Research Report Frank McCullough, and Ned H. Burns

Transcription

1 Te:hni:al Report Do:umentotion Page 1, Report No. 2. Government Aession No. FHWATX Title and Subtitle ANALYSIS OF URLING MOVEMENTS AND ALIBRATION OF PP PROGRAM 3. Reipient's atalog No. 5. Report Date November Performing Organi zrtton ode!-':;:---:--: ' 8. P erform1 ng 0 rgoni z ati on Report No. Authorls) Jose Antonio Tena-olunga, Researh Report Frank Mullough, and Ned H. Burns Performing Orgonizolton Nome and Address enter for Transportation Researh The University of Texas at Austin 1 Austin, Texas ~ ~ 12. Sponsoring Ageny Nome and Address Texas State Department of Highways and Publi Transportation; Transportation Planning Division P. 0. Box 5051 Austin, Texas Supplementary Nates 10 Work Untt No. (TRAISl 11. ontrat or Grant No. Researh Study Type of Report and Period overed Interim 14. Sponsoring Ageny ode Study onduted in ooperation with the U. S. Department of Transportation, Federal Highway Administration. Researh Study Title: "Pres tressed onrete Pavement (PP) Overlay on IH35 in MLennan ounty" 16. Abstrat The long-term work plan at the initial design phase of the MLennan ounty prestressed onrete pavement (PP) overlay onsisted of the determination of the variables that are relevant to design, the development of models and design proedures, and the study of the effet of environmental fators on PP slabs. The present study fouses on the evaluation of the performane of PP. The review of the existing models for Prestressed onrete Pavements are disussed. Next, the use of olleted data in an experimental field setion for omparison against program PSPl is made. A new model for the predition of urling in slabs aused by temperature variations is developed and tested. Program PSP2 is introdued as the result of the upgrading in the models and alibration of models. Finally, onlusions and reommendations based on the instrumentation program, data analysis, and model are outlined. 17. Key Words onrete pavements, highways, prestressed, overlays, stress, frition, urling, warping, elastiity, temperature hanges, gradient, reversal, thermal oeffiient, model, omputer 18. Distribution Statement No restritions. This doument is available to the publi through the National Tehnial Information Servie, Springfield, Virginia Seurity lassif. (of this report) 20. Seurity iani!. (of this page) 21. No. of Pages 22. Pne Unlassified Unlassified 134 Form DOT F (8-72l Reprodution of ompleted page authorized

2 ANALYSIS OF URLING MOVEMENTS AND ALIBRATION OF PP PROGRAM by Jose Antonio Tena-olunga B. Frank Mullough Ned H. Burns Researh Report Number Researh Projet Prestressed onrete Pavement (PP) Overlay on IH35 in MLennan ounty onduted for Texas State Department of Highways and Publi Transportation in ooperation with the U.S. Department of Transportation Federal Highway Administration by the ENTER FOR TRANSPORTATION RESEARH Bureau of Engineering Researh THE UNIVERSITY OF TEXAS AT AUSTIN November 1989

3 The ontents of this report reflet the views of the authors, who are responsible for the fats and the auray of the data presented herein. The ontents do not neessarily represent the offiial views or poliies of the Federal Highway Administration. This report does not onstitute a standard, speifiation, or regulation. ii

4 PREFAE The report presents an analysis of vertial slab movements to haraterize the urling and expansionontration behavior of the slabs. Finally, an analytial model is developed and alibrated to the field data. This work is part of Researh Projet 556, "Prestressed onrete Pavement (PP) Overlay," onduted as a part of the overall researh program at the enter for Transportation Researh (TR), Bureau of Engineering Researh, The University of Texas at Austin. The work was sponsored by the Texas State Department of Highways and Publi Transportation (SDHPf) and the Federal Highway Administration under an agreement with The University of Texas at Austin and the Texas SDHPT. Thanks go to Ken Hankins and Humberto astedo for assistane with planning, arl Bertrand for assistane with data aquisition, and all the personnel in the projet and in the enter for Transportation Researh who were so helpful during the projet. Jose Antonio Tena-olunga B. Frank Mullough Ned H. Burns LIST OF REPORTS Researh Report 556-1, "Prestressed onrete Pavement: Instrumentation and Data olletion for Its Analysis," by Jose A. Tena-olunga, Elliott D. Mandel, Ned H. Burns, and B. Frank Mullough, desribe the planning and organization for Researh Projet 556 and presents field data in tabular and graphial form. Results of defletion tests and ondition surveys are also presented. June Researh Report 556-2, "Prestressed onrete Pavement: Instrumentation, Behavior, and Analysis," by Elliott David Mandel, Ned H. Burns, and B. Frank Mullough, desribes the instrumentation for Researh Projet 556 and presents an analysis of horizontal data in tabular and graphial form. July Researh Report 556-3, "Analysis of urling Movements and alibration of PP Program," by Jose Tena olunga, Ned H. Burns, and B. Frank Mullough, presents the final results of vertial displaement data, reviews the analytial model PSP-1, and desribes the alibration of the model with the olleted field data. November ABSTRAT The long-term work plan at the initial design phase of the MLennan ounty prestressed onrete pavement (PP) overlay onsisted of the determination of variables that are relevant to design, the development of models and design proedures, and the study of the effet of environmental fators on PP slabs. The present study fouses on the evaluation of the performane of PP. The review of the existing models for Prestressed onrete Pavements are disussed. Next, the use of olleted data in an experimental field setion for omparison against progam PSPl is made. A new model for the predition of urling in slabs aused by temperature variations is developed and tested. Program PSP2 is introdued as the result of the ugrading in the models and alibration of models. Finally, onlusions and reommendations based on the instrumentation program, data analysis, and model are outlined. KEY WORDS: onrete pavements, highways, prestressed onrete pavements, overlays, stress, frition, urling, warping, elastiity, inelastiity, temperature hanges, temperature gradient, temperature reversal, thermal oeffiient, mathematial model, omputer program. iii

5 , SUMMARY This report presents an analysis of the data olleted from the MLennan ounty Prestressed onrete Pavement (PP) in detail. The objetives of the data olletion and program alibration are outlined. The fators and variables in omputer program PSPI are analyzed. An analysis of the vertial displaement data is performed and related to the horizontal displaement data and temperature. Final results from the displaement data are presented in tabulated and graphial form. Results of an analysis of vertial displaements are outlined and disussed. The onsisteny and auray of the data are then addressed. omparisons between the olleted data and previous models used in PSPI are made. Then, the models for urling are reviewed. An analytial model for the vertial displaements of PP slabs is developed. The bakground and theory of the model is desribed. Then, the model is tested. The use of the model is presented along with user guidelines. The omputational operation of the model is outlined in the appendix of the report. Finally, onlusions are presented, followed by reommendations based on the instrumentation program and the field data analysis, and, the model studies are outlined. IMPLEMENTATION STATEMENT This report desribes the analysis of urling and the proedures followed for the alibration of omputer program PSPI and the subsequent development of program PSP2 using data olleted from the MLennan ounty Prestressed onrete Pavement (PP). A field data analysis and the development and use of an analytial model for PP are also presented. Data for a wide range of temperatures are reported. The method of olletion of field data is quite suessful. The information presented in this report an be used as a guideline for future programs and analysis. Field measurements that an be used for the design of Prestresed onrete Pavements onstruted on a polyethylene surfae or for the alibration of other mathematial models are provided. Testing of program PSP2 shows that it an be used for the analysis and design of PP. t iv,

6 HAPTER 1. INTRODUTION This hapter presents the purpose of this report. A brief bakground on prestressed onrete pavements (PP) is given, along with a review of some assoiated former work at the enter for Transportation Researh (1R) of The University of Texas at Austin. The relation of this projet to previous studies of PP and with other 1R work is disussed. The objetives and the sope of this work are inluded at the end of the hapter. The study of various types of engineering strutures and a translation of their behavior into models permits optimization of design. This is true for all the diverse areas inluded in the field of ivil Engineering. Behavior analysis is progressive in nature, yielding better models as it evolves. Modeling is a powerful tool in deision making as it allows the hoie of more eonomi alternatives. The lak of models would make the deision-making proess a umbersome task at the levels of planning, design, and onstrution. This lak would also lead to inauraies and waste of resoures. Models are of speial importane in road onstrution. Presently roads are the preferred means of transport, and enormous resoures ~e alloated for their development and maintenane. The PP slab is one of the pavement types more reently available. The study and alibration of a model for PP slabs will provide neessary information for the weighting of this option. An eonomi analysis using a model will help to improve use of the human, material, and eonomi resoures. RELATION OF THIS REPORT TO PREEDING REPORTS The development of a design proedure for PP was onduted at 1R under Researh Projet 40 L In that projet, aspets of early post-tensioning, new onepts in prestress, subbase frition, prestress frition losses, fa Ligu life, monitoring instrumentation, and PP design were studied. The design proedures were developed from design reommendations proposed in the literature together with the experiene gained from the MLennan ounty Projet. Researh Report ontains the development of the original program, PSPl, whih is alibrated herein. This report inludes the ontinuation of the physial measurements of the behavior of the PP slabs in the MLennan ounty Projet ( 1985). It also initiates the determination of the long-range harateristis of PP slabs in servie. Information for the validation or orretion of the observations made during the preeding work, on Projets 556 and 401, is supplied here, together with a more aurate model for future PP design. This model also offers a basis for better maintenane predition and for deisions about design details on future projets. The data olleted during this projet, and the planning and desription of the instrumentation employed for the monitoring of the test setions, are presented in Researh Reports and Researh Report deals mainly with the analysis of the horizontal data for the development of a finite element program. This report fouses mainly on the analysis neessary for the alibration and testing of program PSP2. Researh Report 556-4F presents a general summary and the main results of this study. OBJETIVES The objetives of this study range from general objetives to those whih are more speifi. In this order, they are to: (1) Study the harateristis and behavior of a prestressed onrete pavement overlay, monitoring the performane of the prestressed onrete pavement overlay to build statistial support for the alibration of the program. (2) Perform a statistial analysis of the olleted data. (3) alibrate the program PSPl for the omputation of values more representative of the physial reality. (4) Revise the models proposed in program PSP1 as neessary. The work was performed taking into aount the existing available data and the new information that was produed in this study. SOPE The present work is divided into seven hapters. The first three hapters present the neessary bakground on Prestressed onrete Pavement (PP) slabs, and the work arried out for this study is presented in hapters 4 to 6. hapter 7 ontains the onlusions and reommendations. hapter 1 is a introdution to PP, its bakground, and the bakground of this projet. The onepts for analysis and alibration are disussed in hapter 2. At the end of hapter 2 a haraterization of the input data is presented. This disussion provides the basis for the presentation and later analysis of formulas used in program PSPl. In hapter 3, the analysis of the data is presented. The omparisons between predited and measured behavior are presented in hapter 4. Hypotheses for the modeling and alibration of PSPl are also developed in hapter 4. The testing of the hypotheses is presented in hapter 5 along with their introdution into program PSP2, whih is an upgraded version of program PSPI. The final onlusions and reommendations from this study are presented in hapter 6.

7 HAPTER 2. PLAN FOR ANALYSIS AND ALIBRATION HARATERIZATION OF INPUT FOR THE PP MEHANISTI MODEL Before the plan for the alibration of the model is disussed, a listing of the input for program PSPl is provided. It desribes the role of eah variable in the program (see Table 2.1 ). For a lear and simple representation of the program and its alibration, the onepts of "blak box" and flow hart are used. In Fig 2.1, the blak box of program PSPl is depited. On the left side of the figure, the input with a onstant value for a speifi slab is shown under the lassifiation of "Historial Data"; the letters "b, d, f, and g," et., orrespond to the letters of Table In the seond input group, the values that have to be updated for eah program run are shown. The output of the program is on the right-hand side of the figure. Figure 2.2 is the flow hart of the alibration proess. Figure 2.3 illustrates the onepts of Fig 2.2. The proess starts with the monitoring of PP slabs. Next, the predited values of program PSP 1 are ompared to the olleted data. If the differene between the values is higher than the aepted range for design, the alibration proess ontinues; otherwise the alibration is not neessary. Step three is the analysis of the differenes. In this part all the fators that might affet the values are weighted. Hene, statistial omparisons are performed between the data for dry and wet onditions, edge and interior, different slab lengths, et. One the analysis is finished, models and onstants in PSPl are reviewed; here onstants are orreted and updated if neessary. Next, a hypothesis for the orretion of the problem is expressed. In step six, the neessary orretions and improvements to the models are introdued and heked for validity. The degree of auray is a funtion of the data base available. Then, the auray obtained is heked against the existent data base. Here the proess an follow either of two paths: (1) if more data base is required for this hek, further monitoring of PP is pursued and the proess an go another yle; or (2) if the auray ahieved is reasonable for design or if it is not possible to obtain the required data, the proess is terminated. This proess is systemati and an be repeated until the desired auray is reahed. Eah time the required data base will be more broad, extensive, and expensive. Therefore, it is usually pursued until pratial values for design are ahieved. EFFET OF MODELS IN PP In this setion, the effet of models and input in PSPl is inspeted. For this purpose, the models presented before are onsidered. The input is also taken into aount. The purpose is to outline the best path for the alibration. The riterion followed is that for those models with higher effet, the results are alibrated first. Next, the alibration proeeds with models of seondary importane for the output. Then the alibration is gradual. The models that have a higher effet on the input are those for the determination of the stresses and movements due to frition and urling. They are followed by the funtions for the predition of post-tensioning, steel,and onrete. The inputs for the k-value for soil, reep and shrinkage then follow. Inputs of importane are the Input Program PSP 1 Output Historial Data (Mat and Geom onstants) [b.d,f,g] Atual Data TimeEnvironment Parameters Mid-Temp TopBottom Temp Differential Elapsed Time in Days ,..,... Predition Data... Pavement Stresses Initial Period lntermediallfinal Period Longitudinal Movements - Frition oeffiient Prestress & Frition Stress urling Defletion urling Stress Fig 2.1. Input and output of Program PSPI (blak box onept). 2

8 3 strength of onrete, modulus of elastiity, Poisson's ratio, and thermal oeffiient of soil. The k-value is a funtion of the range of the other values. Thus, its effet on the output in some ases will be important while in other ases its imjx>rtane will be marginal. The frition model is iterative. This an indue roundup or trunation errors, as expressed in Table 2.1. The funtion of this model is a numeri integration of a polynomial funtion. The degree of the polynomial is given by the profile of the oeffiients of frition. The magnitude of an error due to a mis-modeling of frition is illustrated with Fig 2.3(b) (6), whih shows the differene between two JX>Iynomials of different order. For this determination, the employment of the orret oeffiients and profiles of frition is neessary. urling is important in the output, too. Temperature gradients, thermal oeffiients, modulus of elastiity, Poisson's modulus, and the k-value take part in its determination. Beause of the arhiteture of PSPl, urling is determined almost independently; it is affeted also by the frition profile. Therefore, its effet is easy to detet one the frition model an simulate the physial phenomena. Figure 2.3(a) (3) is a shemati representation of the interation of all the fators that intervene in urling and frition. The effets of data input were mentioned in the preeding setion. Some of the effets have already been mentioned here. Their main effet is to modify the rate of hange in the values of the output. Some seondary effets are hanges in the range of predition values. Figure (1) Monitoring PP Slab Behavior No Yes No (8) Data for Maintenane Design Manuals End of alibration (6) Introdution of orretions to Program for Preditions (3) Analysis of Differenes (5) Hypothesis of Problem for orretion (4) Revision of Models & orretion Update of onstants Fig 2.2. Flow hart of alibration proess for Program PSPl.

9 4 Predited 1) 2) e A Underpredited 0 B Overpredited 3)... :: a> E a>..e (.).. (I) i5 ~ 0!-#.~---- Atual Measured Dry ---Edge ~--- Interior Temp Temprnme 440ft 240ft... :: a> E a> ~ i. :? l Dry ilmprnme Temp Temp Fig 2.3(a). Graphial desription of alibration proess.

10 (4) 5 Determined e A Underpredited 0 B Overpredited Measured Temperature Y = b' + mx 0 : :e ::::1 0 b' Y =4aX Temperature Temperature (5) (6) (7) Models 1.1 Dx < E? ! Time Measured Fig 2.3(b). Graphial desription of alibration proess (ontinued). ' 2.3(b) ( 4) is a shemati representation of this effet. Among these input variables, we an point out reep and shrinkage and the thermal oeffiient. The magnitude of error is small for the majority of the variables beause of the range and magnitude of their own values. Therefore, the reommended order for the alibration of the program is ( l) model and oeffiients for frition, (2) model and oeffiients for urling, (3) model and oeffiients for onrete properties, (4) model and oeffiients for steel, and (5) model and oeffiients for post-tensioning. The proess of alibration is iterative. Figure 2.2 illustrates the flow proess that is followed for the alibration of eah model and for the general alibration. The proess stops one the values are reasonable for design purposes or when a higher degree of auray is not possible with the available data base and resoures.

11 6 Data (a) Tag problem indiator (b) Geometri harateristi of slab () Iteration values of tolerane (d) oeffiients of onrete -reep -Shrinkage, Subsequent -Thermal oeffiient -Speifi weight -Poisson's ratio (e) onrete strength development (f) Frition modes -Linear -Exponential -Multilinear (g) Steel properties (h) k-value for soil (i) Set of values for post-tensioning hange in the thermal oeffiient of onrete and soil for wetdry onditions Aging in steel and onrete Errors introdued by alibration proedure TABLE 2.1. HARATER OF DATA Problems that May Derive From Wrong Values Wrong solution path Proportionalitydifferenes Round up errorstrunated number errors Raterange value error mis-modeling Rangeratemis-modeling Initial range Range Ratemis-modeling Range error umulative deviation Rate predition at early agesrange at higher ages Introdution of errorrate of inrement Same as onrete Introdution of errorrate of inrement Initial rate errordiminishing subsequent error Range of flutuations Errors derived from prestress loss *Differene between measured and representative temperature slab *Thermal expansionontration of measuring devies, ;

12 HAPTER 3. ANALYSIS OF THE DATA This hapter inludes a summary of the results of the statistial analysis of the horizontal and vertial data, followed by an analysis of the relationships between vertial displaements and horizontal displaements with temperature. REVISION HORIZONTAL DATA The results of the statistial analysis of the horizontal displaement data are given in Table 3.1. The quality of the data and the uniformity in behavior between slabs an be seen in Table 3.l(a); the average satter for the entire set of data was for the 240-foot slabs and for the 440-foot slabs. RELATIONSHIP BETWEEN HORIZONTAL DISPLAEMENT AND TEMPERATURE Table 3.l(b) shows the values for the relationship between the horizontal displaement and the temperature. Here, the displaement is around inh per 0 F. The oeffiient of Thermal Expansion omputed in Researh Report (Ref 5) was 4.80 x IQ-5 ( Finhinh) for the 240-foot and 440-foot slabs. Additional work on the analysis of the horizontal displaements is made by Mandel et al in Researh Report (Ref 5). VERTIAL DATA For the analysis of the data, the quality and uniformity of the sample were tested. First, a regression analysis was performed. The purpose of this regression analysis was to hek whether the data olleted from eah joint showed the same trend and whether the data from all the joints belonged to the same population. This analysis was performed for the two omponents of the url-unurl yle of the PP slabs; the first part takes plae generally from sunset to sunrise while the seond part (unurl) orresponds to the period when the sunrays are heating the slab surfae. Illustrative graphs of the analysis of url and unurl are in Figs E.l through E.l2 of Appendix E. In general, the results of this analysis showed a high uniformity within the sample population, with regression oeffiients between 0.85 and 0.95 for the urling portion of the yle and between 0.95 and 0.99 TABLE 3.1. SUMMARY OF ANALYSIS FOR THE HORIZONTAL MOVEMENTS (a) Standard Deviation Between Slab Displaements (Average Value) Slab Tri Date Length (ft) July 25 Au~ust 5 Au~ust 26 November 5 Januar~ 21 January 22 February (b) Displaement in Temperature Ratio (inb F) Slab Trip Date Length (ft) July 25 August 5 August 26 November 5 January 21 January 22 Febru~ () alulated oeffiient of Expansion (OFinblinh) to-s Slab Trip Date Length (ft) July 25 Au~ust 5 Au~ust 26 November 5 January 21 January 22 February Average: 4.70 Std Dev: Average: 5.40 Std Dev:

13 8 for the unurling portion of the yle. This unifonnity in the data allowed the use of a representative set of data for the further study of the behavior. Figures E. I through E.6 show the harateristi trend of the sample for the part of the yle when the slab is undergoing urling. For this part of the yle, the best fit was obtained with seond-degree equations for all the joints. Figures E.l and E.4 show a high degree of orrelation and show that all the joints behave in the same manner, with a nonnal satter among them. The reasons for this satter lie in the intrinsi variability of materials, onstrution, loal partiularities, et. For the part of the yle where the slab unurls, the trend showed a linear behavior. The urves and equations for this part of the yle are shown in Figs E.7 through E.l2. Again, there was some slight satter between the joints, whih is refleted in the slight differenes in the slopes of the urves, but this satter was due to the satter onsidered as "normal" in statistis for population samples. Slab Another analysis performed was between the two sizes of slabs monitored. Figures E.l and E.4 are illustrative of the regression equations for the 240-foot slabs, while Figs E.7 and E.9 depit the harateristi equations for the 440-foot PP slabs. The general trend showed a slight differene between them for the urling and unurling portions of the yle, and the differenes were different between seasons; a more detailed analysis is summarized and presented in Table 3.2. A third omparison was made between the different regression urves for the urling experiened along the slab length. Figures E.l through E.l2 depit an example of these urves for the 240 and 440-foot slab lengths at the edge and of the sixth and third points along the length of the slab. Figures E.l through E.6 show the url period and Figs E.7 through E.l2 show the unurl period. This analysis showed that although there was a relationship between them, urling was not proportional between the third, sixth, and edge portions of the slab. TABLE 3.2. SUMMARY OF ANALYSIS FO~ THE VERTIAL MOVEMENTS (a) Standard Deviation Between Slab Displaements (Average Value) Tri Date Length QL July 2S AU1;1USt s AU1;1US1 26 NovemberS January 21 Janu!!:! 22 Febru!!;!! O.oi (b) Displaement to Temperature Ratio (inhi F) Slab Tri Date Length QL July 2S Au~ustS AU!;1USt 26 NovemberS January 21 January 22 Februarz: S () Ratio Between Displaement and Top-Bottom Temperature Differential (inhi F) Slab Tri Date Length (ft) July 2S Au~ust S Ausust 26 NovemberS Janu!!:! 21 January 22 Febru!!;!! S f? f (d) Ratio Between Inrement in Displaement and Inrement in Temperature (AYAT) (inh F) Slab Trip Date Length QL July25 Au&ust S Ausust 26 NovemberS Janu!!:! 21 Janu~22 Febru~9 240 o.ooss

14 9 Table E.l shows the regression equations for these sample data. A review of the eentriities of the regression urves for urling and of the slopes for unurling reveals the orrelations and differenes stated above whih were typial of eah data set This analysis provided the neessary grounds for the simplifiation of the data sample for eah data set. For this purpose the average value was obtained for eah data set for the 240 and 440- foot slabs, as illustrated in Figs E.l3 and E.l4 for the ftrst set of data. The use of representative data for eah data set enabled an in-depth study of the behavior of the slabs and the determination of general trends; a good example is Fig E.15, whih depits the urling at the different lengths of the slab for eah set of data. This urling is given with respet to the total urling at the edge of the slab; the right end of this graph shows the average proportional urling measured during the data ouetion period. The results of the statistial analysis for the vertial data are shown in Table 3.2. The standard deviation of data between slabs is given in part (a); parts (b), (), and (d) ontain the omputed values for the analysis of the different relationships between vertial displaements and temperatures. The quality of the data and the uniformity in behavior between slabs an be notied in Table 3.2(a). The exeption among these values might be the set of data for the trip made on July 25. The deviation values are and for the 240-foot slabs and for the 440- foot slabs. These values are reasonable if we onsider that they are good for data olleted in the field; nevertheless the satter is two times higher than the satter for the remaining sets of data. The reason for the wider range of values in this ftrst trip was a problem experiened with the small surfaes that served as reation surfaes for the Lips of dial gauges and LVDTs. This problem was orreted on the next trip and never happened again. The average satter for the remaining sets of data was inh for the 240-foot slabs and inh for the 440- foot slabs. These values are slightly higher than the values for the standard deviation of the horizontal movements, but it is understandable to obtain higher sauer sine more fators affet the vertial displaement of the slabs. We an state that the overall range of values is exellent for data obtained in the field. Therefore, all the sets of data olleted are uniform and are representative of the behavior to be studied. RELATIONSHIP BETWEEN VERTIAL DISPLAEMENT AND TEMPERATURE For the final version, the ratio of the maximum inrement in urling to that of the temperature in the middle depth of the slab was alulated. The yle patterns for urling were also studied. The results are in Table 3.2 and in Figs 3.1 and 3.2. It is interesting to note that, although the tabulated values are not onstant throughout the sample, they are fairly similar between seasons. Also, it is lear that urling has some relationship to temperature but is more strongly related to temperature differential ~0.1 ;:: ::I <...:> Temp at Middle Depth of Slab ( F) Fig 3.1. urling versus temperature at middle depth of slab ;:= Temp at Middle Depth of Slab ( F) Fig 3.2. urling versus temperature differential between top and bottom of tbe slab.

15 10 DISPLAEMENT AND TEMPERATURE RATIOS The relationship between the displaement and the temperature of the slab at middle depth is shown in Table 3.2(b). The relationship between the venial displaement and the temperature differential between the top and the bottom of the slab is shown in Table 3.2(). A definite trend ould not be deteted with these values sine the trends are different for the 240 and 440-foot slabs. Table 3.2(d) ontains the values of the ratio between the inrements of displaement and the inrements of temperature. Here, a better orrelation was obtained for 240 and 440-foot slabs. However, this orrelation holds only for the same season. YLE PATTERNS The pattern of the temperature-displaement urves was studied for the temperature in the middle of the slab and for the temperature differential that develops between the top and the bottom of the slab. These patterns are depited in Figs 3.1. and 3.2. The pattern that turned out to be more signifiant is the one traed by the venial displaement versus the temperature differential. The general Lrend of the phenomena an be traed in Fig 3.2. From the figure, the general trend for urling an be asertained. There is a path for the ooling period of the slab (I) and a path for the heating period (II) in whih the flattening of the slab takes plae after the urling period. The behavior between the slabs is more uniform for the ooling yle sine the slabs follow very lose paths, and more satter is observed during the heating yle of the slab. A onlusion from this analysis was that the urling behavior is more affeted by hanges in the tempemture differential than by hanges in the temperature in the middle of the slab. Also, the satter between slabs is less for the 240-foot slabs than for the 440-foot slabs. Researh Repon (Ref 3) has illustrations of the profiles for eah set of vertial data. RELATIONSHIP BETWEEN HORIZONTAL AND VERTIAL DISPLAEMENTS The alulated values for the ratios between the horizontal and vertial displaements are shown in Table 3.3(a). A onstant ratio ould not be established for them. The relationship between the magnitudes of the vertial and the horizontal displaement diminishes with older weather, and this relationship is higher for the 240-foot slabs. However, the relationship between the mtio of the 240 to the 440-foot slabs remains fairly onstant for any weather ondition. TABLE 3.3. RATIO BETWEEN HORIZONTAL AND VERTIAL MOVEMENTS AND SIGNIFIANE TEST BETWEEN LVDT AND DIAL GAUGE DATA (a) Ratio between horizontal and vertial displaements Slab Tri Date Length (ft) July 25 August 5 Au~ust 26 November 5 Janu~21 Janu~ 22 February (b) Results of signifiane test between LVDT and dial gauge data: t-values Slab Tri Date Length (ft) July 25 AU1jUSt 5 Au~ust 26 November 5 January 21 January 22 Febru:!!i: , () Signifiane (%) Slab Tri Date Length (ft) July 25 Au~ust 5 Auiust 26 November 5 January 21 January 22 Febru:!!i: <

16 11 VAUDITY OF THE DIAL GAUGE DATA AND LVDTDATA The omparison between the dial gauge data and. L VDT data in the final version was heked statistially. The purpose of this test was to determine whether the data olleted with both instruments belonged to the same population. For this test, the LVDT values were ompared against the average values of the dial gauge set. The results of this signifiane test are shown in Table 3.3(b). The level of signifiane for the data is between 0.09 and The only exeption is the 0.6 obtained for August 5, and the soure of error arne from the LVDT on the 240-foot slab. This LVDT reorded errati readings throughout the period. The other higher values (0.1 and 0.12) orrespond to data for July 25, when some problems ourred with the leveling of the ontat surfaes for the instruments, and to data for February 9, when the LVDTs ould not be zeroed and registered readings in the boundaries of their linear range. Nevertheless, all the values determined from this hek produed a value of signifiane higher than the 36 perent for a t distribution with 4 degrees of freedom (in the ase of edge monitoring) and the 33 perent for 2 degrees of freedom (ase of the interior monitoring). From this test we an draw the following onlusions. (1) All the data reorded belong to the same population. (2) The uniformity of the data is exellent, being higher for the dial gauges than for the LVDT. (3) A set of data omposed of the average values for eah set of readings an be onsidered as representative of that set. The last onlusion is an important one, sine it enables us to redue the number of data values to use from six to one for eah slab length. Thus, for the remaining portion of this report, a unique set of values for eah set of data will be used. This set is formed by the average value of the dial gauge data and is truly representative of the average slab behavior. These sets of data are shown in Table 3.4. TABLE 3.4. SUMMARY OF REPRESENTATIVE DATA SETS (a) For horizontal displaements in 240-foot slabs (inb) Tri Date Hour July 25 August 5 August 26 November 5 Janu~21 January 22 February 9 14: : : : : : : : : : : : : (ontinued)

17 12 TABLE 3.4. SUMMARY OF REPRESENTATIVE DATA SETS (ONTINUED) (b) For vertial displaements in 240-foot slabs (inh) Tri Date Hour July 25 Au~ust 5 Ausust26 NovemberS Janu~21 Janu~ 22 Febru~9 14: : SS O.OOS : S O.OSS S 20: S :00 0.1SSO S S 0: S S 2: S8 0.1S1S : OJS :00 0.1S S 0.09S : S : O.OS4S O.OS S SS 12: S : S O.OOS () For horizontal displaements in 440-foot slabs (inh) Tri Date Hour July 2S AugustS August 26 NovemberS Janu~21 Janu~22 Febru~9 14: : : : : : : : : SO 8: S : S S : O.OOSO 14: (d) For vertial displaements in 440-foot slabs (inb) Tri Date Hour July 2S Ausust s Au~ust 26 NovemberS Janu~ 21 Janu~22 Febru~9 14: : O.OOS4 18: : S S :00 0.1SOO S : S : S : S S : S3S : S : O.OlSO : :

18 HAPTER 4. OMPARISON OF PREDITED AND MEASURED PERFORMANE Analysis of the olleted data presented in the preeding hapter set up the bakground for the development of a hypothesis for the alibration andor the introdution for improving the models in PSPI. As a preliminary step for the omputation of the predited values by program PSPl, the input values were alulated and ompared to the weighted field values. In all ases the omputed values were found aeptable and in agreement with field values for this projet. Part of the PSP model is devoted to the frition between the PP and the underlying pavement layer. For frition, Mendoza et al (Ref 1) reported expansion of the slab in the neighborhood of 0.10 inh. The value of frition reported by them for displaement of this magnitude was The values reorded here under similar onditions were within the same range. Therefore, a oeffiient of frition of 0.2 was used in the input. For the ompressive strength of onrete, the mean value of the final strength determined by laboratory testing of the field samples was alulated. All the other input values were taken from field reords. The input values are given in Appendix A. The numbers turned out by program PSP1 are shown in Table 4.1. Figures 4.1 through 4.6 show the shape of the displaement urves with time for eah type of weather for the 240-foot slabs. These figures are also representative of the urve shapes for the 440-foot slabs. The urves orresponding to the horizontal values are shown in Figs 4.1 through 4.3. They show good agreement between the predited and the measured behavior. This is espeially true for the fall, while slight satter is present for the hot and the old weather. The urves orresponding to the vertial values are shown in Figs 4.4 through 4.6. They show learly that the omputed values do not follow the measured behavior. All the predited values lie far below the values of the field data. An initial omparison between the alulated values of program PSP 1 and the field reords gave good agreement for the horizontal data. However, this did not hold true for the vertial data. PROEDURE FOR ALIBRATION The alibration of the horizontal and the vertial displaements is made in sequene, beause the output of the horizontal values affets the alulation of the values for urling. Hene, the models used in the predition of the horizontal displaements are reviewed flist. ALIDRATION OF THE HORIZONTAL DISPLAEMENTS The determination of the degree of orrelation is used as an initial tool. It tells us how far is the model from omputing representative values. The degree of orrelation between predited and field values for eah set of data is shown in Figs 4.7 through 4.9, for representative onditions of hot, old, and random weather. The field TABLE 4.1. SUMMARY OF VALUES FROM PROGRAM PSPI (a) For horizontal displaements in 240-foot slabs (inh) Tri Date Hour July 25 Au~ust 5 August 26 November 5 January 21 January 22 February 9 14: : : : : : : : : : : : (ontinued) 13

19 14 TABLE 4.1. SUMMARY OF VALUES FROM PROGRAM PSPl (ONTINUED) (b) For vertial displaements in 240-foot slabs (inh) Tri Date Hour July 25 Au~ust 5 Au~ust 26 November 5 January 21 January 22 February 9 14: : : : : : : : : : : : () For horizontal displaements in 440-foot slabs (inh) Tri Date Hour July 25 Au~ust 5 Au~ust 26 November 5 January 21 Janu~22 February 9 14: : : : : : : : : : : : (d) For vertial displaements in 440-foot slabs (inh) Tri Date Hour July 25 Au~ust 5 Au~ust26 November 5 January 21 January 22 February 9 14: : : r 20: : : : : : , 8: : :

20 Field H PSP1 H ::: Field H D 815 PSP1 H Q) Field H E Q) PSP1 H (.) ~ a. <J) 0.1 i!9 D ~ ::: g Iii a I + 8 ~ e 0 i + Fig 4.1. Horizontal g I i e + displaements (measured e i bot season for the Hour and omputed during the 240-foot slabs. :::: Field H 115 PSGP1 H Q) E Q) (.) "" i 0.1 <J) 6 "'@ ~ N ;:: g 0 g I Fig 4.2. Horizonal -0.1 displaements (measured ~ (] and omputed) during the ~ (] fad season for the!!!! (] foot slabs Hour Field H 121 PSGP1 H Field H ::: a 1122 PSGP1 H - e 219 Field H -::: Q) PSP1 H E Q) (.) "" <J) i 0.1 i5 "'@ ~ ~ e ~ 0 I 1!1 ~ Fig 4.3. Horizontal t! B 8 i ii a displaements (measured -0.1 B 8 (] and omputed) during the (] (] old season for 240-foot slabs Hour

21 16 Fig 4.4. Vertial displaements (measured E (!) Field V 726 PSP1 V Field V a 815 PSP1 V e 826 Field V, ::: + o 826 PSP1 V - 'E Q) + u.$! en 0 + -ro (.) t ~ and omputed) during the hot season for foot slabs. 0.0 ' Hour ' ::: 'E (!) Field v 115 PSP1 V 0.12 E (!) u 0.10.$!.. en 0.08, "' u t Fig 4.5. Vertial 0.04 ~ displaements (measured 0.02 and omputed) during the fall season for foot slabs Hour ' Field v 1121 PSP1 V Field v a 122 PSP1 V e 219 Field V :: 0.14 o 219 PSP1 V -:: E (1) u.$! en 0 -ro u I I Fig 4.6. Vertial t 0.04 ~ old season for + displaements (measured and omputed) during the 240-foot slabs. -O.D1 D D D D D ~ ft fl! D D D D Hour '

22 17..- a.. u () a # ro -; U":) N,... -Ol :: (I) 0.1 ::;:) 0.1 (I) Q) ::::::1 ~ -0.1 y e x r:jl RA2 = r:j!.b a 1D o Horizontal Field Values 725 Vertial Field Values (a) Horizontal displaements. (b) Vertial displaements. Fig 4.7. Degree of orrelation between predited and field data for July 25-26, 1988 (hot weather). G -; 0 U":) -,... N = :: (I) ::;:)..- a.. u () a.. 0.4, 0.2 ~ 0.3 G -; (ti rfl > 0.0,...- a a.. u () ~ a....- N ;::::: Ol 0.1.s: (I) J1 ::::> 0.1 (I) Q) ::::::1 0.0 ~..- a.. D u () ' y = e x a R-" y = e e-2x ~D RA2:: f:sjd -0.2 IJ ae Horizontal Field Values 121 Vertial Field Values (a) Horizontal displaements. (b) Vertial displaements. Fig 4.8. Degree of orrelation between predited and field data for January 21-22, 1989 (old weather).

23 ('Q -('Q 0 ~ :n.~ <n ::::> 0.1 <n <1) :::1 ~ u () Q.1 y = e x A"2 = u () 0... y = e e-2x A"2 = Horizontal Field Values (a) Horizontal displaements Vertial Field Values (b) Vertial displaements. 0.2 Fig 4.9. Degree of orrelation between predited and field data for November 5, 1988 (data set hosen at random). values are shown on the horizontal axis, and the vertial axis orresponds to the predited values. For a perfet orrelation, the predited and field values should be idential. In these ases, the plot lies on an imaginary straight line with a slope of 45 degrees. When the values are overpredited the points fall above this imaginary line; they fall below this line if the values are underpredited. The regression oeffiient indiates the degree of dispersion and agreement of the model. Figures 4.7 through 4.9 show the orrelation for the horizontal displaement, and part (b) of eahshows the values for the vertial displaement. The predited horizontal displaements are in lose agreement with the measured values. The oeffiients of partial determination in parts (a) of Figs 4.7 through 4.9 show that the orrelation values for program PSPl range between 0.99 and 0.87 for the horizontal movement. This range of values is proof of a high level of reliability in the model, and the value of the oeffiient of frition agrees with the one proposed in Researh Projet 401 (Ref 1). In this ase, it is reasonable to assume that the model aurately predits the displaements and no further alibration is neessary. The only area in whih better alibration would be possible is a higher preision in the determination of the value of the oeffiient of frition. However, this should be onsidered with aution, sine the output of the program is highly sensitive to the value of the oeffiient of frition and the frition oeffiient itself is affeted by different fators, suh as humidity, wear, et. HYPOTHESIS FOR THE ALIBRATION PROEDURE FOR THE VERTIAL DISPLAEMENTS For the vertial displaements, PSPI underpredits the behavior. A omparison between the values of Table 3.3(b) and (d) and those of Table 4.l(a) and (d), shows that all the predited values are far below the field values. These omparisons are onfmned by the general appearane of Figs 4.4 through 4.6 and parts (b) of Figs 4.7 through 4.9. A misleading fator is the high values reported for the oeffiients of partial determination in parts (b) of Figs 4.7 through 4.9, and these values should not be onsidered. The lak of auray of the models for urling in PSPI demand a more in-depth review of the model. The models should be ompared and the values ontrasted to the patterns determined in the analysis of the field data. The ause of disagreement between the values should be asertained and the determination of a more adequate model arried out.,

24 ., RITIAL REVIEW OF MODELS WESTERGAARD MODEL The model suggested by Westergaard is used in PSPl for the predition of displaements due to urling. Its main equations for displaements are where y = Ye = I = (1+ n) I ai 6TD I t2 h Ye = defletion at the edge, L = slab length, a = onrete thermal oeffiient of ontration and expansion, TO = temperature differential between top and bottom, k = k-value of soil, n = Poisson's ratio of onrete, Mx = bending moment in x diretion per unit h E width, thikness of slab, and modulus of elastiity of onrete. (4.2) (4.3) An inspetion of the model reveals that it is not sensitive to the following fators: (l) The funtion is ylial. That is, the values are repetitive and they always vary in the same fashion. Another onsideration is that, within the effetive range of the model, the displaements between slabs of different dimension in proportion with the dimensions of the slabs. A further onsideration is that the predited values are always larger for larger slabs. (2) Equations 4.1 and 4.2 are not sensitive to the geometry of the slab. That is, the equations do not onsider the relationship between length, width, and slab thikness. Therefore, the slab will have the same displaements notwithstanding how wide it is. An impliation is that, aording to this model, there is only a ertain width of the slab that experienes urling and the remaining portion of the slab width is not affeted and does not affets the values for urling. (3) Relative ranges of temperature should be inluded. That is, the effet of term DT is the same whether or not the range of temperatures happens during hot and dry weather or in old or wet weather. Another onsideration is that it does not onsider the effet with respet to the temperature at the asting of the slab. ( 4) The oeffiient produed on this model provide very low values only. Thus, the range of predited values is always restrited to a narrow band. These assertions an be heked numerially by inputting values in the funtions and using the values obtained from program PSPI for urling. When we relate this model to the observed behavior, we note the mismath of the model. A summary of the data reords for the field visits is given in Table 4.2, a quik analysis of whih reveals the following observed behavior trends: ( 1) The displaements are not in relative proportion to the slab length. As a matter of fat, urling was higher for the 240 foot slabs than for the 440 foot slabs. (2) The geometry of the slab has an effet on the displaement values. This is a natural onsequene of the observation stated above. (3) There is a non-linear effet on vertial displaement when the temperature is inremented. Displaement will not inrease proportionally to the inrement of temperature. Apparently there is a ombined effet. That is, the effet of the inrement is relative to the values of the maximum and minimum temperatures themselves. Table 4.2 shows that the differenes between the inrements of temperature for eah trip varied within I0 F. However, the measured displaements were quite different, and these displaements were related to the relative values between temperatures, e.g., displaements were higher for temperatures around l06 F (July 25) than for 63 F (February 9) despite the fat that the temperature inrement was higher for the latter (32 F versus 30 F). We an onlude then that Westergaard's model is not appropriate for the predition of urling in PP slabs. Also, sine the nature of the model is not onduive to revision, a different model is required. OTHER MODELS The available literature was reviewed and few models are available for the predition of displaements due to temperature in onrete slabs. Westergaard's model is the most aepted. Other models for rigid plates from the strutural field were tried to see if a ombination of them with the Westergaard model ould help desribe the objetive funtion. The majority of the models ontained some empirial relationships and not one of them ould desribe the phenomena. Other equations tried, were those from elasti foundations. They proved to be similar in behavior to Westergaard's model. Therefore, the need for the development of a new model arose. 19

25 20 HYPOTHESIS FOR THE NEW MODEL The problem with available models is the lak of fidelity with atual events whih our in the field. The model to be developed should onsider and reflet atual data. The hypothesis then is based on a model refleting the onepts expressed in hapter 2. That is, we will onsider the slab as the interfae of the exhange of temperature between two soures: the sun and the soil. From here we will develop the effet that temperature has on the material and the interation of the other fators. A desription of the general phenomena was desribed at the beginning of hapter 2, and the modeling for this phenomena is arried out in the next hapter. The planning and work developed for the olletion of data resulted in very aeptable sets of data for the projet. Analysis of these data sets revealed that values predited for horizontal displaements in program PSPI are reasonable. The opposite is true with the predited vertial values, whih are out of range. Further analysis revealed the neessity to develop a new model for the predition of displaements due to urling. Due to the importane of this model, its development is presented in a separate hapter before the model is introdue~ it into the general body of omputer program PSP2, m hapter 6. Thus, the next hapter is devoted to the development and testing of the new model for urling. f,

26 HAPTER 5. DEVELOPMENT OF A MODEL FOR THE EFFET OF TEMPERATURE GRADIENTS IN SLAB PAVEMENTS BEHAVIOR The hange of temperature in the environment develops a temperature gradient in the slab in onjuntion with the thermal harateristis of the slab omponents. This gradient produes differenes between the displaements that take plae on the top and the bottom of the slab. The result is a relative ontration or expansion of one surfae with respet to the other, that is, ontration displaements in one surfae and expansion in the opposite surfae of the slab. The interation of the fores produes urling. In this hapter, the neessary frame for the theoretial and mathematial work is developed to ahieve the model and its objetives, and the model is tested against the olleted data from the field. GENERAL MODEL Figures 5.I(a) and (b) ontain the general representation of the model. Highway strutures expose large surfaes to the environment. The struture is subjeted to temperature variations and solar radiation by absorbing heat energy from the sun. Soils are, likewise, subjeted to sun rays and temperature exhange. In this way, a system for heat exhange is formed. The ative soure of energy is the sun, and a seond (passive) soure is the soil. The soil ats as a "thermi battery" and the pavement onstitutes the interfae for the exhange of heat with the environment. A general statement ould be that radiation absorption auses the expansion of materials. Similarly, the transfer of this heat to the environment has the opposite effet. Sine temperature hanges follow a yli pattern, pavement slabs are subjeted to daily and seasonal variations. Generally, the priniples of thermodynamis apply to the rate of heat exhange. The general state of the fores is shown in the free-body diagram in Fig 5.1(b). The symbols used are defined in the symbols list at the beginning of this report. From Fig 5.1, we an see that three fators (temperature, frition, and stored energy) produe strains and stresses. These three fators affet PP in the following manner: Temperature. Environment an be haraterized in terms of moisture and temperature. Moisture and temperature in turn are affeted by the daily and seasonal yles of our planet. The daily temperature variations in lhe pavement are determined by heat gained and lost, plus other limati onditions. The number of hours and the intensity of sunrays determine the seasons and the peaks in temperature for eah season. All these fators SUN (Heat Soure 1) # ~~~t SUBGRADE (Heat Soure 2),.. dx..,. 1 1 fx (om pression) fx(tension) + ky (a) Top: Free body diagrams for the slab representation. (b) Bottom: Free body diagram of fores. Fig 5.1. Shemati representation for warping and urling. 21

27 22 H do r- dx.. I H do otda h6 2hl3 () Free body diagram for temperature gradient. ~::1m (d) Free body diagram for volumetri thermal hange and frition. h6 h6 2h3 h (e) Free body diagram for stored deformation energy and dead weight. Fig 5.1. Shemati representation for warping and urling (ontinued). h6 h produe a new set of onditions for pavements in a repetitive fashion. This ylial behavior produes periods of expansion and ontration under different servie periods of the struture. In terms of onditions that affet the pavement struture we know that: (a) expansion happens during the hours of major traffi volume and (b) ontration takes plae at night, with a lesser volume of vehiles. Then, temperature, as a parameter of environment, affets the behavior of pavement. Frition. Variations in moisture and temperature ause minute volume hanges in the pavement However, this expansion and ontration are different from there that might our in the subbase. Therefore, movement develops, with frition ourring between slab and subbase. The nature of the frition fore is not ompletely known; however, it is assumed to be produed by two fators: (1) moleular attration and the nature of the surfaes in ontat and (2) irregularities between the surfaes in ontat. For PP, the development of frition has a benefiial and detrimental effet. The type of effet depends on the diretion of movement in relation to the prestressing as the frition develops. A detrimental effet results during the ontration of the slab. Frition introdues tensile stresses on the bottom of the slab. A benefiial effet is produed by the expansion of the slab, beause of the ompressive stresses that develop, also due to frition at the slabsubbase interfae. The movements normally vary from a maximum at the edge to the smallest at the enter of the slab. Therefore, the maximum frition fores develop at the ends and derease toward the enter. onrete stresses resulting from the aumulation of frition fores grow from zero at the end to a maximum at the enter. harateristially, in a daily yle, two movement and frition resistane reversals happen. The reversals our within a few degrees of the maximum or the minimum slab temperatures. The magnitude of the frition restraint stresses depends primarily on three fators: (1) the onrete oeffiients of ontration and expansion, (2) the onrete modulus of elastiity, and, (3) the frition fore versus movement relationship. The tensile stresses are the most important sine they result in unfavorable onditions for the PP. Stored Energy. An inreasing amount of deformation energy is stored in the slab with time. This energy is onsidered to be stored beause it is within the elasti range of deformation of the slab and the strains indued are suseptible to being released in ertain onditions. A more extensive summary of the effet of fators affeting the performane of PP is presented in Researh Report 556-4F (Ref 4). t

28 23 GOVERNING FATORS AND ASSUMPTIONS OF THE MODEL In this setion, the sheme and theory framework for the development of the model is introdued. An outline of the prinipal onsiderations, assumptions, and onventions for the development of the model is presented. GOVERNING FATORS AND MODEL OUTUNE Different fores at while urling is produed. These ations affet urling. Then, the main fores that ontribute to the development of urling must be onsidered. For the present researh, urling is onsidered to be governed by the following fators: (1) Stresses that are the produt of the temperature gradient developed between opposite faes of the slab. (2) Stresses that are the produt of frition between the subbase and the slab produed by the thermi ontration of the slab. (3) Stored deformation energy of the slab. (4) Strutural stiffness of the element. Sine we are mainly dealing with fores, the additive and assoiative properties of fores apply. Then, eah omponent an be determined separately and their effet an be added at the end of the proess. ASSUMPTIONS OF THE MODEL These assumptions for this model: (1) The vertial reation to any setion is diretly proportional to the defletion, Yk. The proportionality onstant k being the modulus of subgrade reation. (2) Zero defletion is at the position of rest on the level subgrade from the initial defletion. (3) onrete is a homogeneous, linearly elasti material. (4) Temperature or moisture differentials from top to bottom produing upward defletions are negative. (5) Upward defletions are positive. ( 6) Tensile stresses are positive. (7) The origin of the oordinates of movement is taken at the midslab. (8) In general, the priniples of elastiity, frition, and energy apply to the model. DEVELOPMENT OF THE MODEL The different steps in the development of the model are disussed herein. They are (1) solution of the model for the predition of urling at the slab edges, (2) generalization of the mxlel for predition of urling along the slab, (3) solution for the transition during gradient reversal, and (4) solution for the omputation of stresses. For larity, expressions from the general theory are designated using a letter as extension, while the partiular expressions developed for the mxlel are numbered in the extension. GENERAL STATE OF FORES The general state of fores in the body during urling was analyzed for eah one of the governing fators. Figure 5.1(b) an be onsidered as a departing point for this development Figure 5.1(, d, and e) are diagrams representing the effets of temperature, frition, and stored energy in a slab element. Development of Equations From Fig 5.1, we an see that three fators (temperature, frition, and stored energy) produe strains and stresses. Temperature. The strains and stresses produed by temperature, frition, and stored energy translate into bending moments ating on the slab. The basi equation for stress from the elasti theory an be employed for the analysis of these bending moments, that is; where s = Ee (5.1) 0' = stress in the material, E = modulus of elastiity of the material, and e: = strain of the material. and for thermal hanges, strains an be expressed as where e: = a ~T a = thermal oeffiient of the material and ~T = inrement in temperature. (5.2) For the determination of the expressions for the bending moments, a differential element of one setion was studied and the neessary relations and algebra worked out. Finally, the effet along the slab width was onsidered. The developed equations for eah fator are presented below. Temperature Gradient. Assuming the temperature is DTo degrees higher at the top of the pavement than at the bottom, the strain prxlued will be a produt of the gradual work performed by the effetive temperature inrement. The state of strains, stresses, and fores produed is depited in Fig 5.l(). Then, the equation for stress for an effetive inrement of temperature gradient DTo is (5.3)

29 24 where a = onrete thennal oeffiient of ontration and expansion, and E = onrete modulus of elastiity, psi. O.To' = effetive inrement of temperature, or n Jo.To (5.4) 0 From the energy theory and from the onepts expressed before, we know that this funtion is ontinuous and that a ontinuous reord of alorifi energy would be neessary for the aurate detennination of the work done. The input of suh set of data would be impratial and uneonomial. Therefore a disrete equation is sought. From the work equations, we know that a good approximation for disrete inrements is ahieved using the average value of the disrete inrements. Then, the adopted equation is 1 n O.To = 2n l:o.tm 0 (5.5) The bending moment Mro produed by this fore is onstant throughout the slab. If now we onsider the slab width in the integrated expression along the slab profile we have where B = width of slab, in., and D = slab thikness, in. (5.6) Volumetri Thermal hange and Frition. The strains indued in a slab element are shown in Fig 5.l(d), assuming a thennal derement ATM. Here, the equation of stresses indued by the slab ontration has the form of Eq 5.3, but, for the tenn ATD', whih is replaed by DTM' that represents the inrement in temperature with referene to the original asting temperature. Then, a virtual moment is produed by a ontrating fore and a frition fore, where the frition fore is of the same magnitude but opposite in diretion to the ontrating fore. The equation for the onstant bending moment MTM produed by slab ontration and frition is Eo. o 2 O.TM' MTM= 4 (5.7) where and O.TM' O.TMi = inrement of temperature at time i, F, (5.8) TMi = temperature at middle depth of slab, F, and TMo = temperature of slab at initial uring, F, and all other terms are as defined above. Stored Deformation Energy and Dead Weight. Assuming that the slab deflets gradually when subjeted to its dead weight, an inreasing amount of defonnation energy is stored in the slab with time. The effet of this defonnation in the slab element is depited in Fig 5.l(e). This energy is onsidered to be stored beause it is within the elasti range of defonnation of the slab and the strains indued are suseptible to be released if the onditions are given. This energy is proportional to the amount of strain eu indued in the slab as it deforms. Then, the strain produed is related to the defletion of the slab, that is where eu = f(yw) ro L2 Yw=- k where eu = strain due to stored defonnation energy; Yw = slab defletion after period T, inh; (5.9) ro = unifonn distributed weight of onrete, lb inh; L = total length of the slab, inh; and k = relative value of soil support (k-value), psi inh. Defletion Y an be related to the strain eu produed by means of the Poisson's modulus. As stated, the stored strain affets the development of ongoing temporal strains in the slab sine it is already stored in the slab. Therefore, it must be aounted for in the development of stresses. The final relation for a unifonnly distributed load is where (5.10) v = Poisson's ratio for onrete and all the other terms are as defmed above. SOLUTION OF THE MODEL In the frrst part of this setion, the partiular solution for the model at the edge is determined separately, f

30 25 affeting it by a parameter representative of the funtion for intermediate points along the slab. In this way the handling of the integration operations is learer. One the initial partiular expression is produed, the generalization of the model is ahieved by the introdution of the parameter developed for the intermediate points along the slab. For larity, the presentation of the development of the solutions follows the same order. Sine the development of the model is based on an established temperature gradient is ating on the slab, the model does not hold for a transition period when the temperature gradient experienes a reversal. In this portion of the yle, the slab experienes a reovery of its former shape. For a better desription of the entire yle it is neessary to introdue a model desribing this reovery period. In the last part of the setion, the solution for the omputation of the stresses indued by urling is introdued. Solution for the Slab Edge Vertial Displaements. A relation developed from the elasti theory is used for the determination of the displaements due to urling. From the theory of elastiity, we know that the seond derivative of Y with respet to X is equal to the bending moment divided by the produt of the modulus of elastiity times the moment of inertia of the element. Then, the following relation is established: the expression for slabs is where (5.11) (5.12) (5.13) and all others are as defmed above. In Eq 5.12, the moment of inertia I is replaed by the flexural rigidity of the slab D, whih differs from the rigidity of the onventional beam by the fator 1 (1 - v2). The determination of Y implies the double integration of the funtion in Eq 5.12, that is, 1 n Y=-J fmdx2 Eio o (5.14) From the preeding setion we got the expressions for the moments due to temperature and frition. Then, the total displaement for urling "Y" will be equal to the sum of both moments, that is: (5.9) Now, we an onsider that the displaements are symmetri for eah half of the slab. Then, if we substitute Mro and MTM for their expressions in Eqs 5.4 and 5.5, and if we perform the integration from the enter of the slab to the edge, the fmal result is Y =! [6.To a+ v ro L3 )x2 $ (BL)2 (1 - v2) 2 \ 15 E2k D + 0TM' a+ 1~ EJ3k) x2 dl (BL)2 (1 - v2)l o2 J (5.15) where all the variables have already been defined, exept f, whih is a parameter representing a restriting fator affeting the level of displaements along the slab. For slab edges the value of this parameter is equal to unity. Equation 5.16 is the funtion that desribes the behavior of the orner of a the PP slab as it urls due to temperature differentials. As we an see, the model onsiders (I) the relative effet of temperatures, (2) frition, (3) slab geometry and (4) the value of the soil support. Now the next step is the determination of the expression for intermediate points and their stresses. They are disussed below. Solution for Vertial Displaement of Intermediate Points Along the Slab. The magnitudes of the urling displaements at different points of the slab length are a funtion of the inreasing stiffness of the slab as it approahes the enter. For the determination of the urling displaements along the slab, a regression analysis using the olleted data was performed. From this analysis, the funtion determined for the model has the form

31 26 (5.17) Then, Eqs 5.16 and 5.17 onstitute the generalization for the predition of urling displaements in slabs. Solution for the Transition of Temperature Gradient Reversal. A little after sunrise, when sunrays start heating the pavement again, the top of the pavement beomes hotter than the bottom. This heat produes a short transition and a reversal of the temperature gradient, and onsequently the diretion of the fores beome opposite to the diretion shown in Fig 5.1. It is during the transition moment when the slab flattens again. In fat, the slab is reovering its horizontal shape before undergoing the opposite part of the yle, in whih the slab stores strains and stresses. From Figs 4.8 and 4.9, we learned that the path desribed by the slab is different from the ooling path. The heating path is steeper in relation to the ooling (or urling) path. The reasons for this influene is that the slab is helped by its own weight, its stiffness, and, in general, the release of energy stored during urling. An in-depth analysis of this transition path is of no use sine the values of interest for design are those of maximum urling and stresses aused by urling. In addition, the omplexity of the data required for the study of this transition is beyond the sope of this researh. However, for a better desription of the slab behavior from yle to yle, this transition of temperature gradient reversal was studied. Regression analysis tehniques were used to develop a parametri equation for this model. The fmal model for the transition of temperature gradient reversal is where p = ATDi = Ymax = parameter in funtion of the history of intensity of the heat radiation and the thermal effetive oeffiient of the pavement - for the present researh, the values ranged between 18.5 for hot weather, and 13 and 10 for old weather; as defined before; maximum url experiened by the slab before the transition, inh; ATDtmax = maximum positive temperature differential produed during the transition, F; and ATDtmax = maximum negative temperature differential produed during the urling yle, 0 F. Solution for the omputation of Stresses. As mentioned earlier, materials dissipate energy by means of deformations. Thus, when a slab is urling, it is dissipating energy and, in onsequene, the stresses to whih it is being subjeted. The opposite happens when the material deformation is restrited The restrition auses the material to start building up stresses. In this ase, stresses due to a temperature differential build up in those portions of the slab that are restrited to deformation or url. Therefore, the level of stresses is maximum where the stresses are fully restrited. For slabs, the full restrition ours at the enterline of the slab. One the displaements have been modeled, we an see that urling is fully restrited at the enterline of the slab. From there, stresses derease gradually and reah a value of zero at the slab edge. The loation of this point is a funtion of the frition, stored energy, partiularities of the slab, onstrution, et. From the engineering point of view, the maximum stress is of interest for design of the slab. This is beause no further eonomy is ahieved in pratial terms sine slabs are designed with a onstant thikness and the speifiations of steel reinforement from design is mainly uniform for the entire slab. From the work developed above, a relationship an be stated for the determination of stresses based on the relationship of Eqs 5.11 and This expressions overs only the effet of urling, sine the effets of temperature with frition are overed in the model for determination of stresses due to slab frition. Then, the expression for the stresses produed by the temperature differential has the form tt v rol3 J r=e 6: D'a+ 15 E2 k (5.19) A onservative and more pratial value for the determination of the maximwn stresses of the slab is given by the relationship a = E {ATD' a) (5.20) This value is onstant at the slab enterline and it is added to the stresses due to frition along the slab. TESTING OF THE MODEL The validation of the model is made omparing the predited values of the model just developed with the atual olleted data. The values for the horizontal displaements were employed as input for the model for urling for the validation. In this way, the error of the predited data was

32 27 restrited to only the error of the model itself and the impliit error of the olleted data. The predited values are in Tables 5.1 to 5.3. Table 5.1 ontains the values for the predited vertial displaements at the edge, the results for urling at the one-sixth point from the slab omer are in Table 5.2, and the values for the predited vertial displaements at the one-sixth point from the enter of the slab (one-third point from the orner of the slab) are in Table 5.3. Then, the predited values were ompared with Table 4.15, whih has the summary of the representative values from the data olleted in the field, and with Tables 4.7 to 4.10 for omparison with intermediate data points. Figures 5.2 to 5.9 depit the degree of orrelation between the field and the predited data for the data sets olleted on July 25, November 5, and January 22. These data are representative of hot, random, and old weather. The orrelation obtained is good. The values ranged between 0.86 and A look at the figures reveals that the degree of dispersion is higher for the initial readings of eah set of data and the orrelation points approah the forty-five degree line as the amount of data aumulates. This behavior is expeted sine the model uses a disrete approximation funtion for temperature. Therefore, the model yields reasonable preditions for use in design. With the satisfatory testing of this model, the proess for the development of a new model, required for the more aurate estimation of urling values in PP slabs, has been ahieved. The neessary assumptions were presented and the solutions for the model were worked out. In the following hapter, program PSn, the upgraded version of PSPI, is introdued along with a summary of the prinipal fators and onepts involved in the alibration. TABLE 5.1. SUMMARY OF PREDITED VALUES FROM NEW MODEL FOR URLING (a) For vertial displaements at the orner of a 240-foot slab (inh) Tri Date Hour July 25 AusustS Ausust26 NovemberS Janu~21 January22 Febru~9 14: : : O.OS S8 0, : : : : S : S : S : : S : : (b) For vertial displaements at the omer of a 440-foot slab (inh) Tri Date Hour July 25 AuestS Au~t26 NovemberS Janu~21 Janu~22 Febru~9 14: : : : S : S : S : : B:l~oY 6: S : : : :

33 28 TABLE 5.2. SUMMARY OF PREDITED VALUES FROM NEW MODEL FOR URLING (a) For vertial displaements at the sixth point of a 240-foot slab (inh) Tri Date Hour July 25 Au~ust 5 Au~ust 26 November 5 Janu~21 Janu~22 Febru~9 14: : O.Q : : : : : : : : : : : O.Q (b) For vertial displaements at the sixth point of a 440-foot slab (inh) Tri Date Hour July 25 August 5 August 26 NovemberS Janu~21 January 22 February 9 14: : : : : : : O.Dl76 4: : : : : :

34 29 TABLE 5.3. SUMMARY OF PREDITED VALUES FROM NEW MODEL FOR URLING (a) For vertial displaements at the third point of' a 240-f'oot slab (inh) Tri Date Hour July 25 Au~ust 5 AU USt 26 November 5 Janu!!l: 21 Janu!!l: 22 Febru!!l: 9 14: : : : O.Dl : : : : : : : : : (b) For vertial displaements at the third point of' a 440-f'oot slab (inh) Trip Date Hour July 25 Au12ust 5 Au12ust 26 NovemberS Janu!!l: 21 Janu!!l: 22 Febru~9 14: : : : : : : : : O.Gl : : : :

35 30 w 0.2 -=:!" N > : 0 :~ "0 0.1 Q) 0:: Q) "0 0 :::!: 1..(') N ~ y = e-3 = x R" Field V24 Fig 5.2. Degree or orrelation between model and field vertial displaements at the orner predited for a 240-root slab for hot weather onditions. 0.2 w - -=:!" N > : 0 ~ "0 0.1 ~ a.. Q) "0 0 ~ 1..(') y = e x... ~ R"2 =0.910 a Field Data Vertial Movement 725 Fig 5.3. Degree or orrelation between model and field vertial displaements at the orner predited ror a 440-foot slab for hot weather onditions. 0.2 Q) ' 0 :::!: >...0 ' Q) -u '6 a: Q) -: Q) E 0.04 ~ 0 :::!: ni u y = e x ' 0.02 R"2=0.935 Q) > "iii ' 0 ~ l; - ' Q) u '6 Q)... a.. -: Q) E ~ 0 :::!: B ' ~ 0.00 _..._..._..... ""'-....,,,.,.,--J Field Data- Vertial Movement -115 Fig 5.4. Degree or orrelation between model and field vertial displaements at the orner or a 240-root slab for a set or weather onditions hosen at random O.Q lit.-.. a y = e x R 11 2 = o...._..., Field Data - Vertial Movement Fig S.S. Degree or orrelation between model and field vertial displaements at the sixth point or a 240-root slab ror a set or weather onditions hosen at random. a o

36 <» -g 0.04 ::!: ~ "! '6 ~ a.. -: <D E 0.02 ~ 0 ::!: ~ -e 0.01 ~ a a y = e..J x RA2= l'------' ' 0.00 O.o Field Data- Vertial Movement -115 Fig 5.6. Degree or orrelation between model and field vertial displaements at the third point or a 240-foot slab for a set or weather onditions hosen at random. a a a 0.03 <» " 0 ::!: 0.02 ~ a " <D 0 '6 e ~ O.Q1 a : <D E <D > 0 ::!: ~ 0.00 t a a a a a ir i ~ y = e x ~ ~ RA2 = o.oo 0.01 o.o2 o.o3 Field Data- Vertial Movement -122 Fig 5.8. Degree or orrelation between model and faeld vertial displaements at the sixth point or a 240-foot slab for old weather onditions a a 0.05 <D -g 0.04 ::!: ~ " s.~ 0.03 " e a.. E <D ~ 0.02 > 0 ::!: B -e ~ O.Q1 a y = e x RA2= , , 0.00 o.o1 om om o~ o~ Field Data- Vertial Movement -122 a a a y = e x RA2 = _.L ~ Field Data- Vertial Movement -122 Fig 5.9. Degree or orrelation between model and raeld vertial displaements at the third point or a 240-foot slab for old weather onditions. Fig 5.7. Degree or orrelation between model and field vertial displaements at the orner or a 240-foot slab for old weather onditions.

37 HAPTER 6. ALIBRATION OF THE PROGRAM PSPl In this hapter, the main fators and the proedures exeuted for the alibration of the program are desribed and the prinipal onepts that resulted from this proess are disussed. Program PSP2 is introdued, and the orrelation between the values predited by the program and the field values are disussed, along with some onsiderations for better use of the program. PRINIPAL FATORS IN ALIBRATION In hapter 2, an analysis of the prinipal variables was performed to detet the possible fators that ould have a role of importane in the behavior of PP slabs and onsequently in the models used in the program. Table 2.1 and Fig 2.3(a) and (b) presented graphially the information produt for this first analysis and served as a departure point for the data olletion planning. The analysis beame more objetive in hapter 3, where the analysis of the data for the models was disussed. At the stage of researh previous to the olletion of data, it was stated that a prime part in the alibration would be the data olletion. The input haraterization made it lear that there were some fators whih made it neessary to arefully plan for the olletion of signifiant data under different onditions for their analysis. Also, a sequene for the alibration proess was suggested. Afterwards, the required data were gathered and the analysis was onluded. Also it was possible to establish the relative weight of eah fator in the alibration. From the work done up to this point, the fators that are onsidered to have a major role in the alibration of the program are: (1) Environment. The driving fore for urling is environment. The hanging onditions of its prinipal parameters, moisture and temperature, provide a different set of onditions eah time, making the pavement slab behave within different ranges of values for all the other fators (frition, thermal ontration, et.). The olleted values and the measured behavior onfirmed the importane of stresses indued by environmental hanges. (2) Frition. The seond fator for the alibration of the program was frition. The measurement of horizontal displaements permitted the analysis of the frition under the P slab. The models turn out to be espeially sensitive to the frition values and profiles. From the analysis it was stated that the model in PSPI rendered reasonable results for the horizontal displaements, and the alibration proeeded to the adjustment of values determined in previous researh projets. The results indiated that the values previously reported were somewhat high for the displaements experiened by the PP slabs monitored in this study. (3) Temperature Gradients and Thermal oeffiient of the Slab. A third fator for the alibration of the program was the analysis and determination of the effet of temperature gradients in the slab and their ombination with other fators. The effet of the thermal oeffiient of the material and its variations were studied along with frition and temperature gradients for the determination of the basi fators in urling. A model that onsidered this and other fators was developed and tested for auray. Then, the new model was implemented into program PSP2 with the hanges neessary to reflet the build-up of stresses due to urling in the slab. It is important to point out that during the analysis and testing of program PSP2 it was possible to observe the range of variations that these three fators experiened during the olletion period. These variations onfirmed the range of variations suggested in manuals. It is also interesting to note the sensitivity refleted by the models to these variations. Variations in the values of the oeffiients must be onsidered in the determination of values for design. An exemplifiation of this responsiveness is presented later on in this hapter. (4) K-value and Strutural Stiffness of the Slab. An aspet brought up by the analysis was that the variations in mehanial strength of the soil represented by the soil support value (k-value) do not affet to a large extent the behavior of the slab at this stage of the slab life. The k-value and the stiffness of the slab work in onjuntion as a strutural unit. The model developed in this respet is somewhat different from the Westergaard model and more losely related to the models used in soil mehanis. One the strutural stiffness effet was measured, it was translated and introdued in the model in terms of strain energy. The alibration proess ended when program PSP2 was tested. From the test it was lear that it was not neessary to pursue a further alibration sine the range of values predited by program PSP2 is aurate enough for use in design. Thus, the alibration did not have to further onsider the remaining fators (suh as models and oeffiients for onrete and steel), whih were mentioned for the tentative alibration proedure in hapter 2. ALffiRATED PROGRAM PSP2 Program PSP2 onstitutes the upgraded version of program PSPl. The flow harts of the model for the predition of urling and its loation within the general flow hart of program PSP2 are presented in this setion. For a more thorough disussion on the flow hart of program PSPl, the reader should refer to Researh 32

38 33 Report 401, by Mendoza-Diaz et al (Ref l). hanges in the program input data and odifiation of data input an be found, along with an example of the data used, in Appendixes A and B. Appendixes and D ontain an example of the output data and the listing of the program. In this setion, the results from the PSP2 orrelation with the data olleted in the field are presented for the horizontal and the vertial data. The program has been maintained and it is upgraded with the following hanges: (1) A new model for urling was developed. This model allows the predition of vertial displaements due to urling andor warping and introdues geometry fators that were not onsidered before in the alulation of this type of displaement. The model for urling is presented and disussed in hapter 5. (2) An equation developed at TR for the omputation of the modulus of elastiity for onretes made in Texas was added. This equation onsiders the type of aggregate in the omputation of the modulus of elastiity. The general flow hart diagram for program PSP2 is shown in Fig 6.1. This flow hart is essentially idential to the one of program PSP1 (Ref 2) exept that subroutine PREDMD is added (to alulate the modulus of elastiity) and the newly developed model was installed in subroutine URL. The flow hart for this new model for urling is presented in Fig 6.2. The flow of the program in subroutine URL with the model installed proeeds in the following manner. First, the values for the average temperature gradients are determined as well as the maximum and minimum temperature gradients for the yle. Next, the program determines for eah reading whether or not the slab is undergoing urling. If it is, the value for urling is determined for eah time interval until the program detets a hange in the gradient. One the program establishes that the slab is experiening a reversal in the temperature gmdient. the program predits the transition part of the yle, whih ours when the slab is flattening. For eah part of the proess, the stresses that build up in the slab are omputed after the displaements are alulated. Then, the ommand of the program returns to the main routine. The program an also predit the build up of stresses from warping. The differene in this ase is that the gradients to be input are the equivalent ones indued by the moisture differentials developed between the top and the bottom of the slab. Additionally, a version of program PSP2 was made for a personal omputer (P). This P version of PSP2 ahieves the objetives of lower osts and higher availability of the program while keeping effiieny. The average time for one run is less than two minutes in a omputer with a speed of 10 MHz. This running time is redued to seonds for a 25 MHz mahine. These running times allow the onsideration of several design possibilities in a short period of time. The general auray of the program was tested using data from the field. Previously, some orrelation values were determined for the horizontal data in hapter 4 and orrelation values for the vertial displaements were presented with the testing of the model for urling. Here, the fmal orrelation values for horizontal and vertial displaements from the interation of models in PSP2 were heked. HORIZONTAL DISPLAEMENTS In all ases the omputed values were found aeptable and in agreement with field values for this projet. A oeffiient of frition of 0.2 was used in the input. For the ompressive strength of onrete, the mean value of the final strength determined by laboratory testing of the field samples was alulated. All the other input values were taken from field reords. Input values are in Appendix B. The numbers turned out by program PSP2 are the same as those presented in Table 4.1. The urves orresponding to the horizontal displaement values are shown in Figs 4.1 to 4.3. They show good agreement between the predited and the measured behavior. Parts (a) of Figs 4.7 to 4.9 show the orrelation for the horizontal displaement The predited horizontal displaements are in lose agreement with the measured values. The oeffiients of partial determination in parts (a) of Figs 4.7 to 4.9 show that the orrelation values for program PSP1 ranged between 0.99 and 0.87 for the horizontal movement Figure 6.3(a) and (b) onfrrms the level of orrelation between the field data and the data predited by PSP2 for the edge points of the data set olleted on July 25. Figures 6.4 and 6.5(a) and (b) onfrrm the level of orrelation between the field data and the data predited by PSP2 for the interior points of data sets olleted on November 5 and January 22 for 240 and 440- foot slabs. VERTIAL DISPLAEMENTS For vertial displaements, the predited values are summarized in Tables 6.1 and 6.2. Figures 6.6 to 6.8 show the orrelation between the field and predited vertial displaement data for the edge and interior loations of the data sets for July 25, November 5, and January 22. The orrelation in these figures is slightly lower than the orrelation obtained for the model alone in Figs 5.2 to 5.9. This is due to round-off and to trunation errors introdued by the interation of the models and possibly to the variations in the value of the thermal oeffiients of the materials with the seasons.

39 34 Read to. to and Temperature Data for Initial Period and Times and Amooot of Prestress for eah Post-tensioning Stage Subroutine TIDEVAR ompute Strainlnrement for Time Inrement D t ex= a DT + D Zt Subroutine PREDMD ompute Profiles of Movement and Frition Restraint Stresses from Frition Submodet 1 uperimpose Post-tensioning Stress Applied, If any, to Frition Stresses ompute Profiles of Movement and Frition Restraint Stresses from Frition Submodel 2 Redefine Newlnilial ondition after Reversal for Movements and Stresses: lxi & fxi E Subroutine FREST Determine urtlng Stresses and Defletions Print Profiles of Slab Stresses, Movements and Vertial Defletions No ~s A ~~~ ~~ Fig General now diagram of omputer program PSP2. (ontinued)

40 Read Number of Days after uring noa when Analysis Period Starls 35 'res... Read Temperature Data for Period onsidered Detennine Prestress Level after Losses for Period onsidered Redefine Newlnitial ondition after Reversal for Movements and Stresses: Zxi & fxi E ompute Profiles of Movement and Frition Restraint Stresses from Frition Submodel1 No Print Profiles of Slab Stresses, Movements and Vertial Defletions Fig General now diagram of omputer program PSP2 (ontinued).

41 36 Subroutine url ilmperatures and Slab Data ompute Disrete Aproximation of Effetivelnrement of Temperature Gradientfor All Time Intervals ompute Values for T~ansition Parts of yle loop for Slab length Inrements ompute Relative Values of Stiffness 'lt!s alulate Displaements During Transition of yle alulate Displaements Due to urling alulate Stresses Indued Fig 6.2. Flow diagram for subroutine URL.

42 -:: Q) E Q) (.) <' S.0.1 : :: 2 0 :X: I' a I' I' I' a I' r I' a I' a I' I' a a I' I' al' I' a I' I' I' I' I' a I' a I' I'.0.2, _ Q Horizontal Displaement from 725 Field Data, (a) At the omer or a 240.root slab. 0.1 ~ u (f.) a >-.Q " Q) ~ " ~ -Q.1 a.. -: Q) E ~..5!! -Q.2.. l.) 0 <'0 E : 0 :X: a a a I' I' I' 1:1 I' I' a I' I' a I' a a I' I' I' I' I' I' I' I' a I' 1:1 I' I' I' I' -Q.4 IIL-~---JL.----' ,, -o.4 -o.a -o.2 -o.1 -o.o 0.1 Horizontal Displaement from 725 Field Data (b) At the omer or a 440 root slab. 37 Fig 6.3. orrelation or program PSP with raeld data ror horizontal displaements at the omer or a 240 and a ~root slab under bot weather onditions I' I' I' (\J (\J I' a.. I' a.. u 0.00 ~ u (f.) 0.00 (f.) a.. I' a.. I' >-.Q I' >-.Q I' ~.0.02 I' a ".0.02 I' I' Q) (.) - I' -...sa 'S I' I' e I' a " g:.0.04 m a '4 I' : a I' : Q) Q) I' I'. E ~ -o.os I' ~ -o.oo I' (.) <'0 I'.. I' <'0 I' l.).. I' l.) I' 0 a 0 I'.0.08 I' s -o.08 a I' a s I' : :: 0 I' I' N I' a 0 -~.0.10 I' ' -Q.10 I' a 0 :X: I' :X: I' I' I' a.0.12 &..._..._...JI...J &-...t IlL-_....._ , -o.12 -o.1o -o.os -o.oo -o.o4 -o.o2 o.oo o.o2 -o.12 -o.1o -o.os -o.oo -o.o4 -o.o2 o.oo o.o2 Horizontal Displaement from 115 Field Data (9N6) Horizontal Displaement from 115 Field Data (10N3) (a) At the sixth point or a 240 root slab. (b) At the third point or a 240 root slab. Fig 6.4 orrelation or program PSP2 with field data ror horizontal interior points or a 240 root slab ror a set of weather hosen at random (a = 5 X ld-6).

43 38 N.. 0 ~ >-. ' ~ u '0 Q)..... "E Q) E Q) u ~ ".. () i5 1 0 N ;:: 0 :I: 0.02 ; 0.04 ; N.. O.Q1 0 () a:» ~ 0.00, a ' 411 Q) ~ -{).01 a Q)... -{).02.. a g -{).02 ~ E -{).04 a ~ Q) a (,) ~ a ~. -{).03 a en a -{).06 a i5 ~ a n; a ~ "E -{),04.. -{).08 a ~ 2 a r:: ~ a ~ -{).05 a ~ ID -{).10 a &t ~ -{).12 K--...a-...&....._ _-.l -{).06 -{).12 -{).10 -{).08 -{).06 -{).04 -{) {),06 -{).05 -{).04 -{).03 -{).02 -{), Horizontal Displaement from 122 Field Data (10S6) Horizontal Displaement from 122 Field Data (1 OS3) (a) At the sixth point of a 440-foot slab ~ # ~ N N ~ ().. ~ ~ ~ a >- a. >- a~ a.t::l a ' a -o 0.1 Q) 0.1 a Q) t) -u '0 a '0 Q) ~ Q)... a a..... a... a a "E ~ Q) - Q) ~ E a a E Q) a a Q) (,) (,) e ~ a ".. ~ 0.0 en 0.0 ~ i5 a i5 a ~ a n; a 0 ~ a ~ (b) At the third point of a 440-foot slab. Fig 6.5. orrelation of program PSP2 with field data for horizontal displaements at interior points of a 440-foot slab under old weather onditions (a = 5 X 10-6). :e a t, ~ ~ ~ ~ -{).1 -{).1 -{) {) Vertial Displaement from 725 Field Data Vertial Displaement from 725 Field Data (a) At the orner of a 240-foot slab. (b) At the orner of a 440-foot slab. Fig 6.6. orrelation of program PSP2 with raeld data for vertial displaement at the orner of a 240 and a 440-foot slab under hot weather onditions (a = 5 X 10-6).

44 39 TABLE 6.1. SUMMARY OF PREDITED VALUES FOR URLING FROM PROGRAM PSP2 (a) For vertial displaements at the orner of a 240-foot slab (inh) Tri Date Hour July 25 Au&ust 5 Au&ust 26 November 5 Janu!!X 21 Janu!!X 22 Febru!!:!19 14: : : : : : : : : : : : (b) For vertial displaements at the orner of a 440-foot slab (inh) Tri Date Hour July 25 August 5 Au~ust26 November 5 January 21 Janu!!:!122 February 9 14: : : : : : : : : : : :

45 40 TABLE 6.2. SUMMARY OF PREDITED VALUES FOR URLING FROM PROGRAM PSP2 (a) For vertial displaements at the sixth point of a 240-foot slab (inh) Tri Date Hour July 25 Au~ust 5 AU USt26 November 5 Janu~21 Janu~22 Febru~9 14: : : : : : : : : : : : (b) For vertial displaements at the sixth point of a 440-foot slab (inh) Tri Date Hour. Julv 25 August 5 August 26 November 5 Janu~21 Janu~22 Febru~9 14: : : : : : : : : : : : f

46 "'I N a.. (..) (.) a.. >...0 -o a:>.2 -o a:>... a.. a:> E a.:> u ('tl is. (J) i5 ('tl :2 ~ " " ", &.- 1:1 " " -' a 1:1 ""..., ~_ "" " a "" " 1:1 1:1 : " a-',~, " " _..._......, Vertial Displaement from 115 Field Data (1 ON6) N 0.06 ~ 0.05 (.) a.. >...0 -g 0.04 t5 '6 a:> a: 0.03 a:> E a.:> u ~ 0.02.~ l " t 0.01 ~ a "" " a -' " "" " " "" "" " 0.00 &-_;...!. a..,... --'-~...J a a " a -' a-' I -'a -' a a " 41 " " Vertial Displaement from 115 Field Data (1 ON3) (a) At the sixth point of a 240-foot slab. (b) At the third point of a 240-foot slab. Fig 6.7. orrelation of program PSP2 with lield data for vertial displaements at interior points of a 240-foot slab for a set of weather onditions hosen at random (a= S X 10-6) oo 0 N a.. 0 (..) (.) a.. 0 i:; 0.02 "" -o 0 a.:> 0 a-' "" -u '6 e ; a.. " 0.01 a.:> -'o " E Q' a.:> u g.. "" (J) i "" -;;; u -'o t ; " 0 ~ "" " "" U Vertial Displaement from 122 Field Data (10S6) (a) At the sixth point of a 440-foot slab. N a.. (..) (.) a.. >...0 -o a.:> ~ -o 2: a.. -: a.:> E a.:> u ('tl is. (J) i u t "' ~ " " " " " -'o " 1:1 " 1:1 " 01:1 " R " "" " " "". "" "" "" _...,.....J Vertial Displaement from 122 Field Data (10S3) (b) At the third point of a 440-foot slab. Fig 6.8. orrelation of program PSP2 with lield data for vertial displaements at interior points of a 440-foot slab under old weather onditions (a = S X 10-6).

47 42 TABLE 6.3. SUMMARY OF PREDITED VALUES FOR URUNG FROM PROGRAM PSP2 (a) For vertial displaements at the third point of a 240-foot slab (inh) Tri Date Hour July 25 Auiust 5 Auiust 26 November 5 J anu!!;!x 21 January 22 February 9 14: : : : : : : : : : : : (b) For vertial displaements at the third point of a 440-foot slab (inh) Tri Date Hour July 25 Au~ust5 August 26 November 5 January 21 January 22 February 9 14: : : : : : : : : : : : ,

48 ..: a a N l. <..::> () l. t N l. a <..::> () l.. >- a. >- a "0 "0 (I) s.52 "0 (I) "0... (I) l. t a 'E a E (I) (I) E E (I) (I) u m u m " a " ~ en 0 a ' ~ u -e (I) (I) > > -0.1 IE _ D -m u t -0.1 E---~---' r ' -om o.o9 o.19 -o Vertial Displaement from 725 Field Data Vertial Displaement from 725 Field Data (a) At the orner of a 240-foot slab. (b) At the orner of a 440-foot slab. Fig 6.9. Best orrelation of program PSP2 with field data for vertial displaements at the orner of a 240 and a 440-foot slab under hot weather onditions (a=... 7 X 10-6). (\J l. <..::> () l. >-. "0 (I). "0 D... a.. 'E (I) E D u m "0.. en i5 ~ u -e fj; N 1 l r <..::> () 0.06 l. >- I:V. ~Y.. ; " s.52 " ; (I) [J ; ~ a.. ; 'E (I) E 0.02 (I) 0.02 u ~... "0.. en 0.01 a ; -m 0.00 u ; t 0.00 d. fj; ; -o.02 -o.01 -o o.01 -o.oo Vertial Displaement from 115 Field Data (10N6) Vertial Displaement from 115 Field Data (1 ON3) (a) At the sixth point of a 240-foot slab. (b) At the third point of a 240-foot slab. Fig Best orrelation of program PSP2 with field data for vertial displaements at interior points of a 240-foot slab for a set of weather onditions hosen at random (a = ""5.5 X 10-6).

49 44 URUNG An outome of the analysis was that urling in PP slabs is speially sensitive to hanges in the thermal hanges of the slab along with some other fators, and as frition and rate of development of the temperature gradient. This range of values is onsistent with values shown in the literature for similar aggregates. An illustration of the effet of the variations of oeffiients of frition and thermal ondutivity is shown in Figs 6. 9 to The data were generated by running PSP2 several times employing adequate values for the thermal oeffiient. Figures 6.9 to 6.11 show the improved orrelation ahieved with these values. Therefore, the designer should onsider this fat when seleting input values in order to ahieve adequate values. This preaution will prevent the use of input data that might predit undesirably low results, leading to unonservative designs. WARPING Program PSP2 an be used to predit warping in the slab. In this ase, the gradient indued by the moisture differential has to be input as the equivalent temperature gradient. The nature of the data olleted and the dry weather onditions prevailing during the period of the data olletion prevented any suh use of the program. Therefore, the use of PSP2 for the predition of warping is beyond the sope of this paper. Nevertheless, the program is designed and an be employed for this purpose in the future if needed. Summarizing the statements earlier in this hapter, program PSP2 provides satisfatory results with the alibration and upgrading introdued in the original program. The output values for the horizontal and vertial displaements were tested and proved to be satisfatory o [J N [J a. u [J (.(} a >- []. "0 (1) t) [] : [J - : (1.) E (1.) 0.01 u.!!:! _,i 0.. Vl a: (1.) Vertial Displaement from 122 Field Data (1 OS6) (a) At the sixth point of a 440-foot slab QJ[J N [J a. u [J (.(} a. [J o.o2 "0.s [J.52 [J "0 (1.)... a. : O.Q1 (1.) ~ E (1.) u (t:\ i5.. Vl ~ [] u -e [J ~ -{),0 1 -{).01 -{) Vertial Displaement from 122 Field Data (10S3) (b) At the third point of a 440-foot slab. Fig Best orrelation of program PSP2 with field data for vertial displaements at interior points of a 440-foot slab under old weather onditions (a= -3.5 X 10-6).

50 HAPTER 7. SUMMARY, ONLUSIONS, AND REOMMENDATIONS The fmdings of this study are presented in this hapter, along with the onlusions and, some reommendations for the design and onstrution of PP slabs are made. SUMMARY This report has disussed the neessary bakground on PP slabs and the analysis arried out for this study in the first three hapters. The olleted data and the program alibration are disussed from hapters 4 to 6. hapter 5 is devoted to the development of a new model for urling. Some of the more important onepts presented are summarized here. PP is an attrative alternative to onventional pavements. The study of PP and its materials and the study of the pavement as a unit are important for understanding PP. For this study, the data olleted in the field for the one-mile-long experimental setion showed a high degree of uniformity. A orrelation test showed that models for horizontal displaements of PP slabs are reasonable but this did not hold for vertial displaements using the program PSPL A new model for the behavior of PP slabs under temperature gradients is from existent models and has shown good agreement with the observed performane of PP. Use of revised program PSP2 an help in the study of P and in design appliations. ONLUSIONS The following onlusions were reahed in these study. FROM THE VISUAL SURVEY (1) No signs of major distresses are present in the PP slabs, and, no progress in the existent raks was deteted during the monitoring of the experimental setion in Wao. (2) The strutural apaity in the experimental setion is uniform and similar to that observed soon after the P onstrution. (3) The seleted joints have performed satisfatorily. No signs of spalling, raking, or warping were notied, whih indiates the good protetion and anhoring of the joints. However, the rate of debris aumulation leads to filled joints. (4) If suffiient debris is present in the joint, the slabs an be prevented from expanding freely. This ondition is more frequent during summer, when the slabs reah maximum expansion. (5) If suffiient debris aumulates in a partiular setion of the joint, that setion of the slab's joint will be prevented from expanding freely. This ondition was notied during this study and it indues development of uneven stresses in the slab. No distress in the slab was observed from this ause. (6) The observed raks running between the stressing pokets loated at the enter of the PP slabs did not show appreiable signs of progress, remaining hairline tight. The same ondition ourred with a pothole whih probably resulted beause of a lay ball in the aggregate. However a seond pothole with the same harateristis as the ftrst one was deteted towards the end of the monitoring stage. (7) In general terms, all the slabs are in good ondition and from the visual survey it was lear that they are behaving normally. FROM THE DATA ANALYSIS (l) The analysis of the data olleted in the field showed it to be highly onsistent and uniform. (2) The displaements of the slabs are uniform with no slabs showing abnormal displaements. (3) The horizontal data were very onsistent for all kinds of weather. The data are strongly related to the temperature and frition in the slab. (4) The magnitude of the vertial displaements is higher than the amount all the previous models for urling ould predit. The revised model predits values that are similar to the vertial displaements measured. (5) Vertial displaements in the slab are more related to the temperature gradient that develops between the top and the bottom of the slab than to the temperature at the middle of the slab. However, there is a orrelation between this kind of displaement and the range of variations of the temperature at the middle of the slab. (6) There are variations in the thermal and frition oeffiients between seasons. These variations should be onsidered during the design stage for an optimum range of design parameters. FROM THE PREDII'ION MODELS (1) The atual level of frition between the slab and the subbase is lower than that reported by previous studies but agrees with the bak alulated oeffiient in the parent researh projet, 401. A reason for this differene might be the time the experimental setion has been in servie, onsidering the ontinuous daily movements as temperature hanges our. 45