H I L L INO I S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007. UNIVERSITY OF ILLINOIS BULLETIN ISSUED WEEKLY Vol. XXXII September 11, 1934 No. 2 [Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, under the Act of August 24, 1912. Acceptance for mailing at the special rate of postage provided for in section 1103, Act of October 3, 1917, authoriied July 31, 1918.] LABORATORY TESTS OF THREE-SPAN REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS A REPORT OF AN INVESTIGATION CONDUCTED BY THE ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS IN COOPERATION WITH THE UNITED STATES BUREAU OF PUBLIC ROADS BY WILBUR M. WILSON AND RALPH W. KLUGE BULLETIN No. 269 ENGINEERING EXPERIMENT STATION PUSLISHED YT THB UNIvSITrrY OF ILLINOIS, URBANA PuI: ONE DOLLAR T HE Engineering Experiment Station was established by act of the Board of Trustees of the University of Illinois on De- cember 8, 1903. It is the purpose of the Station to conduct investigations and make studies of importance to the engineering, manufacturing, railway, mining, and other industrial interests of the State. The management of the Engineering Experiment Station is vested in an Executive Staff composed of the Director and his Assistant, the Heads of the several Departments in the College of Engineering, and the Professor of Industrial Chemistry. This Staff is responsible for the establishment of general policies governing the work of the Station, including the approval of material for publication. All members of the teaching staff of the College-are encouraged to engage in scientific research, either directly or in cooperation with the Research Corps composed of full-time research assistants, research graduate assistants, and special investigators. To render the results of its scientific investigations available .to the public, the Engineering Experiment Station publishes and dis- tributes a series of bulletins. Occasionally it publishes circulars of timely interest, presenting information of importance, compiled from various sources which may not readily be accessible to the clientele of the Station, and reprints of articles appearing in the technical press written by members of the staff. The volume and number at the top of the front cover page are merely arbitrary numbers and refer to the general publications of the University. Either above the title or below the seal is given the num- ber of the Engineering Experiment Station bulletin, circular, or reprint which should be used in referring to these publications. For copies of publications or for other information address THE ENGINEERING EXPERIMENT STATION, UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION BULLETIN No. 269 SEPTEMBER, 1934 -LABORATORY TESTS OF THREE-SPAN REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS A REPORT OF AN INVESTIGATION CONDUCTED BY THE ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS IN COOPERATION WITH THE UNITED STATES BUREAU OF PUBLIC ROADS BY WILBUR M. WILSON RESEARCH PROFESSOR OF STRUCTURAL ENGINEERING AND RALPH W. KLUGE SPECIAL RESEARCH ASSISTANT IN CIVIL ENGINEERING ENGINEERING EXPERIMENT STATION PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 5000-7 34-6165 UNIVERSITY Or IulNO.0 .. Pa-. " CONTENTS PAGE I. INTRODUCTION . . . . . . . . . . . . 9 1. Object and Scope of Investigation . . . . . 9 2. Acknowledgments . . . . . . . . . . 10 PART I. TESTS OF SINGLE-SPAN ARCH RIB II. DESCRIPTION OF SPECIMEN AND APPARATUS . . . . 11 3. Description of Specimen . . . . . . . . 11 4. Analysis of Specimen . . . . . . . . . 12 5. Description of Apparatus. . . . . . . . 13 6. Sensitiveness and Accuracy of Apparatus . . . 19 7. Loads . . . . . . . . . . . . . 20 III. DESCRIPTION OF TESTS . . . . . . . . . . 22 8. Dead-Load Abutment Reactions. . . . . . 22 9. Dead-Load Strain in Concrete . . . . . . 28 10. Elastic Properties of Arch Rib . . . . . . 30 11. Influence Ordinates for Abutment Reactions by Unit Load . .. ...... . . . 39 12. Vertical Movement of Load Points Due to Change in Span . . . . . . . . . 39 13. Vertical Movement of Load Points Due to Settle- ment of One Abutment. . . . . . . . 43 14. Vertical Movement of Load Points Due to Rotation of One Abutment . . . . . . . . . 45 15. Comparison of Values of Dead-Load Abutment Re- actions Determined by Various Methods. . . 48 16. Test to Failure. . . . . . . . . . . 52 PART II. TESTS OF THREE-SPAN STRUCTURE HAVING RIB WITHOUT DECK IV. DESCRIPTION OF SPECIMEN AND APPARATUS . . . . 61 17. Description of Specimen . . . . . . . . 61 18. Analysis of Specimen by Elastic Theory . . . 66 19. Description of Apparatus. . . . . . . . 75 V. EXPERIMENTAL DETERMINATION OF ELASTIC CONSTANTS 80 20. Description of Tests . . . . . . . . . 80 CONTENTS (CONCLUDED) Spread of Abutments . . . . . . Rotation of Abutments and Pier Tops . Settlement of Piers . . . . . . Average Values of Elastic Constants PAGE . . . 81 . . . 83 . . . 83 . . . 89 VI. INFLUENCE ORDINATES OBTAINED BY UNIT LOADS 25. Description of Tests . . . . . . . . . 26. Influence Ordinates Obtained from Movement of Pier Tops. . . . . . . . . . . . 27. Influence Ordinates Obtained from Measured Re- actions 28. Comparison of Deflection Diagrams and Influence Lines . . . . . . . . . . . . . VII. DESIGN-LOAD TESTS . . . . . . . . . . 29. Description of Tests . . . . . . . . . 30. Modulus of Elasticity of Concrete in Ribs . 31. Position of Thrust Line. . . . . . . . 32. Deflection Due to Load . . . . . . . . VIII. TEST TO FAILURE 33. Description of Test 34. Position of Thrust Line . . . . . . . . 35. Deflection Due to Load . . . . . . . . 36. Unit Strength Developed by Concrete in Arch Rib IX. DISCUSSION OF RESULTS 37. Influence of Variation in Modulus of Elasticity of Concrete upon Stress Distribution . . 38. Accuracy of Tests . . . . . . . . . . 39. Verification of Elastic Theory 40. Influence of Pier Deflection upon Stress Distribution X. SUMMARY OF RESULTS . . . . . . . . . . 41. Summary of Results . . . . . . . . . LIST OF FIGURES NO. PAGE 1. Dimensions of Arch Rib . . . . . . . . . . . . . . . 12 2. Stress-Strain Diagrams for Control Cylinders. Single-Span Arch Rib. . 13 3. General View of Single-Span Arch Rib Showing Method of Suspending the Load Blocks . . . . . . . . . . . . . . . . . . 15 4. Weighing Apparatus and Supports for Abutments . . . . . . . 16 5. Instruments for Measuring Span of Arch . . . . . . . . . . 18 6. Design Load for Arch Rib . . . . . . . . . . . . . . 21 7. Thrust Line for Dead Load. Single-Span Arch Rib . . . . . . . 27 8. Abutment Reactions Due to Unit Load of One Ton . . . . . . . 42 9. Influence Lines for East Abutment Reactions. . . . . . . .. . 42 10. Vertical Movement of Load Points Due to Changes in Span . . . . 43 11. Vertical Movement of Load Points Due to Vertical Movement of One Abutment Relative to the Other. . . . . . . . . . .. . 46 12. Vertical Movement of Load Points Due to Rotation of East Abutment 48 13. Vertical Movement of Load Points Due to Rotation of West Abutment . 49 14. Effect of Time Yield in Concrete upon Vertical Movement of Load Points Due to Rotation of One Abutment . . . . . . . . . . . 50 15. Relation Between Load and Horizontal Thrust. Test to Failure . . . 56 16. Relation Between Load and Moment at Abutment. Tests to Failure . . 56 17. Position of Thrust Line. Test to Failure . . . . . . . . . . 58 18. Deflection of Arch Axis. Test to Failure . . . . . . . . . . 60 19. Three-Span Structure Having Rib Without Deck . . . . . .. . 62 20. Details of Pier . ...... . . . . . . . . . . . 62 21. Stress-Strain Diagrams for Concrete ... . . . . . . . 64 22. General View of Specimen ..... . . . . . . . . 65 23. Influence Lines for Stress at Load Point C3 . . . . . . . . . 74 24. Influence Lines for Stress at Load Point C4 . . . . . . . . . 74 25. Live Load Producing Maximum Stress at C4 . . . . . . . . . 75 26. Apparatus for Measuring Reactions at Pier Bases . . . . . . . 78 27. Frame for Measuring Horizontal Movement of Pier Bases Relative to Abutment . . . . . . . . . . . . . . . . . . 78 28. Level Bubbles for Measuring Angular Movements of Abutments and Pier Bases . . . . . . . . . . . . . . . . . . .. 79 29. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Abutment Reactions. 20-Foot Piers . . . . . . . . .. 94 30. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Pier Top Reactions. 20-Foot Piers . . . . . . . . . . . 94 31. Influence Lines by Unit Loads from Movement of Pier Tops. Center-Span Pier Top Reactions. 20-Foot Piers . . . . . . . . . . . 95 32. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Abutment Reactions. 15-Foot Piers . . . . . . . . . . 95 33. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Pier Top Reactions. 15-Foot Piers . . . . . . . . . . . 96 34. Influence Lines by Unit Loads from Movement of Pier Tops. Center-Span Pier Top Reactions. 15-Foot Piers . . . . . . . . . . . 96 LIST OF FIGURES (CONCLUDED) NO. PAGE 35. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Abutment Reactions. 10-Foot Piers . . . . . . . . . . 97 36. Influence Lines by Unit Loads from Movement of Pier Tops. End-Span Pier Top Reactions. 10-Foot Piers . . . . . . . . . . . 97 37. Influence Lines by Unit Loads from Movement of Pier Tops. Center-Span Pier Top Reactions. 10-Foot Piers . . . . . . . . . . . 98 38. Influence Lines by Unit Loads from Measured Reactions. End-Span Abutment Reactions. 20-Foot Piers . . . . . . . . . . 101 39. Influence Lines by Unit Loads from Measured Reactions. End-Span Pier Top Reactions. 20-Foot Piers . . . . . . . . . . . 101 40. Influence Lines by Unit Loads from Measured Reactions. Center-Span Pier Top Reactions. 20-Foot Piers . . . . . . . . . . . 102 41. Influence Lines for Terminal Reactions. Vertical Reaction at Abutment 104 42. Influence Lines for Terminal Reactions. Vertical Reaction at Pier Base 104 43. Influence Lines for Terminal Reactions. Horizontal Reaction at Abutment 105 44. Influence Lines for Terminal Reactions. Horizontal Reaction at Pier Base 105 45. Influence Lines for Terminal Reactions. Moment at Abutment . . . 106 46. Influence Lines for Terminal Reactions. Moment at Pier Base . . . 106 47. Thrust Lines for Design Load . . . . . . . . . . .. . 110 48. Vertical Deflection of Arch Axis. Design-Load Test . . . . .. . 112 49. Failure of Arch Rib at C4 . . . . . . . . . . . . . . 114 50. Thrust Lines for Test to Failure. . . . . . . . . . . .. . 114 51. Deflection of Arch Axis Due to Live Load. Test to Failure. . . . . 115 52. Influence Lines for Moment at Springings by Elastic Theory . . . . 116 53. Influence Lines for Horizontal Thrust by Elastic Theory . . . . . 116 LIST OF TABLES 1. Sieve Analysis of Aggregates. . . . . . . . . . . . .. . 12 2. Physical Properties of Concrete as Determined by Tests of 6-in. by 12-in. Control Cylinders . . . . . . . . . . . . . 13 3. Elastic Constants . . . . . . . . . . . . . . . . . 14 4. Dead-Load Horizontal Abutment Reactions; First Series . . . . . 23 5. Dead-Load Vertical Abutment Reactions; First Series . . . . . . 24 6. Dead-Load Horizontal Abutment Reactions; Second Series . . . . . 25 7. Dead-Load Vertical Abutment Reactions; Second Series. . . .. . . 26 8. Modulus of Elasticity of Concrete Calculated from Measured Strain . 29 9. Change in Abutment Reactions Due to Change in Span; First Series . 31 10. Change in Abutment Reactions Due to Change in Span; Second Series . 32 11. Change in Abutment Reactions Due to Rotation of West Abutment . 33 12. Change in Abutment Reactions Due to Rotation of East Abutment . 34 13. Change in Abutment Reactions Due to Vertical Movement of One Abut- ment Relative to the Other . . . . . . . . . . . . . 35 14. Ratio of Values of Elastic Constants Obtained from Measured Reactions to Values Obtained from Elastic Theory . . . . . . . . . . 38 15. Average Values of Experimentally-Determined Elastic Constants . . . 38 16. Change in Abutment Reactions Due to Applying Live Load of One Ton, Successively, at the Various Load Points . . . . . . . . . LIST OF TABLES (CONTINUED) NO. PAGE 17. Comparison of Measured and Computed Abutment Reactions for Unit Load of One Ton ... . . . . . . . . . . . . . . . . 41 18. Vertical Movement of Load Points Due to Change in Span. . . .. . 44 19. Vertical Movement of Load Points Due to Vertical Movement of West Abutment Relative to East Abutment . . . . . . . . . . 45 20. Vertical Movement of Load Points Due to Rotation of One Abutment, the Other Abutment being Fixed. . . . . . . . . . . .. . 47 21. Effect of Time Yield in Concrete upon Vertical Movement of Load Points Due to Rotation of Abutment . . . . . . . . . . . . 49 22. Values of Dead-Load Moment at Springing Obtained by Four Methods . 51 23. Values of Dead-Load Horizontal Thrust Obtained by Four Methods . 51 24. Values of Dead-Load Vertical Reaction Obtained by Four Methods 51 25. Loads Applied to Arch in Test to Failure . . . . . . . . . . 53 26. Horizontal Thrust Observed in Test to Failure . . . . . . . . 54 27. Vertical Reaction Observed in Test to Failure. . . . . . . . . 55 28. Stress in Concrete Calculated by Elastic Theory . . . . . . . . 57 29. Sieve Analysis of Aggregates. . . . . . . . . . . . . . 63 30. Physical Properties of Concrete as Given by Control Cylinders. . . . 63 31. Influence Ordinates for Reactions at Springing, Calculated by Elastic Theory. Piers 20 feet high . . . . . . . . . . . . . 67 32. Influence Ordinates for Reactions at Springing, Calculated by Elastic Theory. Piers 15 feet high . . . . . . . . . . . . . 68 33. Influence Ordinates for Reactions at Springing, Calculated by Elastic Theory. Piers 10 feet high . . . . . . . . . . . . . 69 34. Influence Ordinates for Reactions at Springing, Calculated by Elastic Theory. Piers 20 feet high . . . . . . . . . . . . . 70 35. Influence Ordinates for Stress at Various Sections, Calculated by Elastic Theory. Single-Span Rib Without Deck . . . . . . . . . 71 36. Influence Ordinates for Stress at Various Sections, Calculated by Elastic Theory. 20-foot piers. . . . . . . . . . . . . . . 71 37. Influence Ordinates for Stress at Various Sections, Calculated by Elastic Theory. 15-foot piers. . . . . . . . . . . . . . . 72 38. Influence Ordinates for Stress at Various Sections, Calculated by Elastic Theory. 10-foot piers. . . . . . . . . . . . . .. . 73 39. Stresses at Various Sections Due to Design Load, Calculated by Elastic Theory . . . . . . . . . . . . . . . . . . . 76 40. Stresses at Various Sections, Calculated by Elastic Theory . . . . . 77 41. Measured Abutment Reactions; Elastic Constants Obtained by Change in Span . . . . . . . . . . . . . . . . . . . . 84 42. Measured Pier Reactions; Elastic Constants Obtained by Change in Span 84 43. Summary of Results; Elastic Constants Obtained by Change in Span . 85 44. Summary of Results; Elastic Constants Obtained by Rotation of Abutments 86 45. Summary of Results; Elastic Constants Obtained by Rotation of Pier Tops 87 46. Summary of Results; Elastic Constants Obtained by Settlement of Piers 88 47. Ratio of Values of Elastic Constants Obtained from Measured Reactions to Values Obtained from Elastic Theory . . . . . . . . . . 89 48. Average Values of Experimentally-Determined Elastic Constants . . . 90 49. Influence Ordinates for Moment by Unit Loads, Obtained from Movement of Pier Tons.......................................... of Pier TODS . . . . . . . . . . . . . . . . . . 8 LIST OF TABLES (CONCLUDED) NO. PAGE 50. Influence Ordinates for Horizontal Thrust and Vertical Shear by Unit Loads, Obtained from Movement of Pier Tops . . . . . . . 93 51. Influence Ordinates for Moment by Unit Loads, Obtained from Measured Reactions . . . . . . . . . .. . . . . .. . . 99 52. Influence Ordinates for Horizontal Thrust and Vertical Shear by Unit Loads, Obtained from Measured Reactions. . . . . . . .. . 100 53. Computations for Modulus of Elasticity of Concrete from Strain in Arch Rib, East Span ... . . . . . . . . . . . . . 108 54. Modulus of Elasticity of Concrete in Arch Ribs . . . . . . . . 109 55. Maximum Stress at East Springing of Center Span Due to Design Load, Calculated by Elastic Theory. .... . . . . . . . . 119 LABORATORY TESTS OF THREE-SPAN REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS I. INTRODUCTION 1. Object and Scope of Investigation.-This bulletin contains the report of tests of a single-span arch rib and also of a structure consist- ing of a three-span series of arch ribs on slender piers. The objects of the tests of a single-span arch rib were to try out on a simple struc- ture the apparatus that was being built for use in testing the more complicated three-span structure; and to determine experimentally the elastic properties of a single-span arch rib which was to be used as the basis of comparison in studying the effect of the elastic defor- mation of the piers upon the properties of a three-span arch series. The objects of the tests of the three-span structure consisting of a rib without deck were to determine the load-carrying capacity of the structure, and to compare the values of reactions and strains measured in the laboratory with values of the corresponding quantities obtained by the elastic theory. There are a number of reasons why this com- parison is desirable. The elastic theory, when applied to a multiple-span arch series on high elastic piers, taking into account the deformation of the piers, is complicated. Comparatively few engineers have acquired such a mastery of this method of analysis as would justify them in using it with the complete confidence which they should have in an analytical method to be used in the design of large important structures. It is believed that an experimental verification of this theory will be welcomed by engineers who are called upon to use it. Tests of a multiple-span arch series having a rib without deck are desirable to determine whether or not errors in the assumptions upon which the analysis is based cause errors in the results of the analysis. Engineers who have mastered this method of analysis have the same confidence in it that they have in the methods of analysis used for simpler structures. That is, they are confident that the results will be correct if the assumptions upon which the analysis is based are cor- rect. But the assumptions upon which the analysis of a reinforced concrete arch is based are known to be in error. Tests of single-span arches with fixed ends* indicate that the errors in the assumptions *See Sections 15 and 16 of this Bulletin and Bulletins 202 and 226 of the Engineering Ex- periment Station of the University of Illinois. ILLINOIS ENGINEERING EXPERIMENT STATION upon which the analysis of single-span arches is based do not seriously affect the results. It was hoped that, as a result of these tests, a similar statement could be made relative to multiple-span structures. The three-span structure having a rib without deck can be analyzed by an all-algebraic process (the elastic theory). Tests of this struc- ture were followed by corresponding tests of a three-span arch series having spandrel columns and a deck,t a structure which cannot be readily analyzed by an all-algebraic process. A comparison of the re- sults obtained by algebraic analysis with those obtained by tests, whether the two sets of results are in complete agreement or not, will be helpful in judging of the dependability of the experimental work to follow. This investigation includes tests to determine (1) reactions at the springings due to movement of the abutments and of the tops of the piers, frequently designated as the "elastic con- stants" of the arch; (2) influence ordinates for reactions at the springings by applying a unit load of one ton successively at various load points; (3) vertical deflections of the load points due to movement of the terminals, the abutments and pier bases; (4) reactions at the springings and strain in the concrete at sec- tions midway between the load points due to the design load; and (5) ultimate load-carrying capacity of the structure. 2. Acknowledgments.-The tests described in this report are a part of the investigation resulting from a co6perative agreement entered into by the Engineering Experiment Station of the University of Illi- nois, of which DEAN A. C. WILLARD is the director, and the United States Bureau of Public Roads, THOMAS H. MACDONALD, Chief of Bureau. The tests were planned by the authors in consultation with MR. E. F. KELLEY, Chief of Division of Tests, and A. L. GEMENY, Senior Structural Engineer, both of the United States Bureau of Pub- lic Roads; and with PROF. GEORGE E. BEGGS, E. H. HARDER, A. C. JANNI, and PROF. CLYDE T. MORRIS, members and chairmen, respec- tively, of the Committee on Concrete and Reinforced Concrete Arches of the American Society of Civil Engineers. The experimental work was done by RALPH KLTJGE, Special Research Assistant in Civil En- gineering, assisted by F. B. METTERHAUSEN, JOHN N. PIROK, NATHAN M. NEWMARK, and GEORGE E. JEWETT, Research Graduate Assistants in Civil Engineering, all working under the supervision of W. M. WIL- tThese tests are reported in Bulletin No. 270. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS SON. The preliminary analyses were made by GLEN MURPHY and W. M. HONOUR, Special Graduate Research Assistants in Civil Engi- neering. The computations involving the experimental data were made by E. C. GRAFTON, Assistant Professor of Structural Engineering, Armour Institute of Technology. The tests were made in the Materials Testing Laboratory of the University of Illinois. The direct expenses of the tests were paid from funds provided by the United States Bureau of Public Roads, the American Society of Civil Engineers, the Engineering Foundation, Universal Atlas Cement Company, Illinois Steel Company, American Bridge Company, Jones and Laughlin Steel Corporation, Interstates Sand and Gravel Company, Lincoln Sand and Gravel Company, Neal Sand and Gravel Company, and Fairbanks, Morse and Company. PART I TESTS OF SINGLE-SPAN ARCH RIB II. DESCRIPTION OF SPECIMEN AND APPARATUS 3. Description of Specimen.-The dimensions of the rib and the size and location of the reinforcing steel for the single span are shown in Fig. 1. The concrete was designed to have a strength of 2200 lb. per sq. in. at 28 days. The sieve analysis is given in Table 1. A 1:3:3 mix having a 1.2 water-cement ratio (by volume) was used, the quantities for a batch being determined by weight. In determining the weights correction was made for the moisture content of the ag- gregate. The weights for a batch, based upon oven-dry aggregate, were: cement 34.8 lb., water 27.9 lb., sand 122.6 lb., and gravel 113.3 lb. Each batch was mixed at least four minutes. Eight 6-in. by 12-in. control cylinders were made, one from each of eight batches. The arch was poured October 17, 1931 and allowed to cure in the form, which was entirely closed, until October 27, when the form was removed. The arch stood in the laboratory uncovered and without ad- ditions of moisture until the tests began. The air was dry, and its temperature was fairly constant at about 80 deg. F. The control cylin- ders were stored in the laboratory near the arch. The stress-strain diagrams for the control cylinders are shown in Fig. 2, and the ultimate strength and modulus of elasticity of the con- crete are given in Table 2. ILLINOIS ENGINEERING EXPERIMENT STATION 'est FIG. 1. DIMENSIONS OF ARCH RIB TABLE 1 SIEVE ANALYSIS OF AGGREGATES Single-Span Arch Rib Percentage Not Passing Sieve Size of Sieve Fine Coarse Aggregate Aggregate 1.5............................... 0.0 0.0 0.75............................. 0.0 36.2 0.375............................ 0.0 92.6 N o. 4 ........................... 3.5 99.1 N o. 8 ........................... 16.4 99.5 No. 14.......................... 33.7 100.0 N o. 28.......................... 56.1 100.0 N o. 48.......................... 84.0 100.0 No. 100......................... 97.5 100.0 Fineness modulus................. 2.91 7.27 4. Analysis of Specimen.-The specimen was analyzed and the fixed-end reactions and the elastic constants are given in Table 3. The quantities in one portion of the table are for a value for n, the ratio of the modulus of elasticity for steel to that for concrete, of 12, and the quantities in the other portion are for a value for n of 9. The small effect of the comparatively large variation in n upon the fixed-end reactions is of interest. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 4800 L 4000 83200 4^2400 1~ t/) ) *J WV :t~. '5 7 1 2 7 06'!. / '2p ?r // 7, / ! k / - V 7 / 1 Y 7 tigt' /L/< Iu(, / - - / f / ?/ I I I fI £/Uni/ Deformat/on FIG. 2. STRESS-STRAIN DIAGRAMS FOR CONTROL CYLINDERS. SINGLE-SPAN ARCH RIB TABLE 2 PHYSICAL PROPERTIES OF CONCRETE, AS DE- TERMINED BY TESTS OF 6-IN. BY 12-IN. CONTROL CYLINDERS Single-Span Arch Rib Age 102 days Modulus of Ultimate Elasticity Cylinder No.* Strength in 106 lb. per lb. per sq. in. sq. In. El 3390 3.24 E2 3900 3.17 E3 3750 3.64 E4 3430 3.27 W1l 4660 3.78 W2 3560 3.52 W3 3750 3.34 W4 3740 3.43 *Cylinder numbers indicate position in arch of batch from which cylinder was poured, the numbers beginning with 1 at the ends and increasing toward the center. The influence ordinates for stress at various sections, as computed by the elastic theory and based upon a value for n of 9, are given in Table 35. The most .vulnerable point in the arch rib is at the extrados at the section through load-point 3. 5. Description of Apparatus.-The apparatus was designed and built specially for these tests. The load upon the structure was pro- / 2 -- ýv -0 --f ff- )-< -- - Y V - - - - - / f ! ^ f - A - - - ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 3 ELASTIC CONSTANTS Single-Span Arch Rib Reaction of East Abutment n= 12 n=9 Unit Load at Hori- Hori- Moment, zontal Vertical Moment, zontal Vertical in. lb. Reac- Reac- in lb Reac- Reac- tion, lb. tion, lb. tion, lb. tion, lb. 1.......................... -23.7087 0.1201 0.9780 -23.79363 0.11934 0.978183 2 ........................ -24.7948 0.4256 0.9056 -24.99431 0.42452 0.906284 3 ........................ - 8.9842 0.7894 0.7755 - 9.12423 0.78948 0.776129 4 ........................ +11.2492 1.0270 0.5981 +11.84849 1.03658 0.598312 5 ........................ +25.0286 1.0270 0.4019 +25.70161 1.03658 0.401688 6......................... +26.2682 0.7894 0.2245 +26.34156 0.78948 0.223871 7 ........................ +16.6306 0.4256 0.0944 +16.64177 0.42452 0.093716 8 ........................ + 5.1585 0.1201 0.0220 + 5.13757 0.11934 0.021817 MOVEMENT OF ABUTMENTS Spread 0.10 in .............. -66 937 1018 0 -84 142 1279 0 Settlement 0.10 in........... +10 163 0 63 +12 734 0 79 Rotation of east abutment, top tippingin. 0.001radian... +63 987 669 102 +80 374 841 127 Rotation of west abutment, top tippingin. 0.001 radian.. . +31 058 669 102 +39 116 841 127 A plus (+) moment produces tension at the intrados. duced by suspending concrete blocks of known weight at the load points of the rib. For each load point there was one large concrete block suspended by four steel rods, as shown in Fig. 3, which served as a loading platform on which to place smaller blocks that, in com- bination with the loading platform, constituted the total load to be applied at a particular point. When the arch was unloaded, the load- ing platform rested upon supports provided for the purpose and the loading beam was carried on the steel suspension rods acting as struts. To apply the load the turnbuckles in the suspension rods were turned, shortening the rods, until the loading beam came into contact with the loading shelf. The turnbuckles were then turned, successively, by small amounts in order, back and forth from one end of the struc- ture to the other, thereby transferring the weight of the loading plat- forms gradually from the supports to the arch. Each loading shelf on the arch was capped with a steel plate with a %-in. steel ball located in its top surface and at the point of application of the load. The loading beam had a small steel block attached at the center of its bot- tom flange. A depression in the bottom of this block fitted over the REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS FIG. 3. GENERAL VIEW OF SINGLE-SPAN ARCH RIB SHOWING METHOD OF SUSPENDING THE LOAD BLOCKS ILLINOIS ENGINEERING EXPERIMENT STATION FIG. 4. WEIGHING APPARATUS AND SUPPORTS FOR ABUTMENTS top of the ball in the steel plate on the loading shelf, thus accurately locating the point of application of the load. The load at a given load point for a particular test was obtained by placing concrete blocks of known weight upon the loading plat- form until the desired load had been obtained. Each abutment was supported on two vertical scales as shown in Fig. 4. The load was transmitted from the abutment to the scales by means of jacks, two for each scale, one on the north and the other on the south side of the abutment. The contact between the abutments and jacks was through knife-edges embedded in the abutment so that the line of action of the vertical forces weighed by each scale was accurately known. An abutment could be raised or lowered without rotation by extending or depressing all jacks by the same amount; or it could be rotated about a horizontal north-and-south axis by extending both of the east jacks and depressing both of the west jacks, or the reverse. The vertical scales were mounted on carefully-machined steel rollers 10 inches in diameter that ran upon a carefully-machined track so that the vertical scales offered practically no resistance to horizontal motion. The horizontal reaction of each abutment was measured by means of a horizontal scale, as shown in Fig. 4. This scale consisted of two right-angle levers (bell cranks), one on the north and the other on the REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS south side of the abutment, that received the horizontal thrust through links and converted it into a vertical force that was delivered to a platform scale. The bell crank had a nominal multiplication ratio of 10 to 1, the actual multiplication ratio being determined for each. The link connecting an abutment and horizontal scale had knife-edge contacts at both ends. The link was maintained in a hori- zontal position and the line of action of the horizontal reaction was determined from the position of the knife-edge embedded in the abut- ment. An abutment could be moved horizontally by turning the link, since it had a right-hand thread at one end and a left-hand thread at the other. The strain in the concrete was measured with an 8-in. Berry strain gage. Readings were taken on two gage lines on the in- trados and two on the extrados at the section midway between each pair of adjacent load points. Strains were measured for the design- load tests and for the tests to determine the load-carrying capacity of the structure. The angular position of the abutments was determined with level bubbles attached to the structure at the points where the rotation was to be measured. The bubble tubes were carried on steel bars fastened to steel pins projecting from the rib on a transverse section through the springing, as shown in Fig. 5. In tests for which the abutments were fixed, the bubbles were ad- justed so as to be in their mid-position before the test began and, after the load was changed, the abutments were rotated till the bubbles re- turned to their central position; in tests for which a predetermined angular movement was to be produced, the position of the bubble was read before the test began, and then the abutments were moved by manipulating the jacks supporting them until the bubble had moved by an amount which, as shown by a previous calibration, corresponded to the desired rotation. The vertical movement of the load points and of the abutments was measured with a hydrostatic gage. This consisted of a number of hook gages, one at each load point and one at each abutment, all con- nected to a single pipe line in such a manner that the water surface was at the same level for all gages at any instant. The changes in span were measured by means of two Ames dials attached to long rods, one on the north and the other on the south side of the arch. The east end of each rod was connected to the east abut- ment at the springing of the arch and the west end carried an Ames dial whose plunger bore upon a steel pin projecting beyond the side of the rib at the west springing. The dials mounted in this manner in- ILLINOIS ENGINEERING EXPERIMENT STATION FIG. 5. INSTRUMENTS FOR MEASURING SPAN OF ARCH dicated the changes in span of the arch. Figure 5 shows the Ames dials that indicated the span, the bubbles that indicated the rotation, the hook gage that indicated the elevation of the abutments, and one of the links that transferred the horizontal thrust upon the abutment to the right-angle lever of the scales that weighed the horizontal thrust. The bar carrying the bubble was attached to two pins pro- jecting from the vertical face of the rib, and was located on a trans- verse section at the springing. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 6. Sensitiveness and Accuracy of Apparatus.-Although the most striking feature of these tests is the size of the specimens, the most important feature is the accuracy with which the quantities had to be measured. Although extreme accuracy was not required in testing the single-span structure, the tests upon this structure were considered as a training in preparation for the tests of the three-span structure to follow, for which extreme accuracy was essential. Check readings were taken on all instruments. The tolerance for the hook gages was 0.001 in., and for the 8-in. strain gage it was one division on the dial, or 0.000025 in. per in. The dials indicating the span were read to 0.001 in., and the links could be turned so as to produce movements of the abutments of that order. It is realized, however, that any considerable change in the temperature would pro- duce an error of several thousandths of an inch, but the temperature of the laboratory was very constant, seldom varying more than one degree F. during a test. The vertical scales were calibrated with weights to a load of 60 000 lb. Two 10 000-lb., two 1000-lb., and one 500-lb. standard weights,* were used to calibrate one of the scales for the lower range. This scale was then used to weigh the large concrete loading platforms shown in Fig. 3, and the latter were used with the standard weights to calibrate the scales up to 60 000 lb. It was expected that the most severe requirement of the scales would be to weigh accurately a com- paratively small increment to a large load, but it was found that the scales would detect a 2-lb. weight when added to a 60 000-lb. load. Since the load increments were added without shock or jar, and since the scales were always heavily loaded and there was no chance for any of the parts of the scales to shift, it is believed that the increments in the vertical reactions were weighed with an extremely high degree of accuracy. The horizontal scales were not calibrated. Instead, the multipli- cation ratio was obtained from the measured distance between knife- edges. The main knife-edges were approximately 12 inches apart and the distance between them was measured with a steel scale graduated to 0.01 in. read with a reading glass. Each measurement had a maxi- mum probable error of not more than 0.01 in., or 0.08 per cent. Read- ings were taken on both sides of each lever, and there were two levers, or four readings, for each scale. The probable error is believed to be not greater than 0.05 per cent. *The standard weights were loaned by the Master Scale Depot of the National Bureau of Standards, Harry M. Roeser, Engineer in charge. ILLINOIS ENGINEERING EXPERIMENT STATION The bubbles, used for measuring the rotation of the abutments, were very sensitive. They were calibrated in order that they might be used to measure the magnitude of a rotation as well as to deter- mine when an abutment had been returned to its normal position. The bubbles varied somewhat, but in general one division difference in the readings at the two ends of a bubble corresponded to an angle change of 0.000023 radian. The level bubble is both very sensitive and very reliable. It is possible, however, to introduce an error in its use if, in attaching the bubble, a strain is produced in the case which, in turn, produces a strain in the vial. For if there is a strain in the vial when the bubble is attached to a structure its radius of curvature may not be the same as when the bubble was calibrated. If the bubble is at- tached in such a manner as to avoid straining the vial this possibility of error is eliminated. The tracks for the rollers supporting the vertical scales were leveled with extreme care so that the tendency to roll down hill would not affect the reading of the horizontal scales. Likewise, the links connecting the abutments to the horizontal scales were leveled ac- curately so that the vertical component of the stress in the link would not be of sufficient magnitude to seriously affect the readings of the vertical scales. 7. Loads.-The loads, which were gravity loads, were applied to the rib at eight loading shelves located as shown in Fig. 1. The load was considered as being made up of two parts, a dead load and a live load. In selecting the magnitude and distribution of the load, the specimen was considered as a model, built to a scale of 1 to 5, of an arch rib having nine panels 15 ft. long and a span and rise of 135 ft. and 33 ft. 9 in., respectively. The load which has been designated as the dead load is not the actual weight of the specimen, but the load that produces stresses in the specimen commensurate with the dead- load stresses in a 135-ft. arch. The dead-load panel loads were pro- portioned relatively to each other so as to keep the dead-load thrust within the kern of the arch over its entire length. The load that has been designated as the design dead load is shown in Fig. 6. The weights given are the super-imposed load, and do not include the weight of the rib, but do include the weight of the suspension appa- ratus and loading beams. The selection of the design live load was governed by the standard specifications for highway bridges. The specifications provide that the live load be made up of two parts, a distributed load expressed in REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS .~ ~ N ~4 V~J~ FIG. 6. DESIGN LOAD FOR ARCH RIB pounds per linear foot of roadway, and a single concentrated load on one load point. Since the specimen represented an arch having 15-ft. panels, the relation between the panel load due to the distributed live load and the single concentrated load was the same as it would be for a bridge having 15-ft. panels. The distributed live load may cover any portion of the length of the roadway and the concentrated live load may be applied at any load point, but at only one, and it is in addition to the panel load due to the distributed live load. The load that has been designated as the design live load consisted of a dis- tributed live load equivalent to 960 lb. per panel on any portion of the roadway, and an additional concentration of 1800 lb. on one panel point. The distribution of the live load shown in Fig. 6 produces in the arch the largest stress that can be produced by the dead load and the live load. The maximum stress produced by this load occurred at the extrados at load-point 3. The selection of the dead load and the live load shown in Fig. 6 has been influenced by the following considerations: The ratio of the dead load to the live load should be within the limits that might reasonably be encountered in the design of a high- way arch bridge having a 135-ft. span. The ratio of the distributed live load to the concentrated live load should be within the limits encountered in practice. The dead-load stress and the live-load stress should not differ greatly at the section where the combined stress is the greatest. ILLINOIS ENGINEERING EXPERIMENT STATION The design loads shown in Fig. 6 meet these requirements, but they are subject to the criticism that the combined dead-load and live-load stress is greater than would be permitted for the grade of concrete usually specified for concrete arches. III. DESCRIPTION OF TESTS 8. Dead-Load Abutment Reactions.-Two series of tests were made to determine the dead-load abutment reactions. The dead load was applied and removed a number of times previous to the tests for which data are reported. The procedure for the first series for which data are reported was as follows: The abutments were brought to their normal position with no load on the arch rib except its own weight. The normal span was deter- mined by adjusting the links until the horizontal scales showed a hori- zontal thrust equal to that due to the weight of the rib as determined by the elastic theory. Likewise the normal angular position of the abutments was determined by rotating them until the vertical scales indicated a moment at the springings equal to that due to the weight of the rib, as determined by the elastic theory. The bubbles were then adjusted so that they were in their mid-position. The abutments hav- ing thus been brought to their normal position all readings were re- corded. These readings include those of the dials indicating the span, those of the bubbles indicating the angular position of the abutments, and those of the horizontal and vertical scales that indicate the abut- ment reactions. In the first series of tests the scale readings were recorded before and after each of eight load changes; for four of these the dead load was put on the arch and for the other four it was removed. The time interval between the two sets of readings, one before and the other after a load change, was usually about 2 hours. The dead load was applied gradually, and as the load increased the span would also increase. To prevent the spread of the abutments from cracking the rib, the links connecting the abutments to the hori- zontal scales were adjusted as the load came upon the arch so as to keep the changes in the span small. After all of the dead load had been transferred to the arch, a final adjustment was made to bring the abutments as nearly as possible to their normal position. Perfect ad- justment was not possible, but the difference in dial readings before and after a load change was always small, being of the order of 0.0003 in., and no correction was made for the differences indicated. The REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 4 DEAD-LOAD HORIZONTAL ABUTMENT REACTIONS; FIRST SERIES Single-Span Arch Rib Change in Horizontal Reaction lb. Load Change West East Abutment Abutment Average Dead load on.................................. 26 756 26 500 26 628 Dead load off.................................. 26 700 26 592 26 646 Dead load on.................................. 26 763 26 491 26 627 Dead load off.................................. 26 760 26 518 26 639 Dead load on.................................. 26 730 26 470 26 600 Dead load off.................................. 26 496 26 314 26 405 Dead load on.................................. 26 421 26 350 26 386 Dead load off.................................. 26 588 26 474 26 516 Average .................................. 26 648 26 464 26 556 difference in temperature before and after a load change was also small, and no correction was made for temperature changes. The dead-load abutment reactions obtained from the first series of tests are given in Tables 4 and 5. The results of the various tests of this series agree fairly well, but the differences are greater than might be expected from the per- formance of the scales during calibration. The fact that the sum of the increments of the vertical reactions did not have the same value for all tests indicated that there were some errors in the results. For this reason a second series of tests was made in which the following procedure was used in determining the reactions: With no load upon the arch, the links and jacks were adjusted so as to bring the abut- ments to their normal position, the adjustments being made in such a manner that the horizontal movement of both abutments was to the east. A full set of scale readings was then recorded, and both abut- ments were moved 0.05 in. further east and a second set of scale read- ings was taken. Likewise two sets of readings were recorded, one after each of two adjustments in each of which both abutments were moved west 0.05 in. Thus before and after each load change four complete sets of scale readings were taken, and the structure was rolled 0.05 in. before each set, the direction of rolling being twice to the east and twice to the west. A complete readjustment of the abutments for span and angular position was made after each movement before the read- ings were taken. The changes in reactions due to the application and removal of the dead load are given in Tables 6 and 7. Table 6 gives the horizontal ILLINOIS ENGINEERING EXPERIMENT STATION CO CO C 0 0 0) C) g.0 0"" *.0 0 QQQQQQQQ< C) 0 .0 -1 0 - C) .0 ~1 C)to 00 -'~ 0 C) C) E-tO C) 0 0 0 '0 g-Cto to^ toto t to. I I i+ cq o l om mo mo o toq tto-0tOtotoo +++++T++ oto to No Nooo to to l t o toNO 8 IOOOO 1+t^« to o o oO to toN N- too ti(MotO i(Nto to> t34oo toto t^otototo t-io 00 toto Nto toO N M N to to M to to t®o to toto to .0 .0l 0 '0 0) C) -a 0 i 5 0. 0. 0 -i 0 o. 0) .0 C) I0 1) 0 0s 0 u .1 o t ast o to o ts. - to toto toto 0 ^tototo 0 CI SNNNNNNNNN to to to to to to to to to C)t»tto t~o 0 0 tst ot ot 0 '0 0 0 0 C) .0 0 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 6 DEAD-LOAD HORIZONTAL ABUTMENT REACTIONS; SECOND SERIES Single-Span Arch Rib Each value is the average of four readings Change in Horizontal Reaction lb. Load Change West East Abutment Abutment Average Dead load on.................................. 26 669 26 733 26 701 Dead load off.................................. 26 856 26 875 26 865 Average .................................. 26 763 26 804 26 783 From Elastic Theory, n = 12............... ...... ...... 26 9S7 thrust, and Table 7 the vertical reactions and moments at the spring- ings. The values of the increments in the horizontal thrust are not quite the same at the two ends, and they are not quite the same when the load is added as when it is removed, but the differences are not great. The sum of the increments of the vertical reactions has the same value, lacking one pound, when the loads are added as when they are removed. This is not positive proof but it is a strong indi- cation that the individual vertical reactions, and therefore the mo- ments at the springings, were determined with a high degree of accu- racy. For the horizontal thrust the minimum value was 0.47 per cent less and the maximum value was 0.34 per cent more than the average. This is good agreement, but it is believed that the vertical reactions and moments were measured more accurately than the horizontal thrust. The average of the values obtained for the sum of all of the verti- cal reactions was 54 287 lb. for the second series. The sum of all the dead loads was supposed to be 54 400 lb. This discrepancy is at- tributed to a loss in the moisture content of the concrete blocks after they had been weighed, a loss that continued throughout the tests of the multiple-span arches, and for which corrections were made in the later tests. The value of the horizontal thrust determined in the second series, the average of all values as given in Table 6, is 26 783 lb.; the value obtained by the elastic theory is 26 987 lb. for a dead load of 54 400 lb., or 26 932 lb. for a dead load of 54 287 lb., the sum of the incre- ments of the vertical reactions. The measured horizontal thrust from the second series is therefore not quite one per cent less than the value obtained by the elastic theory. 26 ILLINOIS ENGINEERING EXPERIMENT STATION ~II z 0 0 0 -1 0 z ~ ~ ~ 0.03 03 ~ ~ 0 ~ - .0 El 03 r~ ~1 I 0" *.s .0 0 0 |. 0 03 00 03 0) 430 ~.0 03 0 - I3 03 +3 80 -- .) 03 I a0 'IV ý0 0040co 03303 cqam ýo ti- oq xo o ic 1*0300 N0 l rq c t0300olo cl cl c~l N0 wo © 01:0 p 0 8l Cl Cl C 003 * 0ow 0 ' MM * ' 00003 wo - - C N. N NN Cl Cl Cl 00-4. 30300 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS ~; .~ (.*) K ~9'. I0 ILLINOIS ENGINEERING EXPERIMENT STATION The dead-load moment at the springing is about one-half as great as the value given by the elastic theory. This is a large relative dif- ference, but the absolute difference is not great since the thrust line is near the axis of the rib at the springing. The dead load thrust lines, one based upon computed and the other upon measured reactions, are compared in Fig. 7. Figure 7a is for the first series, the average of eight tests as given in Tables 4 and 5. Figure 7b is for the second series, the average of two tests as given in Tables 6 and 7. The thrust line determined from the measured reac- tions is below the one from elastic theory at the end and coincident with the latter over the central portion of the arch for both series of tests. The trapezoids are strain diagrams whose determination is de- scribed in Section 9. The small circles represent the centers of pres- sure and, for complete agreement between measured strains and meas- ured abutment reactions, should fall upon the thrust line. The lack of agreement is seen to be very small. The close agreement between the values of the reactions obtained by the elastic theory as usually applied and the values actually meas- ured in the tests that have been so carefully planned and executed is of interest considering the errors that are known to exist in the as- sumptions upon which the analysis is based. 9. Dead-Load Strain in Concrete.-The strain in the concrete was measured at a section midway between each pair of adjacent load points. The measurements were taken with an 8-in. Berry strain gage on two gage lines on the intrados and two on the extrados for each section. Readings were taken for five load changes of the first series of dead-load tests described in Section 8. The trapezoids of strain shown in Fig. 7 are based upon the aver- age values of the strains corresponding to the five load changes. The centers of pressure, based upon the assumption that stress is propor- tional to strain* and that plane sections remain plane, are represented by small circles. These centers of pressure fall very close to the thrust line at all sections, an indication that the reactions have been accu- rately determined. The modulus of elasticity of the concrete at the various sections was determined from the measured strains and from the measured reactions in the following manner: The tangential thrust at each sec- tion was determined from the abutment reactions and the loads. The *The fact that the stresses were small, and that the arch had been loaded several times before tests were made, makes it appear probable that stiess was proportional to strain as has been assumed. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS Co 'C '- Co N- Co Co 'C N- Co 'C Co CO Co Co Co Co Co Co Co r- Co Co Co N Co Co~ Co Co Co Co ci CO i0 Co Nýo0 MM- Co (NO GCO 0 Co '- Co1 4OCO O o Co N I'Dloi CO N C 400 Coa M CO O CO tO00 (0- '0 '43 m Co Co Co Co N CA xo mo G0 mO Co Co Co Cox CO Co Co O Co * Co CO Co o Co Co Co Co t0 Co n~~ Co CooM Col Co Co CO CN Co >ra 0 to Co Co Co C' Co> Co 'C 0 N N Co C coo Co Co Co 00 1 b Co Co Co Co Co 'C N^ 0 0 Co 00 'C Co Co Co CoD CO CoI N' Col CO Co Oi - N^ Co 0 CO CO N- 0 CO '- Co 'C< N Co Co Co Co 'C* N Co Co Co N^ 0 Co Co =- Co N- Co C Co Co Co Coq- Co Co 'C Co Co COCo z .0 w 14 P4 % o 3 A4 0 h a rQ ° o d - 4) 01 0 Co 0 a .0 c a : .. a Co * . . * :0 .0 -. I . g . )4 0 o .0 sji ILLINOIS ENGINEERING EXPERIMENT STATION portion of this thrust taken by the steel was determined from the measured strain on the basis that the strain in the steel was equal to the strain in the adjacent concrete. The remaining thrust was taken by the concrete. The modulus of elasticity of the concrete was thus determined since the tangential thrust, the section of the arch, and the average strain were known for each section. The values of E are given in Table 8. The average value for the whole arch rib is 2 730 000 lb. per sq. in. 10. Elastic Properties of Arch Rib.-Tests were made to determine the elastic properties of the arch rib in which one abutment was fixed and the other abutment was given, successively, each of the three components (X, Y, and 0) of motion. The arch supported the dead load during the tests and the magnitude of the movement was small. Two series of tests were made to determine the reactions due to a unit rotation of one abutment. In one series the east abutment was rotated and the west abutment was fixed; for the other series the west abut- ment was rotated and the east abutment was fixed. Two series of tests were also made to determine the change in the reactions due to a change in the span, but only one series was made to determine the effect of a settlement of one abutment relative to the other. In all tests one abutment was fixed while the other was moved, and the abutment that was moved was given only one component of motion (X,Y, or 0) and was restrained against the other two components. During the first series of tests to determine the effect of the change in span upon the abutment reactions readings were taken in duplicate. For all other tests the abutments were adjusted for position, and the scales were read four times before and four times after the movement that was being studied took place. The whole structure was moved, two times to the east and two times to the west, between the various adjustments, in the manner described in Section 8. The average of the four sets of readings was used in determining the change in the reactions due to a movement of an abutment. The changes in the reactions due to a change in span are given in Tables 9 and 10; Table 9 gives the values obtained from the first series and Table 10 those obtained from the second series. The aver- age values for each series are compared with the values obtained by the elastic theory when the latter are based upon values of E of 2 000 000 and 2 500 000 lb. per sq. in. The values of E that make the measured and computed values of the various reactions equal are given at the bottom of the tables. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 31 01 H 00 ECi2 S Qi a 0) ^ I E^ 0 t H *s - 0ýcq rscoc 00)0)0) 00 N 00a)-CO 10) N CC0 00N0100) 11+1 N 00C 0000 2 mcq 0) C; 00 0)-c 000)000000f i^ 0)00000-(i- N 100+0)1 0)^ 00)0 0)00)0)00 0)0)00'-l 0)0)0) 6''O '-' +OS 00 OC '<- 00-. NOS i000 0)00 II II 0 0 II 0 0. 0 C) : :' 'O ^ : : : : :C<0 : : : 0)- 0) N 0)CM 0).0 0o3 0 0 3g00 a'0 : : 0: 0i^. :10:: :: :: :::0 00gs *0 faf I 0 0 S 00 : : :g I:: ;o o* §R° a> i o ! 0. 4 0 0 32 ILLINOIS ENGINEERING EXPERIMENT STATION CO 03 Q rJ 8 0< *-< Co. Z s W Q C) C) "o Z3R H- f~ § Z 0 . 0 Q 0 0U 0 C) .0 0p 03 0 0 .0 0"s S C) 0 0 4* 0 0 CC CC C) Ca~Ca CC 00 ~.0 ~ 05 0 0 .0 C) 0 0 .0 CO.- 0 COC) .~ .~ CaC) 0 .-C) CCC) COC) .~ .~ Ca C) 0CO 0 CC C) Ca ~ Ca ~ 0 CCO 0 C) .0 0 .OM2 CO CtO o O CO COCO CO OCOMCO CO MCO C COO CO 10.0 +ý 10 CO 00 CO CO CO 00 0 C1t -1 +1 1+I + 0 CO CO co CO o 1i- C 000 0 4- ~ CO 4-CO CO CO CO CO CO C CO C COO 00 00 0 0 0 *4 CO CO^ -C :CO iC C C Ca000 000C) : : ) b : fl I00>4 I0:C) 0000 ::::: 0i: :0:: •I" .4 + . . CO 0 o 0 000 0 0 0 0 4 0 0 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 33 ~CC CC o C) C) ~C) o ~C)C) o ,~ CCIC) o ~.C) - C)CC.CC ECC ~, C.) FCC .C) FCC CC R 0 a-C c3 C-C C c 'C~ 01- -C» 0 C) 3 CC 'C ' CS) 1 CCCC C, ^ CC ° .C1 - C) 'C C) C) -C) CC0 *aC N- 'C| CC C) CC + C)^ CC 'CC CC ^ i§ 5C)C - CC) 'C)- ..-C) -CC) CC CC)C) ~I2 0 C) C CCC C) C) CC.CC~CCCC ~ ~C) CC C) CC' 0' 0 CC ÷ I (N i0 N 0 0 I CC C) CC N N CC CC N 10 0 + >0 CC ,-.i N + ÷ CC + ÷CC '0 CC N +» 1-4 0 CO + CC CC +? CC N + C C- 0 N ©0 0CC CC N + - CC CON + 0CC CC -< CC CCC 8C ^-4< pl. NO N 0C So- QO N C '10 ( N CC 0 CCC CC C N NM CC) N N* NI |s|:C CCs CC CC SIi ') C ::s:: a:)CC CCOC) :CCC ;s :s::a CC a 1§§11 Cin C): : : C s' C) CC S-C g C) M 3 A a 8C Ca C) S3 sC C) C- C ILLINOIS ENGINEERING EXPERIMENT STATION 0 c E- o 0 0 o4 Ea z z © o 0 =oA 0 |G F .a rt 0 F. ^ F. § 4, f.l <1 0 ÷ 0, 0 0 i0 GO OG + GO OG 0 GO GO .G 0, 0N, N,0,N, 0,0,0,0,0, :::: : : : 0000 *a'0 : 0: : 0 : '0 a 0 ::S2 I: : §: : :::s :::: : : ::I S: : s: : :::: !3 .§ -4-. J4 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS H 0 H 0 H H H C N m z z s g w SE o S H. * 2;£ CO 000 c o g O COO ^ 3) S 00« H " .S 0 0 0 + CO 00 CO +- 00 00 0 0 0 00 0 CO CO^ 00 00 00 CO '0 0 0 N CO CO 0> >0 00 0 00 0 + 00 00 00 00 0 0 00 CO 00 10 t0 00 00a «0 CO CO 10 00 00 CO 00 coo=C 00 COn cM cq cq 0.0 IC, r 0 0 wo C§§ 0' 00 I 0 Ca 0 I *0. ILLINOIS ENGINEERING EXPERIMENT STATION The increments in the horizontal thrust do not have the same values at both ends, indicating that there is an error in one or both values. The horizontal thrust was approximately 27 000 lb., and the average difference of 175 lb. between the increments at the two abut- ments is only 0.6 of one per cent of the force measured. Nevertheless the percentage error of the increment is so large that the results have little value. It was learned later that, although the track and rollers under the vertical scales were carefully finished, the reading of the horizontal scales was slightly affected by the position of the rollers under the vertical scales. In the tests where the vertical scales had to be moved horizontally, as in this series, an error was introduced. The scales were later calibrated for horizontal position, and a correction applied when a horizontal displacement of the scales occurred. The algebraic sum of the increments of the vertical reactions, which should always be zero, had a maximum value of 20 lb. This is not a proof, but it is an indication, that the measured vertical re- actions from which the moments at the abutments were computed were very accurately determined. The moment was less at the west abutment than at the east abutment for all increments of both series. It seems probable therefore that, due to variations in E along the arch, or to other causes, the moment due to the spread is different at the two ends of the span. From the first series, M/H is 58.43 in. and 64.70 in. for the west and east abutments, respectively; and the cor- responding values from the second series are 64.45 in. and 69.04 in., respectively. The corresponding value from the elastic theory is 65.75 in., and M/H has the same value when E = 2 000 000 lb. per sq. in. as when E = 2 500 000 lb. per sq. in. Two series of tests were made to determine the changes in the abutment reactions due to rotating one abutment without allowing any translation while the other abutment was fixed. The west abutment was rotated for the first series and the east abutment for the second series. The abutments were adjusted for position, the scales read, and the angular position of the rotated abutment was measured four times, in the manner described in Section 8, before and after each rotation of the abutment. In addition the span, the angular position of the fixed abutment, and the relative elevation of the two abutments were checked to be sure that there were no abutment movements other than the prescribed rotation of the one abutment. The changes in the reactions due to the rotation of the west abut- ment are given in Table 11, and the changes in the reactions due to the rotation of the east abutment in Table 12. The average experi- REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS mental values and the values computed by the elastic theory based upon E = 2 000 000 lb. per sq. in. and also upon E = 2 500 000 lb. per sq. in., are all given in these tables. The composite E, the value of E that makes the computed and the measured values equal, is also given at the bottom of each table. One series of tests was made to determine the changes in the abut- ment reactions due to a vertical movement of one abutment relative to the other. Both abutments were fixed against rotation and change in span while the vertical movement occurred. As in previous tests, the abutments were adjusted for position four times before and after each change in the relative elevation of the two abutments, and after each adjustment the instruments indicating the angular position of the abutments and the changes in span were read to insure that the unwanted components of movement did not occur. The results of the tests are given in Table 13. The series of tests that have just been described constitute a very severe check upon the accuracy of the scale, for the changes in the reactions that were measured were in all cases small compared with the reactions themselves, and the quantities desired are the compara- tively small increments. For example, line 1 of Table 9 gives as the change in the horizontal reaction due to a change in span of 0.15 in. a value of 1555 lb. for the west abutment and 1374 lb. for the east abutment, a difference of 181 lb. From statical considerations the two reactions should be equal. The difference represents an error in the scales, or a horizontal resistance in the roller-supported vertical scales. As was learned later, inequalities in the roller or in the track introduced errors in the reading of the horizontal scale if the vertical scales were shifted horizontally, as they were in this test. The rela- tion between this error and the magnitude of the displacement was later determined, and the horizontal position of the vertical scales was noted in the tests so that correction for the error could be made. However, that was not done for this test. Although a difference of 181 lb. between two values of the same quantity is large relative to the increment of about 1500 lb., it is small compared with the total force being measured, which was about 27 000 lb. Unfortunately, the in- crement, and not the total force, is used in interpreting the data. The difference between the increments in the horizontal reaction at the east and west *abutments, in the tests to determine the changes in the reactions due to a rotation of one abutment, is only about 40 lb. The algebraic sum of the increments of the vertical reactions, which should be zero for all tests, varies from 2 lb. to 25 lb., and for ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 14 RATIO OF VALUES OF ELASTIC CONSTANTS OBTAINED FROM MEASURED REACTIONS TO VALUES OBTAINED FROM ELASTIC THEORY E taken as 2.50 in 106 lb. per sq. in. Ratios Elastic Constants from Movement Horizontal Moment Thrust Average Change in span*......................... 1.00 1.02 1.01 Rotation of east abutment ................ 0.92 0.92 0.92 Rotation of west abutment ................ 0.93 0.93 0.93 Settlement of support..................... 1.02 .... 1.02 Average............................. .... .... 0.97 *Average of two series. TABLE 15 AVERAGE VALUES OF EXPERIMENTALLY-DETERMINED ELASTIC CONSTANTS Values are based on Table 14, and are equal to 0.97 times the values of elastic constants obtained by the elastic theory on the basis that E is 2.50 in 106 lb. per sq. in. Reaction of East Abutment Movement Horizontal Vertical Moment Reaction Reaction in. lb. lb. lb. Spread, 0.10 in ....................................... 64 929 986 Settlement of east abutment, 0.10 in.................... 9 858 .. . 60.86 Rotation of east abutment, top tipping in., 0.001 radian. . 62 067 659 98.59 Rotation of west abutment, top tipping in., 0.001 radian . 30 126 659 98.59 most tests has a value of about 12 lb. Considering the fact that the total vertical force being weighed is about 60 000 lb. this represents excellent scale performance, and the probable error is small relative to even the increment of load. This statement is supported by the fact that the various values of composite E given in Tables 9 to 13, inclusive, are fairly consistent. The ratios of the experimentally-determined values of the elastic constants to the values obtained by the elastic theory on the basis that E = 2 500 000 lb. per sq. in. are given in Table 14. The vertical reactions have been omitted from this table, and from the averages, because they are so small that they cannot be determined as accu- rately as the other quantities, and it is believed that their inclusion would cause a greater error in the results than their exclusion. The average of all the ratios given in Table 14 is 0.97. The average values of the experimentally-determined elastic constants are listed in Table 15. These values are 0.97 times the values obtained from the elastic theory based upon a value of E = 2 500 000 lb. per sq. in. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 11. Influence Ordinates for Abutment Reactions by Unit Load.- In the series of tests to determine the influence ordinates for abut- ment reactions by the use of a unit load, a load of one ton was ap- plied and removed, successively, at each load point. The dead load was on the arch while the tests were made, and the thrust due to com- bined dead load and unit live load fell within the kern at all sections of the arch for all positions of the unit load. A full set of abutment readings was taken before the unit load was applied and again after it was removed. The abutments were adjusted for position and the scales were read four times for each set of readings in the manner described in Section 8. The instruments indicating the position of the abutments were inspected before each reading to insure that the abut- ments were in their proper relative positions. The changes in the reactions are given in Table 16 and the meas- ured values are compared with the values computed by the elastic theory in Table 17. The reactions due to the design dead load are given in the last line of the table; one set of values was obtained from the experimentally-determined influence ordinates, and the other set was determined by the elastic theory. The two sets of values agree closely. The diagrams of Fig. 8 show the reactions due to a load of one ton, as listed in Table 16, and the diagrams of Fig. 9 are influence lines for the various reactions. The measured and computed values are compared in the latter figure. The horizontal thrusts given in Table 16 are very nearly alike for the two abutments. Also, the sum of the increments of all the vertical reactions is very nearly equal to 2000 lb. for all tests. The difference between the changes in the horizontal thrust for the east and west abutments is 15 lb. with the load at 7, and 34 lb. with the load at 6. The former is 1.6 per cent of the increment and 0.05 per cent of the total force being measured. The latter is 2.1 per cent of the increment and 0.13 per cent of the total force. For all other load points the dif- ferences are less than those just enumerated. The greater consistency of the horizontal forces in this series as compared with previous series, is attributed to the fact that the rollers under the vertical scales were in the same position before and after load changes, and the error due to a change in the position of these rollers was eliminated. 12. Vertical Movement of Load Points Due to Change in Span.- The vertical movement of points on an arch axis due to spreading the abutments without allowing them to rotate or move vertically may be 40 ILLINOIS ENGINEERING EXPERIMENT STATION t4CO^ WNýM <0tý- 0;;-'t t--- S--C -'§OO - f-- ON X I) 0 I 0 0I)- M C -IV 0- ,-O X C0i 0 0 4.i-i 0s 0)0 0)000 N ^^QS rf^-i-i00<0)0 0000 .0- 0) 00r t OO-0 0)00U) Qi)0)0 0)t^-) 0..... ... ... ... ... ....)0.) 0)) 0 N..oo. ++ +1+ +I+ +1I+ +1+ 1+1 1+1 1+1 CO'"0 0a) OS)0 .1'D OipM -p'0.0 <0)-0^ 00.00 0000) t^...aU )))*4) .^i )00.. 0" .0 to t=- CC -l 00)0 m m t0"0 'S'^'S ~ 0 X 0 00® . CO.O000 000.0C 00S- 000)1 00i)). 0)000 000)0 0)0.0.IO ®1NN. 000.00 00 ) '000 OS COM0 0 000 - ^i ' os~t- oo-i' --® ooiM0) o.io01. c0poio 00000 h- m ..1 1+1 +1 +T +1±+ +1+ +1+ +1+ - U X=5 = mc=) .00 mI + -I, 8 0 ) gm g t.0I÷ 000 00 001000 00 00 +1+ ý 7 +1+ +± + +± ±T+ TIT- +I+~ +I+! § I 0 0 0 0 m 0 0 00 w e 00 Q Q > &< 0 o n © .0 ^2 ^S -S 0 0 a 0 I z * Q I 0 0 0 0 0) .0 0> 0 *0. .0 0 oo00~ o..0-.~ ~ 00~ ~0) ~ o 0 4)4)~ 4)4)0) 4)4)0) 4)4) U) 4)4)0) 4)4)0) 4)4)0) 4)4)0) ~ ~ 0000. 0000. 0000. 0000. 000 00.0) 000) 00~ 00.0) 00.0) 0O~ OO~ 0 .0) Xc -t QOSQ CO MC 0 00 0)05 7ý ' w=0 c00 H +I+ +I+ + + +I+ + + +1+ +I+ + + U)~ 00 M X0 - . -0 +1+ +1+ +1+ +1+ +1+ +1+ +1+ +1+ .-3~ 0)* - CIO a', 0 OO 00 Ms N40 )) +1 + +1 +1 ± 1+ ++ I+ *.4 a 2 -1 ÷ 1 - +o o o +C i+ + i+ ' + ' + + + I + ? + + r + + + ýo : ;5 :: : oý 1 o + ++*+i +1 +1 +i+ +1 |^ + I'+ + + +7'^ '4 + I'+4 + ' S'c40c I ~-o Ci r (OO >c^ »t® ii e a .0 0 0 .0 Nw. -I -- --- O - I 0l- =m0 -OSl0 000. .-.W -I .00.- 0' (OOC -S ~ 0) 0 0<000 0' U)0 )OC Q O U ^CO* N 00)0 0 0) .-100 000 t0 S510 C 00)0 0) +1+ +1 + ++ 1+ 1+ +1+ +1+ +1+ 12, OF 0 N0 0 000 N01 CI t- 00. -w-w"*-. CO,4 .0 ' E2 *OC * *? O »*,g + f *+ I + IT + I- + + | I ^ + I + I N3 NM (m OSo N NOC ooo N-O- 0 i» NI- +1 +1 +1 +1 +1 +1 + + *0 0) .3 0£ 00-0ý oo-ý4 00-14 ooý ooýl 00 00 0, REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 41 z F- 0 0 0 0 Q ~.0 p- Z.0 -~ ~0 0 0 0 0 z 0 0 0 Q E, 0 0g. V 8-0 00 3 2 0^ 00 I 0.0 .0 r 14, 0 0.0 o: 0-a 00 0.0 0~, 0 g-6 I 0 0 0 0 0 0 CO C-O COGO S-CO C'. It"' 1-11 ago 0-0 mmO 0 =o 0 "0 - N NN NN 00 (e 000 *^- 0 0 OC 00 00 00 t-0 to 0 0o Go 0 G 0 0 Q 0 O== " 0 -(0O M 0 CO 00 V0 =00 00W coo mm c o NN as NNr M r( - o> t O M0000 00 00 P 00 00 ~t 00 C 00 t^ 00 00-C 00" M'- 001 h r-t. 00- 0S- 0-t- N0 0^ 0-0t " OS- m2 O 000 ® M000 0^ 00 00 %i0 00 00 00 00 0 01 0 00- OO' 00C COO 0~i ®05 0 CO b--00 M OS O - r"m 0 0M 0c co- o 00 00 o so 0= 00 0 0 ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 26 HORIZONTAL THRUST OBSERVED IN TEST TO FAILURE Horizontal Thrust lb. Date Load West East Abutment Abutment Average 1-21-32, 3:45 p.m ......... Dead load ...................... 26 828 26 836 26 832 1-21-32, 8:50 p.m ......... Dead load + one live load........ 31 597 31 613 31 605 1-22-32, 9:30 a.m. ........ Dead load + one live load........ 31 552 31 581 31 567 1-22-32, 12:00 noon ....... Dead load + two live loads....... 36 302 36 280 36 291 1-22-32, 5:20 p.m. ........ Dead load + three live loads .. .. 41 038 40 991 41 014 1-22-32, 10:55 p.m........... Dead load + four live loads ...... 45 637 45 617 45 627 1-23-32, 1:05 a.m. ........ Dead load + five live loads....... 49 737 49 890 49 814 tute the design dead load, and also serve as platforms for carrying the blocks that make up the live load. The single heavy concentration re- quired by specifications for highway bridges was placed at load point 3, where the arch failed. The various phases of the test are dis- cussed in the following paragraphs. The abutment reactions were determined from the scale readings. The abutments were adjusted for position four times (except when the fifth live load was added) before and after each load change, and the scales were read after each adjustment. The measured reactions re- ported are from the average of the values obtained from the four sets of readings. The horizontal and vertical reactions are given in Tables 26 and 27, and the relation between the load and the reactions is shown in Figs. 15 and 16. For both figures the full line is from the elastic theory and the broken line is from the measured reactions. For the horizontal thrust the measured value is slightly less than the value obtained from the elastic theory. The fact that the thrust varied di- rectly with the load, even after the arch was badly cracked, is of in- terest. For the west abutment, where the moment was large, the measured value was less than the theoretical one. Moreover, the ex- cess of the computed over the measured values increased with load, indicating that cracking at the overstressed sections decreased the stress at those sections, and increased the stress at sections where the stress was small. The position of the thrust line is shown in Fig. 17. The full line represents the position as determined by the elastic theory, and the broken-line the position as determined from the measured reactions. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS )~-~00 0 0,i ce 44 E) CU2 03 0 F-i Co Co 4 04) coo -0 Co 0 Co N Co Co Co G;! 00 Co '0 4) 0 Co 4) Co 0 Co Co 0 Co Co Co Co Co Co '4 Co Co Co 4) 0 0 + 0 Co 4) Co C Co 0 Co Co Co Co Co Co 0 4) 0 Co 0 0 + Co 4) 0 *0 4) CO Co 0 Ns Co CO 1-o CoI Co 4) 0 0 + Co 4) 0 *0 4) Co Co .4) Co Co Co 0 Co 4) 0 4- ~0 + Co 4) Co 4) So 0 Co> Co Co Co Co 0 Co (-1 + Co 4) 1 Co Co Co 00 Co Co Co Co Co 0 N> Co Co Co 4) 0 4- 4- Co + Co 0 Co I ILLINOIS ENGINEERING EXPERIMENT STATION K. K NJ 0 4 8 /2 /6 0O 24 Hor/zonta/ Thr'sf in Thousands of Pounds FIG. 15. RELATION BETWEEN LOAD AND HORIZONTAL THRUST. TEST TO "4 %163 8 0 N 0 80 FAILURE - Ecrsf.-i-' - -- -west --'A / Abume t -/ - Albufmen^t .- -T7-^--------- / ie? / _ ----- From Nea'sured Reactions f / ^ ^ -°-From E/Gst,/'c Theorc, /60 Z40 320 400 480 lomen't / Thousancds of /nch-Poun'ds 560 640 FIG. 16. RELATION BETWEEN LOAD AND MOMENT AT ABUTMENT. TESTS TO FAILURE The shaded portions are strain diagrams based upon the strains measured with the strain gages. The small circles represent the centers of pressure as determined from the strain diagrams. The centers of pressure are shown only at sections where the thrust line is within the kern of the arch, as they cannot be determined accurately at sections parts of which are subjected to tension. The centers of pressure should I If I_ From /leo'sured ,/_ : /PeL'A'c/7ws^ f 8'/-s'c Theory 74 - ---- y > - - - - - - *'^-ZZ -ZZ 1 -__ ./_¢_- „ _ __ _,__ _ REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS a 4) bo *) 04 4)4)0 4). 4 Cl 'ClCCl C-0ClCl I I I l lC l I I I II I III C-C 0 Cl l C -C lC ".00 Ct-l 0 I I I I I I I I I I I++ +++++ I 1p 1 I1++ Cl Q. > l llC I Ioo 1ooooo ClCl0 l l Ca l'0 C C -l Cl C C C C '~ -ClC CllQClC l ClQQ Clfl Cl s In m - E 4). Cloi' )^ *- 4 ) o ffl 4)S *+ I) .0|4 3g ))4 ~ii! 1111 § *K *.i ' ' "-^S Cl El r. 0~ - - ÷+Cl - ÷) Cl Cl - I÷÷÷÷÷÷÷÷÷*l ++++++++++ IIl Cl 0) C-ClfCOO-l C II II IIII 'Cl ClO ~) - 0 ClC-Cl O~i Cl^ Cl C-Cl l t ) >0 C-ClO lll++÷+++++ OC Cl C Cl =O Cl 000Cl Cl - l C -l i-0 C-Cl - Cl iQ C- 0 oCl -Cl Cl »Cl Cl >ClCll- Cl 0 ClO Cl(C *~l II * **t C! 0 Cl Cl C-C4l^ Cl 00C III II I I .00000000.'jl .Cll C '0 ClClC C ILLINOIS ENGINEERING EXPERIMENT STATION Q % jo REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS ILLINOIS ENGINEERING EXPERIMENT STATION ~k. N. N 1.~ t.. Locda Po/nfs on Archl FIG. 18. DEFLECTION OF ARCH Axis. TEST TO FAILURE fall upon the thrust line located by the measured reactions. The fact that it does fall on or near the thrust line is a check upon the work. The vertical movements of the load points due to the increments in load were determined from the hook-gage readings taken just before and just after each load change. The deflection of the arch at the various load points is shown in Fig. 18. These diagrams indicate that the deflection per unit load increased slightly with an increase in load, an indication consistent with the well known fact that the ratio of strain to stress for concrete increases with the stress. All curves except one intersect in a common point. The curve showing the deflection due to the dead load and the fifth live load passes to the right of the com- mon intersection point, a fact that is not surprising considering the wide cracks that had opened at the intrados at the west springing and under load 3, and also at the extrados near loads 5 and 6.* The deflection midway between loads 3 and 4 at the design load, dead load plus first live load, is 0.118 in. This change in the position of the axis of the rib relative to the thrust line causes a stress of 36 lb. per sq. in., a secondary stress not usually considered in the design of an arch. It is so small, however, that no apprehension need be felt because of its omission. The unit stresses due to the loads used in the test to destruction, *The omission of a 500-lb. load from load point 4 when the fifth live load was added would cause a shift in the deflection diagram. See page 52. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS determined by the elastic theory, are given in Table 28. The largest value is at the section where failure occurred, a section through the west edge of loading shelf 3. The stress at this point due to the ulti- mate load was 3674 lb. per sq. in. The unit strength of the control specimens, the average of eight cylinders, was 3773 lb. per sq. in. The abutment reactions were not weighed at the ultimate load, but the stress at the point where failure occurred, computed from the meas- ured reactions due to the dead load plus the fifth live load, the largest load for which the reactions were measured, was 2143 lb. per sq. in., compared with 2520 lb. per sq. in., at the same point and load, as de- termined by the elastic theory. Both stresses were determined from thrust and moment on the basis that the concrete takes tension and that plane sections remain plane. PART II TESTS OF THREE-SPAN STRUCTURE HAVING RIB WITHOUT DECK IV. DESCRIPTION OF SPECIMEN AND APPARATUS 17. Description of Specimen.-The general dimensions and the no- tation for the specimen are shown in Fig. 19. The details of the arch ribs are the same as for the single-span arch shown in Fig. 1. The details of the piers are shown in Fig. 20. The concrete was designed to be the same as that used in the single-span arch described in Sec- tion 3. The sieve analysis is given in Table 29. A 1:3:3 mix having a 1.19 water-cement ratio (by volume) was used, the quantities for a batch being determined by weight, and correction being made for the moisture content of the aggregate. The weights for a batch, based upon oven-dry aggregates, were: cement 73.1 lb., water 57.4 lb., sand 234.4 lb., and gravel 234.1 lb. Each batch was mixed four minutes or more. Twenty 6-in. by 12-in. control cylinders were made, four from the batches that went into each pier and four from the batches that went into each rib. The modulus of elasticity and the ultimate strength of the concrete, determined from these cylinders after the test of the arch had been completed, are given in Table 30, and the stress-strain diagrams are given in Fig. 21. The specimen is shown in Fig. 22. The two piers were poured April 13, and the three ribs April 14, 1932. The forms were stripped during the period from April 23rd to 26th. During the intervening time the concrete was completely enclosed in the steel forms and the surfaces ILLINOIS ENGINEERING EXPERIMENT STATION Load D£'es'ina'ilolps FIa. 19. THREE-SPAN STRUCTURE HAVING RIB WITHOUT DECK Fia. 20. DETAILS OF PIER 7- REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 29 SIEVE ANALYSIS OF AGGREGATES Percentage Not Passing Sieve Sieve Fine Coarse Aggregate Aggregate 1Y in ....................... 0.0 0.0 % in ........................ 0.0 31.2 Ya in ........................ 0.0 81.4 N o. 4 ........................ 2.0 98.1 N o. 8........................ 13.5 99.2 N o. 14....................... 30.9 99.3 N o. 28....................... 68.2 99.3 N o. 48....................... 95.8 99.3 No. 100...................... 98.9 99.5 Fineness Modulus............. 3.09 7.07 TABLE 30 PHYSICAL PROPERTIES OF CONCRETE AS GIVEN BY CONTROL CYLINDERS Location of Batch from which Ultimate Strength Modulus of Elasticity in l06 Cylinder Was Taken lb. per sq. in. lb. per sq. in. East Rib............... ......... . 1 3 585 3.03 2 3 295 3.22 3 3 140 3.33 4 3 380 4.00 Average 3 350 3.40 Center Rib....................... . 1 3 270 3.22 2 3 515 3.12 3 3 125 3.70 4 3 580 2.63 Average 3 372 3.17 W est Rib.......................... 1 3 435 3.70 2 3 210 3.33 3 3 318 3.57 4 3 420 3.44 Average 3 346 3.51 Average for all ribs........... . ..... ........ . 3 356 3.36 East Pier......... . ..... ........... 1 3 040 2.94 2 3 100 3.44 3 3 260 3.23 4 3 470 3.92 Average 3 217 3.38 W est Pier......................... 1 3 448 3.12 2 3 185 3.12 3 2 980 2.70 4 3 440 3.12 Average 3 263 3.02 Average for two piers............. ......... 3 240 3.20 Average for all cylinders ............ ......... 3 310 3.29 ILLINOIS ENGINEERING EXPERIMENT STATION 300 2400 1600 3800 Z 400 1.600 800  Z0 2400 1600 800 0 S2400 /60 8400 0 Co t ro/ Cy/ iders for West ArcfA //C'onr -- r /'de---rs 12for zastl ano Wet PCeerAs V , v I II I I I I I/ m.- UnL/it Deformaf/io17 Fia. 21. STRESS-STRAIN DIAGRAMS FOR CONCRETE REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS ILLINOIS ENGINEERING EXPERIMENT STATION of the ribs and piers were moist when the forms were removed. After the forms had been removed the concrete was cured in the laboratory where the air was dry and the temperature was approximately 80 deg. F. A multiple-span arch bridge differs from a single-span arch in that, for the latter, the ends are fixed, whereas, for the former, the fixed points of the structure are the outer ends of the two end spans and the bases of the intermediate piers. As a live load crosses a mul- tiple-span arch bridge the uneven loading of the two spans adjacent to a pier produces a horizontal thrust and moment at the top of the pier which move the top even though the pier bases and abutments are fixed. This movement of the pier top produces reactions at the end of the span differing materially from the reactions that would be produced by a fixed abutment supporting a similar single-span arch carrying the same load. One of the principal objects of the investigations was to determine the effect of the elastic deformation of the piers upon the stresses in the arches. Inasmuch as the magnitude of this effect depends upon the slenderness of the piers, tests were made upon the structure when the piers had various heights. The effective height of a pier is the ver- tical distance from the springing of the arch down to the point in the pier that is fixed. By attaching the instruments that indicate the position of the pier base at distances below the springing of 20 ft., 15 ft., and 10 ft., respectively, the one structure was used in making tests of a three-span arch series that had these various pier heights. 18. Analysis of Specimen by Elastic Theory.-The influence ordi- nates for the reactions at the springing for each of the three spans are given in Tables 31, 32, and 33. These reactions have been determined by the elastic theory* and are based upon the usual assumptions that plane sections remain plane, that concrete takes tension, and that E has the same value at all sections and at all stresses, both tension and compression. Tables 31, 32, and 33 contain values of the influence ordinates for structures having pier heights of 20, 15, and 10 ft., re- spectively, based upon a value of n = 9; the values in Table 34 are *The three-span structure consisting of a rib without deck was analyzed separately by W. M. Honour and Glen Murphy. Graduate Research Assistants in Civil Engineering, for an assumed value of E of 2 000 000 lb. per sq. in. and using a method developed in a thesis, "A Study of Multiple-Span Arches" by Donald Edward Larson, which is on file in the library of the University of Illinois. The values given in Table 34 are from these computations. Tnh values presented in Tables 31, 32, and 33 were computed by Ralph Kluge, using a method developed by Professor Hardy Cross in, "Continuous Frames of Reinforced Concrete," pub- lished by John Wiley and Sons. These values are based upon a value of n equal to 9, corre- sponding to a modulus of elasticity of 3 333 000 lb. per sq. in., which is consistent with the experimentally determined values of E given in Tables 30 and 54, and with the elastic constants given in Table 47. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS (2 0 0 0 (2 C 0) 00000000 I I I I I 1+ 00 000000 - i-+t-1f -++ + T 00 OOONCOi^ +11^ M -i-+±+X 001 0 -OO L C O:X 000O 0 NOM 00 C 00000000 Ftt++±++ 0000-OOO-00t CO-lO C Nl-~O N -0-0'- 0 Xt I-- 000 N 0000lO'-'- N 0000 O O^^)£ O - CO i CON 0 ++ +iI I I t0 ONi 00 -0 NOt-t 000os-0000fc 0 - - -::?ll 00000 ON C -0 OOOOONC (2(2 oo~ooos'-o-i t++÷++ O CO 000-^C' N N 0000000 0ý C, Ný 000 0 00000000 00000000 00+00++0 00000000 ++ I I I I ON 000 D0 0-0CC 00* 'iOCONNOMCOi- 00000000 00000- -'-000 00000000 117" 1octtttcýll 1 +++++ 000PI0 N; 0 00COO (OCON -000'-^ OCCO0 00 * 0 0 I iI I++++ NO C- 00000^r- ( *-oo Co OOO Cl 00000000 00000000 I+I++++4++ 00 000Oo 00 - ^ 000 00 00000000 00000000 oo0ooooo 00000000 00000000 I+++++++ 00000000 I++t++ ++ NCOOS 000.(MO ON. 00 ^0 0-*Tt 0 N- 0-.<00 N'fO COO~ 0 C0 O C0 *0L; 0c~ 000N0000 I+++++++ 0N0 00 1't SCO 00 00000000^f~ 00000SO - 0 '-C] 00'QCC CTI' COO' I+++++++ C C C: St C 0 C 0 C "C-C- '0" ~0 CO ILLINOIS ENGINEERING EXPERIMENT STATION 00000000 00000000 I I I I I I + 0- ~ -~ 0-0000 0000 ~ 0=0- 0-0000000 00 0~- 0 0 00 00000000 -t + + ++ ++ + 000-. 000C 000-0 ~. - 0-0 ===00 00000 00000000 0 M-000 0-0000 008N0000 C0 t-100 00 00 a, x c 000M 0 0 000000 060006000 00000 00000000 ++++..++ ++. 111111 +1 I 00000000 00000000 ~' 0-0000-00 000-00000 00000000 + +++ + + + + 0000-00000- 00 00000 00000000 00000000 00000000 000000000000 00C'0 000-0 00000000 0 0 0 0 1-1 0 M 0 0 0 I zt 0 0 C- OC00000000 0-0000.-CO 0k0000000 +++++++= ++++++++ ,+++++++ ++++++++ OS 1++++ Q 1 c OS 8 C0 0'C0i -000 0 O -ft-01 c C »i0 00 0 .< - C- 00000000 00000000 00000000 +++++++I +++++++ I+++++++ I' 0+++C++ 0>0-+++0+++++00 +++ 00-0 0iti'tO-i.0 COOCCC- 000OC 00 t'- T000O00 - COOi NC 000 r-i '-i Cc . . :i-11-1 + +I N I +++ 1++++++ i- 0*00 ^ t- SO S'*CO- ^ * OG =000=,O =t' CO00=---0"50.C= 0OdOCO OOd® odC'C>M0O0r 00 * 00. 00 00-00i 0 0 O i0*iC ^O iC O0 -0 S ©CO 40 M M-- 0 C-< CO OS ^Xt Ot-> 0 Nft<( OM0 0.00I^ - 00''tt '-'C0O Ch'f- 00.-0-^t OC 0.220 0 0^4 O 2OSCOO00. C - C 01 OtC- 2020.2 222092 0! 200200000. 20002o 2002.2o 200ooooo o00 1 .++++++II II.++++ I1 ++++++ 00202000 00.0.0200 00 N00020D2OS +++ I I 1 1I1+1++++ II++++ 002020221( COCO002 (0 02h 0 20.0200 C-0 0220 220l -1C ~0~002C >02 N*t 00 -000t00M00C0202 O C 020202MCO1> iQ00*- O O O Cl -- 00OS 2< COC O 0320200C020CO2 CO 022000MCO* *< C' -IC 20 022202 0 ' 0 0020202- -000 I I-I++++ I +++ I1+++ 2000000 2000CO0220 ls* 00 202CCOO00.. h-2022 222 }C 1-01- C 00000000 00000000 00000000 C CCC C C C C 1-0CC C 1-1-C C C C C C C 1-C 000--Coo 00000000 ++++++++ 1-C 1-10CC 0 C OCCCOCCC 0010CC C CO 00000000 00000000 C C 1-CC C CO - C C C 1-COO 1-1-CO1-C 0 C C CCC 1-CO C OC 1-C 1-CC ++++++++ CC1-C1- 1-CCCC1- 1-1-CCC -CC C1-CCCCCC OC OCC'OC CCCCC0CC C0~C0CC 1-C CCCCO ~01-C 0~0C ~-~1-CC C0~C1-CCC ~~101-CC COCOCC1-C 00~10C0 = ~CC I I I I l++++++ I+++++++ 1-C 1l-101001-C C0 0CCSr- CC C0 C^ CCC CC.lilil 1OO^-CQOf C C EnC1001o-C Co 0CC 0 CC CO C 0CC C C CCDCCCC CC 00'GO-1010CC 0 CCC CC-Ct C1 I^Ol Ot-x C 10CCC1-C 4 4 NN A P4 4 a) t a)aalK]ý) .) oo ootj ý&: ý:ýt k K S C S E H 0 H C 0 S H ~ l2~ 0~. 0 00 C.~ 01, z -~ 0 E £ C) &- H (C S ."S Ex 0 0 a S S S z 0 1, REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 35 INFLUENCE ORDINATES FOR STRESS AT VARIOUS SECTIONS, CALCULATED BY ELASTIC THEORY Single-Span Rib Without Deck Stress, in lb. per sq. in., at Section Through Load Point 3 Extrados Intrados -0.0114 +0.0090 -0.0582 +0.0497 -0.1696 +0.1512 -0.0138 -0.0118 +0.0578 -0.0822 +0.0644 -0.0825 +0.0386 -0.0481 +0.0111 -0.0138 Load Point 4 Extrados Intr +0.0042 -0. +0.0011 -0. -0.0458 +0. -0.1774 +0. -0.0171 -0. +0.0430 -0. +0.0382 -0. +0.0128 -0. ados 0070 0108 0278 1529 0080 0619 0483 0157 East Springing Extrados Intrados +0.0644 -0.0754 +0.0672 -0.0797 +0.0198 -0.0338 -0.0418 +0.0278 -0.0815 +0.0695 -0.0816 +0.0732 -0.0510 +0.0468 -0.0157 +0.0145 Based on n = 9. A plus (+) sign indicates tension. TABLE 36 INFLUENCE ORDINATES FOR STRESS AT VARIOUS SECTIONS, CALCULATED BY ELASTIC THEORY 20-foot piers Load of 1 lb. at El E2 E3 E4 E5 E6 E7 ES Cl C2 C3 C4 C5 C6 C7 C8 W1 W2 W3 W4 W5 W6 W7 W8 Stress, in lb. per sq. in., at Section Through Load Point C3 Extrados -0.0001 -0.0003 -0.0003 +0.0001 +0.0008 +0.0017 +0.0020 +0.0015 -0.0150 -0.0675 -0.1843 -0.0304 +0.0438 +0.0567 +0.0375 +0.0133 -0.0020 +0.0014 +0.0080 +0.0138 +0.0158 +0.0130 +0.0073 +0.0021 Intrados -0.0003 -0.0012 -0.0025 -0.0035 -0.0039 -0.0035 -0.0024 -0.0012 +0.0127 +0.0611 +0.1706 +0.0114 -0.0617 -0.0702 -0.0451 -0.0159 +0.0024 -0.0018 -0.0098 -0.0170 -0.0194 -0.0159 -0.0090 -0.0026 Load Point C4 Extrados +0.0023 +0.0081 +0.0145 +0.0181 +0.0165 +0.0106 +0.0035 -0.0009 +0.0019 -0.0134 -0.0760 -0.2181 -0.0567 +0.0155 +0.0269 +0.0128 -0.0022 +0.0032 +0.0130 +0.0216 +0.0242 +0.0197 +0.0110 +0.0032 Intrados -0.0028 -0.0098 -0.0176 -0.0219 -0.0199 -0.0126 -0.0039 +0.0013 -0.0046 +0.0059 +0.0632 +0.2009 +0.0389 -0.0293 -0.0348 -0.0155 +0.0027 -0.0037 -0.0151 -0.0250 -0.0280 -0.0228 -0.0128 -0.0037 B of BC Extrados -0.0055 -0.0188 -0.0333 -0.0404 -0.0352 -0.0201 -0.0033 +0.0053 +0.0626 +0.0823 +0.0610 +0.0194 -0.0178 -0.0340 -0.0284 -0.0124 +0.0021 -0.0038 -0.0143 -0.0234 -0.0260 -0.0211 -0.0118 -0.0034 Based on n = 9. A plus (+) sign indicates tension. Load of 1 lb. at Load Point Intrados +0.0053 +0.0183 +0.0324 +0.0393 +0.0342 +0.0195 +0.0031 -0.0052 -0.0734 -0.0941 -0.0732 -0.0311 +0.0081 +0.0273 +0.0248 +0.0112 -0.0020 +0.0037 +0.0136 +0.0221 +0.0246 +0.0200 +0.0112 +0.0032 Intrados ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 37 INFLUENCE ORDINATES FOR STRESS AT VARIOUS SECTIONS, CALCULATED BY ELASTIC THEORY 15-foot piers Stress, in lb. per sq. in., at Section Through Load Point C3 Load of 1 lb. at El E2 E3 E4 E5 E6 E7 ES C1 C2 C3 C4 C5 C6 C7 C8 W1 W2 W3 W4 W5 W6 W7 WS Intrados -0.0002 -0.0007 -0.0013 -0.0019 -0.0023 -0.0022 -0.0017 -0.0009 +0.0122 +0.0594 +0.1671 +0.0074 -0.0658 -0.0733 -0.0468 -0.0167 +0.0026 -0.0008 -0.0078 -0.0144 -0.0169 -0.0141 -0.0080 -0.0023 Load Point C4 Extrados +0.0020 +0.0068 +0.0121 +0.0149 +0.0133 +0.0081 +0.0020 -0.0014 +0.0028 -0.0106 -0.0707 -0.2112 -0.0498 +0.0207 +0.0299 +0.0138 -0.0029 +0.0016 +0.0101 +0.0180 +0.0206 +0.0170 +0.0097 +0.0027 Intrados -0.0024 -0.0084 -0.0149 -0.0181 -0.0161 -0.0097 -0.0022 +0.0018 -0.0056 +0.0026 +0.0571 +0.1930 +0.0310 -0.0353 -0.0383 -0.0166 +0.0033 -0.0018 -0.0117 -0.0208 -0.0240 -0.0198 -0.0113 -0.0031 Extrados -0.0050 -0.0169 -0.0297 -0.0357 -0.0304 -0.0163 -0.0011 +0.0061 +0.0615 +0.0787 +0.0544 +0.0107 -0.0265 -0.0405 -0.0321 -0.0135 +0.0028 -0.0020 -0.0112 -0.0194 -0.0222 -0.0182 -0.0103 -0.0030 Based on n = 9. A plus (+) sign indicates tension. based upon a value of n = 15. actions at the springings due to notation used in these tables is The smallness of the change in the re- a large change in n is of interest. The given in Fig. 19. The fixed-end reac- tions for a single-span arch due to a unit load and to unit movement of the abutments, are given in Table 3, and the influence ordinates for stress at various sections of the same structure are given in Table 35. The influence ordinates for stress at the critical sections of the three-span structure are given in Tables 36, 37, and 38 for pier heights of 20 ft., 15 ft., and 10 ft., respectively. The values in Table 36 indicated that the maximum stress due to dead load plus one live load, for a structure having 20-ft. piers, will occur at the section through the load point C4. The influence dia- grams for the stress at sections C3 and C4 are given in Figs. 23 and 24, and the live load producing the maximum stress is shown in Fig. 25. This live load is designated, "one live load." The stresses at load points C3 and C4 and at sections at the springings are given in B of BC Extrados -0.0002 -0,0007 -0.0011 -0.0011 -0.0003 +0.0008 +0.0015 +0.0013 -0.0146 -0.0662 -0.1815 -0.0274 +0.0470 +0.0591 +0.0388 +0.0139 -0.0026 +0.0006 +0.0064 +0.0118 +0.0139 +0.0115 +0.0066 +0.0019 Intrados +0.0048 +0.0165 +0.0289 +0.0347 +0.0296 +0.0159 +0.0011 -0.0059 -0.0723 -0.0905 -0.0668 -0.0227 +0.0165 +0.0335 +0.0283 +0.0123 -0.0026 +0.0020 +0.0106 +0.0184 +0.0210 +0.0172 +0.0097 +0.0028 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 38 INFLUENCE ORDINATES FOR STRESS AT VARIOUS SECTIONS, CALCULATED BY ELASTIC THEORY 10-foot piers Stress, in lb. per sq. in., at Section Through Load of 1 lb. at El E2 E3 E4 E5 E6 E7 E8 C1 C2 C3 C4 C5 C6 C7 C8 W1 W2 W3 W4 W5 W6 W7 W8 Load Point C4 Extrados +0.0014 +0.0051 +0.0088 +0.0106 +0.0090 +0.0049 +0.0002 -0.0018 +0.0038 -0.0068 -0.0637 -0.2019 -0.0406 +0.0278 +0.0335 +0.0146 -0.0032 -0.0003 +0.0064 +0.0129 +0.0155 +0.0132 +0.0076 +0.0022 Intrados -0.0018 -0.0063 -0.0110 -0.0132 -0.0112 -0.0059 -0.0002 +0.0024 -0.0068 -0.0018 +0.0489 +0.1821 +0.0202 -0.0436 -0.0425 -0.0176 +0.0038 +0.0003 -0.0074 -0.0151 -0.0181 -0.0154 -0.0088 -0.0026 B of AB Extrados Intrados -0.0041 +0.0039 -0.0139 +0.0135 -0.0240 +0.0234 -0.0283 +0.0275 -0.0232 +0.0226 -0.0109 +0.0107 +0.0016 -0.0016 +0.0067 -0.0065 +0.0602 -0.0712 +0.0738 -0.0858 +0.0449 -0.0577 -0.0020 -0.0106 -0.0393 +0.0287 -0.0505 +0.0431 -0.0374 +0.0334 -0.0150 +0.0138 +0.0034 -0.0032 +0.0001 -0.0001 -0.0071 +0.0067 -0.0140 +0.0132 -0.0168 +0.0158 -0.0141 +0.0133 -0.0080 +0.0076 -0.0024 +0.0022 Based on n = 9. A plus (+) sign indicates tension. Table 39 for dead load, for one live load, and for dead load plus one live load, the latter being the load for which the structure would be designed. The stresses at C3 and C4 and at the two springings of the center span that would be produced by the load shown in Fig. 25 if the ends of the center span were fixed, are given at the bottom of Table 39. The stresses in Table 40 are for the live load distributed in such a manner as to produce the greatest stress in a single span with fixed ends, the section at which this maximum stress occurs being at C3. All of the stresses have been computed from the moment and thrust on the basis that the concrete takes tension. The maximum stress due to the design load shown in Fig. 25 is a compression of 1236 lb. per sq. in. at the extrados at C4. If the ends of the center span were fixed, the design load shown in Fig. 25 would produce a stress of 1034 lb. per sq. in. at the same point. The maxi- mum stress in the single span due to the dead load and the live load shown at the bottom of Table 40 is a compression of 1089 lb. per sq. Load Point C3 Intrados 0 0 -0.0001 -0.0003 -0.0006 -0.0010 -0.0009 -0.0007 +0.0115 +0.0570 +0.1632 +0.0019 -0.0711 -0.0773 -0.0488 -0.0168 +0.0030 +0.0001 -0.0051 -0.0108 -0.0131 -0.0112 -0.0064 -0.0019 Extrados -0.0004 -0.0012 -0.0019 -0.0019 -0.0012 0 +0.0009 +0.0011 -0.0142 -0.0644 -0.1787 -0.0233 +0.0509 +0.0621 +0.0402 +0.0140 -0.0024 -0.0001 +0.0041 +0.0088 +0.0107 +0.0092 +0.0052 +0.0015 ILLINOIS ENGINEERING EXPERIMENT STATION ..4967 ~ C~/ SS~~24S~ REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS Fla. 25. LIVE LOAD PRODUCING MAXIMUM STRESS AT C4 in. at the extrados at the load point 3. That is, the elastic deforma- tion of the 20-ft. piers increased the maximum stress due to the design load as computed by the elastic theory from 1089 to 1236 lb. per sq. in., a change of 13 per cent. 19. Description of Apparatus.-The apparatus used in the tests of the single-span structure and described in Section 5 was used in the tests of the three-span structure. Some additional apparatus was needed, however, and some modifications of the apparatus that had been used seemed desirable. The vertical reactions of the pier bases were measured with ap- paratus similar to that used for measuring the abutment reactions. The horizontal reaction of each pier base was measured with two horizontal scales, one on the north and the other on the south side of the structure, shown in Fig. 26. The scales are capable of measuring reactions in one direction only. To overcome this limitation, a bell crank supported on knife edges, and carrying a concrete block on the the outer end of its horizontal arm, was provided for each pier base. This apparatus produced an initial horizontal thrust of approximately 1000-lb. upon each of the horizontal scales, the thrust acting outward from the middle of the structure for both piers. In the tests to deter- 5" ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 39 STRESSES AT VARIOUS SECTIONS DUE TO DESIGN LOAD, CALCULATED BY ELASTIC THEORY Stresses, in lb. per sq. in., at Section Pier ___________________ Load Height, ft. Load Pt. Load Pt. B of B of C of C of C3 C4 AB BC BC CD All Intrados - 92 -142 -128 -128 -128 -128 D ead load " 1 ------------------------- heights Extrados -546 -500 -342 -342 -342 -342 Three-Span Series on Elastic Piers Live load 20 Intrados +179 + 646 +137 -258 + 24 +190 Extrados -272 - 736 -142 +187 - 76 -197 15 Intrados +157 + 606 +117 -215 + 67 +169 Extrados -256 - 701 -121 +143 -121 -175 10 Intrados +130 + 552 + 88 -153 +130 +134 Extrados -235 - 654 - 91 + 79 -186 -139 Dead load 20 Intrados + 87 + 504 + 9 -386 -104 + 62 + Extrados -818 -1236 -484 -155 -418 -539 Live load 15 Intrados + 65 + 464 - 11 -343 - 61 + 41 Extrados -802 -1201 -463 -199 -463 -517 10 Intrados + 38 + 410 - 40 -281 + 2 + 6 Extrados -781 -1154 -433 -263 -528 -481 Center Span with Ends of Rib Fixed Live load ..... Intrados + 61 + 410 ..... + 27 +328 . Extrados -187 - 534 ..... -110 -391 .... Dead load ..... Intrados - 31 + 268 . -101 +200 ..... + ..... Extrados -733 -1034 ..... -452 -733 ..... Live load Plus (+) indicates tension, minus (-) indicates compression. Stresses are determined from moment and thrust on assumption that concrete takes tension. Based on n = 9. mine the elastic properties of the structure and to determine the in- fluence ordinates by the application of a unit load, the manipulations that produced an inward thrust at the pier bases reduced, but did not reverse, the initial thrust upon the horizontal scales. This method could be used in these tests because the changes in the thrust were small. The design-load tests and the tests to destruction produced a large horizontal thrust on the bases of the piers, but this was an out- ward thrust and could be resisted by the scales. The positions of the reactions for both piers and abutments are shown in Fig. 25. The hook gages, used to determine the vertical movements of the load points of the single-span structure, were sensitive and reliable but they were tedious to read. So, for the three-span structure, the REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 40 STRESSES AT VARIOUS SECTIONS, CALCULATED BY ELASTIC THEORY Single Span Arch With Fixed Ends Under Live Load Giving Maximum Stress, Which Occurs at Load Point No. 3 Stresses, in lb. per sq. in., at Load Load Point Load Point East West No. 3 No. 4 Springing Springing Dead load Intrados - 92 -142 -128 -128 Extrados - 546 -500 -342 -342 Live load Intrados + 464 +158 -223 +306 Extrados - 543 -236 +152 -343 Dead load Intrados + 372 + 16 -351 +178 plus Extrados -1089 -736 -190 -685 Live Load Live Load Distribution Load Point No.... 1 2 3 4 5 Live Load ........ 960 960 2760 647 18 Stresses are determined from moment and thrust on assumption that concrete takes tension. Computations are based on n = 9. A plus (+) sign indicates a tensile stress. vertical movements of the load points were measured relative to two I beams, one on each side of the rib, continuous from end to end of the structure. Two carefully leveled steel plates, one on the north and the other on the south side of the rib, were grouted on the top of each pier and on the west abutment. Steel rollers on the plates supported the I beams without transmitting any horizontal thrust to the piers or abutments. The vertical movement of points on the arch rib rela- tive to the I beams was measured in the following manner: A %-in. vertical rod embedded in the concrete and projecting below the rib on its center line directly below each load point had a small hole in the lower end. The tops of the I beams were connected with steel battens directly beneath each load point and each batten had a small hole at its center directly beneath the rod projecting from the bottom of the rib. An instrument, consisting of an Ames dial having a conical point on its plunger and mounted on a rod with a conical end, was used to measure the distance from the lower end of the projecting rod to the top of the batten plates, thereby giving the movement of the load points relative to the I beams. The relative elevation of the piers and abutments was measured with a hydrostatic gage. This consisted of four hook gages, one at- tached to the structure at the top of each pier and one attached to ILLINOIS ENGINEERING EXPERIMENT STATION FIG. 26. APPARATUS FOR MEASURING REACTIONS AT PIER BASES ro/e, /r-rcr / Ro/er/-- Seal for A 5s ,/"r/ P/lu/nger ,'Ames z/''/ A,1l, me ,-7ber.5 of 17-ane are - /-/. b,,Ztea,ws FIG. 27. FRAME FOR MEASURING HORIZONTAL MOVEMENT OF PIER BASES RELATIVE TO ABUTMENT each abutment, connected by a pipe line so that the water level would be the same for all gages at any instant. The I beams from which the deflections were measured were also used in measuring the horizontal movement of the west abutment, and of the tops of both piers, relative to the east abutment. These beams were attached to a 1-in. pin that extended through the rib at the east springing of the east span in such a way as to cause the beams to move horizontally with the east abutment. The beams were supported at the pier tops and at the west abutment on rollers that enabled the pier tops and the west abutment to move horizontally relatively to the beams. Ames dials attached to the beams with their plungers bearing REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS (al 1.0/ FIG. 28. LEVEL BUBBLES FOR MEASURING ANGULAR MOVEMENTS OF ABUTMENTS AND PIER BASES on the pins through the ribs at the springings indicated horizontal movement of the west abutment, and of the tops of the piers, relative to the east abutment. The horizontal movement of points on the piers at various levels, relative to the east abutment, was measured by means of the frame shown in Fig. 27. The member A B C D consists of the I beams from which vertical deflections were measured. The frame B C F E was suspended from these beams. Ames dials attached to the lower corn- ers of this frame had plungers bearing upon the pins E and F pro- jecting from the sides of the piers, the centers of the pins being the geometrical centers of the bases of the piers. The pins E and F were 20 feet below the springing of the arches; pins E' and F' and pins E" and F" were 15 feet and 10 feet, respectively, below the springing. The horizontal movement was measured at E and F, at E' and F', and at E" and F" when testing the structure at pier heights of 20 feet, 15 feet, and 10 feet, respectively. In tests requiring one pier to be lowered, shims were inserted at, that pier between the beam and its support. When the pier was raised the shims were removed so that points B and C on the beam remained at the same level, the sides of the sus- pended frame remained in a vertical position, and the corners E and F did not move horizontally because of the vertical movement of the pier. Differences in the changes in the readings of the dials at B and E indicate the horizontal movement of the top relative to the base of the pier. The angular positions of the abutments and pier bases were de- termined with level bubbles attached to the structure at the points ILLINOIS ENGINEERING EXPERIMENT STATION where the rotation was to be measured. The bubble tubes at the abutments were carried on steel bars embedded in the concrete, pro- jecting from the rib at its center line, and normal to the axis of the rib at the springing, as shown in Fig. 28a. The bubble tubes for the piers were carried on horizontal bars embedded in the concrete, one at the top, and one at distances below the top of 20 ft., 15 ft., and 10 ft., corresponding to the desired height of pier. The bubble at the top of the pier was used in determining the rotation of the top of the pier and the bubble at the base, shown in Fig. 28b, was used in de- termining the rotation of the base. In tests for which the abutments and the bases of the piers were fixed, the bubbles were adjusted so as to be in their mid-position be- fore the test began and, after the load was changed, the abutments and pier bases were rotated till the bubbles returned to their central po- sition; in tests for which a predetermined angular movement was to be produced, the bubble was read before the test began and then the piers and abutments were rotated by manipulating the jacks support- ing them until the bubble had moved an amount which, as shown by a previous calibration, corresponded to the desired rotation. V. EXPERIMENTAL DETERMINATION OF ELASTIC CONSTANTS 20. Description of Tests.-The elastic properties of the ribs were determined experimentally by measuring the changes in the reactions that accompanied movements of one end of the rib, the other end being fixed. The moved end was given, successively, motions of rota- tion and of horizontal and vertical translation, being restrained against two of the motions when subjected to the other one. The movement, or lack of movement, was determined as follows: The vertical translation with the hydrostatic gage, the horizontal trans- lation with the Ames dials attached to the pier tops and the west abutment, and the rotation with calibrated bubbles attached to the two abutments and the two pier tops. The points moved were the two abutments and the two pier tops, these being the springings for the various arch ribs. The movements were symmetrical about the center line of the whole structure. That is, when the top of the east pier was tipped inward the top of the west pier was likewise tipped inward; when the east abutment was moved outward the west abutment was likewise moved outward. This procedure simplified the operations necessary to the production of the desired motion, and made possible the simultaneous testing of two spans. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS The arch carried the design dead load continuously during the period occupied by these tests, and the movements were so small that the thrust line remained within the kern of the arch. Before beginning a test involving a given movement, half of the anticipated movement was produced in one direction before the initial readings were taken, so that the motion for which the changes in reactions were to be de- termined was a movement from a position on one side of the normal to a position on the other side of the normal. In the case of a change in span, if the anticipated change was to be 0.10 in. per span or 0.30 in. for the three spans, the west abutment was moved outward 0.15 in. thereby increasing each span approximately 0.05 in. A complete set of zero readings was then recorded. The west abutment was then moved in 0.30 in. thereby shortening each span approximately 0.10 in. to a length 0.05 in. below normal, and a second complete set of readings was recorded. The west abutment was then moved outward 0.30 in. increasing each span 0.10 in. to a value of 0.05 in. above normal, and a third set of readings was recorded. A similar procedure was followed for the other movements. That is, changes in reactions were measured for two movements approximately equal in magnitude but opposite in sense; and for each movement the arch went from a position on one side of normal to another position an approximately equal amount on the other side of normal. In taking a set of readings preceding a predetermined movement, the piers and abutments were adjusted for position four times, and the readings of the instruments were recorded after each adjustment. The average of the four readings was accepted as indicating the con- dition before the movement. The same procedure was used following the movement. The details of the various tests are described in the following sections. 21. Spread of Abutments.-Preliminary to the tests to determine the changes in the reactions due to a change in span, the distance be- tween abutments was increased approximately 0.15 in. above normal, thereby increasing the length of each span by approximately 0.05 in. The bubbles indicating the angular positions of the abutments and of the tops of the piers (the springings of the adjacent arches) were ad- justed so that they were in their mid-position. The links connecting the bases of the piers with the horizontal weighing scales were re- moved, allowing the pier bases to be free to move horizontally except as they were restrained by rolling friction. Readings were then re- ILLINOIS ENGINEERING EXPERIMENT STATION corded of the horizontal and vertical scales for the abutments, of the vertical scales for the piers, of the hydrostatic gages indicating the relative elevations of the abutments and pier tops, and of the Ames dials indicating the horizontal positions of the abutments and pier tops. The piers and abutments were adjusted for position four times and these readings were recorded after each adjustment, except that the hydrostatic gage was read after the first and last adjustment only. The abutments were then forced together about 0.30 in. by turn- ing the links connecting each abutment to its horizontal scales. This decreased each span by approximately 0.10 in. The jacks under the abutments and pier bases were then manipulated until all the bubbles at the abutments and pier tops were in their mid-position. A complete set of readings was again recorded, the abutments and piers being ad- justed for position four times and readings recorded for each position, as before. The abutments were then allowed to move apart 0.30 in. and read- ings recorded as just described. The tests thus included one decrease and one increase in span. The reactions of the abutments are given in Table 41 and the re- actions of the piers in Table 42. If the structure were homogeneous and symmetrical about its center and if the supports of the piers were frictionless there would have been no horizontal forces acting on the pier bases and all three spans would have had the same change in span. Because of the heavy load carried by the wheels under the pier bases there was some friction. The magnitude of this force was de- termined in the following manner: The vertical reactions of the piers, given in Table 42, indicate that the bases were subjected to a moment. Assuming the horizontal thrust in the ribs of two adjacent spans to be on the same level, the unbalanced thrust from adjacent spans must balance the moment on the pier base. In the case of the east pier the moment on the base, for a decrease in span, is 53 783 in. lb. The line of action of the horizontal thrust is 389 in. above the top of the rollers supporting the pier bases. For a decrease in span, the unbal- anced thrust on the east pier, due to the east and center spans, is 53 783 therefore =--138 lb. Since the thrust from the east span, given 389 in Table 41, is 1744 lb. the thrust from the center span is 1774 - 138 = 1606 lb., as determined from the east abutment and east pier. The same thrust, determined from the west pier and west abutment is, 90 037 1883- - -= 1652 lb., the two values differing by 46 lb. An 389 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS average value of 1629 has been used. For an increase in span, the 68 207 thrust in the center span is 1820 -= 1645 lb., as determined 389 81 457 from the east, abutment and east pier; and 1864 - 1655 lb., 389 as determined from the west abutment and west pier, the values dif- fering by 10 lb. An average value of 1650 lb. has been used. The moment at the springings of the center span was determined from the horizontal thrust on the assumption that the distance of the thrust line above the springing for the center was the average of the values for the two end spans. The moment and thrust, resulting from the changes in span are given in Table 43. The 5th line of the table gives the average of the values obtained by decreasing and increasing the span, reduced to a basis of a change in length of each span of 0.10 in. The computed value of these elastic constants depends upon the modulus of elasticity of the concrete. The ratio of the measured values to the values computed by the elastic theory on the assumption that E = 2 500 000 lb. per sq. in. is given for the various spans in the last line of Table 43. The line of action of the horizontal thrust is at nearly the same elevation for the two end spans and its position as determined from measured moments and thrust is about 4.5 in. below its position as determined by the elastic theory. 22. Rotation of Abutments and Pier Tops.-The general principles followed in the tests to determine the elastic constants of the arches by measuring the changes in the reactions that accompanied a change in span were used in the tests to determine the elastic constants by rotating one end of a span when the other end was fixed. The tests were made on the end spans only. Two series of tests were made on each end span. For one series, the pier top was fixed and the abut- ment was rotated without any motion of translation. For the other series, the abutments were fixed and the pier tops were rotated. The results of the first series of tests are given in Table 44 and the results of the second series in Table 45. The last line of each table gives the ratio of the measured to the computed value, the latter being deter- mined by the elastic theory on the basis that E = 2 500 000 lb. per sq. in. 23. Settlement of Piers.-As a preliminary to the tests for deter- mining the elastic constants by changing the elevation of one end of a 84 ILLINOIS ENGINEERING EXPERIMENT STATION z z z z E' Ol z 0 z 0 El z a u i" n a z z z U' z z Z 0 C a' o' 0l _0 0 . hC 0 0 a.0 0" Wi9 0 C) 0 00 w C) 20 o m oo 1+ C- 20 ) 000 § REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS a4) *40 §,9 0 5- a 0a ^i ). 44a. 04 i.C 4) -4) 4) 0o *0, os . C3 .s5 44 4) 4) S!) 4)4 ~CC s4) 4)4 4)4) 5^4 4)g) ^CC 0 0s.s 4)4,4 4)444 4))0 4) .4...4- 4) =41404)440 z w z Z 0 0 CC © © CC El 02 -04O * C0 a; a,' -0 00 C 0 cc0.i 040-o 4041.40: gg•. +1+1 4140 N C -ý +1+1 4, N N- +1+1 G00400 +1+I 40C 04C 04 gTgTg 4 •01. C C g IN ,p <0 10 1i0 x cco- m N10 c ++ 4"t;t-040000 1 00001000 1+ + 00000 1+1+1 40 Nt 000X ?+ I+ 4)O 0 000 00000 1+1+ i4tCi4; 4)oQ) 000 r1)+1c+_ OS CO 00 41 04 OS 4)4 'S i 0.5 4) ILLINOIS ENGINEERING EXPERIMENT STATION .0 3 0 tOO- 10 N I N+N I Na CD OIM-N c3 1+1+ -'- HM i N N N 0 ~,0 .0 0 .00 0, ~ .0 0 L. 53 H 0 0 N N m' N O g§ S.0, 0 0, 0 0 0 0 0 N to N GO CO N- 'P N^ N N 1^ 0 41oOOS + i+ 0 NO 00 N N0 +1+1 (t .0 "t, D' * 0 CT ID +1+1 tO3 N t^*' N ID< CD - t +1+1 + 'TI +^ C +1+1 0000< ^ O N1 N C 'p E- +1+1 +1+1T N^ N 00 0 +i1+1 Ns 000 Nt3 000 'P 'P 00 +1+1' C*Cif-f0'0' .~00 00 00 ~.0 N N 0 N N 00' ~-0. ,.~.0 00 0'-' 0.0 0 CD REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS i§ Co +4t+ CoI CO CC CO CC C- C CO .0O 1+T+ CoCCo Co CTo CO 1+1+ aCT> 000C C tCo CoO 0 CO ^OCO CO CCoC 1+1 + 1++ H CD Co .0 S Co Co Co0 Co1~ Co5 CC Co 0. 0 0 z 0 H H 0 Co z H Co Co HCo z 0 0 U H Co -Il Co H Co 0 Co cj~ Co -2 i C1o o Co COCC -1 -11- 4)4 +1+ GO.~ Co-o 0 0 +1+1 cc0000 E- E- E- E- CO Cl CO Co C -Cl -^ CC GO tCo -f 0 CoMC ^-Cl QCO COl 1+ 1+ CCC00- CO CM 1+1 + CCoC..Coý m - CoCCcmcC - CO +tOCO -r Co C*4OS C0 Co C ±1+ 0o 4) 4)4) SCo Co .4). C. 0* 0 ~~)4) C... CC 0 4)0. C. Co Ca C aCo ICCo ~C*Cl -~ II IC... C.- CC'S -D I'S 88 ILLINOIS ENGINEERING EXPERIMENT STATION 0 43 a4 C.1o0*0 m m CO M<-00 0000 - Co tC'4'C4C -t 1+1+ - 1+1+ - 4*O CO ro i Clr- 0 ++ + +1+ C H z fr* z H CCC t Co .443 E-~z 0 0 C~) Co -c Co H 4) Co 0 r12 .0 Co- &0 C4 *40 Coi Cl CO I I .) 4*. I cO 1.1 I41 44 03 cl rs C) 0 C) z 0 'C 0 4* 04 Ci) 0 4* 04 Co 4* +1+ 1+1+ <00 0 0-I t .o 'I - 'm ONO 1-4- - +1+ co- 10Glo So004400 C +1+1 (NOO000 ým 0 0. CWMH +1+1 4141--- +1+ 4--1 4-- 1 4-4--4-)' 04 0 0 S 0 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 89 TABLE 47 RATIO OF VALUES OF ELASTIC CONSTANTS OBTAINED FROM MEASURED REACTIONS TO VALUES OBTAINED FROM ELASTIC THEORY E taken as 2.50 in 106 lb. per sq. in. Elastic Constant for Movement Moment Horizontal Vertical Average Thrust Shear Average Change in span ...................... 1.4014 1.4518 ..... 1.426 Rotation of abutments. .............. 1.4134 1.3652 1.4802 1.420 Rotation of pier tops................. 1.3604 1.3402 1.3532 1.351 Settlement of support................ 1.3904 ..... 1.3892 1.390 Average of averages.................. 1.391 1.385 1.407 1.396 or 1.400 Numerals indicate number of values averaged. span relative to the other, both piers were lowered 0.125 in. and a shim 0.25 in. thick was inserted on the roller that supports the I beams for measuring the span. The abutments and piers were adjusted for position four times and readings were recorded after each adjustment. Both piers were then raised 0.25 in., and the 0.25 in. shims on the rollers supporting the I beams were removed so that the relative ele- vation of the supports for the I beams would be the same when the piers were in their high as when they were in their low position. The piers and abutments were adjusted for position four times and the readings were recorded after each adjustment. The piers were then lowered 0.25 in. and the operations repeated. The results of the tests are given in Table 46. 24. Average Values of Elastic Constants.-The elastic constants for the arch ribs have been determined by the four tests described in Sections 20 to 23, inclusive. The experimentally-determined values have been expressed as the ratios of these values to the corresponding values computed by the elastic theory, on the basis that the modulus of elasticity is 2 500 000 lb. per sq. in. If all of the assumptions upon which the elastic theory is based were true and if the tests were all accurate, all values of the ratio would be equal. The values of the ratio determined by the various tests are given in Table 47. The numeral above each ratio in this table indicates the number of values averaged. The average of all the averages is 1.396, approximately 1.40, a value that will be used. Values of elastic constants 1.40 times ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 48 AVERAGE VALUES OF EXPERIMENTALLY-DETERMINED ELASTIC CONSTANTS Values are based on Table 47 and are equal to 1.4 times the values of elastic constants obtained by the elastic theory on the basis that E = 2.50 in 106 lb. per sq. in. Movement Spread of 0.10 in ......................... Settlement of east end 0.10 in.............. Rotation of east abutment of 0.001 radian, top tipping in........................ Rotation of west abutment of 0.001 radian, top tipping in ........................ Reaction of East Abutment Moment in. lb -93 712 +14 228 +89 582 +43 481 Horizontal Thrust lb. -1425 0 + 937 + 937 Vertical Shear lb. 0 - 88 -143 +143 the values computed by the elastic theory, on the basis that E is 2 500 000 lb. per sq. in., are given in Table 48. These have been considered as the average values of the experimentally-determined elastic constants, and are the values that will be used in Section 26 to determine the reactions of the tops of the piers due to unit loads from the movement of the pier tops. VI. INFLUENCE ORDINATES OBTAINED BY UNIT LOADS 25. Description of Tests.-The influence ordinates for the reac- tions at the springings of all three spans were determined experi- mentally by measuring the changes that occurred when a unit load was applied and removed, successively, at the various load points. The unit load was 2000 lb., and the tests were made with the design dead load on the structure. The reactions were determined in two ways (1) by weighing the reactions before and after the application and removal of the unit load and computing the moment and the H and V components of the reactions from the changes in the scale read- ings; and (2) by measuring the displacement of the tops of the piers due to the application and removal of the unit load and computing the moment and the H and V components of the reactions correspond- ing to these displacements, using the experimentally-determined elas- tic constants given in Table 48, and the fixed-end reactions for n = 9 given in Table 3. The method of making a test was as follows: With no live load on the structure, the abutments and pier bases were adjusted to their normal position except that the span was made REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS slightly less than normal, thereby decreasing the compression at the extrados at the crown and keeping the thrust line nearer the axis of the rib when the unit load was at load points C4 and C5. The initial position of each of the terminals was indicated by the level bubbles, and the Ames dials and hydrostatic gage attached to the abutments and pier tops. The readings of these instruments were recorded to make it possible to return the terminals to their initial position after each load change. The movement of the pier tops was determined by the Ames dials, which indicated changes in span, and by calibrated level bubbles, which indicated rotations. The scales measured the re- actions at the abutments and pier bases. In making a test, readings were taken with only the design dead load upon the arch, after the unit live load had been added, and again after the unit live load had been removed. A complete test for each load point therefore included determining the change in the reactions due to applying the unit load and also that due to removing the unit load. The influence ordinates were determined by both methods for 20- foot piers, but only by measuring the movement of the pier tops for 15-foot and 10-foot piers. In the tests with 20-foot piers the instru- ments indicating the position of the pier bases were attached to the piers at a distance of 20 feet below the springings of the arches. In the tests with 15-ft. and 10-ft. piers these instruments were attached 15 ft. and 10 ft., respectively, below the springings. In the early tests the abutments and pier bases were adjusted for position four times before and after each load change, and all instruments were read after each adjustment. As the operators developed proficiency in adjusting the terminals, the readings after successive adjustments were in such close agreement that four adjustments seemed unnecessary and only two adjustments were made for all but a few of the early tests. 26. Influence Ordinates Obtained from Movement of Pier Tops.- The influence ordinates for moment, and for the horizontal thrust and the vertical shear at both springings of all spans, were determined from the movement of the pier tops that accompanied the application and removal of a load of 2000 lb. at each of the various load points. These tests were made for structures having pier heights of 20 ft., 15 ft., and 10 ft. The method of determining the reactions at the springings from the movement of the pier tops is illustrated in Tables 49 and 50. Each component (H, V, or M) of a reaction is made up of four parts: The reaction due to spread, the reaction due to rotation of the top of the 92 ILLINOIS ENGINEERING EXPERIMENT STATION TABLE 49 INFLUENCE ORDINATES FOR MOMENT BY UNIT LOADS, OBTAINED FROM MOVEMENT OF PIER ToPS 20-foot piers Due to Spread ............. Rotation at B....... Rotation at C....... Fixed-end loading . . Resultant .......... Spread............. Rotation at B....... Rotation at C ...... Fixed-end loading . . Resultant ........ .. Moment, in in. lb., at A of AB -19 117 - 6 148 0 +51 403 +26 138 +18 836 + 6 444 0 -51 403 -26 123 +26 130 +28 042 B of AB -19 117 -12 667 0 +23 697 - 8 087 +18 836 +13 276 0 -23 697 + 8 415 - 8 251 - 5 788 B of BC +15 369 +12 667 - 2 683 0 +25 353 -13 963 -13 276 + 2 257 0 -24 982 +25 167 +23 622 C of BC +15 369 + 6 148 - 5 527 0 +15 990 -13 963 - 6 444 + 4 649 0 -15 758 +15 874 +15 475 Cof CD D of CD +3655 +3655 0 0 +5527 +2683 0 0 +9182 +6338 -4967 -4967 0 0 -4649 -2257 0 0 -9616 -7224 +9399 +6781 +7887 +5862 east pier, the reaction due to rotation of the top of the west pier, and the fixed-end reaction due to the unit load. These components are listed separately in the various lines of the tables. The resultant com- ponent (H, V, or M) is the algebraic sum of the various parts. Tables 49 and 50 show the computations for the change in the reactions due to applying and removing the unit load from E5. The average of these two values for each component reaction is compared with the value obtained by the elastic theory in the two last lines of the table. The reactions obtained from the movement of the pier tops are compared graphically with the values obtained by the elastic theory in the diagrams of Figs. 29 to 37, inclusive. In these diagrams the light full lines represent the values by the elastic theory; the heavy full lines represent the experimental values determined by applying and removing a unit load successively at each load point on the east half of the structure; and the broken lines represent the experimental values determined by applying and removing a unit load successively at each load point on the west half of the structure. The broken-line diagrams have been turned end for end to make them directly compa- rable with the heavy full-line diagrams. A comparison of the broken- line diagrams and the corresponding heavy full-line diagrams shows the agreement between the experimentally-determined values obtained for points symmetrically spaced with reference to the center of the structure. A comparison of the light full-line diagrams with the two experimentally-determined diagrams shows the agreement between Change in Load On at E5 Off at E5 Average, load on and off ....... By Elastic Theory .............. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS C C C C 0 20 20 02 0 j > C! C OOcOOC OOt.-00 1 + + 1 I + + 00C8o0o 0;i.o0 00 20 C M N N N N ++ + I I + + 1 I 1 ++ + 1 I N SN N i N N I ++ + I + + , + +7 000 R0 M 000 00 00 I ++ + I I + + NI ++ ++ m I N N • o 0o ' oo 0 N n P4 9'4,4 i .< 0 4E z &< N (1 zo 8 22 0. 20 22 CO C U C 22 C, CO 22 0H x1 z 0 Ez P 0 ILLINOIS ENGINEERING EXPERIMENT STATION H El z a SEý z z zo z z i 0K REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS z z- § 2 z zc 1^: ILLINOIS ENGINEERING EXPERIMENT STATION z z z 0ýi REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS z z a QE. Z c, Z o- .0 CO z Z;Lq 00 ZO CO 0 z§ ILLINOIS ENGINEERING EXPERIMENT STATION FIG. 37. INFLUENCE LINES BY UNIT LOADS FROM MOVEMENT OF PIER TOPs. CENTER-SPAN PIER TOP REACTIONS. 10-FOOT PIERS the measured values and the values computed by the elastic theory. These agreements are remarkably close except for small portions of a few diagrams, and in no case is the lack of agreement very great. The influence lines for the fixed-end reactions at the east end of the center span are also given in Fig. 31. 27. Influence Ordinates Obtained from Measured Reactions.-The influence ordinates for moment, and for the horizontal and vertical components of the thrust at both springings of all spans, were deter- mined from the measured changes in the loads on the scales that ac- companied the application and removal of a load of 2000 lb. at each of the various load points. These tests were made only for the struc- ture having 20-foot piers. REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS 99 TABLE 51 INFLUENCE ORDINATES FOR MOMENT BY UNIT LOADS, OBTAINED FROM MEASURED REACTIONS 20-foot piers Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 922 23 24 Moment, in in. lb., at Load on at E1 ....... Load off at El ....... Average, load on and off By Elastic Theory... . Load on at E2 ....... Load off at E2 ....... Average, load on and off By Elastic Theory.... Load on at E3 ....... Load off at E3 ....... Average, load on and off By Elastic Theory.... Load on at E4 ....... Load off at E4........ Average, load on and off By Elastic Theory... . . Load on at E5 ....... Load off at E5 ....... Average, load on and off By Elastic Theory... Load on at E6 ..... Load off at E6........ Average, load on and off By Elastic Theory..... B of AB + 1 170 - 2 855 + 2 012 + 5 735 +13 970 -17 479 +15 724 +17 630 +20 257 -22 612 +21 434 +24 970 +16 208 -15 832 +16 020 +17 713 - 6 068 + 6 038 - 6 053 - 5 788 -36 006 +34 637 -35 321 -35 196 B of BC C of BC + 1 973 + 2 428 - 1 053 + 601 + 1 513 + 913 + 3 667 + 2 259 + 9 622 + 6 478 -12 712 - 8 127 +11 167 + 7 302 +12 634 + 7 829 +21 562 +13 870 -23 720 -13 113 +22 641 +13 491 +22 339 +13 977 +29 372 +18 245 -27 458 -15 974 +28 415 +17 109 +27 099 +17 228 +24 886 +15 098 -23 970 -15 727 +24 428 +15 412 +23 622 +15 475 +11 167 + 7 934 -12 004 - 8 999 +11 585 + 8 466 +13 436 + 9 490 The changes in moments due to applying and removing the unit load at points El to E6, inclusive, are compared with the values ob- tained by the elastic theory in Table 51 and the changes in the hori- zontal and vertical components of the thrust are given in Table 52. The influence lines for reactions determined from the scale read- ings are compared with the corresponding lines obtained from the elastic theory in Figs. 38, 39, and 40. In these figures the light full lines represent the values by the elastic theory; the heavy full lines represent the experimental values determined by applying a unit load at points on the east half of the structure; and the broken lines repre- sent the experimental values determined by applying a unit load at points on the west half of the structure. The broken-line diagrams have been turned end for end to make them directly comparable with the heavy full-line diagrams. A comparison of the broken-line dia- grams and the corresponding heavy full-line diagrams shows the agree- ment between the experimentally-determined values obtained for points symmetrically spaced with reference to the center of the struc- A of AB -51 636 +53 191 -52 413 -51 019 -65 806 +62 945 -64 015 -61 876 -44 698 +41 695 -43 196 -39 449 - 5 093 + 3 201 - 4 147 - 2 390 +27 299 -27 006 +27 152 +28 042 +36 156 -39 469 +37 812 +38 462 C of CD + 942 + 1 155 - 106 + 1 173 + 2 123 - 5 750 + 3 936 + 4 059 + 8 102 - 9 483 + 8 792 + 7 224 + 9 628 -10 050 + 9 839 + 8 861 + 8 656 -10 675 + 9 665 + 7 887 + 3 851 - 6 878 + 5 364 + 4 731 D of CD + 618 + 831 - 106 + 873 +1799 -4454 +3126 +3020 +5186 -7863 +6524 +5374 +7036 -8106 +7571 +6590 +6712 -7759 +7235 +5862 +2231 -5582 +3906 +3512 ILLINOIS ENGINEERING EXPERIMENT STATION ++ + +1++ +1++ +1++ +I++ +I++ I I I + I 1+ 1 1 a4 0 Eý PQ ri z 7g Ea t0 a) P4 0 N P4 0 z S a) S 2; O S O 0 H a) S a) H 0 0 HSO^ a) S Si S S+ 1 I +1 ++ +1 I+ 1 +1++ +11 1+ 1 1 +1++ +i++ 0++÷I +++ ++ +1++ +1++ +i+ ++1++ +I++ ++0000 +T++ +1 - a++) + ++ +T+++ + ++ + ++ + ++ +1 ++ + I ++ +1 ++ + ++ +1++ + ++ +1++ +I++ + I++ +++ a) OS a) +1 +++ 0 a) i01 + +++ . 0 . ... . .. . 0 bo I_ V go . 00 :0 0 *0 > 0 a A 4 4 a A +[++ +1++ +I++ *.0 . ^-^l~ - OiOO CO - t^ SO0DCO 00-1f1 0 CObICO- ;C~ : :;C : :CCC 0 . . . 0 , . . .o4"' ý0'6  :§ : : :§ : ;;' : PA Na, g * *a **a 0Sl M-, 's' 8 -'s' 8 00 o' d oo0> hýilifl h]-l3f r i-^f 0Nm0 ala)NOO 00-a) W)ýtatO a3000 a;a N C N) a) a-araC " +1++ +1I++ +1++ 1+11 ++1 I+1 + +++ + ++ 0 + ++ "0 0 0 + ++ REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS ell z 0 0 Z. Z Z 0 as grf Z1 z ILLINOIS ENGINEERING EXPERIMENT STATION C C C C C FIG. 40. INFLUENCE LINES BY UNIT LOADS FROM MEASURED REACTIONS. CENTER-SPAN PIER TOP REACTIONS. 20-FooT PIERS ture. A comparison of the light full-line diagrams with the two experi- mentally-determined diagrams shows the agreement between the meas- ured values and the values computed by the elastic theory. These agreements are remarkably close except for small portions of a few diagrams and in no case is the lack of agreement very great. 28. Comparison of Deflection Diagrams and Influence Lines.- If an arch or arch series is made of a homogeneous elastic material, a diagram showing the vertical movement of the load points due to one component of movement (X,Y, or 0) of a terminal, is also an influence line of the reaction at the terminal moved, the movement and reaction being parallel. Tests were made to determine the vertical REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS deflection of all load points due to giving, successively, each of the four terminals (the two abutments and the two pier bases) the fol- lowing movements: (1) rotation without horizontal or vertical trans- lation, (2) horizontal translation without rotation or vertical transla- tion, and (3) vertical translation without rotation or horizontal trans- lation. All of these tests were made at pier heights of 20 ft., 15 ft., and 10 ft., making a total of 36 tests in all. The arch carried the design dead load and the movements were planned to keep the thrust line within the kern at all sections. The position of the terminals was indicated by the Ames dials, the hydro- static gages, and the level bubbles; and the vertical movement of points on the arch rib was measured relative to the top of the I beams in the manner described in Section 19. For the tests to determine the deflection of the rib due to a hori- zontal motion of a terminal, the terminals were all brought to their normal position and the readings of all instruments were recorded. All observations were made in duplicate. The terminal to be moved was then moved horizontally without rotation or vertical translation a distance of approximately 0.100 in. and all the other terminals were brought back to their initial position, the readings of all instruments being again recorded. The terminal that had been moved was then returned to its original position and another complete set of readings was taken. Thus a test for one component of movement of one termi- nal involved the determination of the deflection due to moving the terminal first in one direction and then in the opposite direction. The results of the tests are shown in Figs. 41 to 46 inclusive, Figs. 41 and 42 show the deflection due to settlement, Figs. 43 and 44 the deflection due to spread, and Figs. 45 and 46 the deflection due to ro- tation. In these diagrams the light full lines represent the values com- puted by the elastic theory and the heavy broken lines represent the deflection resulting from movement of the terminals. For both sets of diagrams the deflection has been converted into ordinates of the corresponding influence line. The light broken line and the light dotted line represent the influence lines for reactions as determined by the ap- plication and removal of a unit load of 2000 lb., successively, at the various load points, as described in Sections 26 and 27, the ordinates for the light broken lines having been determined from the movement of the pier tops, and the ordinates for the light dotted lines from the measured reactions at the abutments and pier bases. All influence lines for the vertical shears, given in Figs. 41 and 42, are so nearly alike that the various lines can hardly be distinguished. The influence ILLINOIS ENGINEERING EXPERIMENT STATION REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS ILLINOIS ENGINEERING EXPERIMENT STATION REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS lines for the horizontal reaction of the abutments, given in Fig. 43, are all in fair agreement, but the influence lines for the horizontal reaction at the pier bases, given in Fig. 44, differ materially; the lines obtained from the deflection due to movement of the terminals differ from the lines obtained by the elastic theory and from those obtained by the application and removal of a unit load. The influence lines for mo- ment are given in Figs. 45 and 46. The statement just made relative to the influence lines for horizontal reaction at the pier bases is equally applicable to the influence lines for moment at both the pier bases and the abutments. Although the deflection diagrams differ materially from the influ- ence lines, they have the same general characteristics and, if used as influence lines, would indicate approximately the same distribution of live load for maximum stress. The values of the live-load stress as determined from deflection lines and from influence lines would agree fairly well, but the values of the dead-load stress as determined from the deflection diagrams and from the influence lines might differ greatly. It is not surprising that the deflection diagrams for the con- crete structure differ somewhat from the corresponding influence lines, since the theorem of reciprocal deflections is based upon assumptions as to the properties of the material which, it is known, are not valid for concrete. VII. DESIGN-LOAD TESTS 29. Description of Tests.-Tests were made to determine the re- actions and the location of the thrust line for the design load, consist- ing of the dead load and one live load, shown on Fig. 25. With no load upon the structure the abutments and piers were brought to their normal position as indicated by the scale readings. The level bubbles on the abutments and on the pier tops and pier bases were adjusted so as to be in their mid-position, and a complete set of readings was taken. These readings included the Ames dials, indicating the spans, the hydrostatic gages, indicating the relative height of the piers and abutments, the Ames dials, indicating the vertical position of the load points, the horizontal and vertical scales.at both abutments and pier bases, and the strain gage at a section midway between each pair of adjacent load points, at the intrados and at the extrados. All readings were taken in duplicate, and the piers and abutments were moved and then brought back to their normal position between readings. A complete set of readings, as described above, was taken after the dead load, and again after the live load had been added, all 108 ILLINOIS ENGINEERING EXPERIMENT STATION -c Ca H CD S 0 S S S S H Ca S C S S H Ct 0 C) CD~d -~ otn H 0 H CD -C 0 CD C S C CD S 0 H *1 H S C C) p0 tt- CDMCDCO C0000 03 ý- Cf (N _f 0 ;; RM CD I MCO CD CO 1 I0 I 00 1 I .0 CD N~ OCDN- .-cCDCCC .ccCD- I D I D I D N CS D CeDC N Cq COCN-t OOXCMcct"'- CD t-COCO CM CDiO CTD N~ ~ C C DN ''' SOOO -imGO MCID I I I Ct- CO - CI ho CD w m ttciOCT CDlOCD'Otf'c^ OCD1-c i< QOOc -OC O CDC E- CDNCHCCE4 53 I I (0C<1-t CO CD I I I CDCCO CD W Ot^-Oi CO tO CD0< M iM COC CO C cc ; . ' . '. ' . ' * * * *a C)O ' . "< Hi H'I n C C ci ci ci C 0 c3 0D C 0 Cfl REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS '0-40 0 COO' 000 O It CCC CCmm -^COm *- 04 4 0C 1'0 - 1-'0 0 * N 0 4 q 4mi 't040. c-q0 . . . lt-i'i. C'CCC t.-COPZ * 00OS c i . - GOý 10 11: 110 CO . . . mC 0m - 4--. . N . . . * 4404 ( t'-"-^® 00140 ** mm400 = to d4 40 : : e! oo i 117 r- 0 11 * . . cl0< N O iC cOI- .... . . :oor icsot cý ® . cý . c.0,0 .0 o .-i C C *0 , Oýt Q *tf^. 1ý lý Cý 1ý . . . 404040 o00440 400.-f' mcomt mMC C^Ci . . . a,0 00 I'0 N m 404 C, 1ý :ý ; cý 1ý : : '. : 0.04 . 0 4 'IV .0. . fýl3ý e3t» c8i t» efl Q Q 0-04 0-400 -4 ca ^ 9Q -0- to > 4) .-14 < ,.1 4444 4444 c4)K 4 ) £ e S u r a g00 ~*-Vesa'd Loaa *~~- -- -^'5ea'adLocad/ /as~ ,L/ive Lo'ad'\ FIG. 48. VERTICAL DEFLECTION OF ARCH Axis. DESIGN-LOAD TEST The diagrams of Fig. 47 indicate that the position of the thrust line, a factor which, with the magnitude of the thrust, largely deter- mines the stress in the rib, is determined very accurately by the elastic theory even though the modulus of elasticity of the concrete varies through a wide range over the length of the structure. 32. Deflection Due to Load.-The vertical movement of the load points due to the load increments in the design-load tests are given in Fig. 48, for structures having pier heights of 20 ft., 15 ft., and 10 ft., respectively. The east span apparently deflected less under the dead load than the center and west spans. The top of the east pier also rotated slightly, the top tipping east. Both of these phenomena might be due to the fact that the modulus of elasticity of the concrete, as given in Table 54, was greater for the east span than for the other two spans. VIII. TEST TO FAILURE 33. Description of Test.-The load-carrying capacity of the struc- ture was determined when the pier bases were fixed at a distance 20 feet below the springing of the arch. The observations made in ad- I o0/ Y)0. 4 REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS dition to those necessary to insure that all terminals were returned to their normal position were as follows: the scale reading at each pier base and each abutment; the strain in the concrete at sections mid- way between each pair of adjacent load points; the rotation and hori- zontal movement of the pier tops; and the vertical movement of all load points. A complete set of readings was taken when the structure carried each of the following loads: (1) its own weight, the zero or basic reading; (2) the design dead load; (3) the design dead load plus one live load; (4) the design dead load plus two live loads; (5) the design dead load plus three live loads; and (6) the design dead load plus four live loads. The arch was subjected to the design dead load plus four and one-half live loads, which it was able to carry, but no read- ings were taken, although the terminals were returned to their normal position. When the load was increased to five live loads (in addition to the dead load) the arch failed at load point C4. It would seem, therefore, that the load-carrying capacity of the structure was between four and one-half and five times the design live load plus the design dead load. The rib was cracked in many places in each of the three spans, but for all sections except at C4 the cracks, although more than hair cracks, were small, and there was no indication of impending failure at any other section. There were also small cracks in the piers near the bases. The failure was a typical flexural failure for a concrete beam; a large crack opened on one side and the concrete spalled on the opposite side. These features of the failure are shown on Fig. 49. 34. Position of Thrust Line.-The position of the thrust line for the various loads is shown in Fig. 50. In this figure the full lines are from the elastic theory and the broken lines from the measured re- actions. The latter were constructed from the east abutment to the east pier and from the west abutment to the east pier, thereby elimi- nating the use of the horizontal reaction of the base of the east pier, which was found to be in error for all loads above the dead load plus one live load. The trapezoids of strain are shown in the figure. The small circles representing the centers of pressure as determined from the strain are given only at sections where the tension was not great enough to crack the concrete. The fact that, even for loads near the ultimate, the positions of the thrust line determined from the measured reactions and by the ILLINOIS ENGINEERING EXPERIMENT STATION FIG. 49. FAILURE OF ARCH RIB AT C4 East CZ C3 C'? C5 C6 Thr~yst L,~e from E/ast,~ Theor~,, ~ - - - - -. Thr~isf Line from Aleawred f'~-c/,'ons Q - Ce?2ter of Pressure ~~om Alea'sured S/rmns - Sirafr' Dia'~'ram from NeGsured ~5~ra'k~s L-~-4-~-~-.-~-J $co'/e of Sfrez'n 1/? - 0 5 /0 Ten - Thous~rnd,'hs o( ~n fr.'.per/n. FIa. 50. THRUST LINES FOB TEST TO FAILURE eO-Foot P/ers WesP REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS a. 0. % 4 2 0 0.4 0.6 .o 0.8 I.O 1.2 - I Two Live Load0o Z I .I I i H--i inrl Io -r ee/ Z IkV< /Z oa 'iv, //ini" h.. 7- TI k :7 7 / - /i - I -J FIG. 51. DEFLECTION OF ARCH Axis DUE TO LIVE LOAD. TEST TO FAILURE elastic theory agree so well, and the fact that the centers of pressure determined from the measured strain fall upon these thrust lines, would apparently indicate that the position of the thrust line is given accurately by the elastic theory even though considerable portions of the rib contain many tension cracks. 35. Deflection Due to Load.-The vertical movement of the load points is shown by the diagrams of Fig. 51. The consistency of the deflections at the various loads is especially interesting inasmuch as the rib contained some cracks when subjected to the dead load plus two live loads and there were a large number of cracks when the rib was subjected to three and four live loads. These cracks increased the magnitude of the deflection, but did not change the general shape of *1 I N. n n ? Ive Load- - 01 2O-/0oo/ A -A/ £>/A /ers 7 N ee - S -b ?Z; - - - -- - - o ILLINOIS ENGINEERING EXPERIMENT STATION REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS the diagrams. The maximum downward movement due to four live loads was at C4 and was 1.1 in.; the maximum upward movement occurred at W5 and was 0.50 in. The secondary stress at C4 due to the deflection of the arch axis produced by the design load was 78 lbs. per sq. in. This secondary stress was compression at the extrados, the same as the primary stress. 36. Unit Strength Developed by Concrete in Arch Rib. -The structure carried the design dead load plus four and one-half times the design live load, but failed when another one-half live load was added. From the values in Table 39 the unit stress due to the dead load plus four and one-half live loads is 3812 lb. per sq. in. This value is based upon the elastic theory, and is subject to two corrections opposite in sense. The secondary stress due to the deflection of the rib produces, at the ultimate load, a compression at the extrados at C4 of about 400 lb. per sq. in. Its omission is offset by the fact that the true position of the thrust line at C4 is somewhat below the po- sition determined by the elastic theory. The strength of the concrete as determined from control cylinders, given in Table 30, is 3310 lb. per sq. in. IX. DISCUSSION OF RESULTS 37. Influence of Variation in Modulus of Elasticity of Concrete upon Stress Distribution.-The structure was analyzed twice, the first time on the basis that E = 2 000 000 lb. per sq. in. (n = 15) and the second time on the basis that E = 3 333 000 lb. per sq. in.* (n ? 9). Influence lines for the reactions at the springings are given in Figs. 52 and 53. The full-line diagrams are based upon n= 15, and the broken-line diagrams upon the basis that n = 9. These diagrams are for structures having 20-foot piers. Similar diagrams were drawn for structures having 15-foot and 10-foot piers. The influence lines for the vertical shear were also drawn, but the two sets of lines, one for n =9 and the other for n= 15, were in such close agreement that they are not distinguishable from each other. For all structures, changing n from 15 to 9 had very little effect upon the reactions. For the structure tested, the modulus of elasticity of the concrete was very much less at the crown than near the ends, for all spans. For the west span, the modulus for the west half was much less than that for the east half. Moreover, the average modulus for the east span was about 40 per cent greater than that for the center and west *See note page 66. ILLINOIS ENGINEERING EXPERIMENT STATION spans. These variations in the modulus do not appear to have in- fluenced appreciably the stress due to the live load. The effect of the variations upon the dead-load stress is not apparent. 38. Accuracy of Tests.-There are a number of considerations which indicate the accuracy of the tests. In general, points represent- ing experimental results fall on smooth curves. Tests on the opposite ends of the symmetrical structure give remarkably consistent results. Influence ordinates determined from movements of pier tops and from measured reactions are in close agreement. Centers of pressure deter- mined from measured strain fall on thrust lines determined from measured reactions. All of these considerations support the belief that the experimental data have been determined with a high degree of accuracy. 39. Verification of Elastic Theory.-The diagrams of Figs. 29 to 40 inclusive that have ordinates obtained by the application of a unit load agree closely with the corresponding diagrams obtained by the elastic theory. This would apparently indicate that the errors in the assumptions upon which the analysis is based do not greatly affect the results of the analysis. The diagrams of Fig. 50 apparently indi- cate that this statement holds even though the thrust line is so far outside of the kern as to crack the rib. In view of these results it ap- pears justifiable in analyzing a multiple-span arch on elastic piers to assume that E has the same value at all sections and at all stresses, tension or compression, and to compute the moment of inertia. of a section on the basis that concrete takes tension. Moreover, the value of E used in the analysis may vary considerably from the true value without seriously affecting the reactions due to loads. 40. Influence of Pier Deflection upon Stress Distribution. -The diagrams of Fig. 31 indicate that the influence ordinates for each of the components (H, V, and M) of the reaction at the springing is greatly affected by the elastic deformation of the pier. The stress at a given section, however, is a function of all three components of the reaction and, except for sections of the rib near the springing, is not necessarily so greatly affected as the individual components. This is apparent from the influence lines for stress at C4 in Fig. 24. The maximum stress in the three-span series due to the design live load, given in Table 39, is 736 lb. per sq. in. at the extrados on the section through C4; and the corresponding stress in a single span with fixed ends, given in Table 40, is 543 lb. per sq. in. at the extrados on the REINFORCED CONCRETE ARCH RIBS ON SLENDER PIERS TABLE 55 MAXIMUM STRESS AT EAST SPRINGING OF CENTER SPAN DUE TO DESIGN -LOAD, CALCULATED BY ELASTIC THEORY Stress Load Pier Height lb. per sq. in. Load ft. Extrados Intrados Dead load.............. ............... All heights -342 -128 Three Span Series on Elastic Piers Live load............. ....... ............ 20 +359 -427 15 +334 -401 10 -228 +186 Dead load plus live load .................. 20 + 17 -555 15 + 8 -529 10 -570 + 58 Center Span with Ends of Rib Fixed Live load................. .............. ............. . - 401 + 345 Dead load plus live load .................. ............ -743 +217 DEsIGN LOADS Load, in lb., at Load Point No. Load Pier Height ft. 1 2 3 4 5 6 7 8 Dead load ......... All heights 10 000 6600 5900 4700 4700 5900 6600 10 000 Live load.......... 20 960 2760 960 939 302 0 0 0 15 960 2760 960 875 161 0 0 0 10 0 0 0 520 960 2760 960 960 Fixed Ends 0 0 221 911 960 2760 960 960 Computations are based on n = 9. Stresses are computed from moment and thrust on assump- tion that concrete takes tension. A plus (+) sign indicates a tensile stress. section through C3. That is, for the design live load the stress near the crown is increased about 32 per cent by the flexure of the pier. This is for a structure having 20-foot piers. For a structure having 10-foot piers the increase is only 20 per cent. Table 55 shows the maximum design load stress at the east spring- ing of the center span, both when it is supported on elastic piers and when it is fixed at the two ends. The maximum stress when the span is supported on 20-foot piers occurs at the intrados, and is 555 lb. per sq. in. The maximum stress for the span with fixed ends is at the extrados, and is 743 lb. per sq. in. That is, for this arch, the flexure of the pier increases the maximum stress near the crown and decreases the maximum stress near the springing of the center span. ILLINOIS ENGINEERING EXPERIMENT STATION Although the flexure of the piers decreases the negative moment at the springing of the center span, it increases the nagative moment at the abutment of the end span. This becomes apparent upon compar- ing the influence ordinates for the moment at A of AB in Table 31 with the corresponding functions in Table 3. X. SUMMARY OF RESULTS * 41. Summary of Results.-The tests described in this report ap- parently justify the following conclusions relative to the behavior of a three-span series of arch ribs on high piers: (1) The elastic theory based upon the usual assumptions gives values for the moment, thrust, and shear at various sections that agree with the measured values within the tolerance of the tests. (2) In analyzing a multiple-span arch series on elastic piers it may be assumed that E has the same value at all sections and at all stresses, tension or compression, and the moment of inertia of a sec- tion may be computed on the basis that concrete takes tension. More- over, the value of E used in the analysis may differ considerably from the true value without seriously affecting the reactions due to loads. (3) For the structure tested, the flexure of the piers increased the maximum live-load stress in the rib near the crown of the center span and near the abutment of the end span, but decreased the correspond- ing function for the springing of the center span. (4) The maximum unit compression due to design load (dead load plus live load) was 13 per cent greater for the three-span series on 20-foot piers than for a similar single span with fixed ends. (5) Considerable cracking of the arch rib did not greatly alter the position or magnitude of the thrust. (6) The concrete in the arch developed approximately the same unit stress as the same concrete in 6-in. by 12-in control cylinders. RECENT PUBLICATIONS OF THE ENGINEERING EXPERIMENT STATIONt Bulletin No. 222. Flow of Liquids in Pipes of Circular and Annular Cross- Sections, by Alonzo P. Kratz, Horace J. Macintire, and Richard E. Gould. 1931. Fifteen cents. Bulletin No. 223. 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