THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING LOSS OF PRESTRESS IN PRETENSIONED PRESTRESSED CONCRETE BEAMS Ramesh M. Patel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Science in the University of Michigan Department of Civil Engineering 1964 February, 1964 IP-659

ACKNOWLEDGMENT The author wishes to express his sincere appreciation to Professor Leo M. Legatski, the chairman of his committee, for suggesting the subject of the dissertation and for his continuous encouragement and valuable suggestions. The author is also grateful to the other members of his committee, Professor Frank Legg, especially for his help during the making of the beams and cylinders for the experiment, and to Professors Lester V. Colwell, Robert M. Haythornthwaite and Lawrence C. Maugh for their continuous encouragement and valuable suggestions. The experimental study of this investigation was made in the Structural Laboratory of the University of Michigan and was partly financed by that Department. The prestressing cables used in the investigation were donated by Lamar Pipe and Tile Companyo The springs used in the testing frame were donated by Eaton Manufacturing Company, Spring Division. The personnel of the Michigan State Highway Laboratory gave a helping hand during the making of test specimens. Mr. George Geisendorfer of the Structural Laboratory helped untiringly in making test specimens. The initial draft of this dissertation was typed by Miss Reta Teachout. The final typing and reproduction were done by the Collegeof Engineering Industry Program of the University of Michigan. To all persons who helped in the preparation of this dissertation, the author is sincerely thankful. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENT..............................................ii LIST OF TABLES.................................................. iv LIST OF FIGURES.................................................. v LIST OF APPENDICES............................................... viii NOMENCLATURE..........i.................................ix I INTRODUCTION.............................. 1 II OBJECTIVE AND SCOPE................................... 6 III NATURE OF PRESTRESS LOSSES............................. 7 Elastic Shortening of the Concrete......................... 7 Shrinkage of Concrete...................................... 10 Creep of Concrete.................................. 10 Creep of the Prestressing Tendon.......................... 13 IV INSTRUMENTATION............................................... 15 Gage Points in Beams.................................. 15 Tensioning the Strands.22 Loadg e Po in Cylinders.................................. 25 Gage Postin ts on Cylnde s.................................. 2 Tensioning the Strands..................................... 22 Loading the Cylinders..................................... 25 Testing Cylinders........................................ 28 Measuring Deflection................................ 28 Testing Springs..................................... 28 Calibrating Dynamometers or Load Cells..................... 28 V TEST SPECIMENS.................................... 31 VI TEST RESULTS AND DISCUSSION...........................38 Concrete......................................... 38 Modulus of Elasticity of Concrete Cylinders................ 38 Elastic Loss of Prestress................................ 41 Shrinkage of Concrete...................................... 54 Creep of Concrete....................................... 61 Anchorage Length of Strands and Bond of Strands to Concrete 73 Creep of Strand..........................8.............. 81 VII CONCLUSIONS AND RECOMMENDATIONS.............................. 84 BIBLIOGRAPHY................................................ 87 iii

LIST OF TABLES Table Page 6.1 Modulus of Elasticity of Concrete....... o....,.. o...... 40 6.2 Measured Average Strain Readings at Transfer in Full Anchorage Zone.......................................... 40 6.3 Concrete Stresses at c.g.s. Immediately After Transfer of Prestress.................................................... 55 6.4 Deflection of Beam at Transfer and Growth of Camber............ 55 6.5 Measured Elastic, Creep and Shrinkage Strains in Beams and Cylinders...................................................... 69 6.6 Ratio (R): of Measured Creep Strain to Measured Elastic Strain for Beams and Cylinders........................................ 69 6.7 Ratio of (R)* for Beam to (R:(- for Cylinders....0.............. 71 iv

LIST OF FIGURES Figure Page 4.1 Whittemore Gages (Two Inches and Ten Inches Long)...... 16 4.2 Supreme Strand Chuck, Brass Plug, Rubber Cork and Strand with Glued Brass Plug........................... 16 4.3 Position of Gauge Points on Beams and Cylinders........ 17 4.4 Beams in Steel Frames.................................. 21 4.5 Tensioning Device...................................... 23 4.6 Hydraulic Jack, Load Cell, Shims and Strand Chuck Arrangement Used in Prestressing the Strand........... 24 4.7 Shim................................................... 25 4.8 Test Rig for Determining Creep of Concrete Under Load.. 26 4.9 Calibration of Dynamometer No. 1....................... 29 4.10 Calibration of Dynamometer No. 3....................... 30 5.1 Modulus of Elasticity - 7/16" t - Seven-Wire Strand.... 34 6.1 Compressometer........................................ 39 6.2 Concrete - Modulus of Elasticity - Test Series No. 1 - Age 25-1/2 Hours................ o................... 42 6.3 Concrete - Modulus of Elasticity - Test Series No. 1 - Age 28 Days........................................... 43 6.4 Concrete - Modulus of Elasticity - Test Series No. 1 - Age 25-1/2 Hours...................................... 44 6.5 Concrete - Modulus of Elasticity - Test Series No. 1 - Age 28 Days.......................................... 45 6.6 Concrete - Modulus of Elasticity - Test Series No. 2 - Age 27 Hours.......................................... 46 6.7 Concrete - Modulus of Elasticity - Test Series No. 2 - Age 28 Days.......................................... 47 6.8 Concrete - Modulus of Elasticity - Test Series No. 2 - Age 27 Hours........................................... 48 v

Figure 6.9 6.10o 6,11 6.12 6.135 6.14 LIST OF FIGURES (CONT'D) Concrete- Modulus of Elasticity - Test Series No. 2 Age 28 Days....................,.............. Concrete - Modulus of Elasticity - Test Series No. 153 Age 26 Hours....,................*.............. Concrete -Modulus of Elasticity -Test Series No, 153 Concrete -Modulus of Elasticity -Test Series No, 153 Age 26 Hours..........,...................... Concrete - Modulus of Elasticity- Test Series No. 153 Age 28 Days..............,.,................ Shrinkage Strain vs. Time - Test Series No, 1...... Page 49 50 51 52 515 57 6.15 6,16 6.17 6,18 6.19 6.20 6.21 6,22 6.235 Shrinkage Strain vs. Time- Test Series No. No. 2...,.... 15,....... Shrinkage Strain Creep Strain vs. Creep Strain vs. Creep Strain vs. Creep Strain vs. Creep Strain vs. Creep Strain vs. Creep Strain vs. vs. Time - Test Series. 0 0 a. a Time - Test Series No. 1........ Time Time Time Time Time Time - Test Series No. - Test Series No, - Test Series No. - Test Series No. - Test Series No. Curve Us ing Shank 1............. 2............. 2............. 3............. 3............. Formula....... 58 59 65 66 67 68 72 6.24 6.25 6.26 6.27 Strand Stress as Computed by "Effective Modulus?? Method Strand Method Strand - Test Stress - Test Stress Series No. 1... as Computed by Series No. 2... as Computed by "Effective. Modulus??.0... 00. "?Effective Modulus?? 74 75 76.. 78 Method - Test Series No. 15,.................. Anchorage of 7/16?? 0 Seven-Wire Uncoated Strand - Test Series No. 1............,.................... vi

LIST OF FIGURES (CONT'D) Figure Page 6.28 Anchorage of 7/16" 0 Seven-Wire Uncoated Strand - Test Series No. 2.................................... 79 6.29 Anchorage of 7/16" 0 Seven-Wire Uncoated Strand - Test Series No. 3................................... 80 vii

LIST OF APPENDICES Appendix Page A DESIGN OF BEAM..,.................... o,,,... o..ooo o. 92 B MEASURED VERSUS CALCULATED ELASTIC LOSS OF BEAMS.......... 95 C WHITTEMORE GAGE DATA SHEET CREEP AND SHRINKAGE............ 98 D DERIVATION OF FORMULA FOR FINAL STRESS IN STEEL (STRAND) IN BEAM AFTER LOSSES ACCORDING TO "EFFECTIVE MODULUS" METHOD......................o....................... 100 E CALCULATIONS FOR FINAL STRESS IN STRAND AS PLOTTED ON FIGURES (6.24) THROJGH (6.26).......,,.............,..o.. 102 F CALCULATED STRESSES AND STRAINS IN BEAMS AND CYLINDERS AT TRANSFER OF STRESS..........o...... o..,o......... o,,, o, 104 G TYPICAL CALCULATION OF CREEP STRAINS USING SHANK FORMULAo... 106 H CALCULATION OF FINAL STRESS AS RECOMMENDED................. 108 viii

NOMENCLATURE Ac Net area of concrete section of beam Ag Gross area of section of beam As Area of steel At Transformed area of section of beam et Eccentricity of c.g.s. from c.g. of section E Modulus of elasticity of steel Ec Modulus of elasticity of concrete (ot Stress in strand just before transfer ai Stress in strand just after transfer n E /E Fo Total force in strands when tensioned Fi Total force in strands just after transfer ft Stress in top fibers of concrete in a beam f, Stress in bottom fibers of concrete in a beam fs Stress in concrete at the c.g.s. in a beam C1 Distance between top fiber and neutral axis of beam C2 Distance between bottom fiber and neutral axis of beam P As/Ac a Final stress in strand after losses F Final force in strands after losses s Shrinkage strain c Creep strain It Moment of inertia of the transformed section ix

f. Fi/Ac M Moment due to dead load Others are defined as they appear Others are defined as they appear~ x

CHAPTER I INTRODUCTION In December of 1951, pretensioning of reinforcing steel in concrete was first used in the United States when a 24-foot span bridge was erected in Pennsylvania. Since then the use of pretensioned prestressed concrete members has increased rapidly. As of December 31, 1959, only 16 prestressing plants were older than seven years but 104 had been in operation less than three years. As of August 1, 1960, the total number of plants in the United States was approximately 300. In these plants 33 per cent of the output was in precast prestressed beams. It is essential that the structural designer should fully know the properties of materials and utilize them in the most effective way possible. As yet the behavior of prestressed concrete members has not yet been fully ascertained. In order to design safe and economical pretensioned prestressed concrete beams, it is necessary to know the prestress losses due to elastic action, creep and shrinkage of concrete and due to creep of the steel. Although many structures have been built in the last ten years, there is no agreement as to the determination of prestress losses. Pretensioned bonded prestressed concrete beams are studied in this investigation. To determine the effective prestress, four major losses are deducted from the original prestresso They are: (1) Loss due to elastic shortening of concrete. (2) Shrinkage of concrete. (3) Creep of concrete. (4) Creep of the tendons. -1 -

-2 - Some of the basic terminology may be defined as follows: Prestressed concrete member: A prestressed concrete member is a structural member in which a permanent predetermined force system is established in such a manner that the stresses resulting from any anticipated condition of external loading are counteracted to a desired degree. Tendon: A tendon is a steel (cable, wire or bar) used to impart prestress to the concrete. Bonded tendons: The tendons are "Bonded tendons" if bonded to the concrete either directly or through grouting throughout its length to the surrounding concrete. Unbonded tendons: The tendons are "Unbonded tendons" if free to move relative to the surrounding concrete, Pretensioning: An application of the prestress to the concrete in which the tendons are tensioned before the concrete is placedo Post-tensioning: An application of the prestress to the concrete in which the tendons are tensioned after the concrete is placed and curedo Jacking force: A jacking force is the temporary force exerted by the jacking device in prestressing the tendons. Transfer: Transfer is the operation of transferring the tendon force from the fixed anchorage to the concrete, Effective prestress: Effective prestress is the stress remaining in the tendons after all losses have occurred, excluding the effect of superimposed loads, but including effect of weight of membero

-3 - Loss due to elastic shortening of concrete: At transfer, an elastic shortening of the concrete takes place which results in shortening of the tendon and thus loss in stress in the tendonso Shrinkage of concrete: Shrinkage of concrete is contraction of con~ crete due to a great variety of causes other than the application of loado Creep of concrete: Creep of concrete is the time dependent deformation which occurs in concrete under sustained, load as distinguished from the elastic deformation which occurs when the load is first appliedo Creep of tendon: Creep of tendon is an inelastic deformation of the steel under stress. Much work(6,7,11,16,41,47) has been done on creep and shrinkage of ordinary sand and gravel concrete, normally used for reinforced. concrete, under sustained loads. But data have been lacking in creep and shrinkage of concrete normally used for prestressed worko Very few investigati.ons(40453) of creep of high strength tendons have been made. Most of the work on creep and shrinkage measurements on concrete has been on concentrically loaded cylinderso Some of the investigations reported are by Staley and Peabody(9) in 1946 on prisms, by Magnel(lO) in 1947, by Shank(ll) in 1949 on cylinders, by Washa and Fluck(12) in 1950 on cylinders, by Fo Eugene Seamen(l6) in 1957 on cylinders, by Ross(l7) in 1958 on cylinders. Inge Lyse(61) reported on creep and shrinkage on full sized beams in 1958. Also Cernica and Charignon(44) studied creep and shrinkage behavior of beams in 1961. Alan Mattock(45) reported on creep and shrinkage studies on precast-prestressed concrete bridges in 1961.

-4 - Recommendations for estimating prestress losses differ in different countries and also from time to time. Magnel recommended in 1947 that the loss due to creep and shrinkage of concrete in design be taken as 15 per cent of the initial prestressing force for concrete of high strength (greater than 5000 psi) made from sand and crushed stone. The ACI-ASCE Joint Committee 323 (1956) Report recommended that the loss in steel stress not including friction may be assumed to be 355000 psi in the case of pretensioning and 25,000 psi in the case of post-tensioning. Alternatively, the loss in steel stress should be computed from the elastic, creep and shrinkage strains of concrete and the relaxation of steel stress, when individual losses can be predicted with reasonable accuracy. The 1963 revision of the ACI Building Code requires the designer to deduct losses due to elastic shortening, creep and shrinkage of concrete and relaxation of steel stress from the original prestress to estimate the effective prestress. No guides are set for the designer to estimate losses. A review prepared by the Prestressed Concrete Development Committee(46) suggested overall losses in the steel, expressed as a per cent of initial stress, be 20 per cent for post-tensioning and 25 to 30 per cent for pretensioning. For pretensioning, they also suggested using a strain of 300 x 10-6 inches/in, for shrinkage and 0.33 x 10-6 inches/in. for creep for each lb/sq.in. of applied stress for concrete of high strength. Most of the data available for creep and shrinkage of concrete has been derived from tests of concentrically loaded concrete cylinders

-5 - without steel, Since actual beams contain steel, there is an interaction between the concrete and steel so as to modify the creep and shrinkage behavior of concrete in the beams. The strain measurements are made on the surface of the beams assuming perfect bond of the cables. This has been questioned but data are lacking in the study of difference in strain readings between the cable and surface of the concrete beam. Also, the effect of creep of steel on creep and shrinkage of concrete and vice versa in the beams are not known from the cylinder tests. An investigation is needed to compare the estimated losses as found from cylinder measurements with that found on beams,

CHAPTER II OBJECTIVES AND SCOPE The purpose of this investigation is: (1) to study experimentally the four major factors which contribute to prestress losses in pretensioned eccentrically prestressed concrete beams; (2) to study experimentally the difference in strain reductions, if any, on cables and the surface of the concrete beams; (3) to compare the losses as observed for the beams with those found from loading cylinders concentrically; (4) to compare the losses as observed for the beams with current design practices; This investigation is limited to: (1) Members with straight tendons. (2) Members with one eccentricity of tendons. (3) One kind. of concrete mix. (4) Loading of cylinders at stress at the level of cable. Also as the specimens were stored in the laboratory without temperature or humidity control, these effects were not studied. -6 -

CHAPTER III NATURE OF PRESTRESS LOSSES Four major losses that affect the design of prestressed concrete were defined in the previous chapter and will now be examined in more detail. Elastic Shortening of the Concrete When concrete is stressed in compression, it shortens. At transfer of prestress, an elastic shortening of the concrete takes place. Due to bonding of tendon and concrete, the steel also shortens. This results in the loss of stress in the tendon. The loss of tension in the tendons upon transfer is computed upon the assumption that the unit strain in the tendon at the position of the center of gravity of tendons is equal to that in concrete at the same position. The effectiveness of the actual bond is reported to depend mainly on the following factors: the quality of the concrete, the working of the concrete in the molds, its strength at transfer, the type of curing before and after transfer, the level of prestress of the concrete, and the diameter and the surface properties of the tendons and how they are distributed in the cross section. If slippage occurs in the tendons at transfer, the strain measurements on the surface of beams will not adequately reflect this. The shrinkage in the concrete itself during curing and a small amount of creep may be included in the strain measurements immediately after transfer. -7 -

-8 - The strain measured on cylinders under concentric loads includes some creep and shrinkage as it is difficult to load cylinders and then measure strain reductions instantaneously. As concrete shortens under the application of load an amount proportional to the load and the elastic modulus, elastic deformation of the concrete can be calculated from elastic theory. Here it is assumed the bond of concrete to steel is complete. Then the two conditions can be applied: (1) Force in tendons is equal to force in concrete; and (2) strain in concrete at c.g.s. is equal to strain in the tendons at CogoSo Condition (1): Force in Steel = Force in Concrete s Fi Fie(C2-Cl) Mg(ClC2) A (31) AAIt 2It C (see Appendix A) Let e(C2-C9)Ac M Mg(C1-C2) 2 2It K1 =1+ i1 K2 = 2/p P1 = P/K1 p = As/Ac Then oiAs = (fi + fill + 12)Ac = fiAcKl + A2Ac

-9 - Therefore Uji L-L K1 + K2 p _=Li + K2 P1 (3.2) Condition (2): Compressive strain in the concrete = change of strain in the steel. F. Fie2 M A I EC Substituting =1+Ace2 It,04=It aot - G (3p3) in Equation (3-3) fi-1, n- n14 = &rot - fi K2 fi(1 + plni35) = pl(cot-K2) + ni4p1 or fi = pl(aot~-K2) + p~ni4 1 + nK5.9 K (35)) G1 ot K2 + n. Y, or 1+nK= + If we neglect the effect of dead load, i4 = K2 = 0 and Got C =1 + nK5 (356) For a concentrically prestressed member the eccentricity "e" is zero and, neglecting dead load, Equation (5) reduces to cyot ( 7 1 + np,

-10 - Elastic Loss in Cable = aot - ai (308) Equations (355), (3.6) and (357) provide a direct method of estimating the steel stress in the tendons after the elastic loss. It is necessary to know the stress in the tendons just before the transfer of stress to the concrete, ioe, it is necessary to know loss of prestress due to creep of steel between anchorage and cutting of the strandso Shrinkage of Concrete Shrinkage of concrete was previously defined as contraction of concrete due to great variety of causes other than the application of loado The phenomenon of drying shrinkage has been discussed by Davis and Troxello(l3) According to them, shrinkage of concrete is d.ue to the loss of adsorbed water through evaporation from the hardened cement gel. The amount of shrinkage depends on many factors such as water-cement ratio, chemical composition of the cement, maximum size, grading and mineral character of the aggregate, size of the member and rate of drying. Much experimental data is available on shrinkage of various mixes and the total shrinkage varies from 0 to 800 x 10-6 inches/in. (In design of Beams, for the concrete normally used in prestressed work, values of 200 to 300 x 10-6 inches/in. are used for the shrinkage strain in pretensioning,) Creep of Concrete Creep of the concrete is the time-dependent deformation under sustained load. Many theories of creep in concrete have been advanced te explain the mechanism of creep(5 l5) but there appears to be no conclusive evidence as to the exact mechanism of creepo The reader is

-11 - referred to a paper "Theories of Creep in Concrete" by Ao Mo Neville(14) for an excellent comparison of different theories of creep of concrete. Sustained load causes the concrete to creep, this creep being due to: (1) Viscous flow of the cement paste (Lorman, Freudenthal, etc. ) (2) Consolidation due to seepage, or flow of adsorbed water from the cement gel due to applied pressure (Lorman and Ross)o (3) "Delayed elasticity" due to the cement paste acting as a restraint on the elastic deformation of the skeleton formed by the aggregate and the cement crystals in the manner of "Kelvin's Sponge" (Cowan, Arnan, Reiner and Teinowitz) (48) (4) Permanent deformation caused by localized fracture. Creep of the concrete is reported to depend on many variables such as magnitude of sustained load; age at which the sustained load is applied; size of the member; composition of concrete and fineness of cement used in it; size, grading and mineral character of the aggregates; amount of pozzolans used; water-cement ratio; volume of cement paste; temperature and humidity during the curing period prior to loading; rate of drying during the loading period; and temperature during the loading period. An article "Creep of Plain and Reinforced Concrete" by Fluck and Washa(47) discusses in general regarding the creep behavior and the factors influencing such behavior of plain and reinforced and prestressed concrete, It lists many references on laboratory and field tests of

-12 - creep of concreteo Very little information is currently available on creep of concrete under normal prestressed conditions, ioeo, with triangular stress distribution and a changing prestress forceo This series of tests would yield. creep data for ore stress distribution. Many mathematical relationships(40) to predict creep of concrete subjected to constant and also to variable loading have been developed but none have been generally accepted. All require some constants to be determined from laboratory tests and some are too cumbersome to be used in practical design. Also, in all structures in use, stress in the concrete varies with time, and the problem becomes that of predicting the creep under varying stress from the results of tests made under constant stresso Three methods of calculating for the effects of creep and shrinkage, "Effective Modulus," "Rate of Creep" and "Method of Superposition" are discussed by Ross.(17) The "Effective Modulus" Method has been used to predict final stress in the cableo (See Appendix D,) Creep strain as measured in the present investigation includes flow strains due to the load and whatever difference there may be between shrinkage of non-stressed and stressed specimenso The rate of creep decreases with time and creep approaches a limit under constant load. The time to approach a limit may vary considerably depending on many variables such as level of stress and type of aggregateo Also, it is noted(ll) that the change in the rate of creep seems to be uniform up to 3/4 or more of the ultimate strength (well through the working range) and that the change is abrupt near the "true" ultimate strength.

-13 - In pretensioning, the loss of stress in the tendon resulting from the creep of concrete is normally assumed to be 150% of the total stress loss due to elastic action. But the creep strain may vary from 100% to 300% of the elastic deformation of the concrete, depending on many factors, such as strength of concrete, temperature and humidity of environment, level of stresso Creep of the Prestressing Tendon Creep of the tendons is an inelastic deformation of the steel dependent on time and initial prestresso The loss of stress occurs without strain reduction. The mechanism of creep has been dealt with elsewhere (5,14) As in creep of concrete, there is no general agreement on one theory. The creep depends on various factors such as stress level of the steel, type of steel, chemical composition, processing and final treatment such as drawn or stress relieved. Some tests on creep of steel wire have been reported.(40~43) Initially the tests measured creep at constant loadso As the development of prestressed concrete progressed, creep at constant strain was suggested as it was a good duplication of actual conditions prevailing in prestressed concrete. Normally the strands are tensioned with a stress level of about 70% of the ultimate tensile strength. In an actual prestressed concrete member, the deformation of the concrete reduces the drop in stress attributable to relaxation. Tests made by Niels Thlorsen(25) showed stress relaxation in 10 days of 5-8 per cent of the initial prestress for stress

levels equal to 70 per cent of the actual ultimate tensile strength for various smooth wireso Relaxation for 7-wire strands are only slightly higher than for smooth wires. The stress-relieved strand would creep considerably more and is not considered desirable. After transfer of prestress subsequent stress relaxation loss is insignificant. In prestress concrete plants, the transfer of stress from steel to concrete takes place within 24 hours after thetensioning of the strands for pretensioned work in order to make maximum use of the prestress beds. As the stress relaxation after transfer of prestress is much less, it seems that there should be less stress relaxation in steel stress for this kind of worko

CHAPTER IV INSTRUMENTATION For strain measurements on beams and cylinders, Whittemore Mechanical Strain Gages, one with a ten inch gage length and the other with a two inch gage length were used (Figure 41).o For strain measurements on cables used. in beams, another ten inch Whittemore Gage was usedo Gage point locations on beams and cylinders are as shown in Figure 4.53 Gage Points in Beams The gage points consisted of 3/8" diao brass plugs tapered to 1/4" diao in 1/4" length as shown in Figure 4,2 for beams and cables. Gage points were located on both sides of the beams at the elevation of the center of gravity of tendons. For prestressed beams, gage points were 2" Co/c, for 24" on each end to facilitate in obtaining anchorage lengtho For the rest of the beam the plugs were 10" Co/Co For shrinkage beams, the gage points were 10" c./co throughout (Figure 4.3)o Numbers on the beams represent the average distance from the end of the beam. The gage points on the cables were only near the center of the specimens. For obtaining reliable and. accurate strain readings with gage points it is necessary that the gage point holes in the plugs are perpendicular to the plane of the beam surface and that the gage point holes are reamed to seat the points of the Whittemore Gage properlyo In Test Series 1, the following procedure was followed to obtain the gage points in the beams: (1) Both the surfaces to be glued - brass plug and wooden form - were cleanedo -15 -

2" Whittemaore~ gauge - 10 stan iara -W Bar 2 Standard Ba: Figuire 41k~. WAhi~ttemore Gages (Two Inches and Ten Inches Long). r ~7/16 11Strand Brass Plug Gluea.to. tran ~Housing.Rubber Cork with:Groove and Center 4Ho1ge R ous ing. C over Superior PI'rand' --.Chuck. Figure 41k2. Supreme Strand Chuck,, Brass Plug., Rubber Cork and Strand with Glued Brass Plug.

48" I 10 20 30 40 / — -—.. —yIA- -y- -I.. — i. —. 4-. - -- + - + + + + + + + + + + + + + + 2 4 6 8 10 12 14 16 18 20 22 24 50 50 + @ + +- <-A i 40 29 Iof 12 AT 2" C/C IOo *+*c --- 'SI -I — D 10" I 4" r- pi 5-" 5 ' *I I 5" 5" I I- I PRESTRESSED BEAMS T 27" 4. N =W - JD O' to.:z N 6" I_ ---- 1 4.10" 10" 10" 4" _- = I I 5" I — " _ SHRINKAGE BEAMS GAUGE POINTS ARE SIMILAR ON BOTH SIDES OF BEAMS + = BEAM SURFACE GAGE POINTS; @ = GAUGE POINT ON CABLE CYLINDERS,, Figure 4.3. Position of Gauge Points on Beams and Cylinders.

-18 - (2) An initial coat of Duco Cement, of the type used for cementing electrical strain gages, was placed on the previously marked gage point locations on the forms. This coat was allowed to dry for 20-30 minutes. (3) Another coat of cement was applied to both the wood and brass plugs and the plugs were then set in place..The cement was allowed to dry for at least one day. (4) The forms were assembled and concreting done. Care was taken not to strike the plugs with the internal vibrator during concrete placement. (5) The plugs were thus embedded in the concrete at the proper locations. When the forms were removed, the exposed ends of the plugs were punchmarked to the proper gage lengths. (6) These points were then drilled with drill No. 56 and reamed. (7) All sets of points were checked with the Whittemore Gage for obtaining consistent readings. Some points were found to be defective and were redrilled and additionally reamed to obtain consistent readings. This procedure required: (1) careful punchmarking to prevent the loosening of the plug from its bond with the concrete; (2) drilling the holes in the brass plugs with hand drill held approximately at right angles to the surface of the beams; (3) hand reaming the holes keeping the tool at right angles to the surface of the concrete and (4) one hour of time after the forms were stripped and before any readings could be taken.

-19 - To save time as well as to get better gage points the following procedure was followed in Test Series 2 and 35 (1) The brass plugs were drilled and reamed prior to gluing onto the inner form surface. This required. very careful positioning of the plugs to insure being within the range of gage lengths readable on the Whittemore Gage. The holes were filled with plastic clay to avoid filling with Duco Cement. Steps (1) through (4) were done as for Test Series 1o (5) The plugs were embedded in concrete at their proper locationso After the forms were removed, the clay from the gage points was removed and the plugs were cleaned with acetone to remove any cement that might have entered the points. (6) All sets of points were checked by Whittemore Gage. All except one performed satisfactorily. Hand drilling and reaming was used to correct the faulty point. Gage Points in Cylinders The gage points consisted of brass plugs 1/2" in diameter and about 3/4" long. They were drilled and tapped so that a 1/8" diao bolt can fit into the holeo Holes of the size of the bolt were made at 10" c./Co in two diametrically opposite vertical rows on the cardboard cylinder moldso The plugs were then held in position by the bolt inserted from outside. After the concrete reached approximately 4000 psi strength as measured on independent test cylinders, the bolts were taken out, and

-20 - the forms stripped. The exposed surface of the 1/2" diameter plug was used for gage point location. As done on the beams, holes at 10" Co/co were drilled in the brass plugs and reamed to fit the Wlittemore Gage points, Gage Points on Cables For measuring strain reductions on the cable, brass plugs of the type used in beams were modified to fit the curvature of the strand on one side. Different glues were tried to attach these plugs on the strand. The results obtained by using epoxy-type "Twin Weld" Metal Mender manufactured by Fybrglas Industries, Chicago 18, Illinois, were satisfactory if they were applied as follows: Clean the cable and plugs with carbon tetrachlorideo Mix well the two constituents of "Twin Weld" in equal proportions. Wait for two to three minutes. Stretch the cable to a load of 1000 lbo or more. Apply epoxy glue at a desired point enough to fill the gooves of the cable. Apply a thin coat on the plug and attach it on the cable. Use C-clamps to keep plugs in their position. For faster drying use the heat lamp as suggested. by Fybrglas Industries for four to five hourso Wait for a week before taking out the clampsO For taking readings on plugs attached to the cable (Figure 4.2), 1" holes as seen in Figure 4.4 were made by placing 1" diameter tapered rubber corks (Figure 4.2) between the formwork and the cable. The corks were grooved to fit the cable and also the plugs. The rubber corks were waxed for easy removal from the concrete. They were easily pulled out after the formwork was removed,

-21 - Figure 4.4. Beams in Steel Frames.

-22 - In the first test series, where the cable was tensioned after the formwork and cork were placed in position, one of the plugs was dislodged due to the shear force created between the strands and the rubber cork during tensioning of the strands. In Test Series 2 and 3, the cables were tensioned to their design limit first and then the rubber corks and formwork were assembled. No plugs were lost in these series. In each series, the brass plugs were punchmarked for proper distance, drilled and reamed properly to fit points of the Whittemore Gage. Tensioning the Strands Each strand used in the prestressed beams was stressed to a load of 18900 pounds in the self-stressing frame as shown in Figure 4.50 This was accomplished as follows: A 30-ton Simplex hydraulic jack (Figure 4.6) was used to stress the strands individually. The force in the strand was measured by calibrated load cell which had a hole in the center through which the cable passed. This eliminated possible eccentricity on the cello "Supreme" brand strand chucks were used. The required force on the strand was obtained by setting the calibrated amount of strain on a SR-4 Wheatstone bridge and by balancing the bridge when the strand was stressed. The relative positions of the jack, load cell, the strand chucks and ths shims used are shown in Figure 4o.6 The sequence of operations to obtain desired force was as follows: (1) After passing the strand through proper holes of the formwork and tensioning frame, the strand was stressed by a jack until the preset Wheatstone bridge was balanced,

-23 - (J-1 Q):> a) 0 co I 0 H Ed U) Irq z -< a. LJ N Od

SUPREME CHUCK SUPRE \ JACKING PLATE LOAD CELL \:ME CHUCK / 7/16" STRAND SIMPLEX JACK SHIMS rONE BRIDGE Figure 4.6. Hydraulic Jack, Load Cell, Shims and Strand Chuck Arrangement Used in Prestressing the Strand. WHEATS1

-25 - (2) The chuck was slid tightly against the end plate of the stressing frame. (3) The jack was released; some slippage of the strand occurred resulting from the chucks "seating" on the cable. (4) The strand was rejacked until the bridge balanced again. This created some gap (approximately 1/8") between the chuck and the end plate. (5) Shims as shown in Figure 4.7 were inserted in the gap. The thickness of thin shims was 0.007". Figure 4.7. Shim. (6) The strand was tensioned a little more so that one thin shim could be inserted in the gap. (7) The jack was released. (8) The strand was rejacked until the bridge balanced again. It was checked to determine whether the shims were loose or not. (9) In all cases, shims were just loose and so the jack was then released to obtain a desired force on the strand, Loading the Cylinders A test rig as shown in Figure 4.8 was used to load the cylinders concentrically to a desired stress. A 30-ton hydraulic Simplex jack was placed between two plates to exert the pressure. The load was measured by two calibrated SR-4 load cells and a Wheatstone Bridgeo The sequence was as follows:

-26 - Figure 4.8. Test Rig for Determining Creep of Concrete Under Load.

-27 - (1) Level the upper base plate by putting shims on the spring tops if required. (2) Place 2 - 6" x 4" dummy cylinders on top and bottom with 3 - 6" x 12" cylinders in between on the upper base plate in the center of the plate. (3) Place the lower jack plate on top of the dummy cylinder and position the cylinders to be in the center of the plate. (4) Place the jack between lower jack plate and upper jack plate. (5) Place two dynamometers between upper jack plate and top plate. (6) Level the top plate. (7) Apply the load slowly, bringing the special nuts down under upper base plate and lower jack plate until the required load is reached, (8) Tighten the nuts above lower jack plateo Release the jack. (9) Apply the load again and check to see if the nuts are loose. (10) Tighten the nuts and release the jack to obtain the necessary load. An improvement in the above procedure would be to take readings on cylinders after loading 10% of the final load and correcting the cylinder position if an eccentric loading has been obtained. After the satisfactory readings for concentric loaing are obtained the final loading can be reached by applying further load.

-28 - Testing Cylinders The cylinders were tested with a 300,000 pound capacity range Tinius Olsen Testing Machine. Strain readings for cylinders under load were obtained using a compressometer (Figure 6.1). One cylinder in each set was preloaded to 1/2 fc before taking strain readings. Measuring Deflection An Ames' dial reading 1/1000" was used to measure instantaneous as well as creep deflections. The dials were set on the stand as seen in Figure 4.4 for creep deflection measurements. Testing Springs Springs used for test rigs were tested for their deflection characteristics. Four springs with nearly the same spring constants were used in each test rig. According to the manufacturer, the springs were not susceptible to creep for static load of up to 12,000 pounds. Calibrating Dynamometers or Load Cells The load cells which were used in measuring force on steel strand were carefully calibrated by applying known load and noting the strain readings as observed by a standard SR-4 Wheatstone Bridge. The load vso strain curves for Dynamometers No. 1 and No. 3 are given in Figures 4.9 and 4.10.

20 16 000 12000 / 14.00 LB /MICROINCH PER INCH D GAUGE FACTOR 1.94 0 Q. 8 000 ~ Q< 0 -_J 4 000 *O ro ti 400 600 800 1000 1204 STRAIN IN MICROINCHES PER INCH Figure 4.9. Calibration of Dynamometer No. 1.

20 0001 16 000 - c) 0 0 L' 12 000 - Z 11.85 LB /MICROINCH PER INCH GAUGE FACTOR = 1.94 0.j 8000 4000 0 200 400 600 800 1000 1200 1400 1600 0 I STRAIN IN MICROINCHES PER INCH Figure 4.10. Calibration of Dynamometer No. 3.

CHAPTER V TEST SPECIMENS In pretensioned prestressed concrete beams, strain measurements give losses due to creep and shrinkage of the concrete and to creep of the steel. To separate these effects, two test beams were cast in each serieso One beam was prestressed and the other was a unprestressed pilot beam used to measure shrinkage strain only. An attempt was made to find creep characteristics of strands stressed approximately the same as the strand in the prestressed beam. As most of the steel creep loss takes place in the early stage of tensioning, it was assumed that all creep loss took place before the strands were cut. Thus creep of concrete in beams was obtained by deducting the shrinkage strain of concrete from total strain reduction in the prestressed beams. Similarly, for compressed cylinders, the creep of concrete was separated from shrinkage strain by deducting shrinkage strain of nonstressed cylinders from the total strain reduction of loaded cylinders. Three identical test series of specimens of concrete were cast for this study. Each series consisted of one prestressed concrete beam, one shrinkage beam, three stressed cylinders, two shrinkage cylinders and seven compression test cylinderso The dimensions of the concrete beams were as follows: Prestressed Beam: 4" wide, 6-1/2" deep, 8'-0" longo Shrinkage Beam: 4" wide, 6-1/2" deep, 4'-6" long. All the cylinders used were of standard size - 6" diameter, 12" long. The two beams and twelve cylinders of each series were cast at -31 -

-32 - the same time. The size of the prestressed concrete beams was partly governed by the readily available prestressing frame. Shrinkage beams were made shorter as they were found satisfactory for this purpose in earlier studies. The concrete mix was selected from the recommended mixes for use in prestressed concrete bridges by the Michigan State Highway Department. The batch weights required per cubic yard of concrete are: Cement - Type III 658 lbs. Sand 1180 lbs. Gravel (No. 4 to one inch) 1850 lbs. Water 235 lbs. Pozzolith No. 8 (Improved) 1.82 lbs. Vinsol (.012% by Wt. of Cement) 0.079 lbs. The sand and gravel used was from Killins Gravel Company, Ann Arbor, Michigan. The concrete was mixed in a 5-1/2 cu. ft. capacity pug-mill type mixer. An internal vibrator with a 1" diameter head was used to facilitate the placing of concrete. When the concrete had initially set, after about five to six hours, the beams were covered with wet burlap. The cylinders were covered with steel plate to avoid moisture loss, All the specimens were cured for approximately 24 hours under wet burlap. When the average cylinder strength was approximately 4000 psi, the forms for beams and cylinders were stripped. The concrete work sheets and test cylinders data are included on pages 35 through 37. At all times, beams and cylinders were cured and stored under similar conditions.

-33 - Two straight, stress relieved, 7-wire, cold drawn, high carbon, uncoated strands of Roebling System 7/16" diameter were used in prestressed beams and shrinkage beams. The strands used in shrinkage beams were not stressed whereas the tension in each strand used in the prestressed beams during anchorage was 18900 lb. The stress was transferred to the beams by bond of concrete to the strands. No other reinforcing was used in the beams. According to the manufacturer, the properties of the cable were as follows: Ultimate strength - 248,000 psi Modulus of Elasticity - 27 x 106 psi Recommended Tensioning Load/Cable - 18900 lb. Stress-strain curves as obtained for two strands 60' long are shown on Figure 5.1. An attempt was made to measure relaxation of steel stress on these cables.

-34 - 165 155 145 135 125 115 105 95 - 85 (n v) v,) L 55 IC) 75 35 25 45 5 25 15 5 FOR 7/16" DIAMETER 7 WIRE STRANDS GAGE LENGTH - 60 FT. CABLE 2 s105 000 26.92 X0'PSI 3.9 X I0' I Es -136000 27.45 X 1PS1. 4.95 X 1-0' 1 2 3 4 STRAIN IN INCHES X 10s PER INCH Figure 5.1. Modulus of Elasticity. S

-35 - CONCRETE MIX WORK SHEET Series No.: One Date: April 3, 1961 Temperatures: 70~F Humidity: Estimated batch weights required per cubic ye Cement 658 lb. (7 Sk/cyd) Sand (Dry) = 1180 lbo Gravel (Dry) = 1850 lb. Water (Net) = 235 lb. Pozzolith No. 8 = 1.82 lb. Vinsol resin = 0.079 lb. (.012% by wt Estimated batch weights required for these te crete including 10% surplus are: Time: 9:30 A.M. ird of concrete are: t. of cement) ~sts for 5 cu. ft. of con Material Dry Weight Sand 218.5 lb. Gravel 338.50 lb. Water 42.25 lb. Cement 121.85 lb. Pozzolith.337 lb. Vinsol o0169 lb. Required Required Water at Initial Added W Total a Surplus Net Wat 337 c.c. of Pozzolith No. 8 at a concentration.001 lb/c.c. 42.25 c.c. of Vinsol resin at a concentration.0004 lb/c.c. Mixer: Unit Weight: weight = 42.25 lb. Tare +Conc. = 92.75 lb. rater = 14.75 lb. Tare = 18.125 lb dded = 57.00 lb. Wt. of Conc. = 74.625 lb = 2.50 lb. Volume of Tare =.499 c.: er used =54.50 lb. Unit weight = 149.55 lb ft. /c.ft. W/C =.447 Slump = 2-1/2 in. Remarks: Mix 1 minute dry, 3 minutes - - I Air Content = 3.0o with water, wait 2 minutes then mix 2 minutes. Age of Concrete Strength Avg. Strength of Cylinders Date No. Load in lb. in psi Cyl. in psi 1 114,000 4040 25 hours 4-4-61 2 120,000 4230 4160 3 119,000 4200 1 182,500 6450 28 days 5-1-61 2 185,000 6525 6490 I,_ I _ _.......

-36 - CONCRETE MIX WORK SHEET Series No.: Two Date: Temperatures: 69~; 58.50 Estimated batch weights require Cement Type III Sand (Dry) Gravel (Dry) Water (Net) Pozzolith No. 8 Vinsol resin Estimated batch weights require concrete including 10% surplus May 9, 1961 Time: 10:15 A Humidity: 49% ~d per cubic yard of concrete are: = 658 lb. (7 Sk/cyd) = 1180 lb. = 1850 lbo =235 lb. = 1,82 lb. = 0.079 lb. (.012% by wt. of cemo ~d for these tests for 5-1/2 cu.ft.M. ent) o of Material Sand Gravel Water Cement Pozzolith Vins Dry Weight 240 lb. 372-75 lb. 47.85 lb. 134 lb..3707 lb..01608 Required 370.7 c.c. of Pozzolith No. 8 at a concentration.001 lb/c.c. Required 40,2 c.c. of Vinsol resin at a concentration.0004 lb/c.c. Water a Mixer: Unit Weight: Initial weight = 47.85 lb. Tare +Conc. = 92.400 1 Added Water - 10.15 lb. Tare = 18,125 1 Total added = 58.00 lb. Wt. of Conc. = 74.275 1 Surplus =.00 lb. Volume of Tare = o499 c Net Water used = 58.00 lb. Unit weight = 148.9 lb W/C =.433 Slump = 2-3/4 in. Air Content = 3.5% Remarks: Mix 1 minute dry, 3 minutes with water, wait 2 minutes then mix 2 mi: ol lb. b. b. b..ft. /c.fto nutes. Age of Concrete Strength in Avg. Strength of Cylinders Date No. Load in lb. psi Cylo in psi 1 110,000 3890 27 hours 5-10-61 2 105,500 3730 3890 3 114,500 4040 1 159,000 5610 28 days 6-6-61 2 165,000 5830 5810 3 169,500 5990____

-37 - CONCRETE MIX WORK SHEET Series No.: Three Da Temperatures: 78~; 61~ Humidi Estimated batch weights required pe Cement = 6 Sand (Dry) = 11l Gravel (Dry) = 18 Water (Net) = 2 Pozzolith No. 8 = 1. Vinsol resin =. Estimated batch weights required fo: concrete including 10% surplus are: Material Sand Gravel Wat Dry Weight 240 lb. 372 lb. 47.85 te: May 25, 1961 ty: 38% r cubic yard of c 58 lb. (7 Sk/cyd 80 lb. 50 lb. 35 lb. 82 lb. 079 lb. (.012% b; r these tests for Time: 9:30 A.M. oncrete are: ) y wt. of cement) 5-1/2 cu.ft. of er lb. Cement 134 lb. Pozzolith.3707 lb. Vinsol.01608 lb. Required 370.7 c.c. of Pozzolith No. 8 at a concentration.001 lb/c.c. Required 40.2 c.c. of Vinsol resin at a concentration.0004 lb/c.c. Water at Mixer: Unit Weight: Initial weight = 47.85 lb. Tare +Conc. = 91.625 Added Water =13515 lb. Tare = 18.125 Total added = 61.00 lb. Wt. of Conc. = 7350 Surplus = 0.50 lb. Volume of Tare =.499 Co Net Water used = 60.50 lb. Unit weight = 147.5 lb W/C =.450 Slump = 3 in. Air Content = 4o5% Remarks: Mix 1 minute dry, 3 minutes with water, wait 2 minutes then mix 2 mi: lb. lb, lbo ft. /cofto nutes. Age of Concrete Strength in Avg. Strength of Cylinders Date No. Load in lb. psi Cyl. in psi 1 112,500 3970 26 hours 5-26-61 2 114,000 4015 4080 3 117,500 4150 1 157,500 5560 28 days 6-23-61 2 165,000 5830 5760 3 167,000 5900

CHAPTER VI TEST RESULTS AND DISCUSSION Concrete An attempt was made to keep the quality of concrete the same for three test series. Cement was procured in sufficient quantity to make concrete for the three test series. Sand and gravel were used in dry condition to avoid discrepancies in the w/c ratio and were obtained from the same source. The average strength obtained at transfer for the three test series varied by less than 6-1/2 per cent and strength of cylinders in the same mix varied less than 5 per cent. This can be considered as good control for concrete mixes. The air content obtained for the three mixes was less than the planned 5 per cent air. Slump was between 2-1/2" to 3". The average strength at 28 days for the three mixes varied by less than 10 per cent and strength of cylinders in the same mix varied by about 5 per cent. All cylinders and beams were kept wet until the average strength of concrete was approximately 4000 psi and then they were air dried, Modulus of Elasticity of Concrete Cylinders Modulus of Elasticity for concrete was found by loading concrete cylinders. Two different procedures were employed in obtaining the modulus. In one procedure, the cylinder was preloaded to 1/2 its ultimate strength twice and then measurements for strains corresponding to certain loads were taken with the use of the compressometer as shown in Figure 6.1. Stress-strain curves for the other procedure were found without preloading the cylinder. Although the first procedure is recommended by ASTM in -38 -

-59 - Figure 6. 1. Compressometer

-40 - TAB,.-LE 6.1 MODULUS OF ELASTICITY OF CONCRETE Age Measured ACI Formula(') Test in f Secant Modulus EC = 33 Vw;fSeries Days in psi @ 1/2 f' in psi in psi Remarks 1 4240(2) 5.906 x 106 3*751 x 106 1 4200(3) 5.78 x 1o6 3.734 x 1o6 1 28 6525(2) 5.054 x 106 4.667 x io6 28 6450(5) 4.54 x 106 4.627 x lo6 416o3.716 x lo6 f' - Avg.(4) 28 649o 4.64 xl06 fT' - Avg(4) 1 4Q45(2) 3.894 x 106 3.664 x 106 1 3730(5) 3.943 x 106 5.526 x i06 2 28 5990(2) 4.65 x 106 4.46 x io6 28 5610(5) 4.675 x i06 4.521 x 106 1 5890 5.595 x 106 fc f- Avg.(4) 28 5810 4.594 x 106 fc - Avg.(4) 31 4015(2).86 x io6 5.653 x 106 1 4150(5) 5.878 x 106 5.711 x io6 5 28 5560(2) 4.649 x i06 4.295 x jo6 28 5900(3) 4.214 x 106 4.425 x 106 1 4085 5.682 x 106 fl - Avg.(4) 28 5760 4.374 x 106 fl - Avg.(4) (1) (2) (3) (4) Value of w in ACI(62) Formula assumed as 145 Cylinders loaded to 1/2 fc twice before obtaining stress-strain curve Cylinders not preloaded Avg. values of fl - See Concrete Mix Work Sheet TABLE 6.2 MEASURED AVERAGE STRAIN READINGS AT TRANSFER IN FULL ANCHORAGE ZONE Test Series Avg. Beam Surface Avg. Strain as Measured No. Strain in ktin./in. on Cable in jin./in. 1 570 575 2 546 550 53 80 590

-41 - obtaining stress-strain curves, it was felt that to obtain elastic loss in pretensioned prestressed concrete, the second procedure might be more realistic. Secant modulus at 1/2 f' did not vary by more than 2-1/2 per cent for concrete cylinders for the two procedures at transfer of prestress and thus it appears preloading of cylinders to 1/2 fc is unnecessary. Modulus of Elasticity as found by the empirical formula Ec = 33 w3f' and by two previous procedures is tabulated in Table 6.1. There appears to be good agreement between the test values and the empirical values although the test values represent only two cylinders for each series. From the stress-strain curves (Figures 6.2 to 6.13) it is seen that for the stress of 1600 psi at c.g.s. or less, secant modulus at 1/2 fc would be a good approximation for modulus of elasticity of concrete. It should also be noted from stress-strain curves that there is a linear relationship between the stress and strain for the stress levels used in the beam. Elastic Loss of Prestress Elastic loss of prestress is the reduction of force in the cables caused by the elastic shortening of the concrete when the prestress force is applied (at transfer). The concrete is considered to be elastic. The strain readings for each test were recorded immediately before and after transfer of prestress on beams and cylinders. The difference between the two readings gave shortening of concrete due to stress imposed by the strand. Strain readings, obtained in the region of full anchorage on the surface of the concrete beams and on the strands at transfer of prestress, were very nearly equal. (See Table 6.2.) Thus the assumption, that the

f c' -a 4 240 41 CONCRETE PROPERTIES CEMENT TYPE M - 7 SK/C.YD. W/C RATIO * 0.447( BY WT.) SLUMP * 2 V/a " AIR CONTENT - 3.8 % (0 z C,) w 1 — C,) f CY/2 U 2 120 PSI Ec - -2120 396xIo S 543 XIO 3.06X 0 S AT 25 V/s HOURS CYLINDER PRELOADED TO fc'/2 TWICE I 54 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.2. Concrete - Modulus of~ Elasticity - Test Series No. 1 - Age 25-1/2 Hours.

-43 - U) Co 34 z U) U) crl I 2( Co 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.3. Concrete - Modulus of Elasticity - Test Age 28 days. Series No 1 -

a. cn 20 z cn C) w r 15 cC,) 0 400 800 1200 1600 STRAIN IN MICROINCHES PER INCH Figure 6.4. Concrete - Modulus of Elasticity - Test Age 25-1/2 Hours. 200 Series No. 1 -

cn fc/2 3225 PSI aCL z 3000 o) C) LJJ C() 2000 3225 Ec 322 4.542 X 10 PSI 710 X 10 -AT 28 DAYS 1000 / 710 / IN/IN 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.5. Concrete - Modulus of Elasticity - Test Series No. 1 - Age 28 Days.

-46 - V) a. z U) C) Cl) 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.6. Concrete - Modulus of Elasticity - Test Series No. 2 - Age 27 Hours.

-47 - U) Co 300C z co w Ir) ~- 2000 co 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.7. Concrete - Modulus of Elasticity - Test Series No. 2 - Age 28 Days.

U) 0, z U) C,) w IC,) 0 400 800 1200 1600 STRAIN IN MICROINCHES PER INCH Figure 6.8. Concrete - Modulus of Elasticity - Test Age 27 Hours. 2000 Series No. 2 -

-".9) 0 -z 5000- fc'/2r 2805 PSI C,) w I2000- 28 05 E c:z4.675 X 10 P Si 600 IT" AT 2 8 DAYS 1000 -600 /J. IN/ IN 0 400 800 1200 600o 2000 STRAIN IN MICROINCHES PER INCH F'igure 6.9. Concrete - Modulus of Elasticity - Test Series No. 2 -Age 28 Days.

-50 - 4000 fc' 4015 PSI. (26 HOURS) 3500 3 000 CONCRETE PROPER CEMENT TYPE N1 2 5 0 0 W/C RATIO 0 SLUMP AIR CONTENT 7 fc/2 2007 PSI. 2000 cn 1500 2007 Ec = 3-860X10' PSI Z 520 X I0 - AT 26 HOURS U) C) CYLINDER PRELOADED TO fc/2 T1 w I — C,) 500 520/1 IN/IN 0 400 800 1200 1600 STRAIN IN MICROINCHES PER INCH Figure 6.10. Concrete - Modulus of Elasticity - Test Age 28 Hours. 2000 Series No. 3 -

-51 - Cl) 0. z Cl) C,) w C/) 0 400 800 1200 1600 2000 STRAIN IN MICROINCHES PER INCH Figure 6.11. Concrete - Modulus of Elasticity - Test Age 28 Days. Series No. 3 -

-52 - 4 0001 3 500 3 000 2 50 0 U) 2 000 0.. z c) ~15 00 C,) LlU 100 0 5 0 0 STRAIN IN MICROINCHES PER INCH Figure 6.12. Concrete - Modulus of Elasticity - Test Series No. 3 - Age 26 Hours.

-535 - 6000 5000 4000 Cf) 3000 az C/) Cf) w 2000 I 0 0 0 STRAIN IN MICROINCHES PER- INCH Figure 6.15. Concrete - Modulus of El~asticity - Test Series No. 3 - Age 28 Days.

-54 - unit strain in the tendons at the position of the center of gravity of tendons is equal to that in concrete at the same position at transfer in the region of full transfer, seems to be reasonableo An expression for the elastic losses was developed in Chapter III, Concrete stresses at c.g.s. immediately after transfer of prestress are calculated for the middle section of beams (Table 6.3). This agrees fairly well with the measured concrete stresses obtained by multiplying strain and elastic modulus of concrete for test series 1 and 3 (Table 6.3). For test series 2, the elastic strain and thus, the elastic stress loss in beam, is less than the value found from the formula. This could happen if the modulus of elasticity of concrete in the beam is higher than the cylinder or the creep loss in steel is much higher than normal before transfer of prestress, This is in part confirmed by the variation in instantaneous deflections observed for beams 2 and 3 (Table 604)) Thus, elastic loss of prestress can be computed fairly well using the equation as developed in Chapter III, It is believed that some creep strain might have been included in values of elastic strains for later parts of the readingso Shrinkage of Concrete Shrinkage of concrete was measured on 4'-0 long shrinkage beams as well as two 6 x 12 cylinders. Also strain in the cable due to shrinkage in concrete was noted on the shrinkage beam. The cables were embedded in a shrinkage beam to simulate the effect of cables in pretensioned beams on shrinkage of concrete. The shrinkage beams were supported at several points to avoid bending stress effect. Shrinkage versus time curves for

-55 - TABLE CONCRETE STRESSES AT AFTER TRANSFER 6.3 c.g.s. IMMEDIATELY OF PRESTRESS Measured Test Avg. Elastic Series Strain in Measured Stress at c.g.s. Calculated stress* No. p.in./in. = EC Ec, psi i EEc, psi at c.g.s., psi 1 380 3.78 x o106 1436 1424 2 346. 394 x 106 1364 1421 3 370 3.878 x 106 1452 1424 * See Appendix F for calculations. TABLE 6.4 DEFLECTION OF BEAM AT TRANSFER AND GROWTH OF CAMBER Test Series Deflection at ~ of No. Beam at Transfer Growth of Camber 2 o081 in..0439 in. (294 days) 3.112 in..0686 in. (278 days)

-56 - the above three are plotted on Figures 6.14 - 6.16. Also on the same sheet values of humidity and temperature are plotted. This was desirable to observe the effects of changes in temperature and humidity on shrinkage. On Figures 6.15 and 6.16, shrinkage of beams is corrected for temperature using a value of coefficient of expansion of 5.5 iin/in. per ~F. It is clearly seen from these curves that whenever the humidity increased there was reduction in total shrinkage value. Thus uncorrected shrinkage curves include the simultaneous effect of temperature and humidity and so fluctuate according to atmospheric conditions. The shrinkage value used in these curves was the average of 12 readings on shrinkage beams, the average of four readings on cylinders and the average of two readings on the effect of concrete shrinkage on cable strain. From Figures 6.14 - 6.16 it appears that shrinkage in beams is more than cylinders in the earlier stage but later cylinder strain surpasses beam strain and becomes greater. This is due to the steel cable resisting the shrinkage after a certain shrinkage has occurred. Cylinders without reinforcement shrunk more than beams due to lack of restraint of steel. The maximum average value for cylinders appears to be.00047 in./in. while for the beams the value is 0.00043 in./in. after correcting for temperature at the end of one year. The ultimate value for shrinkage of the beam would reach 0.0005 in./in. considering other investigators' reports on long term shrinkage. Magnel(1O) reports that for air cured concrete the generally acceptable value for the ultimate shrinkage is 400 ktin./in. out of which 200 lain./in. occurs during the first 28 days.

600 500o I z cr w 0. C() w I 0 z z w 0 UJ z Ir C) 400o x - AVERAGE SHRINKAGE STRAIN ON CYLINDERS + - SHRINKAGE STRAIN AS MEASURED ON STRAND AT CENTER LINE A - AVERAGE SHRINKAGE STRAIN AS MEASURED ON BEAM SURFACE o - TEMP. AT THE TIME OF MEASUREMENTS o - HUMIDITY AT THE TIME OF MEASUREMENTS w ~/ xi- - -RELATIVE HMI RLT TEMPERATURE I /RELATIVE HUMIDITY I -n I 300o 200 100 w LuJ s o a. z o~ w Iz i-I _ > w 2 -85 -80 -75 - 70 -100 - 75 -50 - 25 0 -- -. 1 - - 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 AGE OF CONCRETE IN MONTHS Figure 6.14. Shrinkage Strain vs. Time - Test Series No. 1.

I I I I I I I I II I I 600 X - AVERAGE SHRINKAGE STRAIN ON CYLINDERS d - SHRINKAGE STRAIN AS MEASURED ON STRAND AT CENTER LINE A - AVERAGE SHRINKAGE STRAIN AS MEASURED ON BEAM SURFACE 500 EB - TEMP. AT THE TIME OF MEASUREMENTS _ ---- - I - * - HUMIDITY AT THE TIME OF;D Z MEASUREMENTS a. 400 -, W ^- AS CORRECTED FOR TEMPERATURE VARIATION I C 1CONSIDERING TEMP. AT 3 DAYS AS BASE READING z 0 300 o0 85 w z [ TEMPERATURE so M u 6 80 200 0 9 * 70 100 |RELATIVE HUMIDITY 00 75 Z t0 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 AGE OF CONCRETE IN MONTHS Figure 6.15. Shrinkage Strain vs. Time - Test Series No. 2.

600 X- AVERAGE SHRINKAGE STRAIN ON CYLINDERS -SHRINKAGE STRAIN AS MEASURED ON STRAND AT CENTER LINE - AVERAGE SHRINKAGE STRAIN AS MEASURED ON BEAM SURFACE ] -TEMP. AT THE TIME OF MEASUREMENTS - HUMIDITY AT THE TIME OF MEASUREMENTS " " "X 500 - 0 z "r w 400 0 CL) IU z 0 300 0: z I w 200 <( -rz ') 100 J I J - - AS CORRECTED FOR TEMPERATURE VARIATION CONSIDERING TEMP. AT 3 DAYS AS BASE READING I I In I /F 85 TEMPERATURE I RELATIVE HUMIDITY a- - L ' I — 4< Iu. 0: o 7 z UJ 2 Z I- I-J - w crl 2 2: 80 75 70 -- 100 - 75 - 50 - 25 - 0 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 AGE OF CONCRETE IN MONTHS Figure 6.16. Shrinkage Strain vs. Time - Test Series No. 3.

-60 - In our case, at 28 days, shrinkage is approximately 300 4in./in. both for beams and cylinders. The values of 0.00047 in./in. for cylinders and of 0.00043 in./in. for beams at the end of one year are shrinkage strains measured after transfer of prestress in beams and cylinders. Before this the concrete was kept moist and it is felt that shrinkage, if it occurred, would be very small. Also its effect on prestress loss until the bond is developed would be very little. In prestressing plants, where steam curing is employed and transfer of prestress takes place in about 18 hours, the effect of shrinkage before transfer would be still less. The new ACI Code (1963) removed the recommended value of 0.0002 in./in. to 0.0003 in./in. for shrinkage as given in the Tentative Recommendation for Prestressed Concrete published by ACI-ASCE Joint Committee 323 in 1956. Zuidema(49) even with a w/c ratio of around.25 got shrinkage of about.00032 in four months. In this investigation, as well as in another investigation(50) the shrinkage of.0005 in./in. has been obtained at the end of twelve months. This points out that the value of shrinkage losses should be decided from the type of mix to be used for the member to be designed and the way it is to be cured. Previous investigations(52) show that steam cured concrete shrinks less than air cured concrete. Also concrete with a low w/c ratio shrinks less. Thus, if the precasting plants use a very stiff mix and use the optimum steam curing procedure recommended by PCI, it is probable that the value of shrinkage may be reduced to less than.0003 in./in., as was recommended for use in design by ACI-ASCE Joint Committee 323 in their Tentative Recommendations for Prestressed Concrete.

-61 - In the absence of a stiff mix and optimum steam curing for manufacture of members, a higher value for shrinkage losses is recommended for design purposes - possibly a value of about 0.00045 in./in to 0.0005 in./in. for normal weight concrete with strength of about 4000 psi at transfer and 5000 psi at 28 days. Creep of Concrete In order to separate the effect of creep of concrete from other effects, two beams, one prestressed and another without prestress, i.e., shrinkage beam, with identical cross sections but shorter in length, were used. Also, three 6 x 12 cylinders were stressed and two 6 x 12 cylinders were used as shrinkage cylinders. The cylinders were stressed at the same level of stress at which concrete near c.g.s. in the beam was stressed at transfer. A slightly higher load was applied in cylinders to take care of loss when the load is transferred through nuts to the steel bars. The creep of concrete was obtained in the following manner: The readings immediately after transfer were considered as base readings for the prestressed beam and shrinkage beam. The change in strain at time t in the prestressed beam accounted for creep of concrete, shrinkage of concrete, temperature and humidity effects on concrete. The change in strain at time t in the shrinkage beam accounted for shrinkage of concrete plus temperature and humidity effects on concrete. Thus differences in readings on the prestressed beam and shrinkage beam gave the creep of concrete at time t.

-62 - Creep of cylinders was obtained in the same manner as previously described for beams. Appendix C gives a typical strain data sheet for prestressed beams, shrinkage beams, and cylinders. For typical calculation of creep and shrinkage of concrete for beams and cylinders, see Appendix C. Creep of concrete versus time for three sets of beams and cylinders are plotted in Figures 6.17 through 6.22. Also Table 6.5 summarizes the average creep strains, shrinkage strains and elastic strains as measured on beams and cylinders at the end of one year. It appears from Figures 6.18, 6.20 and 6.22 that creep in cylinders has not reached its ultimate value but in beams it appears that it is closer to the ultimate value. It should be noted that cylinders are stressed at a uniformly higher level of stress than beams, which have a distribution of stress of a low value on top to a higher value on the bottom. Also the cylinders were kept at approximately constant load while in the beams the load was decreasing due to time-dependent losses. It has been proved(56) that there is not much reduction of load on cylinders when they are spring loaded. Also, the load was increased to its original level, first within six weeks and again at about one year. From this experiment the maximum reduction at one year was less than 1200 lbs. or about 3 per cent. The creep in cylinders is more than that in beams as was expected (Table 6.5), the cylinders being loaded at a higher average stress level than beams as losses occurred in the beams. The elastic strain in the cylinders in Test Series 2 seems to be somewhat high. For the prestressed beam in Test Series 2, the elastic strain is less than would be

400 R CREEP IN BEAM Al w r.I UI 0 300 z 0 + ffc' AT TRANSFER 200 2 fc' (28 DAYS w 1 fc AT TRANSFER AT C G o 100- INITIAL AVERAGE STRES, CREEP IN BEAM AT 10" FROM END 0 5 10 15 20 25 30 35 40 45 50 55 60 65 AGE OF CONCRETE IN DAYS Figure 6.17. Creep Strain vs. Time - Test Series No. 1.

1200 I100 1000 900 I 0 z - 800 our w LL - 700 n) w I 0 600 z 0 0 500 Z 400 W 300 Q2: 0 200 100 fc AT TRANSFER 4160 PSI fc' ( 28 DAYS ) - 6490 PSI fc AT TRANSFER AT C.G.S. OF BEAM = 1489 PSI INITIAL AVERAGE STRESS ON CYLINDERS I 489 PSI CYLINDER LOAD ADJUSTED X X AIN IN CABLE DUE TO CREEP IN BEAM 7 /0 /el X X X X X X X 0 CREEP IN BEAM 10" FROM END [, 4 w co W 4 UJ GQ 03 (C) Z >u I x x X X X XI 0 z iJ X z w I)w 18L z LU 18 0 0 0 *- * -- 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 AGE OF CONCRETE IN MONTHS Figure 6.18. Creep Strain vs. Time - Test Series No. 1. 15 16 17

x 3: 0. z U Lcd 2 CL W 2 0 i I 40 45 50 55 60 65 CONCRETE IN MONTHS Strain vs. Time - Test Series No. 2. Figure 6.19. Creep

1200 0 z bi U) QU. z 0 gr z Qa U0 0 0 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 AGE OF CONCRETE IN MONTHS Figure 6.20. Creep Strain vs. Time - Test Series No. 2.

400 -z a. I 300 rCREEP IN BEAM AT 20" FROM END I z a. 200 wL /CREEP IN BEAM AT 10" FROM END n, 100 fc' AT TRANSFER: 4085 PSI fc' (28 DAYS) = 760 PSI fc AT TRANSFER AT C.G.S. OF BEAM 14 8 9 PSI INITIAL AVE. STRESS ON CYLINDERS = 489 PSI 0 5 10 15 20 25 30 40 50 60 70 80 90 AGE OF CONCRETE IN DAYS 11 I Figure 6.21. Creep Strain vs. Time - Test Series No. 3.

1200 1100 1000 I I I I I I I I I fc AT TRANSFER fc' ( 28 DAYS ) fc AT TRANSFER AT C.G.S. OF BEAM INITIAL AVE. STRESS ON CYLINDERS I I I I rI I I I I -1 4085 = 5760 = 1489 = 1489 PSI PSI PSI PSI z aI Z (-) z 5 -0) z aLiJ Or W. LJ L. C) 90C 80C 70C 60C 500 I I, I LOAD ADJUSTED I 400 300 200 k rCREEP IN BEAM 20" FROM END ++ ++ + + rCREEP IN BEAM 10" FROM END a ~'. -... *~ P: $I: z z: x x x x x x x 100 00 LU Im.J U) LU C z C>u x x x x y x x X z I0 U) (n IUJ Ur nLL UV) z r I.I< 0 I 2 3 4 A7 a- I I - -. o r | I 0 11 12 13 14 15 16 17 18 AGE OF CONCRETE IN MONTHS Figure 6.22. Creep Strain vs. Time - Test Series No. 3.

TABLE 6.5 MEASURED ELASTIC, CREEP AND SHRINKAGE STRAINS IN BEAMS AND CYLINDERS Beams Beam Surface Readings Cable Readings Cylinders Test Test Test Test Test Test Test Test Test Series Series Series Series Series Series Series Series Series Remarks 1 2 3 1 2 3 1 2 3 Elastic Strain in kin./in. 380 346 370 390 350 375 452 490 450 At transfer Creep Strain in uin./in. 635 560 640 718 500 675 780 860 875 Values one year Shrinkage Strain in ktin./in. 465 465 435 470 500 420 500 505 495 Maxm Value in one year TABLE 6.6 (R)* = RATIO OF MEASURED CREEP TO MEASURED ELASTIC STRAIN FOR BEAMS AND CYLINDERS Beams Beam Surface Readings Cable Readings Cylinders Test Test Test Test Test Test Test Test Test Series Series Series Series Series Series Series Series Series Remarks 1 2 3 1 2 3 1 2 3 Measured Creep Strain = (R. Values @ end Measured Creep Strain 1.665 1.62 1.73 1.835 1.43 1.80 1.80 1.752 1.94 one year Measured Elastic Strain of one year I oh \o I

-70 - expected. Due to these opposite effects on beams and cylinders, the ratio between the two creep strains became much higher than in the other two series. To include this variation effect, it was decided to compare ratios Creep Loss (R)* of the beams to Creep Loss (R)* Elastic Loss Elastic Loss of the cylinders for the three test series (Tables 6.6 and 6.7). At the end of one year this ratio for the three series has an average of.91. As the creep in cylinders is still increasing at a faster rate than the beams, the above ratio will decrease. If more experiments are made with different stress levels and different stress distribution on beams, it is believed that some kind of relation can be established between creep of cylinders and creep of beam. According to a previous investigation(^) about 75 per cent of 7-year creep in beams was obtained in the first year. Other investigators have reported 80 per cent of 20-year creep in cylinders during the first year. In this investigation the ratio of creep loss to elastic loss at the end of one year is an average of 1.67 for beams. By projecting the creep curves of beams in Figures 6.18, 6.20, and 6.22 the ultimate creep loss for this investigation would reach a value of about 1.80 times elastic loss. This is in agreement with the report(55) by Subcommittee 5, ACI Committee 455. Many mathematical models(40) have been suggested to represent creep curve of concrete. An equation as suggested by Shank(51) is used here to draw creep curve of concrete. It is recognized that this curve could fit only a limited range of test data and that it does not satisfy the requirement that at infinite time there is a finite value of creep per psi. There are other models which can overcome the deficiency as

-71 - discussed above but they are cumbersome to calculate. Almost all require some constant or constants to be found from experiment, TABLE 6.7 RATIO OF (R)* FOR BEAM TO (R)* FOR CYLINDER Test Test Test Series Series Series Remarks 1 2 3 R)* for Beam At the end R)* for Cylinder.925.925.892 of one year I ~e I ~ I ^ -^ II (R)* - From Table 6.6 An equation as suggested by Shank is of the form y = c'(x)q where y is creep in kin./in. per psi, x is number of days after loading and c', q are constants to be found from experiment. Values of c' and q are found to be.1715 and.202 respectively. (For calculations see Appendix G.) From Figure 6.23 it can be seen that the curve of creep 202 obtained from y =.1715(x)~2 fits very well in the envelope of creep data for cylinders. Using the creep per psi as obtained from cylinder creep, the curve of creep for beams was drawn on the same figure. The beam creep data for the three test series was plotted and an envelope of data was drawn, The curve of creep for beam obtained from cylinder creep per psi fitted nicely in the above envelope of beam creep data. From this it can be concluded that creep of concrete is dependent on stress level but specific creep (or creep per psi) is very nearly constant, at least for variation in stress in the beam due to loss of prestress due to shrinkage and creep of concrete.

IOOC ENVELOPE OF CREEP FOR CYLINDERS 900 500 CURVE USING SHANK FORMULA a2 700 * POINTS USINGOF CREEP FOR BEAMS l)J Z 600 z (Y 500 CURVE USING SHANK FORMULA E AGE OF CONCRETE IN MONTHS _ 400 POINTS FOR TEST SERIES I a.POINTS FOR TEST SERIES 2 W 300 POINTS FOR TEST SERIES 3 0 POINTS USING SHANK FORMULA FOR CYLINDERS 200 * POINTS USING SHANK FORMULA FOR BEAMS (SEE SAMPLE CALCULATIONS FOR CYLINDERS AND BEAMS I00 BY SHANK FORMULA IN APPENDIX G) 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 I 18 AGE OF CONCRETE IN MONTHS I R) I Figure 6.23. Creep Strain vs. Time Curve Using Shank Formula.

-73 - Stresses in the steel strands of beams were found using specific creep of cylinders by the "Effective Modulus" method.(17) (See Figures 6,24 - 6.26, Appendix G.) Stresses in strands, as obtained from using measured strains on the concrete beam surface as well as the strand surface, were plotted in Figures 6.24 - 6.26. The agreement between these curves was fairly good indicating that the "Effective Modulus" method can be used for estimating strand stress if specific creep and shrinkage of beams are known. Ultimate magnitude of specific creep of plain concrete can range from 0.2 to 2,0 millionths in terms of length, but is ordinarily about one millionth or less. For this investigation, specific creep for stressed cylinders is approximately.58 millionths at the end of one year. The ultimate value for specific creep would reach 0.7 millionths, Stress in steel strand, using computed elastic loss, estimated creep loss from formula as suggested by El-Darwish,(50) and assumed shrinkage strain of.00050 in./in., is computed in Appendix H and shown in Figures 6.24 - 6.26. Final stress in steel agrees fairly well with the value obtained using the Effective Modulus method. Anchorage Length of Strands and Bond of Strands to Concrete In pretensioned prestressed concrete no end anchorages are used and thus the transfer of prestress takes place through bond of the strand to the concrete. Hanson and Kaar(54) attributed the transfer to these three factors: (1) Adhesive bond of the concrete to the steel. (2) Friction between the steel and concrete. This is due to Poisson's Effect. A reduction in stress of strand

175 170 3 165 1 O Beam Strand Stress Using "Effective Modulus" Method (See Appendix D and E) A Beam Strand Stress a = or - (eb + s + c) Es eb - Measured Elastic Strain in Beam at Transfer, in Micro-Inches per Inch s - Measured Shrinkage Strain in Beam at Time t, in Micro-Inches per Inch 160 155 150 z z n,.Cf) Cl) w cr IL. I..Hn 145 c - Measured Creep Strain in Beam at Time t, in Micro-Inches per Inch X Beam Strand Stress a = aot - (es + ss + Cs) Es es - Elastic Strain as Measured on Strand at Transfer, in Micro-Inches per Inch Ss - Shrinkage Strain as Measured on Strand at Time t, in Micro-Inches per Inch cs - Creep Strain as Measured on Strand at Time t, in Micro-Inches per Inch 1 aot - Stress in Strand Just Before Transfer = 168,347 psi Es - Modulus of Elasticity of Strand \ ^ - - r STRAND STRESS BY "EFFECTIVE MODULUS" METHOD FRM X K -~ t —"-" ----"S- FROM.. AD. P I -—,! 1401 135 - 130[ 125[ Mrnl~, x rI IVTI rri I ni1i^ II 120o 115' 0 - - -. - a -- - - - - A - - I- - - L - a - - - I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 TIME t IN MONTHS Figure 6.24. Strand Stress as Computed by "Effective Modulus" Method - Test Series No. 1.

170 - 165 - O Beam Strand Stress Using "Effective Modulus" Method (See Appendix D and E) A Beam Strand Stress a = a t - (eb + s + c) Es eb - Measured Elastic Strain in Beam at Transfer, in Micro-Inches per Inch - _ M n ~-A uln nam - MIi. A nli m:_ ~ _ _ _ - T _ 160 155 (n 150 z 145 0 z r 140 I() Z 135 ') u) 130 I(I) 'i ~ j - lntca.u-jL olnr11itntag ozrajln L11 Dtin D;m aC Lme ta, in Mic c - Measured Creep Strain in Beam at Time t, in Micrc X Beam Strand Stress a = qot - (es + Ss + Cs) Es es - Elastic Strain as Measured on Strand at Transfer, i SS - Shrinkage Strain as Measured on Strand at Time t, i cs - Creep Strain as Measured on Strand at time t, in aOt - Stress in Strand Just Before Transfer = 168,347 psi ~i Es - Modulus of Elasticity of Strand — "' — --. - - ~ P ~ ~ v - - - -r - - cro-incnes per inch o-Inches per Inch in Micro-Inches per Inch.n Micro-Inches per Inch Micro-Inches per Inch I i I I ~ i - FROM APPENDIX H 1201 115 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 TIME t IN MONTHS Figure 6.25. Strand Stress as Computed by "Effective Modulus" Method - Test Series No. 2.

170 I 165 F O Beam Strand Stress Using "Effective Modulus" Method (See Appendix D and E) A Beam Strand Stress a = Oot - (eb + s + c) Es eb - Measured Elastic Strain in Beam at Transfer, in Micro-Inches per Inch c-a M<-a lcIr~Q Cith? r; ns Q4 —oeirT n T n o-m q4 +. m- +. in M; rrn_T-nr> 'b o -nor Tnrtk 160 155 150 (f) z U) Cl) w C/) -J w w C) 1451 - vicIC-O UCA L t1 U- 11. J t L UJ LXJL1. JL (13 CIX 1V- ALU U, X11~ (1 -'II. L tIL.J Ll.. k c - Measured Creep Strain in Beam at Time t, in Micro-Inches X Beam Strand Stress a = aot - (es + SS + Cs) Es es - Elastic Strain as Measured on Strand at Transfer, in Micross - Shrinkage Strain as Measured on Strand at Time t, in Microcs - Creep Strain as Measured on Strand at Time t, in Micro-Ir aot - Stress in Strand Just Before Transfer = 168,347 psi Es - Modulus of Elasticity of Strand STRAND STRESS BY "EFFECTIVE MODULUS" ^ ^.- -. -r -_ t t -Inches per Inch -Inches per Inch iches per Inch I -4] Cr\ I per Inch 140 - 135 - METHOD 130I FROM APPENDIX H 1251 1201 115 L 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 TIME t IN MONTHS Figure 6.26. Strand Stress as Computed by "Effective Modulus" Method - Test Series No. 3.

-77 - due to elastic loss causes an increase in the diameter of individual wires and thus the increase in the diameter of the strand. (3) Mechanical anchorage due to the lay of the six outer wires. Surface conditions of all strands used in these tests were almost identical. There were no rust spots and all strands were cleaned with carbon tetrachloride before concretingo From Figures 6.27 - 6.29 it can be seen that the anchorage length for the 7/16" - 7 wire strand for the mix as used and at concrete strength at transfer of around 4000 psi is from 22 in. to 27 in. One could also notice that there is no transfer of prestress in the end region of 2 in. or so. This might be due to sudden transfer of stress due to burning of cableso This could result in the breaking of bond and. slippage could occur resulting in zero stress in the cable, In a series of tests conducted at the University of Michigan, it was found that the surface conditions of the strands play an important role in transfer of prestress to concrete, For 7/16 in, strands the anchorage lengths have been reported(49250'57) between 12 in. to 50 in. by various investigations, lower values for the rusted strands and higher values for smoother and shiny strands. A value of 25 in (49) has been reported for normal weight concrete and non-rusty strands. This investigation confirms the findings of previous works, as this value of transfer length is 22 in. to 27 in. From Table6.2, it is seen that in the complete anchorage zone there is no slippage during transfer.

z:: 0 n, w C,) z x C) z 2 (. LU, I(/) -J w 500 400 DISTANCE REQ'D FOR ANCHORAGE = 23.5" wI x 300[ 200O x / *. / REMARKS STEEL / fc' (AT / E c (AT PERCEN SURFAC / STRAND A I a A -. 2" WHITTEMORE GAUGE x I " WHITTEMORE GAUGE I -,, STRESS AT A TRANSFER) TRANSFER) IT AIR:E CONDITION IS WERE CUT I00l NCHORAGE = 3.85 OF STRAND BY TORCH 173,500 PSI 4156 PSI X 10 PSI 3.8 % t NO RUST 5 10 15 20 25 30 35 40 45 DISTANCE FROM END OF BEAM ALONG C.G.S. IN INCHES Figure 6.27. Anchorage of 7/16" 0 Seven-Wire Uncoated Strand. Test Series 1.

I z 500 CL O. c) I: U 400 z 2 DISTANCE REQ'D FOR ANCHORAGE = 21 II z I 0 tI V) 0 FC) -J w 300O 200h /REMARKS x STEEL STRESS A' / fc'(AT TRANSFE / Ec(AT TRANSFE PERCENT AIR SURFACE CONDI1 STRANDS WERE I I.I I — 3 I * 2" WHITTEMORE GAUGE X 10" WHITTEMORE GAUGE x 100 r ANCHORAGE = 173,500 PSI ER) = 3890 PSI:R) = 3.92 X 106 PSI a 3.5 % rION OF STRAND: NO RUST CUT BY TORCH 5 10 15 20 25 30 35 40 45 DISTANCE FROM END OF BEAM ALONG C.G.S. IN INCHES Figure o.28. Anchorage of 7/16" 0 Seven-Wire Uncoated Strand. Test Series 2.

I I I I I I 0 500 U> 400 z o 300 AJ z 0 o 200 z Icc 0 u *00 0 E 4 -J -r I<[ DISTANCE REQ'D FOR ANCHORAGE = 27 x x / x 2" WHITTEMORE GAUGE X 10" WHITTEMORE GAUGE I REMARKS STEEL STRESS AT ANCHORAGE - 173,500 PSI fc' ( AT TRANSFER) = 4084 PS I Ec (AT TRANSFER) = 3.87X 106 PSI PERCENT AIR = 4.5 % SURFACE CONDITION OF STRAND: NO RUST STRANDS WERE CUT BY TORCH 5 10 15 20 25 30 35 40 45 DISTANCE FR OM END OF BEAM ALONG C.G.S. IN INCHES Figure 6.29. Anchorage of 7/16" 0 Seven-Wire Uncoated Strand. Test Series 3.

-81 - From Figures 6.24 - 6.26 it can be seen that strand stress as obtained by measuring strains on the beam surface and strand surface agrees fairly well for Test Series 2 and 3. In Test Series 1, only one strain reading on the strand surface was available as one of the plugs was lost during tensioning of the strand. The value of steel stress, using the average of four strain readings on a concrete surface, was higher than the value of steel stress using one strain reading on a strand surfaceo Creep of Strand An attempt was made to find the difference in relaxation of strands stressed under two different conditions. One strand was stressed between two supports at a constant length. The other strand was stressed between two supports such that measured reduction in length could be made at one end. This was done by putting a known thickness of shims between the end plate and anchorage as shown in Figure 4.6. Due to the availability of only one dynamometer and. jack for both the strands, difficulties were encountered. in removing strand chucks from the cable and the investigation had to be stopped after 10 days. There was no temperature control in the room. As the force in the cable was measured when the shims became slightly loose, a possible personal error in judgment was involved in the readings. In spite of the above difficulties the results obtained were quite indicative of what would normally happen. The maximum stress loss as observed in strand at constant length was 5.95% in 10 days while it was 4-1/2% for the variable length strand. The first value of 5.95% loss agrees fairly well with results obtained for a stress-relieved 3/8"

-82 - strand stressed at 0.7 fs as reported in (40). As these tests were made after the concrete beams were cast, elastic, creep and shrinkage strains as observed for the beams were used in reducing the length of the strand at the respective times. This in effect duplicated the condition of a prestressed concrete beam in steps instead of continuous reduction due to effects of creep and shrinkage of concrete. It is felt from the above experiment that in prestressing plants where cables are cut in approximately 18-19 hours after pretensioning, the creep would be about 2-1/2% before the cutting of the cables. After cutting the cables, elastic loss occurs followed by creep and shrinkage losses in concrete which occur at a rapid rate in the beginning. This reduces the tension in the strand from the initial stress of 70% of ultimate strength down to 55% of ultimate value after which the creep in the cable is found to be very little. Thus the average value of 4% for creep of steel loss is a good average for strands tensioned at 70% of ultimate strength. This is true for stress-relieved, 7-wire, cold-drawn, high carbon, uncoated Roebling System strands as used in this experiment. The characteristics of other strands may differ from the above values. As the strands were cut 26-28 hours after tensioning, 350% creep loss is assumed in computation of stress after transfer of beamso The total loss would probably reach 5% of the initial stress. The 3% creep loss was arrived at by using information from Reference 40 and using their formula fi b r = gfi [t] t1

-83 -where Ar = relaxation stress loss at time tl f. = initial stress 1 f = ultimate strength of steel s tl = time from application of initial stress in hours g, b and d are constants to be found from tests.

CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS Conclusions From this investigation it can be concluded that: 1. Empirical formula for Modulus of Elasticity - E = 35 w3f is confirmed from this investigation. 2. Modulus of Elasticity of concrete is almost equal to secant modulus at half the ultimate strength with the range of concrete stresses in the beam. (Maximum stress 0.6 f'.) 3. Strains as measured on the beam surface agree fairly well with strains measured on the strands in the middle section of the beam. This indicates perfect bond between concrete and strands and hence possibility of slip is excluded for creep mechanism. Thus the use of plugs on the surface is satisfactory for shrinkage and creep studies of beams. 4. An ultimate shrinkage strain of.0005 in,/in, is estimated for beams of which.0003 in,/in. occurs during the first 28 days. 5. The creep and shrinkage as measured on cylinders should be used with caution. The ratio between maximum average cylinder shrinkage to maximum average beam shrinkage at the end of one year is lo093o 6. The creep in cylinders, concentrically loaded at approximately constant load, is greater than creep in beams, stressed initially at the same stress at the c.g.s. of beam. 7. The average ratio of creep strain to elastic strain for cylinders is 1.83 at the end of one year. The ultimate creep loss would reach about 2.25 times elastic loss. -84 -

-85 - 8. For beams, consider loss due to creep of concrete and creep of steel after transfer as "Creep Loss." This "creep loss" is 1.67 times the elastic loss at the end of one year. On the basis of projection of data, the ultimate creep loss would reach about 1.8 times elastic loss. This ratio of creep loss to elastic loss is for: (1) Initial level of stress in concrete at c.g.s. of 1420 psi. (2) Concrete strength at transfer of about 4000 psi. (3) Stress on top fiber of about 450 psi and bottom fiber of about 2100 psi in the beam. For other conditions the ratio of creep loss to elastic loss may differ. 9. The ratio of Creep Loss for beams to Creep Loss for cylinders Elastic Loss Elastic Loss is an average value of.91 at the end of one year. 10o Specific creep is very nearly constant. This is verified for stress range from initial stress level in beams to the final stress level in beams at the end of one year. 11. "Effective Modulus" method can be used for estimating strand stress if specific creep and shrinkage of beams are known (Chapter VI)o 12. The anchorage length for the 7/16" 7-wire strand is from 22" to 27" (for 4000 psi concrete at transfer) when the strand is cut by torch at transfer. The strength of concrete at transfer is an important factor in the anchorage length. Recommendations To find the final stress in the tendons of the pretensioned prestressed concrete beams, the following is recommended to evaluate various losses:

-86 - lo Calculate the loss of prestress due to creep of steel for the time elapsed between tensioning of tendons and transfer of prestress using creep formula for steel (Chapter VI, page 82 ). This value may be around 2-1/2 to 3% for 7-wire strands stressed to.7 times their ultimate strength and where the transfer takes place in 16 to 20 hours. 2. Calculate the elastic loss of prestress using Equation (3.5), (3.6) or (3~7) as suggested in Chapter III. Modulus of Elasticity of concrete can be found using the empirical equation as suggested by the ACI. Modulus of Elasticity of steel can be used as suggested by the manufacturer. 3. (a) "Effective Modulus" method can be used for estimating strand stress in beams using specific creep obtained from stressed cylinders. 85 (50) (b) Equation r 285 ( in which r is the ratio of ultimate 3 fc creep strain to elastic strain at transfer, can be used to predict the concrete creep strain of prestressed concrete beams. This equation was derived from test beams protected from weather and without control of temperature and relative humidity. 4. Shrinkage strain of.0003 in./in. is recommended for normal weight concrete of 5000 psi strength with low w/c ratio and optimum steam curing. In the absence of stiff mix (low w/c ratio) and optimum steam curing for manufacture of members, a value of.0005 in./in. for shrinkage is recommended for normal weight concrete with 4000 psi strength at transfer and 5000 psi at 28 days.

BIBLIOGRAPHY 1o Evans, Ro H., and Bennet, E. W., Prestressed Concrete, Theory and Design, New York, John Wiley and Sons, Inc., 1958. 2, Lin, T. Y., Design of Prestressed Concrete Structures, New York, John Wiley and Sons, Inc., 1956. 3o Keeton, Jo Ro, Study of Creep in Concrete, Technical Report R-082, U.S. Naval Civil Engineering Laboratory, Port Hueneme, California, May 1960. 4. Best, C. H., Pirtz, D., and Polivka, M., A Loading System for Creep Studies of Concrete, ASTM Bulletin No. 224, (September 1957), 44-48. 5. Lorman, W. Ro, "The Theory of Concrete Creep," Proceedings, ASTM, 40, (1940), 1082. 6. Maney, G. A., "Concrete Under Sustained Working Loads; Evidence that Shrinkage Dominates Time Yield," Proceedings, ASTM, 41, (1941), 1021, 7. Ross, Ao Do, "Creep and Shrinkage in Plain, Reinforced and Prestressed Concrete, A General Method of Calculation," Journal, Institute of Civil Engineers (London), 21, (November 1943), 38. 8. Schorer, H., "Prestressed Concrete, Design Principles and Reinforcing Units," ACI Journal Proceedings, 39, (June 1943), 493-528, 9. Staley, Howard R., and Peabody,D, Jro. "Shrinkage and Plastic Flow of Prestressed Concrete," ACI Journal Proc.o 42, (January 1946), 229-244. 10, Magnel, Gustave, "Creep of Steel and Concrete in Relation to Prestressed Concrete," ACI Journal Proc., 44, (February 1948), 485-500. 11. Shank, J. R., "Plastic Flow of Concrete at High Overload," ACI Journal Proc., 45, (February 1949), 493-500). 12. Washa, Go W., and Fluck, P. G., "Effect of Sustained Loading on Compressive Strength and Modulus of Elasticity of Concrete," ACI Journal Proc., 46, (May 1950), 693-700. 13. Davis, R. E., and Troxell, G. E., "Properties of Concrete and Their Influence on Prestress Design," ACI Journal Proco, 50, (January 1954), 381-392. 14. Neville, A. M., "Theories of Creep in Concrete," ACI Journal Proc., 52, (Sept. 1955), 47-60. -87 -

-88 - 15o Ross, A. D., "Loss of Prestress in Concrete," Civil Engineering and Public Works Review, No. 527, 45, (May 1950), 307-309. 16. Seaman, F. E,, "Determination of Creep Strain of Concrete Under Sustained Compressive Stress," ACI Journal Proc., 53, (February 1957), 803-810. 17. Ross, A, Do, "Creep of Concrete Under Variable Stress," ACI Journal Proc., 54, (March 1958), 739-758. 18. ACI-ASCE Joint Committee 323. "Tentative Recommendations for Prestressed Concrete," ACI Journal Proc., 54, (January 1958). 19o Dawance, Go, Tests Concerning Creep and Shrinkage Losses in Prestressed Concrete, Publications, International Association for Bridge and Structural Engineering, 1952. 20. Magnel, Go, Prestressed Concrete, Concrete Publications, Ltd., 14 Dartmouth St., Westminster, S.W. London, 1948. 21. Shideler, J. J., "Lightweight Aggregate Concrete for Structural Use", ACI Journal Proc., 54, (October 1957), 299-328. 22. Carlson, Ro W., "Drying Shrinkage of Concrete as Affected by Many Factors," Proceedings, ASTM, 38, (1938), 493-528. 23, Nordley, G. M., and Venuti, W. J., "Fatigue and Static Tests of Steel Strand Prestressed Beams of Expanded Shale Concrete and Conventional Concrete," ACI Journal Proc., 54, (Aug. 1957) 141-160. 24. Erzen, C. Z., "An Expression for Creep and Its Application to Prestressed Concrete," ACI Journal Proc., 53, (Aug. 1956), 205-214. 25. Thorsen, N,, "Use of Large Tendons in Pretensioned Concretes," ACI Journal Proc., 52, (Feb. 1956), 649-660. 26. Pickett, Go, "Effect of Aggregate on Shrinkage of Concrete and A Hypothesis Concerning Shrinkage," ACI Journal Proc., 52, (Jan. 1956) 581-590 27. Komendant, A. E., Prestressed Concrete Structures, McGraw-Hill Book., Inco, New York, 1952, 28. Middendorf, K. H., "Comparison of Cold-Drawn Stress-Relieved Wire", PCI Journal, 5, No. 1, (March 1960). 29. Godfrey, H. J., "The Physical Properties and Methods of Testing Prestressed Concrete Wire Strand," PCI Journal, 1, No.3, (Dec. 1956). 30. Preston, H. K., "Wire and Strand for Prestressed Concrete," PCI Journal, 1, No.4, (March 1957).

-89 - 31. Preston, H. K., "Proper Use of Wire and Strand in Prestressed Concrete," PCI Journal, 2, No. 3, (Dec. 1957). 32. Lofroos, W. N., and Ozell, A. M., "The Apparent Modulus of Prestressed Concrete Beams under Different Stress Levels," PCI Journal, 4, No.2, (Sept. 1959). 33. Diao, K. K., "Average Concrete Stress Along Prestressing Steel in Prestressed Concrete Beams," PCI Journal, 5, No. 2, (June 1960). 34. McHenry, Do, "A New Aspect of Creep in Concrete and Its Application to Design," Proceedings, ASTM, 43, (1943). 35. Collings, A. R., "Recent British Research on Prestressed Concrete," PCI Journal, 6, No. 2, (June 1961). 36. Kluge, R. W., "Field Studies of Prestress Loss Due to Shrinkage and Creep," PCI Journal, 3, No. 1, (June 1958). 37. Dean, Wo E., "Research in Prestressed Concrete at the University of Florida and Its Practical Application in Bridge Practice," PCI Journal, 6, No. 3, (Dec. 1961). 38. Ozell, A. M., Prestressed Concrete Design, Bulletin No. 74, Engr. and Indust. Experimental Station, University of Florida, August 1955. 39. Freudenthal, A. M., and Roll, F., "Creep and Creep Recovery of Concrete under High Compressive Stress," ACI Journal Proc., 54, (June 1958), 1111-1142. 40. Fisher, Jo W., Kingham, R. I., and Viest, I. M., "Creep and Shrinkage of Concrete in Outdoor Exposure and Relaxation of Prestressing Steel," HRB Spec. Rep. 66, (1961), 103-131. 41. Troxell, Go D., Raphael, J. M., and Davis, Ro E., "Long Time Creep and Shrinkage Tests of Plain and Reinforced Concrete," Proceedings, ASTM, 58, (1958), 1101-1120. 42. Cottingham, W. S., Fluck, P. G., and Washa, G. W., "Creep of Prestressed Concrete Beams," ACI Journal Proc., 57, (February 1961) 929-936. 43. Everling, W. 0., "Steel Wire for Prestressed Concrete," The First National Prestressed Concrete Short Course, Maritime Base, Sto Petersburg, Florida, October, 1955. 44. Cernica, J. N., and Charignon, M. J., "Plastic Strain in Prestressed Concrete Beams Under Sustained Load," Proceedings, ACI Journal, 58, (August 1961) 215-221. 45. Mattock, A. H., "Precast, Prestressed Concrete Bridges: 5. Creep and Shrinkage Studies," Journal,Research and Development Laboratories, Portland Cement Association, 3, No. 2, (May 1961), 32-66.

-90 - 46. "Developments in Prestressed Concrete - A Review Prepared by the Prestressed Concrete Development Committee," Proceedings, Institution of Civil Engineers, 8, (November 1957), 292-322. 47. Fluck, P. G., and Washa, W. G., "Creep of Plain and Reinforced Concrete," Proceedings, ACI Journal, 54, (April 1958), 879-896. 48. Arnan, M. A., Reiner, M., and Teinowitz, M., Research on Loading Tests of Reinforced Concrete Floor Structures, Research Council of Isreal (Jerusalem), (1950), 52. 49. Zuidema, M.., A Study of Losses of Prestress Force in Pre-tensioned, Concentrically Prestressed Members, Structural Engineering Research, University of Michigan, Ann Arbor, Michigan, 1959. 50. El-Darwish, I. A., Prestress Losses in Pretensioned Prestressed Concrete Members with Bent Tendons, D.Sc. Thesis, University of Michigan, Ann Arbor, Michigan, 1963. 51. Shank, J. R., "The Mechanics of Plastic Flow of Concrete," Proceedings, ACI Journal, 32, (Nov.-Dec. 1935), 149-180. 52. Powers, T. C., "Causes and Control of Volume Change," Journal, Research and Development Laboratories, 1, No. 1, (Jan. 1959), 29-39. 53. Cottingham, W. S., Fluck, P. G., and Washa, G. W., "Creep of Prestressed Concrete Beams," Proceedings ACI Journal, 57, (Feb. 1961), 929-936. 54. Hanson, N. W., and Kaar, P. H., "Flexual Bond Tests of Pretensioned Prestressed Beams," Proc., ACI Journal, 55, (January 1959), 783-803. 55 "Deflection of Prestressed Concrete Members," Subcommittee 5, ACI Committee 435, Proc., 60, (December 1963). 56. Compomanes, N. V., The Prestress Losses and the Flexural Strength of Lightweight Prestressed Concrete Beams, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, 1951. 57. Dean, W. E., "Research in Prestressed Concrete at the University of Florida and Its Practical Application in Bridge Design," PCI Journal, 6, No. 4, (December 1961), 60-70. 58, Raphael, J. M., "The Development of Stresses in Shasta Dam," Trans. ASCE, 118, (1953)o 59. Stussi, Fritz, "On the Relaxation of Steel Wires," (in German with English and French summaries), International Assno for Bridge and Structural Engrg., 19, (1959), 273-286,

-91 -60. Bletzacker, R. W., "The Concepts of Rheology Applid to Portland Cement Concrete," Proceedings, ASTM, 62, (1962), 996-1007. 61. Lyse, I., "Tests of Full-sized Prestressed Concrete Bridge Beams," Proc., ACI Journal, 54, (May 1958), 979-986. 62. "Building Code Requirements for Reinforced ACI Committee 318, Proc., ACI Journal, 60, Concrete (ACI 318-63)," (July 1963), 809-815.

APPENDIX A DESIGN OF BEAM Concrete Strength Limiting Stresses At Transfer 4000 psi 2400 psi At 28 Days 5000 psi 2250 psi Load from 2 - 7/16" Roebling cables = 37,800 lbs = 173,554 psi Es = 27 x 10 psi Ec = 1,800,000 + 5000 x 4000 = 3.8 x 106 psi t 6 -l,'3 " q -.71531 4" 3.287".IA 35.2135' I t Section Properties: Ac = 25.7812 in.2 At = 27.33 in.2 C1 = 3.287"; C2 = 3.213" e =.713" It = 92.24 in.4 As =.2178 in.2 0 -.- -,.-I -_ 4" w = 27.08 lb /'; F = 37800 lb O Span = 7' - 6"; Mg = 2284"lb n = 27/3.8 = 7.1 (1) Fi = MgetnAs Fo+ I It ---— __ —. 1 2t 1 + nA+( — +At It 37800 + 2284 x.713 x 7.1 x.2178 92.24 1 + 7,1 x.2178(.1 +.32) 27.33 92.) 35515. 5 lb -92 -

-953 - Elastic Loss = 37800 - 35515.5 = 2284.5 lb - 224.5 = 10489 psi.2178 % Elastic Loss 97x 100 - --------------- - 6.04% (2) Due to Shrinkage of Concrete - Assume.00025"1/" Loss =.00025 x 27 x 106 = 6750 psi %Loss- 6750 x0-.9 % Loss 173,554 x 1 0- - - - - - - - - - - - - -.9 (3) Due to Creep of Concrete Loss = 1.5 x 10489 = 15733 Psi % Loss =l.5 x 6.04 --- —---------— 9.06% (4) Due to Creep 'of Steel - Assume Loss =.04 x 175,554 = 6942 p % Loss = 4% --- - - - - - - - - - - - - 4.oo% Total Losses % Losses F =37800 x -39914 psi 39914 x 100 - 22.99%, 173,554.73 = 27594 lb say., 23% Concrete Stresses at Transfer: f -35515.5 35515.5 x.713 x 3.287 2284 x 3.287 27.33 92.24 92.24 = - 1299.5 + 902,3 - 81.4 = - 478.6 psi fb - 2995 -35515.5 x.713 x 3.213+ 2284 x 3.213. 295-92.24 + 92.24 = - 1299.5 - 882 + 79.5 = - 2102 psi

-94 - f 35515.5 x.7132 = - 129905 - 92.241 2284 x.713 + 92.24 =- 1299.5 - 195.7 + 17.6 =1477.6 psi Load to be put on Cylinder: = 1477.6 x 28.3 = 418i6 lbs Final Concrete Stresses after all Losses Occur: -t - 27594 + 27594 x.713 x 3.287 -2284 x 3.287 27,33 92.24 92.24 = - 1009.6 + 701.1 - 81.4 = - 389,9 psi 27594 27.33 27594 x.713 x 3.213 92.24 -v2284 x 3.213 92.24 = -, 1009.6 - 685,3 + 79,5 = - 1615.4 psi 27594 27.33 27594 x.7132 92.24 - 2284 x.713 + 92.24 = - 1009,6 - 152.1 + 17.65 =-1144.1 psi

APPENDIX B MEASURED VERSUS CALCULATED ELASTIC LOSS OF BEAMS a0= Stress in Steel at Anchorage = 173,554 psi Use 3% Loss in Steel due to Creep of Steel clot= 173,554 x.97 = 168347 psi See Appendix A for Beam Properties. Use formula as given in Chapter III for finding stress after cutting cables. 01. cr ot - K2 + i 1 + nK3 + K2 E = 27 x106psi e (C2 I- Cl)Ac_ i1.713 (3.213 - 2 x?.287)25.781 2 92. I- - = -.007373.24 = 2284", lbs Mg YLxl12=27.o8 x7.5 2 xl.s mg ~8 Mg (Cl - C2) 2284 x.074= 2 x 92.24 =6.1.91617 184.48.2178 = oo8448 p 25.7812 -K1 1 + ~l = 1 -.007373 =.992727 K2 = -.91617 =_o84.oo8448 -184 Ace2 = 1 +r - = 1 +.1421 = 1.1421 3 I p P1 -K.oo8448 -.99727=.008509 K3 = Pl-'3 =.008509 x 1.1421 =.009718 4 Mge _ 2284 x.713 - 17.65 e4 =It 92.24 -95 -

-96 - Test Series 1 n = 27 =7.2658; nK3.716 168,347 - 108.5 + 128.2 1o070609 Elastic Loss = a a. = 168,31 ot 1 % Loss = 11,085 x 100 = 173,554 Measured Loss = 380 x 10 x 27 x % Loss = 10 26 x 100 173,554 3 =.070609; n4 = 128.2 157,262 psi 47 - 157,262 = 11,085 psi --- -- -- 639% 106 = 10,260 psi - ------ 5.91% Test Series 2 n = 27 = 7.511; nK3 =.072992; n14 = 128.2 3.595 168,347.3 =16 9 13 psi 1.072992 si Elastic Loss = 168,347 - 156,913 = 11,434 psi o Loss = 11,434 x 100 ---------------- 6.58% 173,554 Measured Loss = 346 x 10-6x 27 x 106 = 9,342 psi 9342 % Loss = 934 x 100 = --------- -- 5.38% 173,554 Test Series 3 n 27 = 7.333; nK3 =.071265; 3.682 ai = 168,367.3 = 157,166 psi 1.071265 ni4 = 128.2

-97 -Elastic Loss = 168,347 - 157,166 = 11,181 psi % Loss = 11,181 x 100 = 6.44% 173,554 -6 6 Measured Loss = 370 x 10 x 27 x 10 = 9,990 psi % Loss = 9,990 x 100 = -------- 5.76% 173,554

APPEWDIX C WHITTEMORE GAGE DATA SHEETSHRINKAGE Cono, Date Time Temps. -May 25,1961 9.30 a.m. 780 - 610 Date Feb 26,1962 Time Temps. 75.50 - 58'0 Age 278 days Humidity 34% Std. Bars 1 10.-05580 in. (#2. gage-#2 bar, Beam) 2 10.052000 in. (#2 gage-#2 bar-Cable) 10.014.780 in. (#1 gage-#l bar-Cylinders) Beam:- S-3 Cylinders:- S-3-1; S-3-2 Gange Initial Final Shortening Average Average Remarks No. Reading Reading Shrinkage Shrinkage in A~in/ir. in piun/in. NE 36 O 0 3265 32 9NW 4685 4240 445 3 SE 5140 4725 415 SW 5015 4550 465 _____ NE 2600 2145 455 19 NW 5585 5140 445 442 426 SE 2900 2450 -450 SW 3990 3570 420_ __ _ _ _ __ _ _ _ _ _ _ NE 3500 3095 40529SRed 2 NW 3540 310410 4o5 2 WRa 29 SE 3770 3370 400 ing Discarded SW 5615 5300 CbeN 2590 2150 417 417 Cal 4485 4090 395 ____ iN 7275 6770 505 S5315 7380 6890 490 498 498 ___ 7640 -7105 535 2S 6790 6330 46 9.29 19 9 -98-M

-99 - WHITTEMORE GAGE DATA SHEET CREEP Beam:P-3 Cylinders: C-3.A.gre of Concrete 278 days Gage Initial Final Shortening Average Average Average Remarks No. Reading Reading Creep and Shrinkage Creep Shrinkage in ~i in/in, in ~t in/in. in in/in. NE 8220 7665 555 10 NW ~ 5820 5225 595 5646130 SE ~4510 3945 5655542 SW 3560 3050 510 NE 7230 6595 635 20 NW 3645 2730 915724629 SE 448o 3600 880 2o2 9 SW 5350 488o 470 NE 4710 3790 920 30NW 6700 5495 1205954253 30SE 5145 4145 1000954253 SW 5715 5010 705 NE 5735 4670 1065 4o NW 2780 1740 940 1032 426 606 SE 6425 5325 1100 SW 3470 2445 1025 NE 5035 4olo 1025 50 NW 5050 4o45 1005 lo746621 SE 1800 690 1110 14 2 SW 5810 4760 1050 N ] 5875 } 48io 1065 JT1 Cable 4950 J385516 J109514 j 180 417 j 663 C iN 5665 4490 1175 Y iS 6735 5270 1465 i 2N 5835 4780 1055 1346 498 848 n 2S 5535 3825 1710 d e 3N 6440 5315 1125 s 3S 6715 5170 1545 w N w E 0 C E 0 20 t20 10 I 10 -I 20 -140 -- 50 -I,I -- *@eee@@eO@9oO 30. co 0 S0 -TI i 0......e 5 0 4 0

APPENDIIX ID DERIVATION OF FORMULA FOR FINAL STRESS IN STEEL (STRAND) I TB AM AFTER LOSSES ACCORDING TO "EFFECTIVE MODULUS" METHOD I) 1. Force in Steel = Force in Concrete i.e., F F(C2 - Cl) caA + 21t + Mg (Ci - C2)A At jc Similar to solution on pages 8,and 10, Chapter III. f p1 2. Strain in Concrete at C.G.S. = Change of Strain in Steel at c.g.s. F Fe2 AC It M e It orot - a + 5 = c "Effective Modulus" EI is defined as c =E c + c1Ec where el= specific creep under I psi at time t (see page 73, Chapter VI) E C = Elastic Modulus Similar to solution in Chapter III for finding Elastic Loss, fn' '3 - n' + sE5 = aot -f- K pl 2

-101 -i.e., f pi (aot - sEs - K2 + n I i) n=Es 1 + nIK3 Et 3 C i.e., aro SES 2+ni ot +KnK Thus if the shrinkage and. creep per psi are known, the stress in steel can be found.

APPENDIX E CALCULATIONS FOR FINAL STRESS IN STRANhD AS PLOTTED ON FIGURES (6.24) THROUGH (6.26) arot - sEs - K2 + nve4 Final Stress cr + K 2 (Appendix D) 1 + nlK3 Values for a calculated for Test Series 1., 2., and 3 in Tables El, E2, and E3, respectively. Values of aot.1 ES, K, and K2 used from Appendix B. -102 -

TABLE El E @ TRANSFER = 3.716 x 106 PSI; n = 7.2658 Shrinkage Cylinder Creep Shrinf Creep in per psi of Beam in nlin./in. Final Stress Days in./in. = 1 + clE n' = n x (4) 1 + n'K5 s sEs n'4 ot - sEs - K2 + n'4 () a = (11) + K2 Days.iin./in... c c 1491 l (6) in psi (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 15 400 2683 -1.997 14.51 1.141 250 6750 256 161,745 141,757 141,865 30 460.3085 2.146 15.59 1.152 340 9180 275 159,334 138,310 138,418 60 560.3756 2.396 17.41 1.169 365 9855 307 158,691 135,749 135,857 90 620.4158 2.545 18.49 1.180 385 10395 326 158,170 134,042 134,149 120 655.4393 2.632 19.12 1.186 390 10530 337 158,046 133,259 133,367 150 680.4561 2.695 19.58 1.190 375 10125 -346 158,460 133,159 133,267 255 740.4963 2.844 20.66 1.201 450 12150 365 156,454 130,269 130,377 315 760.5097 2.894 21.03 1.204 465 12555 371 156,055 129,613 129,721 365 780.5231 2.944 21.39 1.208 460 12420 376 156,195 129,300 129,408 430 810.5433 3.019 21.94 1.213 430 11610 387 157,016 129,444 129,552 TABLE E2 E @ TRANSFER = 3.595 x 106 PSI; n = 7.511 15 415.2783 2.000 15.02 1.146 265 7155 265 161,349 140,793 140,901 30 490.3286 2.181 16.38 1.159 305 8235 289 160,293 138,302 138,410 60 560.3756 2.350 17.65 1.172 345 9315 312 159,236 135,866 135,974 90 625.4192 2.507 18.83 1.183 355 9585 332 158,986 134,392 134,500 120 680.4561 2.640 19.83 1.193 370 9990 350 158,599 132,941 133,049 235 815.5466 2.965 22.27 1.216 440 11880 393 156,752 128,907 129,015 295 840.5634 3.025 22.72 1.221 460 12420 401 156,220 127,944 128,052 365 860.5768 3.074 23.09 1.224 465 12555 408 156,092 127,526 127,634 400 865.5801 3.085 23.17 1.225 435 11745 409 156,903 128,084 128,192 TABLE E3 E @ TRANSFER = 3.682 x 106 PSI; n = 7.333 15 415.2783 2.025 14.85 1.144 230 6210 262 162,291 141,862 141,970 30 495.3320 2.222 16.29 1.158 299 8073 287 160,453 138,560 138,668 60 590.3957 2.457 18.02 1.175 315 8505 318 160,052 136,214 136,322 78 640.4292 2.580 18.92 1.184 345 9315 334 159,258 134,508 134,616 120 720.4829 2.778 20.37 1.198 350 9450 360 159,149 132,845 132,953 220 830.5567 3.050 22.37 1.217 420 11340 395 157,294 129,247 129,355 278 850.5701 3.099 22.72 1.221 430 11610 401 157,030 128,607 128,715 365 880.5902 3.173 23.27 1.226 410 11070 411 157,580 128,531 128,639 380 885.5936 3.186 23.36 1.227 395 10665 412 157,986 128,757 128,865 H 0 \I I

APPENDIX F CALCULATED STRESSES ANDU STRAINS IN BEAMS AN~D CYLINDERS AT TRANSFER OF STRESS Test Series 1 Fi= 157,262*x x.2178 = 34251.6 lbs =-34251.6 - 4251.6 x.7132 s 27.33 92.24 + 17.65 = - 1253.3 - 188.7 + 17.6 = - 1424.4 at c.g.s. of Beam Calculated Cylinders: Strain in Beam =fs- - 1424.4 — EC 3.716 x 106 =383 x 1 in./in. 42200 Stress = 28.3 = 1491 psi Calculated Average Strain in Cylinders = 1491 -6 3.716 x 106 = 4oi x 10- in./in. Test Series 2 F i = 156,913* x.2178 - 34175.6 lb f = - 34175.6 5 27.33 34175.6 x.713 2 92.24 + 17.65 = - 1250.5 - 188.3 + 17.6 = - 1421.2 at c.g.s. of Beam Calculated Strain in Beam = 122 =39x106in./in. Cylinders: Stress = 1491 psi Calculated Average Strain in Cylinders = 1 4 =- 1 414.7 x 10-6 in./in. From Appendix B. -104 -

-105 - Test Series 3 Fi = 157,166* x.2178 = 34230.7 lbs 2 f 34230.7 = 34230.7 x.713 + 17.65 s 27.33 92.24 - 1252.5 - 188.6 + 17.6 = - 1423.5 at c.g.s. of Beam Calculated Strain in Beam = 14235 = 386.6 x 10i in./in. 3.682 x 10 Cylinders: Stress = 1491 psi 1491-6 Calculated Average Strain in Cylinders = 1491 = 405 x 10 in./in. 3.682 x 106 From Appendix B.

APPENDIX G TYPICAL CALCULATION OF CREEP STRAINS USING SHANK FORMULA Shank Formula: y = c'(x) where = x = c' q = creep in 4 in./in. per psi number of days after loading constants to be determined from experiment Days Creep of Cylinders in 4 in./in. Test Series 1 Test Series 2 90 360...... 620 780 Iw. 625 860 Test Series 3 655 875 Ave. Creep in. in./in. 633 838 Average Cylinder Stress = 1491 Creep per psi Creep per psi 133= 424 (at 90 days) = 1491 = 838 =.562 (at 360 days) = 1491 (360)q = 4q (90)q Y90 Y360.562.424 Y36o Y90 q =.202 202 360: = 3.28. c' _.202.1715 (x) 562 =.1715 3.28 Equation is y = -106 -

-107 - To Find Creep Strain in 4 in./in. for Cylinder and Beam at 180 Days. Cylinder: y =.1715 (180)'202 =.49 Creep strain at 180 days = 1491 x.49 = 731 p in./in. Beam: We need stress in concrete at c.g.s. to find creep strains at c.g.s. Using a. - average value from Appendix B - equal to 157,000 psi and average shrinkage, s, of Beam and creep strain, 2, as obtained from formula for Cylinder (above), stress at c.g.s. is calculated as follows: a = stress in strand = a. - (s + c2)Es = 157,000 - (425 + 731) x 27 =125,800 psi 125,800 x.2178 125,800 x.2178 x.7132 2284 x.713 f - - - + s 25.7812 92.24 92.24 (at c.g.s. of Beam) = - 1134 psi Estimated Creep Strain in Beam =.49 x 1134 = 556 pi in,/inO at c.g.s.

A PPEiN DI X H CALCULATION FOR FINAL STRESS IN STRAND AS RECOMM4ENDED 1.or = 173,554 psi Say 3% Loss in Strand due to Creep of Steel before Transfer of Stress. C t= 173,554 x.97 - 168,,347 2. At Transfer fT = 4000 psi (Design) C E C = 33 54~ x 4000 = 3.645 x lops n= -27 - 7.4074; 3.645 nK 3. 0009718 x 7.4074 =.07198 cx. =168,347 - 108.5 + 128.2 = 1.07198 168.,367 -1.07198 157,061 psi Elastic Loss: 168,367 - 157,061 % Elastic Loss: 11,306 x 100 173,~554 = 11,306 psi --- - -- - - -6.51% 3, r = 28.5 *_ f C 28.5 3 000 28.5 17,1 = 1. 67 Creep Loss: 11.,306 x 1.67 = 18,881 psi % Creep Loss: 18,.881 x 100 173,554 --- - - - - -10.87% 4. Shrinkage Loss: 6.0005- x 27 x 10 = 13,,500 psi % Shrinkage Loss: --- - -- - - -7.77% = ar. - Creep Loss -Shrinkage Loss = 157,061 - 18,881 - 13,500 = 124,680 psi Total Loss = 173,554 - 124,680 = 48,974 psi % Loss = 48,974 x 100 173,554,., --- - -- - - -28.21% * From Reference 50. -108 -

UNIVERSIT OF MICHIGAN 3 9015 03483 1175