face of a longitudinal reinforcing wire; b.) Shear and tensile loads applied at the crack face of a longitudinal reinforcing wire. Adapted from (2). ................................................................................................ The plastic zone at the tip of a fatigue crack in the longitudinal steel. .......................................................................... 66 75 > Shw~rbhxpgfiw?8> e 3e§ %LTE5 %LTEO Pr %TLE WDEC W50 KEY TO SYMBOLS Slab bending correction factor. Spring constant which represents aggregate interlock at a crack or joint. Area of steel per foot width. Diameter of the reinforcement. Subbase/slab friction resistance adjustment factor. Concrete modulus of elasticity. Steel modulus of elasticity. Coefficient of frictional resistance between the slab and the underlying layer. Ultimate concrete compressive strength. Allowable steel stress. Slab thickness. Moment of inertia of the reinforcing wire/bar (I = 1tr4/4). Modulus of foundation support. Distance between two free ends of the slab. Bending moment at the outer fiber of the reinforcement a distance x from the crack face. Bending moment on the reinforcement at the face of the concrete. Applied load. Load transferred by the critical wire. Percent deflection-based load transfer efficiency. Percent stress-based load transfer efficiency. Percent steel required. Maximum load transferred across the joint/crack. Transferred load efficiency. First moment of one-half the cross-sectional area of a single wire with respect to the neutral axis (Q=2r’/3). Radius of the reinforcement. Distance over which debonding took place between the concrete and the longitudinal reinforcement wires. Shear force acting on the reinforcing wire. Total cack width adjustment. Crack width adjustment for differential expansion/contraction. Crack width based on subgrade drag theory. Distance along the reinforcement from the face of the concrete. Deflection in the reinforcement. N ea (“£912.98 M 6a AT ATnu'n K e or E 1 3 2 l1 obear O'bear-al Gem: = 0mm: = 0c Ge: = Gestentp German's! 0L 0': 618M Glens-a1 0' UL Tat TR, 1,). 'Yc Deflection in the reinforcement at the joint/crack. Transverse crack width. Total drying shrinkage. Coefficient of thermal expansion/contraction for concrete. Coefficient of thermal expansion/contraction for steel. Relative stiffness of the reinforcement-concrete system. Shrinkage coefficient for concrete. Strain in the reinforcing at the crack due to uniform thermal and moisture changes. Measured deflection on loaded side of joint or crack. Measured deflection on unloaded side of joint or crack. Deflection measured at the center of a load applied in the center of the slab. Deflection measured away from the applied load at a distance equal to the distance between the sensors used in determining load transfer across cracks and joints. Crack width as determined with the regression equations. Joint opening. Difference between temperature at hardening and present temperature. Minimum annual temperature drop in concrete. Modulus of reinforcement support. Radius of relative stiffness of the concrete. Distance into the concrete, where the second point of contraflexure exists. Poisson's ratio of the concrete. Bearing stress in the reinforcement. Allowable bearing stress in the reinforcement. Bending stress in the reinforcement. Allowable bending stress in the reinforcement. Maximum tensile stress in the concrete. Total stress in the embedded steel due to differential expansion/contraction. Maximum stress in steel away from crack caused by temperature differentials. Stresses in the steel away from the crack due to shrinkage of the concrete. Maximum bending stress on the loaded side of the joint or crack. Maximum stress in the steel due to foundation friction. Tensile stress in the reinforcement. Allowable tensile stress in the reinforcement. Maximum bending stress on the unloaded side of the joint or crack. Allowable shear stress in the longitudinal steel. Frictional resistance between the slab and the underlying layer. Maximum shear stress in reinforcing wire at a distance x from the transverse crack. Unit weight of pcc. m mils inches feet pound-force I112 in in lbl'm2 lb°in lb/ft3 degree Fahrenheit 3 4 xiv CONVERSION FACTORS 19 micron centimeters meters Newton Pa N°m kg/m3 degree Celsius Multiply by 0.2540 2.5400 0.3048 4.4482 6.4516 16.3871 41.6231 6894.76 0.1298 16.0185 Tc = (T; - 32)/1.8 CHAPTER 1. INTRODUCTION Many jointed reinforced concrete pavements (JRCP) have been constructed. The common premature failure of the longitudinal steel in JRCP designed using current design procedures suggests that a more accurate analysis of the actual stress state in the steel must be performed to provide the basis for new design procedures that will prevent premature failures. 1.1 Problem Statement JRCP are designed to crack in the transverse direction. Generally speaking, concrete paving slabs eventually crack to form joints/cracks at a spacing (in feet) of 1.5 to 2.0 times the thickness of the slab (in inches). Transverse cracks caused by curling, warping and drying and temperature shrinkage typically develop within the first few years of the pavements’ service life. Transverse cracks sometimes initiate as contractions of the slab, caused by drying and thermal shrinkage, are restrained by the friction between the slab and the underlying layer. Crack initiation also occurs as the pavement is fatigued under repeated heavy loads, especially in combination with critical curling and warping stresses. Once the crack is present, most JRCP designs rely on aggregate interlock to transfer shear loads across the crack. The cracks deteriorate and open with time and accumulated loadings, thereby reducing the aggregate interlock load transfer capabilities. The inability of the pavement to transfer load across the crack increases slab deflections. As the 2 longitudinal reinforcement fails and the crack opens, debris and water infiltrate the crack, leading to pumping, faulting, spalling and other forms of crack deterioration. Eventually, a progressive loss of foundation support beneath the slab develops as pumping continues. The loss of support greatly increases the magnitude of load-related stresses in the slab, thereby initiating additional cracking. It is important that the transverse cracks remain tight so long-term load transfer capabilities are maintained and rapid deterioration of the pavement does not occur. The purpose of the longitudinal reinforcing steel in a pavement is to keep the transverse cracks from opening so that the fractured faces are held tightly together, thereby maintaining the aggregate interlock load transfer capabilities and keeping excessive moisture and incompressibles from penetrating the crack. Current longitudinal reinforcement design procedures guard only against tensile failure from stresses induced during thermal contraction and drying shrinkage. A closer look at the fracture surfaces of longitudinal reinforcement in both failed field installations and laboratory test specimens suggest that this steel often fails through a combination of bending fatigue and tensile forces (6, 32). Current JRCP reinforcement design procedures underestimate the quantity of steel required for good crack performance throughout the pavement’s expected service life because they typically consider only one component of the total stress state that typically develops in longitudinal pavement reinforcement. The total stress state of the 3 reinforcement must be better defined and analyzed before a more comprehensive design procedure can be developed. 1.2 Objective The objective of this study is to perform a comprehensive analysis of the stresses induced in longitudinal reinforcement for JRCP and use this analysis to evaluate the adequacy of currently accepted JRCP reinforcement design procedures. 1.3 Scope This thesis includes a description of the potential sources of stress in the longitudinal reinforcement of JRCP, an analysis of these stresses, a demonstration of how these analysis concepts can be applied to determine the adequacy of a given JRCP longitudinal steel design and an evaluation of past and current reinforcement design procedures. The sources of stress considered include vehicle loadings and temperature and moisture changes. The discussion of temperature- and moisture-related stress is two-fold, including the effects of gradients in the slab and the effects of differential expansion/contraction between the concrete and steel. Concepts are presented for performing a complete stress analysis of the longitudinal steel. The stresses induced by uniform and gradient thermal and moisture conditions and traffic loads are analyzed to quantify the total stress state. The tensile, shear and bending st allowable of the long Finally. 0 the reinfo described 4 bending stresses that are induced in the steel by these combined loads are compared with allowable values. This procedure is then applied to a typical JRCP to check the adequacy of the longitudinal steel design. Finally, current design procedures are evaluated with respect to their abilities to address the reinforcement stresses identified above. The drawbacks to these design procedures are described along with possible ways for improving them. CHAPTER 2. CURRENT DESIGN PROCEDURES 2.] Traditional Reinforcement Designs The longitudinal reinforcement in JRCP is typically placed 2 to 3 in below the pavement. Placing the reinforcement above mid-depth helps keep the cracks tight at the surface so water and incompressibles can not infiltrate. Welded wire fabric (mesh) consisting of either deformed or smooth longitudinal wire is generally used. Typical reinforcement quantities range from 0.07 to 0.17 percent longitudinal steel by cross-sectional area of concrete. Longitudinal wires are typically spaced at 6 in (center-to—center) and transverse wires are commonly spaced at 1 ft (center-to-center). Transverse wires are generally smooth and are typically approximately 0.30 in in diameter. 2.2 Uniform Temperature and Moisture Changes Current longitudinal reinforcing design procedures for JRCP determine the quantity of steel required to prevent a tensile failure. The tensile stress addressed by these procedures develops when expansion and contraction of the concrete is restrained by friction with the underlying layers. A slab expands or contracts when there is a change in temperature or moisture which is uniform throughout the thickness of the pavement. Concrete expands as the temperature and/or moisture content increases and contracts as the temperature and/or moisture content decreases. Diurnal and seasonal temperature fluctuations cause the slab to expand and contract as the slab warms and cools. Changes in pavement length due to the passing of seasons are much more significant than those due to diurnal 6 fluctuations in states like Minnesota or Michigan, where large temperature changes take place between the summer and winter months. Other climates, such as deserts, have larger diurnal temperature differentials because of the intense heat during the day and the tremendous radiation cooling at night. Seasonal fluctuations tend to produce large uniform moisture and temperature changes in the slab while daily fluctuations produce large gradients, which are discussed later. Drying shrinkage also affects the volume of the pavement. Concrete is typically placed with a higher water content than is needed for hydration. Some of this excess water can be removed by evaporation, causing the concrete to shrink. The amount of drying shrinkage that develops is affected by aggregate type, quantity and gradation, use of chemical and mineral admixtures, water content and wind, moisture and temperature conditions (9). A study by Janssen, et al (16) demonstrated that drying shrinkage is very small in some cases. Several pavements were instrumented in central Illinois so that changes in moisture content through the slab depth could be monitored for a three-month period after construction. The pavements studied exhibited a loss of moisture within 3/4 in of the surface, and experienced increased moisture contents within 3/4 in of the slab bottom. There were no measurable changes in moisture content at distances greater than approximately 3/4 in from both surfaces. This study suggested that drying shrinkage does not significantly affect the average moisture content in some environments, and may 7 produce little uniform expansion or contraction. Shrinkage may be more significant in arid climates. When using the results from this study, one must realize that the conclusions may be slightly misleading since moisture measurements were recorded for only a fairly short time period (3 months) and these measurements are representative of only one season. Expansion and contraction caused by uniform temperature and/or moisture changes is restrained by friction between the slab and the underlying layer. If bonding or grain interlock occurs between the two layers, this friction is referred to as shearing resistance. Restraining expansion and contraction induces compressive and tensile stresses, respectively, in the slab. The tensile strength of concrete is about 8 to 12 percent of the compressive strength (19). Therefore, the critical condition occurs when slab contraction is being restrained by the friction between the bottom of the concrete and the underlying layer, placing the concrete in tension. These stresses in the concrete continue to increase in magnitude as the pavement continues to contract, until the contraction forces are greater than the resisting frictional forces. At this point, stresses decrease slightly and the pavement begins to move (contract), assuming the magnitude of the stress is less than the strength of the concrete. The greatest amount of movement occurs at the free ends (or doweled joints), with theoretically no movement occurring in the center of the slab. The magnitude of stress varies with the distance between the two free ends (or doweled joints), the thickness of the slab and the friction between the bottom of the slab and the 8 underlying layer. The largest tensile stresses are generally located midway between the doweled joints or free edges. The maximum possible stress can be calculated using the subgrade drag formula (9): 7 fl = c 2.1 o c 2 8 8 ( ) where 6,; = Maximum tensile stress in the concrete, psi; L = Distance between two free ends of the slab, ft; “ye = Unit weight of pcc, pcf; and f = Coefficient of frictional resistance between the slab and the underlying layer, radians. The coefficient of frictional resistance, f, represents the frictional resistance between the bottom of the slab and the top of the underlying layer. A major drawback to the subgrade drag theory is that it is very difficult to determine the coefficient of frictional resistance with any degree of confidence. Tests of slab drag resistance have shown that the apparent coefficient of friction is not constant but increases with increasing slab displacement, reaching a maximum value, just before the slab slides freely (7). The coefficient of frictional resistance is used in equations presented in the AASH'TO Design Guide. The AASHTO Design Guide uses the coefficient of frictional resistance and the friction factor, F, interchangeably. Friction factors recommended by AASHTO for various types of underlying materials are given in Table 2.1 (1). Other experiments conducted suggested values quite different from these. It can be assumed that bonding or 9 interlock is occurring if the friction factor is greater than 3. A friction factor should be carefully selected based on local conditions and materials (9). Table 2.1 Friction factors for a variety of materials underlying the slab (l). Type of Material Friction Factor Beneath Slab (F) Surface treatment 2.2 Lime stabilization 1.8 Asphalt stabilization 1.8 Cement stabilization 1.8 River gravel 1.5 Crushed stone 1.5 Sandstone 1.2 Natural gravel 0.9 Equation 2.1 assumes the entire slab is mobilized. In reality, as previously mentioned, often only the ends of the slab are sliding and areas closer to the center of the slab are not developing full friction so this method may overestimate the stress present in the middle of the slab. A set of pavement design parameters were established for the purpose of comparing various stress analyses and reinforcement design procedures in this study; these parameters are listed in Table 2.2. A pavement represented by these parameters is used as a point of comparison throughout this work and is referred to as the "reference pavement. " The tensile stress in the concrete for this pavement based on Equation 2.1 is 20.3 psi. Stresses in concrete caused by uniform changes in temperature need not be 10 calculated separately from those caused by uniform changes in moisture because the mechanism of resistance to expansion and contraction in the slab is the same. A transverse crack develops when the induced tensile stress surpasses the tensile strength of the concrete. Once a slab is cracked, the reinforcement must resist the tensile forces created as the slab contracts. If the quantity of reinforcement is not sufficient to withstand the tensile stress, then the crack width may increase significantly. An increase in crack width allows incompressibles and water to infiltrate the crack, thereby causing further deterioration. Stress in the steel due to subgrade drag can be determined using the following equation: L h S = 'Y c( ) (f) (2.2) 2 Ar where 0, Maximum stress in the steel due to foundation friction, psi; Slab thickness, ft; Area of steel per foot width, in2. 3" 11 ll ll Based on Equation 2.2, the tensile stress in the longitudinal steel of the reference pavement is 12,700 psi. 11 Table 2.2 Reference pavement parameters. PC Concrete Lonfludinal Steel Description 9-in slab thickness; Smooth wire mesh; 27-ft panel length; 6-in x 12-in mesh; 12-ft panel width; cold-drawn steel; doweled joints 0.33 l-in dia. long. wire; Modulus ofElasticity 3.6 x 10° psi 29 x 10‘5 psi Thermal Coefi‘icient 5.5 x 104501“:1 6.5 x 10*5 ”F‘ Drying Shrinkage 400 x 1045 rads Unit Weight 144 pcf Poisson 's Ratio 0.15 0.30 f’c (28-day) 4,000 psi Tensile strength f ’, = 400 psi fy = 65,000JSI Foundation Materials Unbound granular base Coefficient of friction between pcc and base, f = 1.5 Coefficient of restraint between pee and base, C = 0.8 Coefficient of frictional resistance between pee and base, 1;; = 1.26 psi Modulus of subgrade reaction, k = 250 psi/in Environmental Conditions Placement temperature of pcc = 75 °F Average maximum annual pavement temperature = 110 °F Minimum average monthly pavement temperature = 0 "F Average minimum annual pavement temperature = -20 “F 12 2.3 Subgrade Drag Design Longitudinal reinforcement design procedures for JRCP have historically been based solely on the "subgrade drag" theory; they have changed very little in the last 50 years. The subgrade drag theory designs against tensile failure due to restraint of slab movement by friction or “drag” between the slab and supporting layers. The sole function of the steel, according to this theory, is to hold tight the cracks which will inevitably develop due to temperature and drying shrinkage, not to increase the structural capacity of the pavement. Both the Portland Cement Association (PCA) and AASHTO employ the subgrade drag theory in their reinforcement design procedures. Equation 2.3 can be used to determine the amount of steel required to prevent the crack from opening during thermal and drying shrinkage based on the subgrade drag theory. This equation assumes cracking occurs in the center of the slab where stresses are highest. The area of steel, A,, required for the reference pavement, as determined by Equation 2.3, A; 2 f, is 0.049 in2 per foot width. This is equivalent to 0.045 percent steel by cross-sectional area of concrete. These values assume an allowable steel stress, f., of 48,800 psi. The AASHTO Guide suggests using this equation divided by the cross-sectional area of the slab so that the required longitudinal steel for a JRCP is given as a percent of the total cross-sectional area. A nomograph developed by AASHTO to solve this equation is 13 presented in Figure 2.1. The nomograph indicates that 0.045 percent steel is required for the reference pavement. NouooaAPH sow-:3 P,=-§,£.- xioo TL “in HISO . U 0 -' .2 .1 a .. 3 .'1 a: -: E 3: o 1- .2- s -- '1 x J—i ‘- F's; .‘ i .UI” "‘ c I- -4 - -.. § .3; _: .i- in 5.: 3“ .8- .0 ‘20 a ”a 3 « 1.- 2 no 0’ duo toL Figure 2.1 Reinforcement design chart for JRCP from the AASHTO Guide (1). Design procedures based on subgrade drag theory require that the user estimate the allowable steel stress. AASHTO suggests that the allowable stress (working stress) in the steel be equivalent to 75 percent of the yield strength (1). For Grade 60 and Grade 40 smooth bars, the respective allowable stresses are 45,000 psi and 30,000 psi. The AASHTO Design Guide gives a typical yield strength of 65,000 psi for both smooth and deformed welded wire fabric. This corresponds to a 48,800 psi allowable stress. This design procedure is a simplification of the forces acting on the slab and has been shown to be inadequate (9). A major flaw with the design procedure is that it does not 14 account for the fatigue effects of repeated heavy traffic loadings. According to the design procedures based on the subgrade drag theory, a pavement designed to service one truck loading would require the same amount of longitudinal reinforcement as a pavement which will service one million truck loadings, assuming all other variables are the same. 2.4 Modified Subgrade Drag Theory Researchers at the University of Texas at Austin attempted to develop a new formula for determining reinforcement requirements based on results of their studies and a literature review (23). The main task of the study consisted of determining the force required to push a l4-ft long by 2—ft wide by 3.5—in thick slab across various types and treatments of granular materials. The researchers determined the significance of several parameters on frictional resistance as follows: slab thickness, and therefore slab weight, is insignificant; surface texture of the subbase is insignificant; the thickness of the underlying layer is significant. These assumptions were based on the idea that the plane of failure was in the underlying layer when slabs were tested on bituminous- or lime-treated material. If the slab was tested on a untreated clay or bond-breaker, then the plane of failure was found to be at the interface (29). The test results were used to develop an equation to replace the previously-used dimensionless coefficient of friction, f, with frictional resistance, tn. Frictional resistance is the total frictional force applied at the interface divided by the contact area. Recommended frictional resistance values come from a research project conducted at the 15 University of Texas at Austin which measured the force required to push stacked precast concrete blocks across various types of damp, firmly compacted bases. Table 2.3 shows maximum coefficients obtained from this study for several types of base layers (29). The total area of the bottom of the stack of concrete blocks is 870 inz. Table 2.3 Frictional resistance factors for various types of base material (29). Base Type Coefficient Untreated Clay 1.10 3/4 in. Gravel 1.26 3/4 in. Broken Stone 1.00 Asphalt Stabilized 2.20 Flexible Subbase 3.37 Lime Treated 1.70 Cement Stabilized 15.40 This theory assumes there is no relationship between slab weight and total frictional force, whereas the subgrade drag theory assumes the frictional force is proportional to slab weight and the coefficient of frictional resistance, f. Since it is widely accepted that the total frictional force is proportional to the slab weight, the Texas researchers decided to re-evaluate their theory (20). 2.5 Revisions to the Modified Subgrade Drag Theory The investigators made revisions to the modified subgrade drag formula to include slab thickness. Thickness affects the reinforcement design since the weight of the slab affects 16 the magnitude of stress present and slab weight is a function of slab thickness. With the new revisions, the following formula was derived: 6LT R P: = X 100% (2-4) 3 The percent steel required, P,, for the reference pavement, as determined by Equation 2.4, is 0.050 percent, which is very similar to the quantity of steel determined using the original subgrade drag equation (0.045 percent). This formula varies the required reinforcement with slab thickness, unlike the original subgrade drag equation. Figure 2.2 shows the sensitivity of the required steel to changes in slab length and thickness as calculated by the revised formula. The deficiencies of this formula are similar to those for the original subgrade drag theory. a? 0.5 J 0.4 m <1 E 0.3 0 80 100 120 140 160 SLABLENGTH,ft feh=6 e—h=8 -A—h=10-><—h=12-V—h=l4j Figure 2.2 Reinforcement required by the modified subgrade drag equation for pavements of various thicknesses. 17 2.6 J RCP - 5 Computer Program JRCP-5 is a jointed reinforced concrete pavement analysis computer program used to predict pavement performance. The performance prediction is based on mathematical models developed to simulate pavement performance. JRCP-5 is a useful tool, but it must be remembered that the accuracy of performance predictions for various types of pavement conditions is dependent on the source and size of the pool of variables considered when developing the models. These models are also limited by the number of variables which can be utilized. Kunt and McCullough used this program in the development of a set of equations to be used as aides in longitudinal steel design (20). Table 2.4 Variables considered in the development of the regression equations (20). Variable Low Medium High Reinforcement, % by area of concrete 0.01 0.10 0.30 Wire size, gage 3 4 5 Slab thickness, in 6 ll 15 Slab length, ft 20 60 150 Elastic modulus of concrete, 106 psi1 3.0 5.0 7.0 Subbase friction, radians 1.0 2.0 15.0 Thermal coefficient of concrete, 4.0 6.0 8.0 10“ in/inPF‘ Total drying shrinkage, 10" in/in' 2.0 4.0 6.0 Minimum annual temperature, °F -20 -10 0 Curing temperature, °F 50 75 100 Minimum daily temperature, °F 36‘, 232, 153 67, 55 ,48 93, 84, 82 ' Value at day 1 Value at days 2-6 Value at days 7-28 18 Previous formulas for determining required steel are based on a limited number of parameters. Kunt and McCullough picked ten variables believed relevant and performed a factorial set of runs using the JRCP-5 computer program to determine their significance on crack width, joint opening and steel stress. The variables and ranges chosen are shown in Table 2.4 (20). Both short-term and long-term effects were considered. The short-term analysis considered 28-day conditions and the long-term analysis considered the state of the pavement when pavement temperatures drop to their minimum value (20). J SLAB was used to determine which parameters have significant effects on steel stress, crack width and joint width. Each parameter was analyzed at three levels. The relationships developed between each significant variable and the crack width, joint opening and steel stress were identified using the SAS0 statistical package. The following regression equations were then developed with the pertinent parameters for short- and long-term conditions to determine the effects of these parameters (including steel content) on crack width, joint opening and steel stress (20): Short-Term: 894 l. 72.277(I+ i) (1+ 15 )0-"’(1+ 1,0002 [”67 _ 100 10 2 5 03- h ( . ) (1+ P, )4.2I4(1+b)1.595(1+_1_6 )0,“ 19 3.152 0.0012(14-i) (14.31 )2-025(1;1,0002 )2.217(1+b)1.592 A’ = 100 (2.6) (1+ P;)H9922;1—0)0984 2595 0.014(1+—';) (1+1,0002f“‘(1+p,)’-"7(1+i f’”(1+100.000a. f9“ _ 100 10 M‘ 1: (2.7) 1 —'-'- “‘ (+mf Long Term: 64,565(I+1—L—)18(1+:_R )0-346(I;__ ATrnin )0.897 6’: 100 (2.8) (1+P.i"”(1+b)’-Ih‘”(1+ of" 3.12 0.0023(1+-i) (HT—R )’-,8‘(1+1 0002)’,38(1+100000a,)’37 Ax = (2.9) (1+ p, )95’(1+0-)096 2614 L aoas3(1+—) (1+1,0002)’ 7(1+p_,)”’(1+1—h— "Mariam/"(1+ A1,", ) AV: 1002154 (2.10) (1+3 10 20 where 01c = Thermal coefficient of concrete, radians/°F; b = Diameter of the reinforcement, in; Z = Total drying shrinkage, in/in; ATM... = Minimum annual temperature drop in concrete, °F; E. = Elastic modulus of concrete, psi; 0, - Steel stress at the crack, psi; AJr = Crack width, in; and Ax,- = Joint opening, in. The Percent Reinforcement Optimization (PROl) computer program was developed to solve Equations 2.5 - 2.10. The required reinforcement is found through an iterative process which recalculates each of the six equations, increasing the amount of steel each time until all criteria (adequately small crack width or joint opening and steel stress) is satisfied. Acceptable values for the criteria are chosen by the designer based on the conditions present in her/his area. The minimum amount of steel which satisfies all of the chosen criteria is the steel design. If the values obtained by the regression equations are greater than those chosen, the equations are re-evaluated using a larger percentage of steel. The program also automatically checks to insure that all input variables are within the ranges used to develop each equation (20). The longitudinal steel requirement for the reference pavement, as determined by the regression equations, is 0.22 percent. Table 2.5 summarizes the performance values that were obtained for the reference pavement using Equations 2.5 - 2.10. The long-term steel stress was the controlling factor for this pavement. The crack widths are well below 0.025 in which is typically considered to be an accepted maximum allowable crack width for 21 undoweled joints/cracks. Also, sufficient load transfer for a doweled joint is obtained when a joint width of 0.264 in is present. Table 2.5 Results of the regression equation analysis for the reference pavement. Short term: Steel stress at the crack, (6,) 31,500 psi Crack width, (Ax) 0.000797 in Joint opening, (Ag) 0.156 in Long term: Steel stress at the crack, (0,), 43,700 psi Crack width, (Ax) 0.00100 in Joint opening/3y) 0.264 in An advantage of using these equations over subgrade drag theory equations is that more parameters are considered, allowing a more rational determination of steel requirements. The most significant drawback to determining steel requirements with these regression equations is that repeated traffic loadings are not addressed. This is especially significant in a long-term analysis of the crack width, joint width and steel stress. Also, these equations do not consider thermal or moisture gradients in the pavement. The combined effect of repeated traffic loadings and environmental conditions has tremendous implications on the established criteria and should be further quantified. 22 The effect of coarse aggregate properties was considered during the development of the regression equations. This makes the prediction more accurate for concrete made with particular types of aggregate, but the aggregate properties were based on those found in Texas, so the equations may be less valid for pavements considered elsewhere in the United States. Other aspects which should be considered when determining shrinkage are water-to-cement ratios, aggregate-to—cement ratios, aggregate gradation, etc. (20). 2.7 Comparing the Procedures Table 2.6 presents the steel requirements determined using each procedure previously presented and the reference pavement data presented in Table 2.2. Table 2.6 Longitudinal steel required in reference pavement based on various design procedures. Design Procedure Required Steel (% of x-sectional area) Original Subgrade Drag 0.049 Revised Modified Subgrade Drag 0.050 Regression Equations 0.22 Typical Standard Design 0.07 - 0.17 A substantially higher amount of steel is required to satisfy the regression equations than for the original or modified subgrade drag equations. It is worth noting that the typical standard pavement section generally contains more steel than is required by subgrade drag even though these values are obtained using the subgrade drag equation. This is because 23 most highway agencies have decreased their concrete pavement joint spacings in recent years to improve overall pavement performance, but have not generally adjusted the steel design for the reduction in subgrade drag forces that accompany shorter joint spacings. Before committing to any one of these design procedures, it is best to determine the sensitivity of each procedure to various parameters which are known to have a significant effect on the performance of the longitudinal steel. Several graphs are presented to help show the sensitivity of each design procedure to various parameters. All graphs assume a yield stress of 65,000 psi (20). Figure 2.3 illustrates that the subgrade drag equations result in lower steel requirements than the regression equations for slabs longer than 50 ft; the opposite is true for slabs shorter than 50 ft. Figure 2.4 shows that the designs produced using the original subgrade drag equation are not affected by slab thickness. It is also seen that designs derived from the modified subgrade drag equation are more sensitive to slab thickness than those obtained using the regression equations. Figure 2.5 illustrates that frictional resistance has an equivalent effect on both of the subgrade drag equations. These equations are more sensitive to frictional resistance than the regression equation. 2.8 Summary The subgrade drag theory and the regression equations have great importance, but do not adequately address all of the stresses often present in the longitudinal steel. The most significant stresses neglected are those induced by traffic loads and temperature/moisture 24 gradients. The next chapter describes methods of identifying and quantifying these neglected stresses. Once the types and magnitudes of all the significant stresses are known, a design procedure can be developed to address them all. 0.5 3 . [-7 Z 0.4 H 0 E 0.3 E3 . a: 0.2 /A o 0 / h / E 0.1 E 0 W : § % 20 40 60 80 100 120 SLAB LENGTH, ft [-12:— Regression Equa. -e— Subgrade Drag £1— Mod. Subgrade Drag] Figure 2.3 Comparison of effects of slab length on required longitudinal steel for various design procedures. °\° 0.5 Q 1’ g m .. E 0.3 8 .. m 0.2 8 .. 0.1 g 0 i e i i i i 4 6 8 10 12 14 16 SLAB THICKNESS, in te— Regression Equa. —e- Subgrade Drag -a- Mod. Subgrade Drag Figure 2.4 Comparison of effects of slab thickness on required longitudinal steel for various design procedures. 25 \° °. 0.5 E .. E 0.4 0 8 0.3 ‘ g 0.2 o 1» I: 0.1 1 m 0 i : i 4 i r e . 4 t =4 0 1 2 3 4 5 6 FRICTIONAL RESISTANCE, psi + Regression Equation —°— Subgrade Drag —e— Mod. Subgrade Drag Figure 2.5 Comparison of frictional resistance on required longitudinal steel for various design procedures. CHAPTER 3. ANALYSIS OF LONGITUDINAL REINFORCEMENT STRESSES 3.1 Introduction The total stress state of the longitudinal reinforcement is very complex. Each source of stress must be identified before the total stress state can be defined. The effects of uniform thermal and moisture changes were discussed previously along with the manner in which current design procedures account for these stresses. Other effects produced by uniform temperature and moisture changes which are not considered in current design procedures, such as differential expansion/contraction of the concrete and steel, should also be considered. Furthermore, the effects of load-related and curling/warping stresses must be determined. Only after all of these stresses are identified and quantified can a mechanistic-based design procedure be developed to consider the combined effects of these stresses. 3.2 Stresses Due to Differential Expansion and Contraction Stresses in the longitudinal reinforcement at the crack caused by uniform temperature and moisture changes have been described and quantified using subgrade drag theory. Stress is also induced in the embedded longitudinal steel by differential expansion/contraction. Since the coefficient of thermal expansion/contraction for steel, (1., is typically greater than that of concrete, etc, the two expand and contract at different rates when they are unrestrained. Debonding occurs in the smooth longitudinal wires between the two 26 27 transverse wires on each side of the crack so differential expansion/contraction does not induce stress in this area. Bonding is assumed to be present in the remaining portions of the longitudinal steel which are away from the crack. Stress induced in the bonded areas of the longitudinal reinforcement by thermal expansion or contraction of the surrounding concrete is equal to the product of the difference between the unrestrained expansion or contraction strains of the two materials and the elastic modulus of the reinforcement: oDes-temp: AT(a3'Ca¢-)Es (3.1) where 0mm, = Maximum stress in steel away from crack caused by differential thermal expansion/contraction (positive = tension), psi; C = Subbase/slab friction resistance adjustment factor, typically assumed as 0.65 for stabilized subbase, 0.80 for granular subbase; AT = Difference between temperature at hardening and present temperature, (positive = temperature decrease) F°; and E, = Steel modulus of elasticity, psi. Differential expansion/contraction also occurs as the moisture content of the concrete changes because, unlike steel, concrete expands with increasing moisture contents and shrinks with decreasing moisture contents. The swelling which occurs if an unreinforced specimen is soaked continuously for several years is only about one-fourth the shrinkage if the specimen is air-dried for the same period (19). Therefore, the compressive stresses induced in the steel by shrinkage of the concrete are much higher than the tensile stresses induced by expansion. The shrinkage coefficient, 6,, for unreinforced concrete ranges 28 from 300 - 11,00 x 10" in/in (l). A drying shrinkage of 400 x 1045 in/in was assumed for the reference pavement. The majority of the drying shrinkage will have occurred twenty- eight days after placement if the relative humidity is maintained at about 50 percent or lower (19). The stresses in the steel away from the crack due to shrinkage of the concrete, o,,,,,,,.~,,, can be estimated as: 0 esmoist = “CECE; (3.2) The total stress in the embedded steel, o”, away from the crack due to differential contraction can be derived from Equation 3.1 and 3.2 as: 0e: =[AT(0tr'Cac)-CECIE, (3.3) It should be noted that Equation 3.3 is very sensitive to the shrinkage coefficient value, so care should be taken in selecting an appropriate coefficient. The compressive stress in the longitudinal steel caused by differential expansion/contraction in the reference pavement when the ambient temperature is at its average minimum annual temperature is 750 psi. The compressive stress in the steel when the ambient temperature is at its average maximum annual temperature is 8,670 psi. The difference between these two values is large because tensile stresses are induced by the differential change caused when a decrease in temperature occurs and compressive stresses are induced in the steel as the concrete shrinks. In this scenario stresses induced by thermal expansion are counteracted by the stresses induced by drying shrinkage. A 29 temperature greater than that at the time of placement induces a compressive stress in the steel so the total stress is increased. This suggest that, unlike the stresses induced in concrete by a temperature change, the critical stress in the steel will typically occur when the temperature in the pavement is at its maximum. The 8,670 psi maximum compressive stress induced in the embedded steel is less critical than the 12,700 psi tensile stress induced at the crack due to foundation friction (as computed previously using Equation 2.2); therefore, stress analysis at the crack does control, as has been assumed in most traditional reinforcement design procedures. 3.3 Effects of Uniform and Differential Temperature and Moisture Changes on Crack Width The effect of uniform temperature/moisture changes on crack width should be determined since crack width has a large effect on the stresses induced in the steel by traffic loads. When the stress in the reinforcement at the crack face has been determined using subgrade drag theory (Equation 2.2), the resulting effect on the crack width can be estimated. This width can also be adjusted to take into account the differential expansion/contraction of the concrete and steel. Dividing the calculated stress by the steel modulus of elasticity gives the strain in the reinforcing at the crack, 8,, due to uniform changes in thermal and moisture conditions: Lhy f ,=—c 3.4 e ZArE. ( ) 30 It is generally assumed that debonding takes place between the concrete and the longitudinal reinforcement wires over the entire distance, s,, between the transverse wire on either side of the crack when smooth wire mesh is used for reinforcement. Therefore, the wire elongates or strains over this entire distance. The elongation of the steel between transverse wires at the crack can be calculated by multiplying the strain in the reinforcement by the spacing of the transverse wires in the reinforcing mesh: wso = 855: (35) where wsp = Crack width based on subgrade drag theory, in. If deformed longitudinal wires are used, debonding will occur over a much shorter distance (probably equal to the distance between 2-4 deformations). This results in a tighter crack with no increase in steel stress. The narrower crack width also reduces load- related stress in the reinforcing, as is discussed in the next section. The crack width can be adjusted to account for differential thermal and moisture movements of the steel and concrete. For smooth wire mesh, where debonding is assumed to occur over the entire distance between the transverse wires on either side of the cracks, this adjustment due to differential expansion/contraction, ngc, is given as: WDEC = StIAT(Cac'as)+ CBC} (3.6) 31 For deformed wire, s,, should again be reduced to some appropriate value greater than zero but less than the transverse wire spacing. The total adjusted crack width, including components from both subgrade drag theory and differential expansion/contraction analysis, is: w= St[ :17": + [AT(C0tc-01.)+ Cal] (37) S S An adequate quantity of steel should be provided so that the actual stress is lower than the allowable stress and the crack width is less than 0.025 in to maintain adequate grain interlock load transfer. The allowable steel tensile stress is typically taken as 75 percent of the yield stress. As previously stated, the stress in the steel at the midpanel crack due to subgrade drag (Equation 2.2) for the reference pavement is 12,700 psi. This is well below the allowable stress of 45,000 psi given for the reinforcement in the reference pavement. Applying Equation 3.7 to the reference pavement gives a crack width of 0.006 in. This is much less than the 0.025 in crack width criteria previously determined for the reference pavement. 3.4 Traffic Loadings Traffic loads contribute significantly to the total stress state of the reinforcement, but are generally ignored in current analyses and design procedures to simplify matters. However, vehicle loads have been acknowledged as primary sources of stress in concrete pavements since the mid-1920's, when the Westergaard equations were developed. For example, 32 bearing, bending and shear stresses induced by heavy vehicles are considered when investigating dowel bar performance in contraction joints (15, 24). A contraction joint is merely a designed crack, so it stands to reason that these same stresses should also be considered in the design of longitudinal reinforcement where uncontrolled cracking occurs. Furthermore, Zollinger and Barenberg have recently attemped to incorporate bearing, bending and shear stresses in a mechanistic reinforcement design procedure for continuously reinforced concrete pavements (CRCP) (32). The effects of these stresses must also be considered when designing the longitudinal steel in a JRCP if the steel is to function properly throughout its service life. 3.4.1 Load Transfer Concepts The relative portions of applied load transferred across the crack by aggregate interlock and the longitudinal steel must be determined before load-related stresses in the steel can be quantified. The amount of load transferred is dependent upon many variables. Some of the more important parameters include: magnitude of the applied load; stiffness of the foundation; thickness of the pavement; joint spacing; quantity of reinforcement in the slab; crack width and texture of the crack face. Decreasing pavement deflections by providing adequate concrete thickness and placing the slab on a stiff foundation increases the aggregate interlock endurance at the crack by reducing differential vertical movements across the crack. Using greater quantities of steel and/or deformed reinforcing steel helps to keep the cracks tight so the faces of the crack stay in contact. An aggregate type and grading should be selected so that the faces of the cracked slab have good surface texture 33 for optimal load transfer characteristics. In summary, structural and material designs that reduce crack widths and vertical deflections and produce coarser crack face texture provide better load transfer (6). As loads are accumulated across the crack, the crack will begin to open and the surface texture of the crack faces begins to wear down from the abrasion. More of the applied load is transmitted to a smaller area of the foundation (under the loaded slab fragment rather than under both fragments) as the ability of the aggregate interlock to transfer the load across the crack decreases. If the increased stress is beyond the elastic limit of the foundation material, then permanent deformation of the foundation occurs and deflections of the loaded slab increase. Keeping the crack tight ensures that aggregate interlock will transfer a larger percentage of the load for a longer period of time, resulting in better overall long-term pavement performance. Several different approaches can be taken to measure load transfer efficiency across a crack. Calculations based on surface deflection measurements are generally favored because only relatively inexpensive, easily obtainable, nondestructive deflection testing (NDT) is required. The disadvantage of using deflection measurements is that they are difficult to incorporate directly into pavement design (24). The most commonly-used formula for determining load transfer efficiency based on deflection measurements is (24): 34 % LTEs = @9- x 100% (3.8) 5 L where %LTE5 = Percent deflection-based load transfer efficiency, %; 5w. = Measured deflection on unloaded side of joint or crack, mils; and Sr = Measured deflection on loaded side of joint or crack, mils. Although other equations are available, there is a preference for this equation because of its conceptual simplicity and ease of application. Figure 3.1 gives a pictorial definition and calculation of Equation 3.8 as it is applied to the reference pavement by using deflection values obtained using J SLAB-92, a finite element analysis tool for concrete pavements. If the pavement on either side of a crack deflects equal amounts when a load is applied very near to the crack, then each side carries equal loads and is subject to equal stress (24). A 100 percent load transfer efficiency is obtained when the measured loads, stresses or deflections on each side of the crack are equivalent. A theoretical 0 percent load transfer efficiency is obtained when the unloaded side of the crack does not deflect. This implies that none of the applied load is transferred across the crack so the unloaded side is unstressed. Measurement of load transfer efficiency typically falls somewhere between 100 and 0 percent. It is generally accepted that the deflection load transfer should be kept above 75 percent to maintain adequate pavement performance (9). The measured deflection of the unloaded side of the slab is often slightly less than that of the loaded side, even when perfect load transfer exists, due to slab bending. As a result, unadjusted LTE values may underestimate actual LTE. Load transfer efficiency 35 9000 ’b. oenscruou LOAD TRANSFER LT .122; X100 dL LT = 13'9 x100 = 330/. 16.7 ...—J\._i— 90 00 ’b. d._ .__ 15-7miis gm . 13.9 nuts 6 LT =fl x100 dL+dUL JOINT EFFECTIVENESS 2 (13.9 1 =——— 1 = U 13.9mm " 0° 9”" .l_~___ 0 1 dt =16] mils "’0 dut =13.9 mils ' " =125 a q r srness LOAD Tamer-ea = :rELHOO L sraess LT = 2%: “00 a 36% Figure 3.1 Visual definitions of deflection load transfer; joint/crack effectiveness and stress transfer. 36 determined by deflection measurements taken with one point at the center of the applied load and the other 12 inches or more away on the unloaded side of the crack should be corrected for slab bending. Deflections measurements recorded in the center of the slab, as shown in Figure 3.2, can be used with the following equation for estimating the correction factor for slab bending (1 1): where A = Slab bending correction factor; 80 = Deflection measured at the center of a load applied in the center of the slab, mils; and Deflection measured away from the applied load at a distance equal to the distance between the sensors used in determining load transfer across cracks and joints, mils. 5: The deflection basin for the reference pavement is shown in Figure 3.3. The values were obtained for this graph using J SLAB-92. The slab bending correction factor for the reference pavement is 1.03. Unadjusted LTE values should be multiplied by this correction factor to obtain a better estimate of LTE. The effects of slab bending can be ignored if the deflection measurements are taken within a few inches of the crack (1 1). Some researchers prefer to describe the ability to transfer load across a discontinuity as joint effectiveness. The following is used to quantify joint effectiveness, EFF, (14): 25% 5UL+5L) %EFF = X 100% (3.10) 37 CENTER SLAB LOADING SLAB BENDING FACTOR = A = .29. Di A = 6.3 6. A '-'-‘ 1.03 LOADING AT JOINT OR CRACK LTE: 1.03 x ._1_3-_91_ x 100 LTE '2 85% Figure 3.2 Slab bending factor calculations for load transfer adjustment. 38 Figure 3.1 shows an example calculation of Equation 3.10 for the reference pavement. The ratio of stresses on either side of a crack or joint is a useful pavement design and analysis parameter. Load transfer efficiency based on pavement stresses can be calculated, using an equation similar to Equation 3.8, which utilizes measured deflections (11): %LTE., = Mx100% (3.11) 0’1. where %LTE¢, = Percent stress-based load transfer efficiency, %; our, Maximum bending stress on the unloaded side of the joint or crack, psi; and CL = Maximum bending stress on the loaded side of the joint or crack, psi. -2.5 .2. '3 ‘ are- Z .. o ‘4 i. i: 4.5 . U 0 E3” "5“ m -5.5 ‘ Q -6 _ .. 5 1 1 -72 -60 48 -36 24 -1'2 0 12 24 36 48 60 72 86 DISTANCE FROM LOAD, in Figure 3.3 Deflection basin for the reference pavement as determined with JSLAB-92. Load transfer based on deflection measurements is directly but nonlinearly proportional to the efficiency determined by stress. Figure 3.4 depicts the relationship between deflection- based load transfer and stress-based load transfer. Pavement stresses can not be measured 39 directly and must be estimated based on material properties and measured deflections or strains. For this reason, it is generally easier to use pavement deflections for determining either deflection-based or stress-based load transfer efficiency. 100 ‘JS 0 o 00 7O 00 40 20 10 % LOAD TRANSFER (DEFLECTIONI, . P l 1 l l J l l l l O 10 30 4O 50 00 70 IO 00 100 91. 1.0m rnmsren isrnssS). LT a Figure 3.4 Approximate relationship between load transfer efficiency based on deflection and load transfer efficiency based on stress (11). In theory, a crack which is 100 percent efficient will transfer 50 percent of the applied load to the unloaded side of the crack if the applied load is located immediately adjacent to the crack or joint and the slab is not well supported. The theoretical minimum possible transferred load is zero. The actual percentage of load transferred is calculated with the following equation (14): %TLE = x100% I (3.12) 40 where %TLE = Transferred load efficiency; Pr = Maximum load transferred across the joint/crack; and P = Applied load. In practice, the transferred load efficiency is generally less than 50 percent but more than 0. Some of the applied load is transferred to the foundation as the slab deflects. The concrete surrounding the reinforcement often erodes or fails due to high bearing stress, thereby decreasing the potential for load transfer. The magnitude of the applied load also affects load transfer. Larger applied loads produce higher deflections, so the reinforcement is deflected through any looseness and onto sound concrete, producing larger deflections in the unloaded slab fragment. Between 40 and 45 percent of the applied load is presumed to be transferred in dowel bar design, with a value of 42 percent typically assumed (24). Most of this load transfer is attributed to the stiffness of the dowel and its ability to carry the load in shear across the joint. The load transfer mechanism across reinforced cracks is more complex, with both aggregate interlock and reinforcing steel carrying portions of the load. The effect of aggregate interlock is more substantial when analyzing load transfer across cracks in a JRCP because dowels are much stiffer than reinforcing wires and are designed to transfer the entire load. Therefore, when dowels are present, little or no load is transferred across the joint by aggregate interlock. Reinforcing wires in concrete pavements have traditionally been designed only to hold the cracks tight so that loads are 41 carried mainly through aggregate interlock. As loads are accumulated and the faces of the slab begin to wear down due to abrasion, the longitudinal reinforcement will begin to carry increasingly greater portions of the applied loads across the crack. The ability of the wire to help transfer the load will diminish as the concrete surrounding the steel begins to fail due to excessive bearing pressures and/or the steel itself starts to fatigue. Previous investigations have shown large reductions in joint effectiveness provided by aggregate interlock for relatively small increases in joint opening. Figure 3.5 shows the change in joint effectiveness for various joint openings as load cycles are accumulated. Several references suggest that crack widths should be maintained at no more than 0.023 to 0.030 in for aggregate interlock to be an effective and reliable means of load transfer (9, 11). Figure 3.5 verifies these guidelines by showing that a 9-in pcc slab on 6 in of gravel subbase develops significant loss of aggregate interlock when joint openings exceed 0.030 in. Stiffer pavement systems may perform adequately with larger joint openings. For example, a pavement with a thickness greater than 9 in or a stiffer foundation may have a larger crack width and still maintain sufficient load transfer. This is illustrated in Figure 3.6, which depicts load transfer vs. joint opening for slabs of various thickness on a 6-in granular base (32). Assuming that the only function of the reinforcement is to hold cracks tight, and that it does not provide significant additional load transfer capacity, a reinforced crack will behave in the same manner as an unreinforced joint. Therefore, the performance of an undoweled joint will be similar to that of a reinforced crack if the widths of each are small, 42 '00 V i 1 fi fi f V v ‘c' Joint Opening 0.025 in. § 80 . u a. . 3. 60 .0035 . 4 U ._: A g c v . o .2 40 0.045 E . .' O 0.065. . . . a 20 . 0 o . o A A A A O I 2 3 4 5 6 7 8 9 10 Loading Cycles, 100000 1 i Joint Opening 0.0IS-in. ‘ ' 1 8 a) 9 O) O b O Eiiectiveness, percent N , O 0 Loading Cycles, 100 000 Figure 3.5 Influence of joint opening on effectiveness: (top) 9-in concrete slab, 6-in gravel subbase; (bottom) 7-in concrete slab, 6-in gravel subbase (7). 43 such that the load can be transferred across the discontinuity by aggregate interlock. Figure 3.6 (32) shows that an undoweled pavement similar to the reference pavement performed well when the joint width was held to 0.025 in. With a 0.025-in joint, the deflection load transfer was found to be approximately 82.5 percent, so a similar pavement with a Mk of this width would also be expected to have a load deflection transfer of approximately 82.5 percent (32). 0 Load Trsnsisr Etiicisncy, 9i. -- PCA Tsst fissults 80 1. — Extsnosd Results i i Psvsnisnt Thickness 60* 40* 20’ J o l 0 o 1 1 . 0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.07 0.00 0.09 Joint Opening, in Figure 3.6 Load transfer efficiency and joint opening relationship for pavement thicknesses 5 to 13 in (32). 3.4.2 Group Action of the Reinforcing System A portion of the load applied on one side of a transverse crack is transferred across the crack and into the next slab by the reinforcement. Reductions in slab moments and stress due to the reinforcing wire/bars are highly dependent on the location and stiffness of the 44 reinforcement. It can be assumed that the reinforcing wire or bar directly beneath the load (where the maximum moment and stress is located) carries a larger portion of the load than the other wires or bars. The magnitude of the load on this bar must be determined in order to develop a design that guards against shear, bending and bearing failure. Friberg built upon Westergaard's earlier work in his study regarding the group action of dowel bars in a rigid pavement. The same theory Friberg developed for dowels can be used when analyzing the longitudinal reinforcement in JRCP. Friberg concluded that, when dowel bars are utilized, the maximum positive moment occurs directly beneath a load applied at the edge of a slab and the maximum negative moment is located a distance 1.88 from the load. This distance, 1.8i, is referred to as the effective length of the slab, where i is the radius of relative stiffness, a measure of the stiffness of the slab relative to the stiffness of the slab support and calculated with the following equation (13): e = draw 12m - u’) (3.13) where k - Modulus of foundation support, psi/in; and p. - Poisson's ratio of the concrete. Other investigators have re-evaluated the effective length since Friberg’s work with the help of finite element analyses. Researchers currently believe the effective length to be 1.0!} (24). Beyond this distance (1.013), the shear stresses are very small and can be considered negligible (24). The effective length for the reference problem is assumed to be 1.08 = 30.757 in based on Equation 3.13. 45 Based on the preceding information, the wire/bar immediately under the load carries the largest load, and the loads carried by other wires/bars decrease to approximately zero at a distance 1.0! away. Friberg concluded that the load distributed to each dowel within the 1.0! of the applied load could be approximated using a linearly decreasing load distribution. This distribution would assume a unit value at the point of maximum shear (or stress) and decrease linearly to zero at a distance of 1.08 on either side. Therefore, the shear load carried by each bar can be estimated as inversely and linearly proportional to the dowel's distance from the applied load. The number of effective dowels is determined by summing the unit load fractions canied by each individual dowel. The load carried by each bar is then calculated by dividing the unit load fraction carried by the individual bar by the sum calculated above, and then multiplying this quantity by the transferred load. Figure 3.7 illustrates this process for the reinforcing wires in the reference pavement. Multiple applied loads can be addressed using principles of superposition. Friberg's method for calculating the stress in an individual dowel within a group can be applied to the temperature steel provided the following is considered. Aggregate interlock plays a much larger role in transferring loads across relatively tight, lightly reinforced cracks than across relatively open doweled joints. Therefore, a much smaller percentage of the applied load is transferred across the crack by the reinforcement. The portion of the load transferred solely by the reinforcing wires must be determined or estimated before the load can be distributed to the individual wires using Friberg's technique. This 46 1.0 I.= 30. 757” l‘ Longitudinal Joint @000 11.. 9’1I/OOOOOOOOOOO= IV 0.02 4.47 Effective Wires Figure 3.7 Effective wire calculation for the reference pavement. Adapted from (30). determination can be made using available finite element pavement analysis programs, such as J SLAB-92, which was used for this study. JSLAB-92 is a finite element program (originally developed as JSLAB at the Construction Technology Laboratories (CTL) and modified at Michigan State University as JSLAB-92 (13)) capable of estimating pavement deflections and stresses. The program can model two-layered systems (fully bonded or fully unbonded) of up to nine slabs. The foundation is modeled as a Winkler extensionless base. Wheel loads can be modeled as concentrated point loads or distributed area loads. Load transfer across the joints may be modeled as aggregate interlock/keyway, doweled or a combination of the two. Stresses and 47 deflections in the concrete are computed at the nodes of the finite element mesh at the top and bottom of each layer. Shear forces and bending moments in each wire can be obtained at the face on both sides of the crack (32). JSLAB-92 was used to determine the stresses and deflections for the reference pavement. This program treats the aggregate interlock load transfer system as a series of springs. Before J SLAB-92 can be used to predict the deflections which will occur at a crack, a spring constant must be selected to represent the aggregate interlock of the pavement being designed, taking into account crack width and the texture of the crack face. This spring constant is referred to as the AGG value. It can be estimated using Figures 3.6, 3.8 and 3.9 if the crack width is known. As stated previously, the crack width for the reference slab should be limited to 0.025 in. The load transfer efficiency for a 0.025 in crack width in the 9-in reference pavement is estimated as 82.5 percent using Figure 3.6. Figure 3.8 shows that at least one field study also found an average load transfer efficiency of 83 percent for a crack width of 0.025 in. Assuming an 83 percent load transfer efficiency, the AGG/k! value can be estimated using Figure 3.9. The product of this value, the modulus of subgrade reaction, k, and the radius of relative stiffness, I, gives the AGG. The finite element program ILLI-SLAB was used to generate Figure 3.9. The AGG value was backcalculated using numbers selected from best-fit curves through experimental data, so the accuracy of the AGG value has been estimated at :30 percent (20). Results IOO ~ \ - \ - ‘ MEAN RELATIONSHIP 84’ >_. 3 23 80 J 2 ooweuzo JOINTS ii: [Li a: 60 a [LI i5 MEAN RELATIONSHIP 3 non-Downeo .101~rs E 40 7 o 8 —-| -1 20 FLORIDA 001 5 TEST 5an 6 5 j C J | l l l l 0 0.02 0.04 0.06 0.08 0.I0 0.12 100 80 60 LTE 5. 7. 40 20 10'310‘210“ 10° 10‘ 102 103 10 JOINT OPENING, in Figure 3.8 Influence of mechanical load transfer device on joint effectiveness after 12 yrs. of service (31). . Symmetric Edge Lood _ b 4 Nondimensionol Stiffness, AGG/k t Figure 3.9 LTEa as a function of dimensionless joint stiffness (AGG/kl) (25). 49 obtained from ILLI-SLAB are generally very comparable to those obtained using JSLAB- 92 because their structural analyses are similar except for minor differences in their elemental stiffness matrixes. Therefore it is reasonable to use Figure 3.9 to determine the AGG values required by JSLAB-92. Assuming 83 percent load transfer efficiency, AGG/k! = 8.5 (from Figure 3.9). The AGG value for the reference pavement is then computed as 55,300 psi. The accuracy of the AGG value can be checked by analyzing the reference pavement using JSLAB-92 to model a joint/crack which transfers a load applied in the wheel path at the joint/crack by means of aggregate interlock (no longitudinal steel). If the load transfer efficiency calculated using the J SLAB-92 output is greater than 83 percent then the AGG value must be decreased; if it is less than 83 percent then AGG must be increased; if it is equal to 83 percent then the AGG value is correct. Several iterations may be required before the load transfer determined with JSLAB-92 is equivalent to the predicted load transfer. For example, the estimated AGG value of 55,300 psi (obtained from Figure 3.9) can be used as the first trial value for the reference pavement. After several iterations of comparing computed and measured deflections for different values of AGG, the AGG value for the reference pavement was determined to be 36,000 psi, which is 35 percent less than the 55,300 psi estimated using Figure 3.9. 50 When an appropriate AGG value has been determined, the reference pavement can be re- analyzed with the longitudinal steel included. JSLAB-92 was used to analyze the shear induced in each longitudinal wire when a 9000-lb load is applied to the reference pavement at the comer of the crack. It was estimated that 606 lb of the 9000-lb load is transferred across the crack by the reinforcing wires. The critical wire which transfers the largest portion of the load was found to be the wire directly under the applied load. From the JSLAB-92 analysis, the load transferred by the critical wire was estimated to be Pcfit = 252 lb. The Friberg analysis determined Pcm = 187 lb. Figure 3.10 shows a graph of the distribution of shear force in the longitudinal wires at the crack as estimated using both Friberg's technique and JSLAB-92. There is a 26 percent difference between the maximum shears determined by the two methods. Using the maximum of the two (the J SLAB-92 analysis) as the design value is the less conservative approach. Figure 3.10 also shows that the effective length, as determined by both procedures, is approximately 34 in. The deflections and deflection-based load transfer efficiency were determined for each wire of the reference pavement using JSLAB-92 and are plotted in Figure 3.11. A minimum load transfer efficiency of 76 percent was obtained in the first wire from the point of loading; 100 percent load efficiency is estimated beyond the eighth wire. The maximum deflection is 25 mils. 51 a ti 2‘: LL a 1* I 1.0! x -100 i 1 t i 1 i e i 0 5 10 15 20 WIRE NUMBER [-0— JSLAB e Friberg's Method] Figure 3.10 Friberg analysis vs. JSLAB-92 for determining the effective length of the reference pavement. 80 1'25 4O \ 20 O i i i i . i v i 0 5 10 15 20 25 WIRE NUNIBER {-0- Deflection —9— Load Transfer) LOAD TRANSFE , 8 ° 23 f e.‘ DEFLECTION, mils Figure 3.1 1 Deflections and deflection load transfer efficiency determined for the reference pavement using JSLAB-92 and a 9,000-1b load applied at the transverse crack. 52 Figure 3.12 shows the distribution of the bending stresses in each wire on both sides of the crack for the reference pavement, as determined using JSLAB-92. The bending stress is largest under the first wire, as expected, and drops off significantly by the tenth wire. o 5 10 15 20 25 WIRE NUMBER [-e— Loaded Side 4» Unleaded ] Figure 3.12 Bending stress in each longitudinal wire of the reference pavement determined using JSLAB-92 and a 9,000-lb comer load applied at the transverse crack. The preceding analysis of the distribution of the stresses among the wires has shown which wire has the maximum shear and bending stresses. Now it must be determined at what point along this wire the maximum bending and shear stresses are located so that the critical stress state of the reinforcing system can be quantified. Before this can be done, it must be detennined if any looseness has developed around the reinforcing wires due to bearing failures in the concrete. 53 3.4.3 Bearing Stress The magnitude of the bearing stress of the reinforcement on the surrounding concrete can have a substantial impact on the performance of the pavement. This stress is at its maximum at the joint or crack face, resulting in a failure that begins at the joint or crack face and progresses inward when bearing pressures exceed critical values. Such failures provide an effective widening of the joint or crack in the vicinity of the reinforcing wires, resulting in loss of load transfer capacity and increases in steel bending stresses. Bearing stress is primarily a function of the magnitude of the applied load and the spacing and diameter of the reinforcement, but is also affected by slab thickness, modulus of elasticity of the concrete and steel, joint/crack width, modulus of subgrade support, etc. (9). Analytical solutions for calculating bearing stresses have been derived by Bradbury (5) and Friberg ( 12) based on equations developed by Timoshenko and Lessels (26). The work of Timoshenko and Lessels assumes a semi-infinite elastic material extending into an elastic body. When the bar is deflected, the intensity of reaction at any section is proportional to the deflection at that section (16). The analysis of a semi-infinitely long bar is simplified by assuming that the pressures on the reinforcement within the elastic body are negligible beyond some distance away from the point of load application. This simplification is justified provided the length of the steel is greater than the effective length of the dowel- concrete system. Figure 3.13 depicts the load and deflection characteristics upon which Timoshenko and Lessels solutions are based (30). 54 The analytical solution assumes that some fraction of the applied load, P, is transferred to the reinforcement as, P,. The wire carrying the largest portion of P, is the critical wire and the load transferred through this wire, Pm, and should be used in the design analysis. The applied PC,“ results in a deflection of the reinforcement, as shown in Figure 3.13. The transferred load generates a resisting moment, Mo , at the face of the slab equal to one—half the product of transferred load and the crack width. An upward pressure (bearing stress) is exerted on the bottom portion of the reinforcement for a distance it into the concrete, where a point of contraflexure exists. After the point of contraflexure, the bearing pressure is located on top of the reinforcement for a distance (2, where a second point of contraflexure is observed. At this point, the bearing pressure will, once again, be on the bottom of the reinforcement. The bearing stresses are significantly reduced after each point of contraflexure. The equation which models the deflection of the bar (or wire), as derived by Timoshenko and Lessels, is (26): en" . y=——2——[Pc,,-,cosBx-BMJcosBx-smflx” (3.14) 2 B 5,1 where y = Deflection in the reinforcement, in; b = Relative stiffness of the reinforcement-concrete system, in"; I = Moment of inertia of the reinforcing wire/bar (I = 11774), in‘; r = Radius of the reinforcement, in; x .= Distance along the reinforcement from the face of the concrete, in; M, = Bending moment on the reinforcement at the face of the concrete (M.= -P. w). psi; 2 = Crack width, in; and PC... = Load transferred load by the critical wire, lb. 55 The relative stiffness of the reinforcement encased in concrete is given by the following equation (30): Kb = 3.15 B 41/48.! ( > One problem in determining the relative stiffness, B, is in estimating the modulus of reinforcement support, K. The definition of K is the pressure required on the reinforcing wire/bar to produce a unit deflection in the reinforcement-concrete system. Conceptually, K is the same type of spring constant as the modulus of subgrade support, k, typically used in rigid pavement design. Experimentally-obtained K values have varied greatly because of the sensitivity of this parameter to testing procedures, different loads, concrete properties, reinforcement diameters and lengths, concrete support conditions, development of looseness around the reinforcement, etc. (17). K values obtained for dowel bars ranged from 3 x 105 to 2 x 106 psi/in with a value of 1.5 x 106 psi/1n typically assumed when tests are unavailable (24). Although tests have been run to determine K for dowels embedded in concrete, similar tests have apparently not been conducted for longitudinal reinforcement. The modulus of reinforcement wire support should be lower than the modulus of dowel support since the longitudinal reinforcing wires are typically more flexible than dowel bars. For the purposes of this analysis, a constant value of 1 x 10‘5 psi/in is be assumed for K. 56 as fl: igu . O . 57 Fortunately, B varies with the fourth root of K, so the effects of errors in estimating K is relatively small. Figure 3.14 shows the deflections of the critical wire when the reference pavement is loaded with a 9,000-lb load at the corner of the crack. 'v'vv'vvvvvvvv'v'vvvvvvv U: C 1 I I I I 1 1 1 1 1 1 1 I I 1 I I 1 1 1 1 1 1 1 1 1 I 1 I E 1 1 1 \ Max. deflection = 3.4 mils ..................... DEFLECTION, mils 1'6 3. .'.. \. I 1 N 0.) Ur . 1 r . -3.5 1/ i i i i r i i i t 0 1 2 3 4 5 DISTANCE FROM THE CRACK, in Figure 3.14 Deflections in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. Friberg applied Timoshenko and Lessels work to dowel bars, although these equations describe the stresses in the longitudinal reinforcement more accurately because longitudinal reinforcement in a slab is a better approximation of a semi-infinite bar (26). Friberg developed the following equations for bending and shear in the reinforcement based on Timoshenko and Lessels investigations (30): 58 M- e'B‘[ - - ] 318 _-T Pcfi,sran-BM0(Stan+cosBx) (- ) V = 1‘3" [(25 Mo' PCn'r )Si’l Bx" Pen-,cos Bx] (3.17) Plots of these three equations, as applied to the reference slab, are given in Figure 3.15. 10 2 0 '§ -0 V: " 5 -~-10 ; .. ~ - +-20 ,_] 'y E -4000.= 4. : : : 1 : l 809 0 5 10 15 20 25 WIRE NUMBER [-e— Bending _._ Shem + DeflectiQ Figure 3.15 Deflections, shear stresses and bending stresses in the critical wire of the reference pavement with a 9,000-lb corner load is applied at the transverse crack. Given that the concrete slab is much stiffer than the steel wire, the moment at the reinforcement—concrete interface was derived as Mo=-P,zfl (z= the crack width). Substituting Mo=-P,z/2 and x=0 into Equation 3.14, the deflection of the reinforcement at the joint/crack is (30): 59 PCn’: = ___2 3.18) yo 4B2 51( +32) ( (NOTE: After finding a discrepancy while using versions of Equations 3.18 and 3.19 taken from Principles of Pavement Design by E. Yoder and M. Witczak (30), the equations were re-derived. There is an error in the equations given by Yoder and Witczak. Equations 3.18 and 3.19 are the correct versions and should be used as they appear.) The maximum deflection for the critical wire in the reference pavement is 25 mils, assuming a 9,000-lb load is applied at the comer of the crack. The maximum bearing stress on the concrete at the face of the crack can now be calculated as (30): KPCn': 6 car = K 0 = (2+ Z) 3.19) b y 4525:] B ( The maximum bearing stress must be lower than the allowable bearing stress to avoid damaging the concrete around the reinforcement. Damaging the concrete creates looseness which will decrease the load transfer efficiency. The American Concrete Institute (AC1) recommends the following maximum allowable bearing stress (30): M (3.20) cbear-al = 3 60 where ohm.“ = Allowable bearing stress, psi; and f ’c = Ultimate concrete compressive strength, psi. None of the equations presented above account for fatigue damage. Figure 3.16 shows a plot of the deflection and bearing stress in the critical wire of the reference pavement. Since the maximum deflection in the steel is only 3.4 mils, the maximum bearing stress is only 3,410 psi. This is well below the allowable 4,890 psi. BEARING STRESS, psi DEFLECTION, (mils) 0 1 2 3 4 5 DISTANCE FROM THE CRACK, in [—o— Bearing Stress + Deflection ] Figure 3.16 Deflection and bearing stress in the critical wire of the reference pavement when a 9,000 -lb comer load is applied at the transverse crack. 6 1 3.4.4 Shear Stress Equation 3.17 can be used to calculate shear a distance, x, away from the crack face. The shear stress in the steel can then be computed as: . - Q " M (3.21) where 1,, = Maximum shear stress in reinforcing wire at a distance x from the transverse crack, psi; V = Shear force acting on the reinforcing wire, 1b; and Q = First moment of one-half the cross-sectional area of a single wire with respect to the neutral axis (Q=2r’/3), in3 ; Figure 3.17 shows shear stresses calculated using Equations 3.17 and 3.21 in the critical wire directly under a 9,000-lb load applied to the reference pavement at the crack comer. The maximum shear stress in the reinforcement is 3,900 psi and occurs at the crack face. The first point of zero shear stress is located approximately 0.5 in into the concrete; this is also the location of the maximum bending moment, as described below. The shear again approaches zero approximately 2.5 in into the concrete. 3.4.5 Bending Stresses Field and laboratory studies indicate that the longitudinal steel in concrete pavements often exhibits bending fatigue damage and/or failure. For example, Zollinger and Barenberg (32) discuss the presence of bending stresses in the reinforcement at cracks in their paper on mechanistic design considerations for continuously reinforced concrete pavements. In addition, a close look at the reinforcement failures in JRCP sections being removed for 62 SHEAR STRESS, psi t. a. § § \ |/ Max. Shear Stress = 3,900 psi -3000 [ 4000 ' i i i i i ' fi‘ i i i 0 1 2 3 4 5 DISTANCE FROM THE CRACK, in Figure 3.17 Shear stresses in the critical wire of the reference pavement when a 9,000-1b comer load is applied at the transverse crack. full-depth repairs has shown that, when corrosion failures have not yet occurred, the fracture in the steel is predominately caused by bending fatigue and not tensile stresses. The observed steel fractures closely resemble those associated with bending fatigue failure (see Figure 3.18). It is also interesting to note that the failures do not occur at the crack but at some short distance (typically about 0.5 in) into the slab. This was also observed in a study recently completed at Michigan State University, where large-scale reinforced concrete slab test specimens were subjected to repeated applications of loads simulating the passage of heavy truck traffic (6). All loads were applied across transverse cracks which were induced shortly after the test specimens were cast. A close look at the reinforcement after the test specimens failed indicated that the failures were initiated at the wire nearest the edge of the specimen closest to the applied load. An examination of the wire fractures showed classical bending fatigue failures. Frequently, the reinforcing wires 63 would fail in l- to 2-inch fragments, with each end of the fragments exhibiting bending fatigue failures (see Figure 3.19). Thus, bending stresses have been shown to be critical stresses in the reinforcement, even though current longitudinal reinforcement design procedures fail to address these stresses. The magnitude of the bending moment in the reinforcement at a distance, x, from the crack can be determined using Equation 3.16. The maximum moment can be computed by determining the distance from the crack face at which the shear force is equal to zero, and then solving for the moment at that distance using Equation 3.22 (30). . fix M = -M€-—./I+ (1+ [32 )2 (3.22) 213 Stress in the steel generated by these moments is determined by substituting the moment into the following equation: M 0...... = —5 (3.23) I where 61,.“ = Bending stress in the reinforcement, psi; and M = Bending moment at the outer fiber of the reinforcement, in-lb; Figure 3.20 shows the bending stress in the critical wire of the reference pavement. The bending stress at the crack face is only approximately 880 psi, but reaches a maximum 64 stress of 15,900 psi 0.5 in into the concrete. The bending stress approaches zero approximately 3.5 in from the crack. High Nominal Stress Low Nominal Stress Figure 3.18 Schematic of marks on surfaces bending fatigue fractures produced under loading conditions at high and low nominal stress (3). 11141111 Figure 3.19 Photo of a bending fatigue failure found in the longitudinal wire of a specimen tested during the referenced research project (6). 65 The bending stress and shear stress curves are superimposed in Figure 3.21, which shows that the magnitude of the bending stress is much greater than that of the shear stress. As expected, the shear stress is equal to zero at the same point where the bending stress is at '5 =- .. M---l a -2000 ' / iii a: / a -7000 - ........ U Max. Bending Stress = 15,900 E T a.12000 ..... Z J' U {a} Ila-17ooo ,,.,.;.,. 0 l 2 3 4 5 DISTANCE FROM THE CRACK, in Figure 3.20 Bending stresses in the critical wire of the reference pavement when a 9,000- lb comer load is applied at the transverse crack. g -2000 ° / " m a. s. / v a -7000 H .. E. I / m-IZOOO v -l7000 : i 1 i : 1 1 i l 2 3 4 5 DISTANCE FROM THE CRACK, in C... Shear Stress + Bending StreQ O Figure 3.21 Bending and shear stresses in the critical wire of the reference pavement when a 9,000-lb comer load is applied at the transverse crack. 66 its maximum, approximately 0.5 in into the concrete. This explains why bending fatigue failures in the steel were found approximately 0.5 in from the crack in the Michigan State University study (6). 3.5 Curling and Warping Curling of the slab develops when there is a temperature gradient through the slab profile, whereas warping develops when a moisture gradient is present. Either type of gradient will cause a change in slab length which varies with slab depth, inducing a degree of curvature in the pavement. The resistance to curvature by the slab weight creates curling and warping stresses in the pavement and the steel. Stresses induced in the steel are especially high across cracks where the steel must bridge discontinuities in pavement slope at the ends of the slab, as shown in Figure 3.22. The effects of curling and warping on pavements and the longitudinal steel are discussed below. Point of Maximum Bending Stress l Figure 3.22 Effects of curling and warping on steel bending at transverse cracks of a jointed concrete pavement. 67 3.5.1 Curling Due to Daily and Seasonal Environmental Changes Daily and seasonal temperature fluctuations induce temperature gradients in pavements. Variations in the ambient air temperature and intensity in sunshine produce relatively rapid fluctuations in the surface temperature of the pavement while the temperature at the bottom changes much more slowly. The slab surface expands as it warms (positive gradient), causing the slab comers to curl downward, (negative gradient) leaving the center of the slab unsupported (or less supported) by the foundation. Stresses at the bottom of the longitudinal edges are greatly increased. Nighttime temperatures often cool the slab surface (negative) gradient, causing it to contract. This forces the slab corners to curl upward, often leaving them unsupported. A positive gradient produces compressive stresses at the top of the slab and tensile stresses at the bottom, whereas a negative gradient produces compressive stresses at the bottom and tensile stresses at the top. Figure 3.23 illustrates the effects of curling on the slab shape. Temperature gradients are generally more pronounced in the spring and summer. In the spring, ambient temperatures are warm but ground temperatures are still cold from winter freezing, so a much larger temperature gradient may exist. Late summer gradients are more pronounced because there is a larger variation between daily and nightly pavement temperatures. Gradients generally range between -1.5 to 3.0 °F per in. A typical maximum daytime gradient, when the pavement surface is warmer than the bottom, is +2.0 °F per in. of slab thickness. A typical maximum nighttime gradient when the top of the slab is cooler than the base of the slab is -1.0 °F per in. (9). 68 {ya-”y NEGATIVE GRADIENT (Top of the slab is dry or cool) POSITIVE GRADIENT (Top of the slab is wet or warm ) Figure 3.23 Effects of curling and warping on the shape of a jointed concrete pavement. The changes in the slab shape induces stresses in the reinforcement. The more critical stresses induced in the steel occur when the steel must bridge the slope discontinuity across the cracks (see Figure 3.22). The magnitude of the stresses in the steel caused by curling is dependent on the degree of curvature of the slab. A finite element analysis can be performed to determine the magnitude of the bending stresses induced when the slab curls. 69 J SLAB-92 was used to analyze the effects of various gradients on the reference pavement. Three gradients were used in the analysis: +2, +1.5 and -2 °F/in. Figure 3.24 shows how the deformation of the slab varies with temperature gradient along the crack face of the reference pavement. The ends of the slab curl upward in response to negative gradients so deflections are shown as positive values. Conversely, positive gradients deflect the ends .p O U C A I A DEFLECTION, mils O 20 40 60 100 120 140 160 DISTANCE FROM THE OUTER EDGE, in [-9— +2 Gradient —v— -2 Gradient ~8— +1.5 Gradierfl Figure 3.24 Effects of various thermal gradients on the deflections of the reference pavement when no load is applied, as determined using JSLAB-92. of the slab downward. Deflections caused by positive gradients are generally smaller than those induced by negative gradients of the same magnitude because upward deflections are not restricted by the foundation. This is why the peak deflection for the -2 °F/rn gradient is 35 mils, while the peak deflection for the +2 °F/in gradient is only 26 mils. Figure 3.24 also shows that slab deflections increase by approximately 20% when the 70 gradient is increased from +1.5 to +2 °F/in. This illustrates how the degree of curvature increases with increases of the magnitude of the gradient. Although curling/warping does not have a significant effect on shear stress in the longitudinal reinforcement, it does induce bending stress. Figure 3.25 shows the effects of various gradients on the bending stresses in the reference pavement. It can be seen that the magnitude of bending stress in a pavement with a +2.0 °F/in gradient is over 1,000 psi ....... . STRESS,psi 3’ c 3’ E E ING 4000 ------- WWW—W“ a - 1 1 n 1 1 1 1 1 1 a 6000 . . . . . . . , . 0 5 10 15 20 25 WIRE NUMBER Ee— +2 Gradient —v— -2 Gradient —a— +1.5 Gradient] Figure 3.25 Effect of various thermal gradients on bending stress in the reference pavement, "as determined using JSLAB-92 greater than that found in the same pavement with a +1.5 °F/in gradient. This is reasonable because, as seen in Figure 3.24, the deflections are also much greater for larger gradients. A comparison of identical pavements with equal but opposite gradients shows bending stresses are larger when the gradient is negative because the foundation restricts 71 the corners of the Slab from curling when a negative gradients is present (see Figure 3.25). 3.5.2 Construction-Related Warping When the moisture content of the slab decreases from the bottom of the slab to the top (negative gradient), the slab comers will warp upward. The slab corners warp downward when a positive gradient exists (see Figure 3.23). A negative moisture gradient induces tensile stresses at the top of the pavement and compressive stresses at the bottom, while a positive gradient produces compressive stresses at the top and tensile stresses at the bottom. Stresses are induced in the reinforcement when the concrete changes shape, but more critical stresses occur as the steel bridges discontinuities in the pavement slope, as described previously for concrete curling. Two primary sources of moisture gradients are drying of the concrete after placement and seasonal changes in the environment. As discussed previously, Janssen et al. showed that the upper 3/4 inch of the pavement typically experiences a loss of moisture while the lower 3/4 inch experiences an increase in moisture content. There were no measurable changes in moisture contents at depths greater than approximately 3/4 inch from either surface ( 16). Figure 3.26 depicts an example moisture gradient from this study. The duration of this study was only 3 months but it is reasonable to assume that the top of the pavement 6- 3 Months DEPTH IN PCC, in h 6 _ 41Month Grimm \__J 1 8 0‘ i; i is” 10 MOISWRECONTENT,% Figure 3.26 Example moisture gradient in pcc (6). will generally have a lower moisture content than the bottom of the pavement, which will be at, or near, saturation. Therefore, there can be significant changes in the moisture gradient after placement. 3.5 .3 Warping Due to Seasonal Environmental Changes Moisture gradients also occur due to seasonal changes in the environment. This phenomena can be seen to a greater extent in drier climates (30). A study conducted in 73 the 1930's by the Bureau of Public Roads noted considerably more warping during a Phoenix winter, when the relative humidity was fairly low and the surface of the pavement was extremely dry, then in other seasons. It was also observed that, as warping occurs, slabs settle into underlying, unstabilized granular layers. This helps reduce some of the curling and warping stresses in the pavement by providing a larger contact area between the bottom of the slab and the underlying layer. In wet climates, the moisture content remains fairly constant throughout the depth of the slab (9). Only a very limited amount of data has been collected regarding stress or strain in concrete slabs due to warping. Determining the concrete moisture content and the amount of slab movement are just some of the difficulties which arise. For this reason, warping stresses are often estimated by treating moisture gradients as “effective thermal gradients” of -0.5 to 0.5 °F/in. These valves can then be added to or subtracted from actual temperature gradients to obtain a composite value that represents the combined effects of both temperature and moisture. The analysis can then proceed using the finite element programs and techniques described previously. The combined effects of the curling and warping are reduced since they typically counteract each other. Cyclic curling and warping stresses (fatigue) are not considered in the development of reinforcement designs. The effects of this omission are probably negligible since the number of stress reversals are small compared to load-related stress cycles. 74 3.6 Combined Stresses Now that the stresses induced by each source have been identified and quantified, they must be combined so the total stress state can be determined. All stresses are linearly additive accept the bending stresses induced by traffic and curling and warping. Once the total stress state is defined, the combined effects of axial tension and shear and axial tension and bending can be evaluated. 3.6.1 Combined Bending Stresses The critical bending stress condition generally results when the stresses due to curling/warping and loading conditions are additive. An example of this would be when the slab comers are curled downward (positive gradient) and the load is placed at the comer of the slab. Figure 3.27 shows the combined effects of a 9,000-1b load applied at the comer of the crack and curling/warping for the reference pavement. The maximum bending stress in the steel at the crack face with the applied load and no gradient is 2,800 psi; with a +2 °F/in gradient and a load, the stress is 7,100 psi; and with a combination of the -2 °F/in gradient and a load, the stress is -l,000 psi. The maximum bending stress at the crack face is in the reinforcing wire directly under the applied load for both +0 and +2 °F/in gradients, but in the wire furthest from the applied load for the -2 °F/in gradient. This is because stress induced by the load is in the opposite direction of the stress induced by curling/warping. As the distance from the load increases, the stresses induced by the load decreases, so the effects of curling/warping dominate. 75 E. 8000 0 vi 6000 ~ gal 4000 ’ m . r- 2000 ~ m .. e 0 0 E -2000 0 E 4000 g -6000 : i : i i + + i = 0 5 10 15 20 25 WIRE NUMBER [-9— +2 Gradient —v— -2 Gradient —a— +0 Gradient] Figure 3.27 Bending stress induced in the reference pavement for various thermal gradients when a 9,000-lb comer load is applied at the transverse crack, as determined using J SLAB-92. These combined bending stresses can be reduced significantly through the use of tied concrete shoulders. Tied shoulders restrict the edges of the pavement from coming off the foundation and provide better support for edge loads. 3.6.2 Combined Axial Tension and Bending The combined effects of axial stress (induced when the pavement contracts uniformly) and bending stress (induced by traffic loads and curling/warping) must be determined In most JRCP pavements, the maximum tensile stress can be obtained by applying the principles of superposition to the bending and axial tensile stresses. The allowable combined shear and axial stress can be considered to be the following, based on the AISC Steel Construction Manual (21): 76 cbend + O’tens 51 (3.24) O.bemi-al O' tens-a! The combined stress for the reference pavement is 0.67, so a sufficient quantity of steel is present to withstand the combined extentional and bending stress. 3.6.3 Combined Axial Tension and Shear The combined effects of axial stress (induced when the pavement uniformly contracts and shear stress (induced by traffic loads) must also be considered. A large axial tensile stress can significantly reduce the steel fatigue life since it pulls the faces of the crack in the steel apart, thereby reducing the friction between the crack faces and increasing the stress concentration at the crack tip tremendously (see Figure 3.29). An axial compressive stress increases the fatigue life of the steel by forcing the faces of the crack together, thereby increasing friction between the two faces and reducing the damage caused by shear stress. The allowable combined shear and axial stress can be considered to be the following, based on the AISC Steel Construction Manual (21): o - 0.3312, (3.25) tens-a1 — 77 1: a, = 0.2213, (3.25) The tensile stress in the reference pavement is only 51 percent of the maximum allowable and the shear stress is only 29 percent of the maximum allowable. Therefore, the quantity of steel in the reference pavement should withstand the combined effects of shear and tensile stress. a. l Shear Longitudinal b . Reinforcing Wires l Shear /| Tension ‘— é _> f Figure 3.28 a.) Shear load applied at the crack face of a longitudinal reinforcing wire; b.) Shear and tensile loads applied at the crack face of a longitudinal reinforcing wire. Adapted from (2). CHAPTER 4. CONCEPTS FOR THE MECHANISTIC DESIGN OF THE LON GITUDINAL STEEL IN A JRCP 4.1 Introduction As stated in chapter 2, current longitudinal design procedures are based on subgrade drag and only protect against tensile failures due to uniform changes in temperature. Chapter 3 showed that other stresses which may affect pavement performance include bearing stresses on the concrete, and bending and shear stresses in the steel. Therefore, a complete mechanistic analysis of the longitudinal steel stresses must consider the maximum stresses determined by subgrade drag theory along with the bearing, bending and shear stresses induced by traffic loads and the bending stresses induced by curling and warping. The stresses in the reinforcement were quantified based on several assumptions: Adequate drainage is provided. QA/QC checks are made on the construction materials. Proper construction techniques are utilized. Joint seals and drains are installed and maintained throughout the life of the pavement. 0000 If any of these assumptions are incorrect, the stress and crack width analyses must be adjusted accordingly. For example, if a known portion of the pavement structure is weaker because poor construction techniques or weaker construction materials were utilized, then this should be reflected in the analysis. 78 79 A step-by-step description of how to perform a mechanistic analysis on the longitudinal steel in JRCP is provided to help designers check the adequacy of their designs. The first step is to determine the geometric parameters, environmental and loading conditions and material properties for the pavement design being analyzed. The tensile stresses in the bonded and unbonded portions of the steel can then be checked. A finite element analysis must be performed before the critical bearing, bending and shear stresses can be determined. After each stress has been evaluated, it should be checked against its allowable value. If all design criteria are satisfied, then the design is adequate. Otherwise, the design should be adjusted and re-checked. 4.2 Mechanistic Analysis of Longitudinal Steel The mechanistic analysis can be broken down into seven steps. A brief description of each step is presented below, along with an example calculation for the reference pavement. STEP 1. ESTABLISH DESIGN PARAMETERS The information required for the mechanistic analysis can be divided into four major categories: geometric design; material properties; environmental conditions and loading conditions. Values used for these parameters-should be representative of those typically found or used in the area where the pavement is being constructed. A list of the required design parameters is given in Table 4.1: 80 Table 4.1 Information required for mechanistic analysis of longitudinal steel in jointed concrete pavement. Geometric Properties Pavement Structure: Steel: Slab thickness Wire diameter Slab width Trans. and long. wire spacing Slab length Deformed or smooth wire Joint design Material Properties Concrete and Steel: Modulus of elasticity Poisson’s ratio Thermal coefficient of expansion Concrete: Steel: Compressive strength Yield strength Unit weight Drying shrinkage (unreinforced) Drying shrinkage (normally reinforced) Foundation Material: Modulus of subgrade reaction Subbase/slab friction factor (See Table 2.1) Subbase/slab frictional resistance adjustment factor (0.80 for unstabilized subbase, .65 for stabilized subbase) Environmental Conditions Temperatures: Average maximum annual temperature of pavement Average minimum annual temperature of pavement Temperature of pavement at final set Load Conditions Design load Design tire pressures Distribution area of the applied load Critical location of the applied load 81 In most states, vehicle loads are restricted to 20 kips per single axle and 34 kips per tandem, with the gross vehicle weight not to exceed 80 kips. Since the axle lengths and the distances between axles are generally greater than the pavement radius of relative stiffness (typically two-to—four ft for all but the thickest slabs), the distributions of stresses induced in the slab by adjacent axles or tires do not overlap. Furthermore, stresses induced in the slab by maximum legal single axle loads are generally higher than those induced by tandems, even though the allowable load is higher for tandems, because the single axle loads are more concentrated. Therefore, the design load for longitudinal reinforcing in JRCP is one half of the 20-kip single axle, or 10 kips. The critical load location is at the transverse crack near the edge of the panel. Typical maximum tire pressure and tire width values are 110 psi and 8 in, respectively. With this information the dimensions of the load distribution area can be determined. These are typical design load conditions; each agency should use the load conditions applicable to their area. STEP 2. DETERMINE THE MAXIMUM TENSILE STRESS IN THE REINFORCEMENT AT TIIE CRACK The primary source of tensile stress is uniform temperature changes. Subgrade drag theory is used to determine the tensile stress in the longitudinal reinforcement at the crack by evaluating Equation 2.2. The allowable tensile stress can be calculated using Equation 4.1 (21). o......=(0.751f, (4.1) 82 where om..-“ = Allowable tensile stress in the reinforcement, psi; and fy = Yield strength of the reinforcing steel, psi. If the calculated stress is not within the allowable limits, possible design modifications include: the use of closer wire spacing and/or a larger quantity of steel (larger diameter wires); use steel with a higher yield strength; reduction of the joint spacing; or reduction of the friction between the subbase and slab. Example calculation for the reference pavement: Applying Equation 2.2 to the reference pavement gives a tensile stress of 12,700 psi. The allowable stress is 48,800 psi. Therefore the tensile stress is within the allowable limits. STEP 3. DETERMINE THE STRESS AWAY FROM THE CRACK IN THE BONDED STEEL Stress is induced in the steel by differential expansion/contraction and concrete shrinkage in areas away from the crack where the concrete and steel are bonded. The magnitude of this stress can be determined using Equation 3.3. The maximum allowable stress is computed using Equation 4.1. If the stress due to differential expansion/contraction and drying shrinkage is not less than the allowable stress, then the steel quantity must be increased (as described previously) or a mix design must be developed with a lower shrinkage coefficient and/or better thermal 83 compatibility. The shrinkage coefficient can be reduced by lowering the water-to-cement ratio, lowering the paste-to-aggregate ratio or using more dense coarse aggregates. The concrete thermal coefficient of expansion can be varied by changing coarse aggregate types. Example calculation for the reference pavement: The coefficients of thermal expansion/contraction typically used for steel and concrete are 6.5 x 10°6 "F1 and 5.5 x 1045 °F', respectively. The shrinkage coefficient has a greater impact on the total stress induced in the bonded steel. Since the concrete is "lightly" reinforced (compared to a member which is reinforced for structural applications), the shrinkage coefficient is probably greater than the 250 x 10‘ inlin used for "normally" reinforced members. A typical shrinkage coefficient for unreinforced concrete is 600 x 1045 inlin. Therefore, shrinkage coefficients for most JRCP will be between 250 x 1045 inlin and 600 x 1045 inlin. The magnitude of the stress in the bonded steel of the reference pavement when the minimum annual pavement temperature occurs is given in Table 4.2 below for various shrinkage coefficients. The stresses evaluated for a shrinkage coefficient of 400 x 1045 inlin are being used in the analysis for the reference pavement, although all values are within the allowable tensile limit of 45,800 psi evaluated in step 2. The thermal coefficient of expansion for steel is greater than that for concrete so a compressive stress is induced when the temperature of the pavement structure rises above the hardening temperature. Therefore, stresses induced are greater when pavement 84 temperatures are above the hardening temperature because the thermal and drying shrinkage stresses are both compressive and therefore additive. Table 4.2 The magnitude of compressive stress in the bonded steel of the reference pavement for various shrinkage coefficients. Shrinkage Coefficient Pavement Temperature (10‘ inlin) Ave. Min. Annual (.20 "10 Ave. Max. Annual (110 T) 600 8,140 psi 16,100 psi 400 3,490 psi 11,400 psi 250 15 psi 7,930 psi STEP 4. DETERMINE THE CRITICAL WIRE Before the critical wire (i.e., the wire carrying the greatest portion of the transferred load) can be identified, several variables must be evaluated. First the load transfer efficiency across the crack must be obtained based on the width of the crack in the pavement. The load transfer efficiency is then used to determine the AGG value, which is needed for the pavement finite element analysis programs. Once the AGG value is known, a finite element analysis program can be used to determine which longitudinal wire is carrying the largest portion of the load across the crack. The critical wire is generally the one closest to (or directly beneath) the applied load unless the radius of relative stiffness is very large and loads applied by nearby tires produce higher shear loads on nearby wires. 85 Determine the crack opening The crack opening is dependent on the differential expansion/contraction between the unbonded concrete and steel, the concrete drying shrinkage and uniform temperature changes. The size of the maximum crack width is dependent on the spacing of the transverse wires for smooth reinforcement and the spacing and type of deformations for deformed reinforcement. The crack width adjustment can be detemlined using Equation 3.7. Load Transfer Efficiency Load transfer efficiency for various crack widths is dependent on many factors. A primary factor is the texture of the crack face. Unfortunately, there is currently no method available for correlating crack face texture to load transfer efficiency based on the aggregate/aggregate blend properties and the concrete mix design used for the pavement. Cracks propagating through a concrete with a weaker aggregate, such as slag or recycled concrete aggregate, will tend to crack through the aggregate, rather than around the aggregate, leaving a smoother crack face with decreased load transfer potential. The angularity and abrasion resistance of the aggregate also affects the load transfer potential at the crack (6). Until further research is completed and the load transfer characteristics for various crack widths can be quantified based on the aggregate and mix design characteristics, Figures 3.6 or 3.8 can be used to estimate the load transfer efficiency for a pavement based on the crack width without consideration for variations in crack texture between mixes, as described in chapter 3. 86 Determining the AGG value The method for determining the AGG value was described previously in chapter 3. An iterative approach must be used to determine the AGG value. An initial AGG value (based on crack width) can be estimated using Figure 3.6 or 3.8 and 3.9. A finite element program can then be used to determine the load transfer efficiency of an unreinforced version of the design pavement (with the assumed AGG value) when a load is applied at the crack, 18 in from the center of the tire to the outer edge of the pavement (in the outside wheelpath). If the load transfer efficiency detemiined by the finite element analysis is greater than that obtained from Figure 3.6 or 3.8, then the AGG value must be decreased; if it is lower, then the AGG value is increased. The finite element analysis is repeated using the revised AGG value and the load transfer efficiencies are compared again. The correct AGG value is obtained when the load transfer efficiency determined by the deflections obtained from the finite element analysis is approximately equal to the load transfer efficiency obtained from Figures 3.6 or 3.8. Determine the critical wire Once the correct AGG value has been determined, the pavement is re-analyzed with the finite element program, only this time the reinforcement is included and the design load is placed at the critical location. The reinforcing wire transferring the largest portion of the load across the crack is the critical wire and the load carried by that wire, Pm, is used in subsequent analyses. 87 Example calculation for the reference pavement: First the width of the crack in the reference pavement is determined. A coefficient of shrinkage of 600 x 10'6 °F1 is used in this analysis because the shrinkage is not restrained by the steel near the crack since the steel is debonded from the concrete in this area. Based on Equation 3.7, the maximum crack width for the reference pavement is 0.006 mils. A maximum allowable 0.025 mils design crack width was chosen for the reference pavement. A load transfer efficiency of 83 percent is obtained from Figure 3.6 and 3.8 for a 0.025 in crack width. The AGG value which provides a load transfer efficiency of 83 percent for an unreinforced version of the reference pavement when analyzed with J SLAB-92 is 36,000 lein/in. Table 4.3 shows the percent load transfer obtained for the AGG value used in each iteration while determining the correct AGG value. Table 4.3 Percent load transfer obtained for the AGG value tried in each finite element analysis performed. AGG Load Transfer Efficiency (%) 71,000 91 45 ,000 86.3 35,000 82.8 36,000 83.2 The reference pavement is re-analyzed with the reinforcement included to determine which longitudinal reinforcing wire transfers the largest portion of the load across the crack. 88 This wire is called the critical wire. The critical wire for the reference pavement was determined to be the first wire on the outside edge of the pavement. STEP 5. DETERMINE THE MAXIMUM SHEAR STRESS The finite element analysis performed to determine the critical wire can be used to obtain the maximum shear force in the critical wire at the crack face. The maximum shear load ' transferred through larger wire is converted into a shear stress using Equation 3.21. The maximum shear stress must be less than the allowable shear stress, tar, calculated using Equation 4.2 (21). a.1=(0.41f, (4.2) The crack width is dependent on the quantity and configuration of the reinforcement. It must be reduced until the bearing, bending and shear stresses are less than their respective allowable values. The quantity and configuration of the steel will control the magnitude of the stresses resulting from a design load applied at a crack of a given width. For example, when equivalent loads are applied, the stresses in larger diameter wires spaced further apart can be smaller than those found in smaller wires spaced closer together even though the total cross-sectional areas of steel are equivalent. Evidence of this is provided by the results of accelerated pavement loading tests performed at Michigan State University, as was discussed previously (6). Repeated loads were applied to a pavement slab specimen with No. 6 bars placed at the transverse crack 89 and spaced such that the total cross-sectional area of steel was 0.27 percent of the crack face. A slab specimen which contained 0.394 diameter wires spaced 6 in apart across the slab such that the total cross-sectional area of steel was 0.23 percent of the crack face was also tested. The specimen with the No. 6 bars was subjected to more than 7 million loading cycles without failure. The other specimen failed before 1.5 million loading cycles were applied. The shear and bending forces distributed across the larger diameter wire produced a lower stress state even though larger forces acted on the bar because the increased bar spacing. This also means that a larger crack width may be acceptable if the reinforcement design is capable of carrying a larger portion of the load without exceeding the allowable bearing, bending or shear stresses, since less load will be transferred across the crack by aggregate interlock. The same study showed the performance of the slab specimens to increase with increasing longitudinal steel quantities, as would be expected. Example calculation for the reference pavement: The finite element analysis determined the maximum shear force in the longitudinal steel of the reference pavement to be 252 lb. Using Equation 3.21, a shear stress of 3,900 psi is obtained, which is well within the allowable value of 26,000 psi calculated using Equation 4.2. STEP 6. DETERMINE THE MAXIMUM BEARING STRESS The maximum bearing stress in the critical wire is obtained by evaluating Equation 3.19 for the maximum shear force in the critical wire. If this value is greater than the allowable bearing stress determined with Equation 3.20, then the bearing stresses can be decreased 90 in the same manner that the shear stresses are decreased. The allowable bearing stress can be increased by increasing the strength of the concrete. Example calculation for the reference pavement: The maximum bearing stress in the reference pavement is located at the crack under the critical wire. The bearing stress is 3,420 psi which is less than the allowable bearing stress, of 4,890 psi. STEP 7. DETERMINE THE MAXIMUM BENDING STRESS The maximum bending stress occurs in the critical wire a short distance, x, into the concrete. The maximum bending stress can be obtained by plotting the bending stresses determined using Equation 3.16 for various distances, x, away from the crack face along the reinforcement. The maximum value will be within the first few inches from the crack into the concrete. If the maximum value is greater than the allowable bending stress, 0mg“. calculated using Equation 4.3, then the bending stresses can be decreased in the same manner that the shear stresses are decreased. obend-al = (0-6)f , (4.3) Example calculation for the reference pavement: The maximum bending moment acting on the longitudinal wire is 57 in-lbs and is located 0.52 in from the crack face into the concrete along the critical wire. The moment induces 91 a bending stress of 15,900 psi, which is less than the maximum allowable bending stress of 39,000 psi. 4.3 Summary of the Mechanistic Analysis Table 4.4 is a summary of the results of all the mechanistic analyses performed on the reference pavement. All individual stresses in the longitudinal steel of the pavement were determined to be less than their respective allowable stresses (see Table 4.4). This mechanistic analysis approach was used to analyze several other longitudinal pavement designs which were based on subgrade drag theory. As previously mentioned, the quantity of steel, in most pavements currently in service (which are overdesigned compared to designs based on subgrade drag theory), is sufficiently large to keep the stresses induced by all forementioned sources below their allowable values. Thus the question still remains as to why JRCP pavements are failing prematurely. As previously described, the allowable stress criteria are based on those used in the design of steel structures. Unlike, many steel structures, the loads a pavement endures are cyclic. Therefore it can reasonably be assumed that, if pavement reinforcing is failing due to fatigue (as has been seen [6, 32]), then using allowable stresses established for more static load conditions would underestimate the service life of the longitudinal steel. 92 Table 4.4 Results of the mechanistic analysis of the reference pavement. Type of Stress in the Wire Maximum Allowable Maximum (%) Stress (psi) Stress (psi) Allowable Tensile stress at the crack ..... acting alone 12,700 48,750 26 ..... acting with shear 12,700 24,750 5] Differential expansion/contraction 11,400 48,750 23 stress away form the crack Shear stress ..... acting alone 3,900 26,000 15 ..... acting with tension 3,900 14,000 28 Bending stress 15,900 39,000 41 Combined extensional and bending 2(Oepphw/Gmowed) 67 Bearing stress 3,400 4,892 70 4.4 The Effects of Fatigue on the Longitudinal Steel Metal fatigue is defined as "a process which causes premature failure or damage of a component subjected to repeated loading" (4). A static analysis of the reinforcement assumes the stresses are distributed across the cross-section of the wire. In reality, there are areas where the stresses are concentrated. Stress concentrations can be caused by imperfections in the steel, initial stresses in the wire which result from cold rolling the metal, discontinuities in the contact pressures between the steel and the concrete created by voids in the concrete or the aggregate particles, corrosion, etc. The areas of increased stress are referred to as points of high localized stress. Even if the stress distributed 93 across the majority of the wire causes elastic strain, the localized stresses may exceed the yield stress. Once the yield stress is exceeded, plastic deformation occurs and there is potential for cracks to initiate. After the crack is initiated, a plastic zone forms near the tip of the crack where the yield stress is exceeded (see Figure 4.1). As the crack grows, the total energy of the system is decreased by the propagation of the crack (4). Longitudinal Rein orcin Wire Concrete /— Crack tip é/ '\ Plastic Figure 4.1 The plastic zone at the tip of a fatigue crack in the longitudinal steel. Factors such as wire diameter, surface texture, the manufacturing process of the wire and the stresses being applied to the wire have significant effects on the fatigue life of the wire. The effects of each of these factors are discussed briefly below. The most significant effect of using a larger wire or bar diameter is a decrease in stress, since the area over which the force is distributed increases. The reduction in stress significantly increases the fatigue life of the wire. It is also interesting to note that a wire 94 with a larger diameter has an increased volume of material being stressed. Since steel fatigue failures develop most quickly at the weakest points in the material (i.e., at flaws), the use of a larger cross-sectional area of steel increases the probability that a flaw will be encountered in the area where the stress is at a maximum. Thus, it is possible to see more rapid failures with the use of larger wires. However, this phenomena does not occur to a substantial degree for wire or bar diameters less than 2 in (2). The surface texture of the wire also affects the fatigue life. Pits and scratches in the steel are added stress concentrations. Sources of these imperfections include corrosion and machining marks. These imperfections act as initiated cracks so the fatigue life of the specimen is significantly reduced. This shows the importance of using a corrosion inhibitor. Concentrated stresses also occur at the comers of deformations on deformed steel. Although, the total stress in the wire is reduced since the deformations also play a significant role in keeping the crack tighter. This reduces the stresses in the wire by increasing the aggregate interlock potential. The method used to manufacture and process the reinforcing wire also has an impact on the fatigue life. Most reinforcing wire (smooth and deformed) used in the United States is cold rolled. Cold rolling the steel produces residual compressive stresses and work- hardens the reinforcing wire surface (2). The fatigue life of a component subjected to stresses can be increased if the steel has been cold rolled since the compressive residual stresses at the steel surface will counteract the high tensile stresses at the outer fiber of the wire induced by bending stresses. This is relevant to the fatigue life of the longitudinal 95 steel, since the mechanistic analysis showed bending stresses to be comparatively large in the reference pavement. There are currently no design procedures for pavement reinforcing that consider fatigue effects. Developing a fatigue-based design for the longitudinal reinforcing wires of a JRCP is extremely difficult for several reasons. One reason is that it is difficult to determine the fatigue life of the longitudinal reinforcing wire when multi-axial loading is present and the amplitudes of these loads are varying: The total stress/strain state, which varies with varying loads and axle configurations, must be understood before the fatigue damage can be quantified. The stress/strain state is also affected by uniform changes in pavement temperature, which affect the tensile stresses in the steel. A high axial tensile stress can significantly reduce the steel fatigue life by pulling the faces of the crack in the steel apart, thereby reducing the friction between the crack faces so the stress concentration at the crack tip increases tremendously. In summary, a comprehensive design procedure for the longitudinal steel in JRCP must be able to account for fatigue. However, the complexity of the stress state in the longitudinal steel and the large variability in the loads that the pavement will experience make the development of such a procedure extremely difficult. These problems must be overcome if a truly comprehensive mechanistic design procedure is to be developed. 96 CHAPTER 5. CONCLUSIONS AND RECOMNIENDATIONS 5.1 Conclusions from an Assessment of the Current Design Procedures for the Longitudinal Steel in J RCP Three design procedures are currently available for longitudinal reinforcing in JRCP: 0 Original Subgrade Drag 0 Revised Subgrade Drag 0 Regression Equations The original and revised subgrade drag design procedures give very similar results for pavements greater than 10 in thick, since they are based on the same theory. For pavements less than 10 in thick, the revised subgrade drag procedure requires an increasing amount of steel as the slab thickness decreases because designs based on the original subgrade drag formula are not affected by changes in pavement thickness. Longitudinal steel designs obtained by using the regression equations are fairly consistent with those obtained using the revised subgrade drag procedure for slabs of varying thicknesses. Since more parameters are used when determining the steel requirements using the regression equations, the steel requirements obtained from them vary more with the geometric and material properties and the environmental conditions for the pavement being designed. The longitudinal steel requirement determined for the reference pavement using subgrade drag theory is less than half the quantity typically used in most states, yet pavements constructed using these methods often fail prematurely in the field. The regression 97 equations produce designs with slightly higher amounts of steel than are typically used. Although increases in steel from the quantities typically used to the quantities required to satisfy the regression equations will most likely prolong the life of many pavements, the suitability of these equations for design purposes is somewhat questionable because they have been used to design only a few pavements which have been constructed only recently. Current design procedures account for some of the stresses present in the longitudinal steel, but other significant stresses, including those induced by heavy vehicle loads, are not considered. The premature failure of the transverse cracks in many JRCP indicates that there is a strong need to develop a design procedure that considers these additional stresses. Before a new design procedure can be developed the stress state in the longitudinal wire must be more clearly defined. This entails quantifying not only the stresses induced by uniform temperature and moisture changes, as is done by subgrade drag theory, but also those induced by differential movements due to thermal expansion/contraction, shrinkage and traffic loads. 5.2 Conclusions from the Mechanistic Analysis of the Longitudinal Steel in a JRCP To perform a mechanistic analysis of the stresses in the longitudinal steel in a JRCP, the following must be determined: 0 the axial stresses in the steel at the crack, 0 the axial stresses in the steel away from the crack, 0 the crack width, 0 shear stresses at the crack, 98 o the bearing stresses of the steel on the concrete, and o the bending stresses. The first part of the analysis consists of determining the axial stresses in the steel at the crack caused by uniform temperature and moisture changes. The axial stresses in the bonded portion of the steel away from the crack are caused by the differential expansion/contraction of the steel and the concrete, and the drying shrinkage of the concrete. Crack width is influenced by uniform temperature and moisture changes in the pavement, differential expansion/contraction and the drying shrinkage of the concrete, reinforcing steel type, strength and quantity and friction between the slab and the subbase. Equations for quantifying these values are given in chapter 3. A finite element analysis program is required to determine the effects of traffic loadings and curling and warping on the reinforcing stresses. Curling and warping of the pavement induces bending stresses in the steel. The maximum stresses in the steel due to the effects of curling and warping are at the cracks, where the steel must bridge the discontinuities in the slopes of the two pavement slabs (see Figure 3.22). The combined effects of curling/warping and traffic loads applied at the crack greatly increase the stress in the steel. These effects can be quantified by using a finite element analysis program to determine the shear being transferred across the crack by the reinforcing wire. The equations provided in chapter 3 can be used to determine the bending and bearing stresses present based on the maximum shear in the steel at the crack. 99 Once the maximum stresses are quantified, they can be checked against their respective allowable values. The allowable values can be evaluated using the equations in chapter 4. If any of the stresses or the crack width are greater than their respective allowable values, then adjustments should be made to the pavement design, as described in chapter 4, so that all calculated values are within acceptable limits. The accuracy of this analysis is dependent upon how representative the material and geometric properties and the environmental and loading conditions used in the analysis are of actual field conditions. It is also dependent upon how well the models used to obtain the stresses and crack width represent actual conditions. 5.3 Recommendations for a New Design Procedure for the Longitudinal Steel in a JRCP There are three major sources of stress in the longitudinal steel of a JRCP: 0 Uniform temperature or moisture changes in the pavement, 0 Differential expansion and contraction between the concrete and steel away from the crack and 0 Traffic loads. As previously mentioned, the quantity of steel, in most pavements are currently in service (which are overdesigned compared to designs based on subgrade drag theory), are sufficiently large to keep the stresses induced by all forementioned sources below their critical values. This does not explain why pavements with apparently adequate 100 reinforcement designs have been failing in the field. One reason for this may be because neither current design procedures or the proposed mechanistic analysis consider fatigue. As repeated heavy vehicle loads pass over the crack, abrasion of the crack face reduces the surface texture. The decrease in surface texture decreases the load transfer efficiency, so deflections increase as loads are applied and larger stresses are induced in the steel. The larger stresses cause plastic deformations and the cracks begin to open. As the width of the crack increases, so do the shear and bending stresses in the steel induced by traffic loads. As the crack opens and the shear and bending stresses induced by the repetitive traffic loads increase, the bending and shear stresses become much higher than the tensile stresses caused by uniform temperature and moisture changes. The analysis performed in chapter 4 on the reference pavement showed the bending stresses induced by traffic loads to be the critical stresses for a crack opening which is considered "acceptable" (9, 7, 11), yet these stresses are not considered by any of the design procedures currently available. It was shown in chapter 3 and 4 that even when the bending stresses are within the allowable limits it is still possible for a bending fatigue failure to occur. A design procedure which accounts for this fatigue induced by the cyclic traffic loads must be developed. A mechanistic design procedure which accounts for fatigue can be developed after the effects of multi-axial loading produced and the varying load magnitudes are determined. 101 To do this, a mechanistic analysis of the longitudinal steel must be performed to determine the effects of various load magnitudes on the stress state of the longitudinal steel. The detrimental effects of the critical stress state of the multi-axial stresses must be determined so that the total fatigue life of the specimen can be defined. Once this is done, a truly comprehensive mechanistic design procedure can be developed. LIST OF REFERENCES 10. LIST OF REFERENCES American Association of State Highway and Transportation Officials, AASHTO Guide for Design of Pavement Structures, Washington, DC, 1986. Bannantine, J .A., J .J . Comer and J .L. Handrock, Fundamentals of Metal Fatigue Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ, 1990. Barsom, J .M. and ST. Rolfe, Fracture and Fatigue Control in Structures- Applications of Fracture Mechanics, Second Edition, Prentice-Hall Inc., Englewood Cliffs, NJ, 1987. Boresi, AP. and OM. Sidebottom, Advanced Mechanics of Materials, Fourth Edition, John Wiley & Sons, Inc., New York, NY, 1985. Bradbury, R.D., Reinforced Concrete Pavements, Wire Reinforcement Institute, Washington, DC, 1938. Bruinsma, IE, 2.1. Raja, M.B. Snyder and J .M. Vandenbossche, Factors Affecting the Deterioration of Transverse Cracks in Jointed Reinforced Concrete Pavements - Final Report, Michigan Department of Transportation, Lansing, MI, March 1995. Darter, M.I. and EJ. Barenberg, Design of Zero-Maintenance Plain Jointed Concrete Pavements - Vol. I . - Final Report, Federal Highway Administration, Washington, DC, June 1977. Dong, M., “User’s Manual of Computer Program JSLAB-92,” Federal Highway Administration, Washington, DC, June 1992. ERES Consultants, AASHTO Design Procedures F or New Pavements - Participant's Manual, National Highway Institute, Federal Highway Administration, Washington, DC, 1994. ERES Consultants, Inc., Concrete Pavement Design Manual, Second Revision, National Highway Institute, Federal Highway Administration, Washington, DC, 1990. 102 ll. 12. 13. 14. 15. 16. l7. 18. 19. 103 ERES Consultants, Inc., Techniques for Pavement Rehabilitation - A Training Course Participant’s Notebook, Third Revision, National Highway Institution, Federal Highway Administration, Washington, DC, October 1987. Friberg, B.F., “Design of Dowels in Transverse Joints of Concrete Pavements,” Transactions, Vol. 105, American Society of Civil Engineers, New York, NY, 1940. Guo, H., T.J. Pasko and MB. Snyder, “Maximum Bearing Stress of Concrete in Doweled Portland Cement Concrete Pavements,” Transportation Research Record 1388, TRB, National Research Council, Washington, DC, 1993. Ioannides, A.M. and GT. Korovesis, “Aggregate Interlock: A Pure-Shear Load Transfer Mechanism,” Transportation Research Record 1286, TRB, National Research Council, Washington, DC, 1990. Ioannides, A.M., Ying-Haur Lee and M.I. Darter, “Control of Faulting Through Joint Load Transfer Design,” Transportation Research Record I 286, TRB, National Research Council, Washington, DC, 1990. Janssen, D.J., B.J. Dempsey, J .B. DuBose and A.J. Patel. Predicting the Progression of D-Cracking, Civil Engineering Studies, Transportation Engineering Series No. 44. Illinois Cooperative Highway and Transportation Series No. 211, UIUC-ENG-86-2005, University of Illinois Department of Civil Engineering, Savoy, IL, 1986. Kilareski, W.P., M.A. Ozbeki and DA. Anderson, Rigid Pavement Joint Evaluation and Full Depth Patch Designs, Fourth Cycle of Pavement Research at the Pennsylvania Transportation Research Facility - Final Report, Vol. 4, FHWAlPA-84-026, Pennsylvania Department of Transportation, Harrisburg, PA, December 1994. Korovesis, GT. and A.M. Ioannides, “Discussion of ‘Effects of Concrete Overlay Debonding on Pavement Performance’ by T. Van Dam, E. Blackman and M. Y. Shahin,” Transportation Research Record 1136, TRB, National Research Council. Washington, DC, 1987. Kosmatka, SH. and WC. Panarese, Design and Control of Concrete Mixtures, Thirteenth Edition, Portland Cement Concrete Association, Skokie, IL, 1990. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 104 Kunt, M.M. and BF. McCullough, Improved Design and Construction Procedures for Concrete Pavements Based on Mechanistic Modeling Techniques, Research Report ll69-5F, Center for Transportation Research, University of Texas, Austin, TX, 1992. Manual of Steel Construction, Eighth edition, American Institute of Steel Construction Inc., Chicago, IL, 1980. Poblete, M.P., J .Ceza, J. David, R. Espinosa, A. Garcia and J. Gonzalez, “Model of Slab Cracking for Portland Cement Concrete Pavements,” Transportation Research Record 1307, TRB, National Research Council, Washington, DC, 1991. Rivero-Vallego, F. and BF. McCullough, Drying Shrinkage and Temperature Drop Stresses in Jointed Reinforced Concrete Pavement, Research Report 177-1, Center for Transportation Research, University of Texas, Austin, TX, 1977. Snyder, M.B., Dowel Load Transfer Systems for F ull-Depth Repairs of Jointed Portland Cement Concrete Pavements, Ph.D. thesis. University of Illinois, Urbana, IL. 1989. “Steel Reinforcement Properties and Availability,” ACI Committee 439, AC] Vol. 74, No. 10, Detroit, Michigan, 1977. Timoshenko, S. and J .M. Lessels, Applied Elasticity, Westinghouse Technical Night School Press, Pittsburgh, PA, 1925. Van Wijk, A.J., Rigid Pavement Pumping: (1)Subbase Erosion (2) Economic Modeling, PhD. thesis, Purdue University, West Lafayette, IN, 1985. Westergaard, H.M., “Analysis of Stresses in Concrete Pavement Due to Variations of Temperature,” Proceedings - Volume 6, Highway Research Board, Washington, DC, 1927. Wimsatt, A.J. and BF. McCullough, “Subbase Friction Effects on Concrete Pavements,” Proceedings of the Fourth International Conference on the Design and Rehabilitation of Concrete Pavements, Purdue University, West Lafayette, IN, 1989. Yoder, El. and M.W. Witczak, Principles of Pavement Design, Second Edition, John Wiley & Sons, Inc., New York, NY, 1975. Zollinger, D.G. and EJ. Barenburg, Background for Development of Mechanistic Based Design Procedures for Jointed Concrete Pavements, Interim Report, Illinois Department of Transportation, Springfield, IL, May 1989. 32. 105 Zollinger, D.G. and EJ. Barenberg, “Mechanistic Design Considerations for Punchout Distress in Continuously Reinforced Concrete Pavement,” Transportation Research Record 1286, TRB, National Research Council. Washington, DC, 1990. MICHIGAN STRTE UNIV. LIBRQRIES 1111111"111111“1111111111111111111111 31293014111110