Concrete Stress Distribituion in Ultimated Strength Design

PORTLANDCEMENT AND ASSOCIATION LABORATORIES . -, ,/ J \ J RESEARCH DEVELOPMENT I \j ,“ CONCRETE STRESSDISTRIBUTION IN ULTIMATE STRENGTH DESIGN ~ ~ :i . ; ..; .., i By E. Hognestad, N. W. Hanson and D. McHenry I !: !“ Bulletin D6A 1 1’ ,, ;. Supplemental to Bulletin D6 ,, ,; :f., I ,, i~, I Authorized Reprint from Copyrighted Journal of the American Concrete institute December 1955, Proceedings Vol. 52, p. 455 Bulletins Published by the Development Department Research and Development Laboratories of the Portland Cement Association .&D1 –“Influence of Soil Volume Change gineering,” by E. J. FELT. and Vegetation on Highway En- Reprinted from Twenty-Sixth Annual Highway Conference of the Universityof Colorado, January, 1953, 5%76. D2 —“Nature of Bond in Pre-Tensioned JANNItY. ceedirzgs, Prestressed Concrete,” by JACK R. 19.54);Z%-o- Reprinted from ~ournal of the American Concrete Znstitute 50, 717-736 (1954). (MzIy, D2A–Discussion of the paper “Nature of Bond in Pre-Tensioned Prestressed Concrete,” by P. W, ABELES, K. HAJNAL-KONYI, N. W. HANSON and Author, JACK R. JANNEY. Reprinted from Journal of the American Concrete Institute (December, Part 2, 1954); Proceedingsj 50, 736-1 to 736-11 (1954). D3 –“Investigation of the Ivfoisture-Volume Units,” by JOSEPHJ. SHIDELER. Stability of Concrete Masonry Published by Portland Cement Association, Research and Development Laboratories, Skokie, lllinok, March, 1955. D4 –“’A Method for Determining the Moisture Concrete in Terms of Relative Humidity,” Reprinted from Proceedings, (1955). D5 -“ Factors Influencing EARL J, FELT. American Condition of Hardened by CARL A. ME~ZEL Materials, 5% 1-26 Society for Testing Physical Properties of Soil-Cement Mixtures,” by Reprinted from Highway Research Board Bulletin 108, 138-162 (1955). D6 –“Concrete Stress Distribution in Ultimate Strength HOGNJESTAn, N. W. HANSON and D. MCHENRY. Reprinted from .lowwd Of the American concrete Institute Proceedings, 52, 455-479 (1956). Design;’ by E. (December. 1955): D6A Stress Distribution (pztrt of D6)—Discussion of :he paper, “Concrete m Ultimate Strength Design,” by P. W. ABELES, A. J. AsMDOWN, HOMER M. A. L. L. BAKER, ULF BWXREN, HENRY J. CowAN, HAD~EY, SMITH, L. KONRAD E. HRUBAN, A. J. J. N.f. PREN~IS, E. ROSENBLUETH, G. MYOUNG, TAYLOR, and Authors, E. HOGNESTAD, N. W. HANSON, and D. MCHENRY. Reprinted from Journal of the American Concrete Institute (December, Part 2, 1956); Proceedings, 52, 1305-1330 (1956). CONCRETE STRESS DISTRIBUTION IN ULTIMATE STRENGTH DESIGN By E. Hognes+ad, N. W. Hanson and D. McHenry Including ~uggren, discussion Hlenry by P. W. Abeles, M. A. J. Ashdown, Konrad J. Taylor, A. Hruban, and L. L. J. Baker, M. UIf J. Cowan, G. M. Smith, and Homer Hadley, A. Prentis, E. Hog- E. Rosenbleuth, nestad, N. W. L. E. Young, D. McHenry. authors, Hanson PORTLAND RESEARCH AND CEMEN’T ASSOCIATION LABORATORIES DEVELOPMENT 5420 Old Orchard Road Skokie,Illinois Title No. 52-28 Concrete Stress Distribution Design* in Ultimate Strength BY EIVIND I+ OGNESTAD, N. W. HANSON, and DOUGLAS McHENRYt SYNOPSIS Test data are presented which demonstrate the reality and validity of the fundamental plasticity concepts involved in ultimate strength design themies such as those presented by Whitney, Jensen, and othem. A review of earlier experimentalinvestigations regarding the stress distribution in the compression zone of structural concrete flexural members revealed that, though many test methods have been tried, very limited direct test data are avai[able. On the other hand, considerable information has been derived inc[irectly from strength and behavior observed in tests of reinforced beams and columns. An eccentrically loaded specimen and a test method were developed which permit the flexural stress distribution to be measured. Complete information regarding the flexural stress distribution, including stress-straingraphs with a descending curve beyond the maximum stress, is reported for w/c ratios of 1.0, 0.67, 0.50, 0.40, and 0.33 at test ages of 7, 14, 28, and 90 days. INTRODUCTION The distribution of concrete stress in the compression zone of reinforced concrete memlbers subject to flexure is of fundamental importance in theories regarding the ultimate strength and behavior of such structural members. This stress distribution, often referred to as the stress block, was therefore discussed in the early papers which in about 1900 led to the development of Though the techmathematical design procedures for structural concrete. nical details emphasized have changed from time to time, periodic attention has been devoted to the stress block ever since, Nevertheless, it has been possible to gather information only slowly, principally because it h difficult to measure stresses in concrete, although strains may bc measured easily and accurately. on ultimate The formation in 1952 of the joint ACI-ASCE committe~ strength design was a result of the growing recognition of the significance of the ultimate strength of structural members. To aid the committee in its assignment “to evaluate and correlate theories and data bearing on ultimate *Premnted at the AC I Eighth Regional Meeting, Atlanta, Ga,, Oct. 31. 1955. Title No. 52-2S is a part of copyrighted JOUENAL OF THE AXENICAN CUNCX~T~ l~smmrr~, V. 27, No. 4, Dec. 1955, Proceedings V. 52, Separate (copies in Triplicate) should reaoh the Institute not later than Apr. yi:$56~ available at 50 cents each. Discussion Address 18:263 W. McNichols Rd., Detroit 19, Mich. ‘ +Members American Conorete Institute, Manager, Structural Development Section, Associate Development Eu~ineer. and Director of Development, respectively, Research and Development Divmion, Portland Cement Ass”. , Chicago, 111. 455 456 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 load design procedures with a view to establishing them as accepted it has again become desirable to improve and extend our knowledge the stress block. Historical back!~round practice, ” regarding Several early studies of reinforced concrete beams, such as R. M. von Thullie’s flexural theory of 1897 and W. Ritter’s introduction of the parabolic stress block in 189!), were aimed at developing a theory to explain and permit prediction of ultimate strengths observed in tests, To find a stressStrain re]ation for concrete, these early investigate m-s turned to concentric compression tests of prismatic plain concrete specimens. They observed deformation and load from zero to the maximum load, and the prism stressstrain relation was then applied to the beam problem. Such studies and the corresponding emphasis cur ultimate strength were discontinued about, 1900, at which time the elastic straight-line theory and the concepts of working loads and working stresses became accepted in design Since the modular ratio n is prominently used in the throughout the world. straight-line theory, extensive researches were devoted to the modulus of elasticity of concretes at low loads. Effects of many variables were investigated in concentric compression tests, and various expressions for the modulus of elasticity of concretes and the modular ratio were suggested. A renewed interest in ultimate strength of structural concrete began about 1930, initiated by F. von Emperger’s critical studies of the modular ratio and working stresses as used in design. Since then, a large number of ultimate strength theories involving a variety of hypothetical stress blocks have been developed. Several theories based on stress-strain relations obtained in concentric prism tests rarried beyond the maximum load have also been presented, and some investigators have attempted directly or indirectly to mcamrre the distribution of flexura] stresses in tests of reinforced beams. The number of theories suggested between 1930 and 1950 became so large that in 1951 critical reviewslfz were published in both European and American literature. American studies of ultimate strength of structural concrete have now reached such an advauced stage of development that detailed information regarding the stress di~tribution in flexure is urgently needed to formulate new design procedures based on ultimate strength and to gain their acceptance The investigation reported herein was undertaken, therefore, to in practice. evaluate previous findings of factual nature and to contribute new data regarding the properties of the stress block. Obiect and scope of investigation This investigation was conducted at the Research and Development LaboThe objectives of the study ratories of the Portland Cement Assu. in 1954. are: (1) to evaluate previous methods and results in experimental investigations of the stress block, and (2) to develop a test method leading to an improved and quantitative understanding of the stress block. irnate bending moment M ‘o P P q !7’ = K’ A.(2) f. A. . =Pti. e’=k.. . ..S– The equations for pure flexure are identical. .. 0. . . Notation The letter symbols A. with the right-hand member of 13q. ~uu = ultimate load reinforcement ratio tension reinforcement index compression reinforcement index inders = average compressive compression zone stress in concrete % % Q % f *U = stress in tension reinforcement at ultimate load FLEXURAL STRENGTH = strain in concrete = strain in reinforcement at ultimate load = ultimate concrete strain in flexure = steel strain at initial yielding CONCRETE OF STRUCTURAL Basic ultimate strength equatians To illustrate the properties of the stress block that are most important for practical purposes.. =k. .’ bc+A:~.G Stress conditions at the ultimate load capacity of a rectangular structural concrete member subject to combined flexural and axial load are shown in Fig.’f. 0. 14. Equilibrium of moments and forces is expressed by Mu. . kl.k. . . .k.ULTIMATE STRENGTH DESIGN 457 An eccentrj tally loaded specimen and a test method were developed.k7.fs”.. (2) equal to zero. . =&. .0.” ‘ .50. forcement to compression edge of member = distance between centroids of tension and compression reinforcements = modulus of elasticity of concrete to = ewentrici ty of load with respect centroid c~ftension reinforcement = compressive stress in concrete = compressive strength of 6 x 12-in. bd f. .. f.67. . and 90 days.. (1) equal to Mul~ and the right-hand member of Eq. . An analytical approach originated by F. = Load.. . .’ a b C c d used herein are defined below for convenient reference: = concrete gross area = area of tension reinforcement = area of cc~mpression reinforcement = eccentricity of load = width of rectangular member = total internal compressive force in cOncrete = distance from neutral axis to compression edge of member = distance from centroid of tension rein.4 as well as to prestresscd concrete. 28... . +A.’d’f’. E.’ bc(d-k.’ f.and . 0.= coefficients related to magnitude and position of internal compressive force in concrete compression zone (Fig. .f. . and the method was used to measure the properties of the stress block for five concretes with w/c ratios of 1. A. MULL = Ult.. . A.%. ‘— ‘— bdf. 1) k. modified moment term d’ Ec e’ f. . . . Stiissi’ is used. . which through refinements contributed later by others has been made applicable to flexure of reinforced concrete with and without axial load2.. .33 at test ages of 7. . cyl- f’8w = stress in compression reinforcement at ultimate load = yield point of reinforcement f. .t.(1) . 1.c) A. = c/d ratio indicating position of neutral axis at failure M = bending moment. the basic equations for the flexural strength of structural concrete are reviewed.’ A. Pu. . . . and 0.40. f. .’ h the equations for tension failure. 1 lf..’ bd –=f. 1—Conditions mate load at ulti- . . . . the tensile stress f~u as well as Pult and M. (1) to (5) apply also to T-sections and to hollow sections. . = “f: A. . . (3) reinforcement Similarly.. . . . =. q = bdfc’ . Assuming a linear distribution of strain. . provided that the neutral axis at ultimate load is not below the compression flange. equals the yield point . For prestressed concrete.& q = bdf.. the only property the stress block needed is the ratio lcJlc. ”s Fig. we obtain with the notation of Fig. Eq. . . f. determine the stress in the tension reinforcement at failure. . in which the indexes are +. .’ q’ = — bd f ‘. It is then necessary to know eu and independent values of kz and klk~.. reinforcement we obtain by solving Eq. not only the ratio k2/lCllG3.. .458 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 If failure is initiated by yielding of the tension minf orcement. (5) with Eq. .. . ..T”U. . . .... .. . . . (5) Combining Eq.’ j. . an equation similar to Eq. .kS.it can be determined.. (1) and (2). . (1) and ~~) in which %=q(’”-$q) the tension f.. if both tension and compression trically loaded member are yielding at failure.. .. =-~ %:+6. . and assuming that the compression reinforcement is yielding at failure. (3) and (4).’””””””””””””’””” .. the steel stress and with no compression at the ultimate moment f. .’ reinforcement wc obtain of an eccen- ’’-’+( $-)+d(:+)+%(q)$ +:’’q+))l(’)))l(’ and A. crushing of the concrete takes place before yieldIt is then necessary to consider strains to ing of the tension reinforcement. (5) can be used in conjunction with the stress-strain relation for the prestressing steel to determine the steel stress at failure. . Pu. . Eq. . . index . If it is assumed that the concrete carries no tension below the neutral axis. . of For compression failures. . . Experimental studies of stress block It is difficult to measure stresses in concrete at high loads. It has been shown that the sudden failures observed in compression tests are related to the release of energy stored in the testing machine. 8-11 An example of such a relation is given in Fig. Many methods have been tried. and e.006 Strain .C02 . If the entire shape of the stress block is known.X34 Concrete . 2. Strains can of course be measured easily by various types of gages.ULTIMATE STRENGTH DESIGN 459 Therefore. since final collapse of compression test specimens often takes place shortly after the maximum stress is reached. which in turn is related to bhe stiffness of the machine..’ By using suitably stiff testing machines.. any shape of section and nonsymmetrical bending of rectangular sections can be analyzed. S. Most of these expressions consider the range from zero to the maximum stress only.. an analysis of this type is a powerful design tool provided that the properties of the stress block are known at least to the extent of k.oot3 . Bureau of Reclamation . 2—Tests of 3 x 6-in. concrete cylirrders—U. or by surrounding the concrete specimen by a system of steel relations have been observed beyond the maximum springsl stress-strain load. therefore. to obtain experimental data regarding the distribution of concrete stresses in flexure. but stresses at high loads are not proportional to strains. ‘~ Fig. that an ultimate strain can be developed which is greater than the strain at the maximum stress in a conventional compression test.~ o .k. Prism stressstrain relations have been applied to bending of reinforced concrete by assuming a linear clistribution of strain in the compression zone.010 . Others have recognized. however. Concentric compression tests—A great number of mathematical expressions have been developed for the stress-strain relationship for concentrically loaded prisms. Some authors have assumed that the extreme “fibers” in a beam at the ultimate moment are subject to a maximum stress and a corresponding strain which are both similar to those determined in a simple compression test. k. Tests finding are carried out by increasing the load in stages. research with the machine is being continued . Only limited results have been published. and the data obtained must be interpreted cautiously. Stress meters—An attempt has been made to measure stresses by embedding a stack of eight steel plates in the compression zone of a reinforced concrete beam. In this manner the machine has been used to study the shape of the stress block. but that the general characteristics as shown by Fig. A more satisfactory method was developed by the U.15 ~hotoelastic methods—The distribution of concrete stresses has also been studied by embedding a piece of special glass in the compression zone of patterns obtained by passing reinforced COIV. involved in this method are therefore considerable. ~Z Is A plain concrete the maximum load in concentric strain response to load is highly therefore.3H2k! beams.’8 Test results indicate that although strain is linearly distributed even near failure. concrete strained beyond the maximum point of the stress-strain curve is useless for loadcarrying purposes in beams as well as in columns. concentric compression to flexure challenged. that. Bending-simulation machine—A special testing machine has been constructed at Imlperial College. Bureau of l<eclamation. The action of compressive stresses in the concrete compression zone of a prismatic specimen are ~imulated by six pairs of hydraulic jacks operated by six independent pump systems. sensitive to time. 14 to study the behavior of reinforced concrete members. the distribution of compressive stress is curvilinear with the maxi- . a straight-line stress The principal weakness of this distribution was found even at high loads. An opposite view has been expressed by some who point out that special testing techniques are required to determine the true stress-strain relationship in concentric compression. 2 are applicable to flexure of reinforced concrete. while the central section where strain gages were located was reduced to give a “modulus” similar to that of the concrete. London. Small pressure cells were embedded in reinforced concrete beams. It was found. that variations in moisture and humidity conditions greatly affect the observed stress distributions since volume changes of the The experimental difficulties concrete influenced stresses in the glass plate. loads corresponding and at each stage to a linear dis- by trial the distribution of jack tribution of strains in the specimen as indicated by strain gages. Interpreting the strain readings in the individual steel plates in terms of concrete stresses. experimental method is the unknown extent to which the introduction of the steel plates locally changes the stress conditions in the concrete. S.’7 At both ends the plates had the same width as the beam.460 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 Application of stress-strain relations for of reinforced concrete has been vigorously specimen which has been strained beyond compression is generally cracked. and the It has been asserted. however. lb Photoelastic polarized light through the glass plate at various beam loads gave information from which the distribution of compressive stresses was determined. 5 to 0. . fvj and beam dimensions are known. On the other hand. . . . ..~. .625 + ~:_ 600 1500 . . is measured. Concrete does not resist tension.stress is cOn~tantand equal tOiv. .. .. . Concrete compressive stressis a function of strain only. expressions that have been suggested for the ultimate strain cti at the compression face vary considerably. . . . . . . .(7) Prentis.. and Lee24-20 have derived further information regardThe ing the stress block from beam test data by numerical differentiation. .. 4. is proportional to compressive strength. . In this respect most investigate ors have agreed. . . Linear distribution of strain. . . j = F (e). . .’. . has generally been derived indirectly by computing from the requirements of equilibrium those properties of the stress block which were consistent with the observed behavior and strength of reinforced members. . methods used 10Y these three investigators are related. .. Effects of time and of strain gradient are neglected. Saliger” indicated that ~.0034 to 0.. The reinforcement is elastic to the yield point. . . .’ ““’’’ ”’’”””’”’”’””””’’””’”’””””” ““””””” (6) and for f.’ The value lclk~ has been evaluated from depth c to the neutral axis at the ultimate k~. . . . The use of such pressure cells is a promising approach to studies of the stress block. . In recent American tests’s . .6 have u~ually been suggested. . . . Appleton’ found 3000 + 0. 3.5 f: k. Values from 0.. . . (3) when Muz~. Then beam test data by measuring the moment and assuming a value of T= C=~~=k IkJ(b c. . . The ratio k21’lcllcz has been studied extensively since this ratio can be determined by Eq.’. Strains have been measured directly with the aid of strain gages.. For instance.. Beyond the yield strain ~~. . 1..(la) Since Mul. . . . . and Lee’s derivation Four major assumptions were made: typifies the principles involved. .. A large number of tests have indicated that the distribution of strain in structural concrete is essentially linear even near ultimate load. .lc3 = —— klk~ can be evaluated. . . . however. . Tests of reinforced concrete—Most of the available data regarding properties of the stress block have been obtained from tests of reinforced concrete members. . ..ULTIMATE STRENGTH DESIGN 461 mum measurecl stress equal to the cylinder strength. . . . values of 0.23 it has been fo~nd that CUis largely independent of compressive strength. Information regarding stress distribution.ks = 0. . = E. ‘“ ROS. In this manner Billet and 1. . . . f.500+ f. . . . while Brandtzaeg.0038 have been suggested.G.. The need for fairly complex and expensive instrumentation is probably the principal reason that the use of this approach has been limited. 2. .2’ and Jensenz’ indicated a decrease of CUwith increasing strength. and average e.’ greater than 2000 psi Gaston23 obtained lc. Harnann. . differentiating with respect to c. . . ... and strain cC may be established by Eq. 4). . .. . . . . . . . . . 2 . however. ./AcC Each of the.(10) .. . (8) into Eq. . . . . .s c c for O<c$ ) < 6. . If e. and some tensile stresses are carried by the concrete even at high loads.. three investigators presented only a single application of the mathematical methods to test data. . particularly by avoiding the rornplications introdrrced by the presence of reinforcement and tensile stresses iu the concrete. .— ~. . METHODS OF TESTING AND ANALYSIS Prentis applied numerical differentiation to test data from an unbended prestressed b(earn. At low loads the col]mete is I]ot. (9).. (8) by Substituting arranging f.. By further applying a minor thrust Pz that could be varied independently of PI.. steel strain q are influenced by the position of strain gages relative to cracks in the ~onrxete tension zone. . (10) and (11) when the differentials de. cracked and Eq. (10) and (11) do not apply. ..+. . Eq. . Certain experimental errors.. the position c . and re- ( () ~c~+zes~+. .]dee are replaced by finite differences Ac. . stresses in the concrete were eliminated. The same principle was used in the present investigation by applying a major thrust PI by a testing machine (Fig. . . . tend to be strongly amplified in nuFor instance. the neutral axis could be maintained at a face of the test Thereby complications resulting from tensile specimen throughout the test. . . and q are observed in tetit beams loaded in small increments... ... . + % of the neutral axis is then given .. .. . It was felt that these deficiencies could be overcome by modifying the testing procedures.. .. . measured values of the merical cliff erentiation of this type... . . a relation between stress ~. .. . ..469 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 ‘k=H E7-BPF(&x) “g”’-Ben’i with the notation of Fig. . . . . ... ... . 3. .. which permits more accurate steel strain measurements than are possible for embedded bar reinforcement. .(11) fc=flfd 1+: for % > % . . .. . . and promising results were obtained.. ULTIMATE STRENGTH DESIGN 463 Fig. 4. face was also recorded. and actual areas rather than the nominal area of 40 sq in. The central unreinforced test region has a cross section 5 x 8 in. P. dimensions of the test region were measured carefully by vernier calipers. 5. rod force was adjusted continuously so that the strain at the neutral face was maintained at zero within 5 millionths. the deflection at mid-height was measured by a dial gage so that the eccentricities of the thrusts were known at all load levels. was introduced by a hydraulic jack through one or two tie rods. which tends to amplify experimental scatter. All of these strains as well as the output of the tie rod transducer were recorded continuously throughout the Transverse strain on the compression test period by an electronic recorder. The specimens were carefully leveled in the machine. . electric strain gages at the neutral surface. The main thrust PI was applied at a steady rate from zero to failure during By operating a hydraulic pump. the major thrust PI was applied by a testing machine. Strains were measured by two 6-in. steel roller. the tie a testing period of about 15 min. Method of testing Suitable tension. and one gage at mid-depth of each of the two side faces (Fig. all testing operations were carried out with unusual care. The sole purpose of the reinforced brackets at the ends of the specimen is to accommodate the two thrusts. These considerations led to a specimen with dimensions as given in Fig. As shown in Fig. All such reinforce ment ended at least 4 in. from the test region.. and shear reinforcement was placed in the end brackets to obtain failure in the central unreinforced test region. and the load was measured by a calibrated electric strain gage transducer. the relative positions of PI and P2 as well as the stiffness of the tie rod by which Pz was applied were so chosen that no uncontrolled release of strain energy could take place. Since numerical differentiation. compression. was to be applied to the test data. 4). After curing. and is 16 in. long. two gages at the compression surface. The minor thrust and the load was applied through a 3/16-in. Finally. were used in all calculations. 4—Tesil specimen 1 Considering the stiffness characteristics of the testing machine used to apply the major thrust PI. cylinders were tested with each specimen. 4) k. P1al + Pfat (1~. 5—Testing arrangements Three or fcmr 6 x 12-in. Analysis Without assumptions moments (Fig. +r2)c in which the eccentricities al and az include both the initial values and those In this manner the important properties due to deflection of the specimen. The complete stress-strain able assumptions are made: curves in flexure may be obtained if two reason- 1. of the stress block kllcs and l~z/klks may be measured directly from zero load to failure. ‘Ilk has been well verified by numerous earliertests and was confirmed also by the tests reported herein. (12) bcf.k. . Two 6-in. Linear distribution of strain across the ted section.464 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 Fig. electric strain gages were mounted on one cylinder of each group. =—=— be jc’ k2=l–– of any kind we obtain by equilibrium of forces and c l’.’ ““’ ””’”’””’’”’””’”’’””””””’”””” “’’’””””” ““’’”””” (13) ”””””’”’””’’’’’’’”’’”””””” ‘“’’’’”’”””’’””’”’”’’”””’ + P. and strains were recorded as a function of load in a testing machine with known stiffness characteristics. .=6. . . (16) and Eq. (1. respectively.(15) P. crete stress is a function of strain only. I 0 c . Three pairs with w/c ratios of 1. 3 and 4) . . . by differentiating . f = F(c). . 28. and 90 days from the casting date. Eq. wc obtain ~+2rno.. and strain e.. . and rearranging Eq. The ages at which tests were made were 7. 0. Thereby the cylinder strength was varied from about 700 to 7600 psi. . . . e~~entriciti& U1 and az stron”gly affect Eq. 14..67. . Therefore. . . . . All (‘fibers]] follow one and the same stress-strain curve. .. + P2 hc Plal + P2az bc’ “ in which f. . ..67. (17). while these numerical quantitles are not involved in Eq. . .. .40 were tested at ages The test data for duplicate specimens of 14. ~c F(6. . and 0. con- 465 2.. = — By ditierentiating 1 — ~ and m“ = the third and the fifth term of Eq. to e. 1.40. . . The maj orilty of the tests involved one specimen representing one of the 20 combinations of w/c ratio and age. Five water-cement ratios were used. . (16). VARIABLES Variables AND MATERIALS The variables included in this study were quality and age of the concrete. . . 14. . . . . . and 0. . by calculating in small increments. (16) and (17) give concrete stress j“. (14) with respect .50. . Thus. . 0. .33 by weight. In the initial stages of the investigation. +%=d+ e.0.4) into (14a) o. . (15) . were within 4 percent of one another in all respectsj even after cliff erentiation. the investigation was continued with single specimens. as computed independently by Eq. .. . . .(16) Similarly f. . . .. .(17) Eq. the differentials may be closely approximated by finite differences Afo/Ae. . .0. (17) are compared.)de. . .. We then obtain by equilibrium of forces and of moments (Fig. . and Amo/Ae. . ..ULTIMATE STRENGTH DESIGN In other words.(14) . specimens were cast in pairs to study the reproducibility of the test results. 0. . accuracy of the test data is checked since several experimental sources of error affect For instance. and 7 days. as a function of the continuously measured strain G and the measured quantities ~o and rno. . . 0. . (14a) Substituting fc=cc~+j’ . It should be noted that when relations between stress j. errors in measurement of the the two equations differently. 5 40 38 3. and a.5 3 3 3 3 3 3 3 1. forms and in the 6 x 12-in. Mix proportions are given in Table 1. After about 18 hr. TEST RESULTS Ted data recorded for each specimen gave strain in the prismatic test region and the tie rod force PA as a function of the major thrust PI (Fig. I 775 1420 1650 2175 1670 2870 3050 3710 2950 4595 5170 5470 5350 6070 6720 6480 5715 6480 7540 7610 I . psi 1 Slymp. The size.: 90 7 14 28 90 ——— 7 14 28 90 I —. CYlind. and the specimens and their companion cylinders were placed in a moist room for the remainder of the curing period.’. The concretes were mixed in either a 2-CU ft tilting drum mixer or a 6-CU ft nontilting mixer. and was later lightly troweled. Percent sand w weight of tOtaI aggregate w/c. PA was initially increased with PI until a maximum was .16 222 0. a.466 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 TABLE l—CONCRETE MIXTURES Cement. III.40 ———— 28 7. face of the specimen. cylinder molds by an internal spud vibrator. by weight — 2.6J 239 0. drying was started by storage in the laboratory at a temperature of about 75 F and a relative humidity of 50 to 80 percent. Ill.? 3 3 3 3 3 1. maximum The gravel was separated into three fractions and recombined. Forms were removed two days after casting.00 A B c D E 7 % 90 7 M 90 7 . 4). sacks Per CU yd Water. The surface was screeded about 15 min after casting. f. This period of drying was necessitated by instrumentation requirements. Specimens were The concretes were placed in plastic-coated cast in a horizontal position.67 35 4. Five days before testing.5 2 3 3 1. Coarse aggregate was a gravel of 1~-in..50 _—— __ 32 6.33 The cement used in making the concretes was a blend of Type I cements. were so chosen that PI always increased until a The eccentricities To maintain the neutral axis at a maximum value was reached at failure.01 227 0. Arength. lb per (3Uyd concrete Age at days I——l test.72 255 1. Fine aggregate was Elgin sand modified by a timaIl amount of lake sand to improve the grading. the specimens and cylinders were covered by moist burlap for initial curing. and all specimens were tested dry.75 225 0. aggregates arc a partly siliceous and partly calcareous material of glacial origin from Elgin. release of strain energy took place.ULTIMATE STRENGTH DESIGN 467 . There are strong reasons to believe.6 “ / .15 of the equip- Fig.3 “ + — := 1. therefore./’+--+ . by Eq.001 . 6 shows the values of ?c.9 Age= ./’ . that the observed phenomena are related only to properties concrete test specimen.lc. but the nature of the curves obtained at other test ages was similar. These values represent the average for the entire depth from the neutral surface to the extreme fiber. Fig. ~ Concrete .+ . the piston of the hydraulic jack in the tie rod system was conIn this manner no uncontrolled tinuously extended throughout every test. At low loads k.002 . not to the elastic properties of the loading ment and the testing machine. after which Pz decreased again. is near the value 1/3 corresponding to a triangular stress distribution. The smooth and steady increase of the k1k3 curves in Fig. though P2 decreased deflection of tlhe specimen shortened the tie rods. With increasing load. of the tie rods were always so chosen that PQ tended to decrease rapidly as Thus. 6—klks and kj versuscompressive strain The eccentricity a. and k. and stiffness reached.003 Strain.7 . (12) and (13). the position of the resultant internal compressive force changes. at the compression face. however.4 / . and . 6 indicates a rise of the average concrete streiss to about 90 percent of the cylinder strength for a high water-cement ratio and to about 65 percent for low w/c ratios. These curves are given only for the 90-day specimens. computed as a function of the strain e./’+ . 6= ?./ —-—-+------~+ ~ ~ I [3 / .5 ‘ . near failure.0 :F 0 .8 90 days .. W/c = 0.’ regardless of concrete age. which were derived indirectly from beam tests. Ultimate strength design values Three stress block properties of fundamental importance in ultimate strength design—lc.0 Age= 7days .k. = —— 3200 + .. a variation whi~h may be expressed by k. and E. . Strain distributions across the test section of four typical specimens are given in Fig. and various curves representing equations given by previous investigators are shown. 28 days 1 .. .40 Age= i4day:j .003 . . These points form a rather narrow band in the lower scatter fringe of the points derived by Billet and Appletons and Gaston23 from reinforced concrete beam test data.—are given in Fig.50 Age..005 INSTITUTE December 1955 I La ~1= Wj =1.’ ..001 - [.CK3 I o ( &-in. . . 7—Typical strain profiles AA $ w/c = 0.lc. 8a reveal that lc. .004 c .35 . (7). (18) Eq.004 ... Jet/k. A decrease of hJc8 with increasing cylinder strength is evident. (18) gives lower ?cIlcgvalues than those obtained by Billet and Appleton’s Eq. since a slight decrease in lclh~ and an increase in 1c2 at times took place near failure while visible crushing was in progress.f...002 4 $ .. A linear distribution of strains was found even at high loads. .fC’ 3900 + 0. 8 as a function of cylinder strength and with test age as a parameter.f 6 .. .053 w~c = 0. kl?c~values—-The 20 points of this investigation shown in Fig.468 JOURNAL OF THE AMERICAN CONCRETE . All quantities correspond to the maximum values of klk~ obtained in each test. is a function of j. as has been shown by numerous previous tests of reinforced concrete beams and columns.k.002 . 0 Fig. ‘Values of klk~ derived from other tests of reinforced concrete as well as values of eu measured in such tests are also given in the figure. 90 days i for the high w/c ratios lLZ reaches the value 1/2..33 Age.. which value could also be obtained by a rectangular stress distribution. (6) and Gaston’s Eq.?c%. 7. 8—Ultimate strength properties of stress block .Hognestad —= “-$~+~%~~~n \ . o A A Age II II II 7-days 14 U 28 90 of “ “ Reinforced . A.d.’r-Hognestad -+’L ——————— —________ >--–7––6 /.Hognestad . 0 ----i~-. . .~---J&nsen -—____— . Fig. psi Z. \\” \\. . . .g~= ‘:- k. .k3= o A o -A-— ——— 3900 +o. \ (c) _ \“ \ --?&A +: .35f: /’ —-_ ~~.00( .\ % \ \ ) . .\. References ~7(-JyJ~g(yyJ f:. .. Tests concrete.( (0 +——’— a —— _______ A __-. ‘.’ /’ /’ “-kfi.Gaston (a) \ —j’---- ~ -’Jensen < \“ ~’ . . —————— —— ___ —— (b) ..) L. . +/ )’ +— -- ————- ‘~u= ~ .35f: + .~-Whi+ney .:~. ——— .. .k3= 3900+ ~2m+f: o._ A :~=>= .\ r * ULTIMATE STRENGTH DESIGN 469 1 \ .00-4-f:/6. 5$23 looo20003m34ay3~ Concrete Cylinder Strength.5x\06 + 0 .. ..Jen~en . For low concrete strengths. .. he considered k.. Ultimate sh-sins—The ultimate strains eu measured in this investigation are comparedl in Fig.fo’ ““’ ””””’””’”’””’”””””’””””’ f. . . he found low values of kl~63as compared to Eq. ..ld. Concretes with a slump of about 6 in. cylinders for testing./k. lcJkjks groups of beam test data.5/0. (18).lc.46 fc’2/80. All failures occurred in the upper half. . . Compressive strength of the top cylinder was about 25 percent lower than that of the bottom cylinder. For practical design purposes Whitney’s choice of constant value k2/klk.=~..’” k. In investigations of the ultimate strength has usually been derived by studying test results of large of beams. ... 1600 + 0. high which were sawed into four 8-in. and water gain probably led to a reduction in strength in the upper part of the columns. . .. On the basis of found an increase of ever. . .470 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 However. . . = 1. .59. . Fig. . 8a.35. and concentrations of curvature and strains were found near the tops of the columns./klk. were used in the latter tests.65. assuming that kz/klkt is independent of concrete Fc. . = 3900 +o. . 8b) indicate function of concrete strength regardless of age: k. . .000 L+. .k.. . . . . . (18) and the k~k~ values obtained on a theoretical basis by Jensen. 8C to those of numerous earlier tests of reinforced con- . lcp/itlkt vatues—The ultimate moment of beams failing in tension is relatively insensitive to variations of il. . a clo~se agreement exists between Eq.. 1+8 . experimental scatter will be amplified strongly. ./k~k.. . . .""" `"'"`"""""'''"'"'"'"'"'"'""'"'"''"""' () (20) A curve representing lcJcs values derived by Hognestad2 from tests of eccentrically lc~aded reinforced columns is also shown in Fig. .(19) in which the plasticity of reinforced concrete b= 1+ ratio D was derived beams as from measured ultimate moments ~. If this ratio is computed from individual measured ultimate moments by Eq. 8b. therefore. Jensen27 with cylinder strength as shown ir.. . . tests Hognestadz found k2/klk8 = 0. . ..85 = 0“59 (Fig” strength. . column Data obtained in the present investigation (Fig. ... .. A similar bleeding effect was observed by Kennedyzs in vertically cast cylinders 32 in.0). which gives satisfactory agreement with beam tests.jk.” By amuming a trapezoidal stress block with a maximum stress equal to the cylinder strength (k. 8b). howit sufficiently accurate for practical purposes to take value of 1/2 has been much used in foreign literature. Whitney2g found kz/klk3 = 0. should suffice.. Jensen obtainecl . . . . . = 1/2. It is believed that the principal reason for this difference is that the prisms of the present investigation wel e cast in a horizontal position whereas the colums of Hognestad’s investigation were 6 to 7 ft high and were cast in a vertical position. (3). The k.’ – is a (21 ) = 0.r instance. . to properties c}f the testing is therefore substantiated..U + 6% . .ks ti> ~ .(5a) and C=k. .. = 0. . 9.. . is ewident. . . the maximum Stress-strain. ....003 is believed to be sufficiently the computed ultimate strength of reinforced concrete relatively insensitive to the numerical value of ~.(23) (23a) ~b= ~b g f. . ... . . .. our points form a rather narrow band in the lower scatter fringe of the reinforced of c. . . . . .004- 10s..5X concrete test data. . and ultimate strength design may be carried out by Eq... . . fe’kUbd =l’=A. .. . The stress-strain curves obtained are shown at the left in Fig..=k. . . By assuming that strain is distributed linearly and that concrete stress is a function of strain only. fv . Failure is then initiated by yielding of the tension reinforcement. . . . The applicability of Eq. the cylinders failed suddenly shortly after the maximum load was reached. pb.. (3). .. Except for low concrete strengths. . 9. . . Eq. ku=i . ... to cause a cc)mpression failure. which may be expressed A variation by cu=0. . . and it is seen that the sudden failures always took place when the cylinder stress-strain curve reached a slope equal to that of the machine that sudden failure of cylinders is related curve.. . . or .. . 'klh`~g +~u'" ''"'"'"""""'''"'"'""'''""""''"""'"""'""""'"'"' % Stress-strain culrves Ultimate strength properties of the stress block were computed by equilibrium of forces and moments. concrete Again. with concrete age is found. . with fJ/6. . Whitney’sT suggestion. . ... . . curves obtained in concentric compression tests of 6 x 12-in. . . (16) and (17). (16) and (17) can be used to derive complete stress-strain relationships f or concrete in flexure. . .. .. relatively low reinforcement ratios are generally used in reinforced concrete beams..(22) For design purposes conservative since members is usually a constant value c. . we have for rectangular beams (3) ceases when sufficient reinforcement is present To evaluate the balanced steel ratio....k.ULTIMATE STRENGTH DESIGN 471 mete columns variation strength and beams. . . . . machine rather than to properties of the concrete.. Lines related to the stiffness of the testing machine used are shown in Fig. . control cylinders are shown at the right of Fig. . . . . and no systematic of c.(lb) which gives p. .. . 4 Balanced steel ratio in beams—In practice. ... .. The curves represent the average of values obtained by Eq. + % f. . difference being about 3. Results obtained by the Itwo equations usually differed by about 2 percent. . 9..5 percent. ... . .. / -->. . ‘“% ~ -+ . cylinders> stress-strain relations . .479 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 .... .’. .LOl (6 ~ . .s JM4 . ~ 4: ~ $.T~ .050’. .~XUrSl Tests (5 by 8 tnj16-in Prisms) Fig. -. 9—Concrete ..Compression Tests IZ-in..X)4 o .0..+..0 ..J ~ ..0 Strain ..X1..(332 . —~ o 6 Age-90 days . Yc-a33 .%~~ .->.7 days .CQ2 LOS .=im _ F...9 c r’ I Age... . . . . . . as would be expected. .000 . . 10 as a function of cylinder strength... 12 as functions of cylinder strength j. and k.(24) ka=0. . 9 to 11 indicate that the true general characteristics of stress-strain relationships for concrete in concentric compression are indeed applicable to fiexure. . ..~ylinders _j’day-o —0 —v A—78 1 t 7—90 I! m Concrete 4c00 5000 6000 Cg I inder Strength -f. The similarities shown in Fig. 10-Modulus . were obtainec[ by Eq. (18) and (24) then give k. corresponding values from flexural and cylinder tests differ relatively little..’/80.j//26. le. . of elasticity 7000 psi 1 I mm 9003 Fig... Straight lines define k. . may then be obtained from the measured values of lcllc.000 p5i+4G0 f~=EC 1 “2 IM I I 1- Prisms oo~J ‘r Icno mm . Values of k. are summarized in Table 2 for convenient reference.E =3 . Individual values of kl. ..’.. and k.35 j: 3000 -t. II —~ . . (13)..000 .. 9) give individual values of kt. and k.g-4 c 0 .94–f. . Though age at test affects the modulus-strength relationship. .. kg.f. 5 ...= 0.50– j. 11.””” ”””””””” ““’ ””””””””””””””’””””””””’””””’ (18) and (21) (26) to (26) Values of ultimate strength factors as given by Eq. The two moduli are compared directly in Fig.(25) Eq..800..’ -. In this manner k. 9 shows a striking similarity. The initial moduli of elasticity obtained from the two sets of curves are given in Fig. ./26.=— 3900 + 0.0..ULTIMATE STRENGTH DESIGN 473 Comparison of the flexural and the cylinder stress-strain curves in Fig.— ~1~ ) ~ /@’ / . were plotted in Fig. and k3 The maximum stresses obtained in flexural tests divided by the corresponding average cylinder strength (Fig.82. . ..— I —14 .~1:--Jensen ‘--1. k. 58 0. .41 -. .75 0.70 0.65 0.39 0. 0. . 1 EII. = .49 0.43 0.80 0.60 E%% Eq:(22) 0.46 0.54 0.– n -0 5 E= fro’m Cyl. . .. The load at which the derivative of volume strain with respect to load is assumption zero was referred to as the “critical load.42 0.45 0.0037 0.0034 0.--.44 0.54 0.—E4 ~ 3 X3 al c 1l—Flexural moduli versus compressive of elasticity - : U* .0031 0. . .49 0. however... .66 0.01 0.92 0. and c~ are numerical values of measured strains in longitudinal and transverse directions.67 — E~. . where c. respectively. . .40 . .— 50 ksi I 60 ksi E$$8] 1.66 0.48 0. .39 0.25) E&26) 1.65 0.94 0.03 0.0032 0. . . ::. .474 c JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 19s5 Fig. Eq.74 0. .42 0.0028 Eq. ..si 0.51 0. (27) may be used as an approxiare possibly not equal. For a small element at the compression face of a flexural member.97 0.. . (23a) for f.37 —. .48 0. psi Phenomena of failure Important (observations regarding the failure phenomenon in concentric compression tests of concrete were made by Brandtzaeg$O through studies of volume strains .!.’ ‘“’ . ” It is a fundamental in Brandtzaeg’s theory of failure for concrete31 that a progressive internal splitting is initiated at the critical load on minute sections scattered throughout the concrete mass.65 0. e! being the transverse TABLE strain in the plane STRENGTH of the compression face.. .61 0.67 0.45 0.12 0.90 0.58 0. At 77 to 85 percent of the maximum load. 2—ULTIMATE FACTORS I 40 k. At low loads he found that the volume of test prisms decreased roughly in proportion to the applied load. . and strains in the two transverse directions Nevertheless.57 0.69 0.f 0. of longitudintd strains is present.0038 0.62 0. . .43 0.89 0. . .k\24) 0...0035 0.53 0. “g 5 c o . . a gradient.(27) ev=ec — 26. volume started to increase at an increasing rate so that a volume expansion finally was present at failure.35 0.nders : millio.37 0.41 0.46 0.45 0.!36 (l::.64 0.. mation. .49 1. .61 0.0029 0.93 0.92 0. .94 0.63 0. which were longitudinal tension cracks on the compression face of the specimen. The transverse strain data were then not significant at high loads. All low concreltc strength specimens failed by crushing to a depth of 1/2 to 2/3 of the total depth of the cross section.0. The magnitude of the critical stress varies from 71 to 96 percent of the maximum stress.m 20 “ “ 2000 Concrete Wxl Cylinder 4000 5000 fj.4 /’ A. Two examples of successful measurements are shown in Fig. large deformations took place after the critical stress was reached and before failure took place (Fig.f. 9. In some cases a large piece of coarse aggregate was located adjacent to the gage and near the concrete surface.90 11000 Strength. . E#J psi 7um mx) +o ‘.. For low concrete strengths. Even so.ULTIMATE STRENGTH DESIGN 475 0 + 0 -m-~ A . beams and eccentrically loaded columns Hognestad2 observed critical loads at columns. Similarly. .~50Q. Points corresponding to critical loads are indicated on some flexural stress-strain curves in Fig. 12—Ultimate strength factors In this manner BrandtzaegZO found critical loads at 70 to 90 percent of the ultimate load for reinforced concrete failing in compression. the first visible cracks. “-iy-— /m.& & + A .50. + 0 0AA- Age ‘t ‘1 7 days 14 u . which was 58 percent over the critical load. for instance. Fig.’ : .. . 13. the strain corresponding to the critical stress was 21 percent of the ultimate strain. . electric gage mounted on the compression face of each flexural specimen. 9). were not observed before a load 99 percent of ultimate was reached. reported herein by loaded 80 to 95 perce:nt of the ultimate for eccentrically Transverse strains were measured in the investigation one l-in. For the 28-day specimen with w/c = 1. 79 percent of the cross section was stressed beyond the critical stress. At the ultimate load. therefore. the concrete stress distribution deviates considerably from the triangular distribution used in This is in complete accord with earlier indirect the straight-line theory. N. Structural concrete design The tests reported here show beyond reasonable doubt that the flexural stress-strain relation of concrete possesses a descending curve beyond the maximum stress. In tests of reinf omed concrete beams. 37 Jensen. so that strains greater than those corresponding to maximum stress were developed before cracks appeared. these principles involved tests verify the reality and validity of the fundamental in ultimate strength flexural theories such as those presented by Whitney.001 — Unit Strain Behavior of the high-strength specimens differed somewhat. and the ultimate load was 66 percent over the critical load. and crushing took place only to a depth of 1/5 to 1/4 of the total depth. if strained beyond the maximum load. a redistribution of stress may have occurred to less strained material closer to the neutral axis. Concentrically loaded cylinder specimens will usually crack considerably In the flexural tests reported herein. the critical strain was 46 percent of the ultimate strain (Fig. may be helpful in considering design values suit- . and by a number of invetkigators later. 27 ancl others. Baumann in 1934. 9). which characterize the properties of the stress block (Table 2).476 JOLK!NAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 — Fig. 13—Typical volume change curves ---/ Critical Stress Unit Volume — . findings in which the shape of the stress block was deduced from observed behavior ancl strength of reinforced structural members. The first cracks were seen at ultimate load. this important phenomenon was observed by A. !l’albot in 1906. For the 28day specimen with w~c = 0. Therefore. The numerical constants obtained. therefore. by O. Near the ultimate load.33. the present study represents the most complete investigation yet reported of measured properties of the stress block. strengthen knowledge regarding the fundamental principles involved in ultimate strength flexural theories. strengthen our knowledge regarding the stress block in flexure. however. and 0. which characterize the properties of the stress block. 28. Nevertheless. 0. and the centroid of the stress block coincided with the eccentricity of the total applied load. Jensen. to simplification of routine design methods. S. 14. A striking similarity was found between flexural stress-strain relations and those obtained in concentric In both cases a descending curve compression t(mts of 6 x 12-in. and others. and the test data obtained demonstrate the reality and validity of the fundamental plasticity concepts involved in ultimate strength flexm-al theories for structural concrete such as those presented by C.0. it is believed that the tests have disclosed both general principles and quantitative data c)f some significance in the field of ultimate strength design.”’and 90 days. are in general agreement with values derived earlier from tests of reinforced structural members. therefore. The numerical values obtained. to construction practices.33 at test ages of 7. and to many other matters beyond the scope of the present paper. one maximum aggregate size. a joint ACI-ASCE committee .ULTIMATE STRENGTH DESIGN 477 able for practical ultimate strength design. strong consideration must be given also to tests of reinforced concrete beams and columns. To make definite design recommendations. Furthermore. and one size of specimen withouh compression reinforcement. though many test methods have been tried. The tests reported herein. The average compressive stress in the concrete then always equaled the total axial load divided by the section area. . A study of previous experimental investigations regarding the stress block revealed that.50.33 SUMMARY This investigation ing a test method was conducted with the primary objective of developleading to an improved quantitative understanding of In other words. cylinders. by numerical differentiation the flexural stress-strain relationship of the concrete was determined from zero load to failure. it was our goal to concrete stress distribution in flexure. unreinforced concrete section was loaded with an eccentricity that was varied during each test in such a manner that the neutral axis remained at a face of the section throughout the test to failure. Such tests vvere made for concretes with w/c ratios of 1. but they are also fully aware that the tests cover only one type of aggregate. V. As far as the authors are aware. P. Such recommendations are available in the report of Committee 327.40. to overload factors. On the othel hand. very limited direct test data are available. 0. A test method was developed in which a 5 x 8-in. 0. Ultimate Loacl Design. considerable information regarding the stress block has been derived indirectly from strength and behavior observed in numerous previous tests of reinforced structural members. was observed beyond the maximum stress. Whitney.67. Blanks. 1953. 103. 1951. 4. [‘Ultimate Flexural Strength of Prestremed and Conventionally Reinforced Concrete Beams.. 1932. and Its Application to Design. 2.. No. T.F.. “The Plasticity crete and Its FMect on the Ultlmate Strength of Beams. 20. 5842 to 584-6. “Recent Research in Reinforced Concrete. Apr. 18. “Bruchzustand und Sicherheit im Eisenbetonbalken. 317-320 and 33%346. 35. and Maldari.. Nov. 4. V. Supplement. M. [‘A Study of Combined Bending and Axial Load in Reinforced Concrete Members. 8. Llniversity of Illinois Engineering Experiment Stiation.. D. 39. Jllne 1952.. Part III. “Studies of Compres~ivc Stress Distribution in Simply Reinforced Concrete Near the Point of “Failure. L. Proc.me pout. Gilkey.” Wisconsin.j “Ueber die Sicherheit des einfach bewehrtcn Eisenbeton-llechteckbalkens. A. 4. Proc. Jun. Stress Ratio of ConSupple- CiVrZEngineering. D. V. 17. E. L. for Bridge and Structural Zurich. ‘‘ Proceedings. 12..” Betcm uv. rechtecklgen Stahlbetonbalkens. ” Bulkk. 16. O. V. Berlin. Bachelor of Science Thesis. 39. pp. SF-12. Baker. E. Baker. Billet.. 5. 1938. Pro. 1950. V. J. ” Ci. U. 7.. ” Proceedings. 1!3. Discussion of a paper by C. V. pp. 6. Haberstock. 5846 to 584-15. 50.. Hognestad. ” ACI JOURNAL. ” Bulletin No. V. V. Con- 5. F.” Journal of the In. 8$7-854. d’.” Laborator~ Report N’o. AC1 JOURNAL. No. K. ‘[The Plasticity and Its Effect on the Ukimate Strength of Beams. for Bridge and Structural Engineering. London. 114-125. 1951. Discussion of a paper by V. and Vandegrift. B. Parme. 128pp. V. PPOC.. 14. NOV. pp. Jensen. 1941. ‘[Plastic Flow of Concrete Relieves High-Load Concentrations.. M:~Y 1949.d Eisen. 30. Siess. V. “Fundamental Concepts in Ultimate Load Design of Reinforced crete Members.” Publications.. 64 pp.. Feb. Ilrice.. L. “R. E. L.. A. ‘[Stress-Strain Curves for Concrete Strained Beyond Denver. Kiendl. A. 11.. ” ACI JOURNAL. 269-310.23 pp.” 1943. V. 864. 249-251.re Engineering. Ott. Mar.4. 160pp. 1950. . Whitney. PP. 10.. No. Aug. J. 1952. 3. C. 1. 13. 17. 35.V. I’.” A(X JOURNAL. pp.. . 15.. D. “Review of Research on Ultimate .tiMion of Civil Enginws. 262-298. and McHeruy. ment. H. ” ACI JOURNAL. 1950. P. “Further Research in Reinforced Concrete.. Institution of Civil Engineers.” June 1954. ACI JOURNAL. J. 10. D. and McHenry. International Assn. pp.n No. ” Publications. L. and Its Application to Ultimate Lcmd Design. [(A Comparison of physical Properties Of Con~ret~ 9. Hognestad. Discussion of a paper by V. 48. Highway Research Board. 2. Berlin... Herr. F. I. Jenscm. Discrete. pp. D. 1-9.. 862Strength of Reinforced Concrete Members. Saliger. 1936. S. ‘[When Concrete Becomes Apr. pp. Proc.V. Bureau of Reclamation. Deutscher Auwchuss fur Stahlbcton. P. London. A. 399. Janney. pp. Urbana. E. pp. Stusei. Ramaley. ‘ ‘Flexural Strength of Prestrcssed Concrete Beams. 809-828.. P. 19 and 20. Zurich. G. International Assn. No. the Ultimate Lo~d. and McHenry.478 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE December 1955 REFERENCES 1.. awaiting publication. L. H. pp. ‘I!ksaie de mesure dcs contrainbes dam une section fl&hie en b4ton arm6. pp. L.. Ratio of Concrete 1943. University of Made of Three Varieties of Coarse Aggregate. 487-495.il Engineering. pp. and Appleton.. Nov. Hadley. “Die n-freien Bereehnungsweisen des einfach bewehrten. 1947... 48. Hognestad. 320-322. Nov. S. R. V. H. Brandtzaeg. “Failure of a Material Composed of Non-Isotropic Kgl. 44. on Prestressed Concrete. Eisenbeton- 22. ASCE. 7’0. pp. 30. A. 33. Massachusetts 126-135.J. Material—l?rufungsanstalt. 29. 251-282. C. 80. Urbana. 483-498. Oct. 20. ” Avhandlinger til 25-iirsjubileetr Norges Tekniske H$iskole. Thesis. 32. 1950.” Proceedings. V. 8. 68 pp. pp. Norske Videnskaber.” ACI JOURNAL. London. pp. “An Investigation of the Load-Deformation Characteristics of Reinforced Concrete Beams up to the Point of Failure. Zurich. “The Failure of Plain and Spirally Reinforced sion.. 1955. “Design of Reinforced Concrete Flexure and Direct Compression. June 1943. 809. 1951. Dec. Brandtzaeg. 23. “Festigkeit und Verformung von auf Biegung beanspruchten balken. V. [(plastic Theor3. 1942. V. C. &7elslcabs Skrifter.” BuZletin No. Whitney. Cambridge. ASCE. Members Under Flexure or Combined 1937. No. No. L. Brandtmeg. Saligcr. 26 pp. “Berechnung schnitt. 5.V. [(Inela~tic Behavior of Reinforced Concrete Members Subi ect tO ShOrtTime Stat/c Loads. Jensen. University of Illinois Engineering Experiment Station. 1951. 1940. Urbana. V. No.” Det 31. Zurich.Jube 1943. Mar. S. S. 1927. V.. 1. ASCE. Separates No. Proc. 345. N.” Magazine of ConcreteResearch. Kennedy. 132 pp.ULTIMATE STRENGTH DESIGN 479 19. Nov. 39. 26. 73-77.. R. Proc. 1956.” Bericht No.~enkche Bauzeitung. 1942.. 1952. 677-764. 1935. 565-582. Conference Institute of Technology. Elements. P. 107. A. pp. Aug. 72 pp. 28. Lee L. V. Ros. Discussions: V. 60 pp.. E[. A... Trondheim. “The Plasticity Ratio of Concrete and Its Effect on the Ultimate Strength of Beams. 27.. “High Strength Concrete. Whitney. Sept. S. “Report of ASCE-ACI Joint Committee on Ultimate Strength Design. PP. Concrete in CompresStation. ” ProccedirLgs. 79. 33. 1953. ” tkhwei. pp. J. Vienna. “Der Bruchspannungszustand und der Sicherheitsgrad von rechteckigen Eisenbetonquerschnitten untcr Biegung oder aussermittigcm Druck. of Reinforced Concrete Design.. 24... .. M. ” PrOceeding~. p. 21. University of Illinois.pp. Jan. Civil Engineering Department. Urbana. 505. Separate No. V. Apr. ” AGI JOUENAIJ. ) .” Bulletin No. P. H. Die neue Theorie des Mahlbetons auf Grund de.” Proceedings of the First U. ” M. 429 and 530. University of Illinois Engineering Experiment 1929. 190. Hamann. 68 pp. 81. Transactions. Prentis.. “Ultimate Strength of Reinforced Concrete Beams as Related to the Plasticity Ratio of Concrete. Trondheim. der Druckspannungs-Kurve in Stahlbeton-Biegequer 25. 286. M. Bddsamkeit uor dem Bvach. Paper N’o. 3rd Edition. 2. (Committee 327 report will also be published in ACI JOURNAL. Gaston. 1952. “The Distribution of Concrete Stress in Reinforced and Prestrcssed Concrete Beams When Tested to Destruction by a Pure Bending Moment. oct. Dec. 141. 271 pp. Jensen. Eidg. Franz Deuticke.Jan. H. R. -Apr.. 629-630. p. authors. Abeles. J. W.Development Department Bulletin D6A DISCUSSION of the paper CONCRETE STRESSDISTRIBUTION IN ULTIMATE STRENGTH DESIGN By P. 1305 Dec. Konrad A. J. 1956I. J. Hanson and D. Ulf B@ggren.. Cowan. Institute 52. W. E. N. Smith. J. and Prentis. Hadley. Ashdown. McHenry. L. L. M. Henry Rosenblueth. E. . Young. Hruban. M. Authorized Journal Reprint from Copyrighted of the American Concrete 2. G. Baker. Part . E. A. Proceedings Vol. A. Taylor. L. Horner M. Hognestad. question on numerous tests. 28. .. J. ASHDOWN. .5. E. . J. and Douglas McHenry: Concrete Stress Distribution Strength Design* in Ultimate By P. SMITH and L. . . nor even over a correspondingly short gage length. W. .bftialbdz f: =q(l–o. A. E.l. KONRAD HRUBAN. L. it seems of interest to compare 1he data The writer has invetitigated this given in the paper ]vith actual test results. ~cc. 6.yrightcd *AC1 CONCRErE INST:TI:. 1956. In view of these considerations. . 52.(a) In this equation a limit of ku = 0. consequent 1y there will be in many cases an excess of strength as compared with the calculated value based on the yield point. G. Z’roc. This can be coneludeci from a great number of test results showing higher It may be pointed out that so ultimate bencling moments than calculated. is well bonded. ABELES t The extensive tests carried out by the authors represent a valuable iuvest. 52. 52-28 Discussion OFa paper by Eivind Hognestad. TAYLOR. This critical titrem appears to be i rnportant to ensure safe values when the compressive resistance of the conrrcte is of direct intlueucc upou failure. HENRY J. crack distribution with rather fine cracks in ]vhich necking of the steel will be less than in wide cracks. . far nobody has been able to measure the strain of steel ~vithiu a narrolv rrack. HADLEY. . THE . . . . PRENTIS. ..5 has been taken into account for a rcrtaw tt’hen comparing IZq. Pmcccdzrws V. however. (a). V.Disc. when JOURA7AL.TOU=XAL o. tConsultmg F. 52-28 is a part of m. “rritical” strtx+s in the stress-strain diagrams for compression and flcxurc m-as interesting. London. . and AUTHORS By P. L. Dec. (3) and Table 2. England. . ~~-cfind that the ratio kz/klks agrees approximately only for a concrete strength of 1000 psi with the factor of 0. A. 1 and all approximation.igation of the stress distribution in compression of concretes of different strength The method of testing was ingenious and the introduction of the properties. ROSENBLUETH. .5. M. . N. A. and safe values. W.5 q). there is ncI need for an additional factor of safety het~veen actual failure load and calculated load.kcnlCAX 1305 . . Part 2.nmneer. authors’ Eq. p. This will result in a good particularly it’ the reinforcement. HOMER M. BAKER. V. With regard to tensile failure. YOUNG. J. R = . W. Hanson. COWAN. ABELES. No.E. limited to high-strength materials in ordinary 13q. as long as the margin bctw-ecm actual test results and calculation is not too great. has ~iven satisfactory and prestressed concrete. . 435. l>iac. . 1955. (a) ~vith the gular stress ‘block with k2/lcIks = 0. ULF BJUGGREN. M. 5 (t/d)’]}/ (1 + f]). .5 q) – f. Table IX. According to Table 2 for this strength the following data apply: k. . ors’ data given for j. = 0. Thus the equation equivalent to Eq. .5 according to Eq. qb becomes 0.029 + 0.. which results generally in R = Mu. This equation due to yielding of For balanced design.67 (for f. (a) becomes R= MUl.5. = 0..(d) and much lower results are obtained than from Eq. lcJc.’./bd’f. and (23a) are investigated for concrete of strength j. f. . (c) In the following. . .48. apparently due to reduced necking of the steel in the cracks. .’ = 8000 psi). (3). 1956 it is 0. . (23). considering that high-strength steel is used and prestressing may be applied.957q – 0. .375. (b) In Eq. can be taken into account by considering an equivalent cooperation of the concrete in the tensile zone. . The actual test values investigated by the writer in reference 1 were higher than those computed according to the writer’s Eq. .’ = {Q(1 – 0./lc. while with higher concrete strengths it increases up to 0. Eq.65.24. and e. . . . (a).65. .. extending uniformly over the depth (t — c). (a). (b).’ R = MUt.1 fU = values may be compared Beam NTO.481q2. .. .1. (a) and results in h?b = 0./bdZf: = 0./bd’f. . according to Eq.[(t/d – 1) g – 0. (b) Rb is only slightly increased to 0. to the cylinder strength gives higher R values in the range in which failure occurs the steel and thus agrees better with the test results. . with some test results investigated by the of the writer’s tests with high-strength ma- terials in Vienna (1935) resulted in a relatively higher value R = 0.05 and the ratio t/d = 1. for a factor f I= 0. These writer. based on his suggestion in the discussion to Cox’s paper.’ = 6000 psi. and thus the ultimate bending moment would become less than the values obtained from Ilq.’ = 6000 psi are evaluated for E = 30X 10 Gpsi for various steel stresses.387 for the same qb = 0.246 for 91 ksi and a cube strength of 8500 psi. Consequently he showed in reference 2 that this excess in strength. .it. which has given safe In the following table the authvalues at least for high-strength materials. ~1 is the ratio of the equivalent concrete tensile stress.65q) . . (a). =q(l– 0. .0031.1306 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part 2 Dec. = 0. corresponding to fi = 6000 psi. first the appropriate steel stress jti must be selected. below pb = 2. When comparing these values with those shown in the table on p. which is much in the above table for fv = 90 ksi. however. since 9 to 11 tensioned wires 0. Leeds. the R values are reduced to 0.08 in diameter were placed in the compressive zone. as Janney has clear] y shown. related to the ultimate elongation of the steel. the prestress relaxes considerably and may even become zero in a crack. which is less easily obtained with unpensioned high-strength steel than with prest ressed steel. According to one school of thought. 10-. These relate to tests carried out by Stott. = 2. 4 It may be pointed out that the results obtained by the authors and sum- marized in Table 2 give important data about the compressive resistance of However. if instead of the difference of elongation the entire elongation of the wire were taken into account. stressed concrete beams will have to be investigated by further tests in which the actual resistance of the tensile reinforcement is taken into account and not based on the assumed elongation at failure. computed f or a steel stress of 100 ksi. Rfi = 0. in fact. 1306.2 percent. then according to the above table.47 percent. and the writer would be glad to hear the authors’ opinion on this question particularly in view of the above examples.19 per- Another comparison may be made with regard to prestressed concrete. the entire elongation applies. is correct but that an intermediate solution applies. mainly depend on forced.18 of Tables I and III of reference 1 will be considered. to be considered for ten- sioned steel in prestressed concrete. This discrepancy would become even much more pronounced.352 and 0. 1/2 in.ULTIMATE STRENGTH DESIGN 1307 the percentage cent computed being only 1. approximately jc’ = 6000 psi. According to the other school. whereas in fact R values 0.77 and 1. resulting in a difference of 80 ksi.08 percent varied between 0.316-0. insufficient data are available about the actual elongation of the steel in a crack. and beams No. when failure approaches.425.2 percent would apply. only the difference between the elongation of the steel at failure and that of the effective prestress should be taken into account. the bond resistance. For ordinary reinforced concrete with mild steel . This is based on the assumption that. The R values obtained for these tests of percentages between 0.386 were obtained for percentages much below 2.270 and p. using concrete of a cube strength of 8260 psi. below top surface. a certain amount has to be deducted. If in the special case an effective prestress of 140 ksi and a yield point of 220 ksi is considered. There are different views about the elongation e. It would appear that neither of the two assumptions mentioned above. As already mentioned.386. the strength of the wires being 268 ksi and the initial prestress varying between 171 and 182 ksi. These remarks refer solely to high-strength materials. If the ultimate bending moment is reduced by the resistance of the compressive reinforcement. their application to the design of reinforced or preconcrete. whether the steel is tensioned or not.315 to 0. which solely influences failure conditions if the beam is under-reinThe efficiency of the tensile resistance will. This would partly of reinforced concrete beams. and in my opinion this is due to insufficient appreciation of what occurs between the steel and concrete in beams. London. It is gratifying to note that the strain profiles are essentially linear. The differences between these test results and those obtained from beams are quite wide as revealed by Fig. so that for a thircl-point loaded beam. the concrete drags on the steel (especially noticeable with deformed bars).. May JOURNAL. 80-6. England. ACI JOURNAL. even taking into consideration the narrow width employed. one should expect that the ultimate compressive strength values on the much longer length to be rather lower than those obtained in these tests. ASHDOWN* The method of determining the stress distribution in the compression zone of a concrete beam put forward by the authors represents a distinct. P. The wide scatter of values for the modulus of e~asticity of concrete is well known. 717-736. which would require an effective depth of at least double that of the specimen for an equivalent compressed area. by the interaction between concrete and steel. respectively.. *Imperial College o i Science and Technology. 1956 the data ultimate given in Table 2 will most certainly give satisfactory results for load design. REFERENCES 1. W. 3. p. F?oc. 38. It may be noted. “The Use of High-Strength Steel in Ordinary Reinforced and Prestressed Concrete Beams. turd 13ngineering. Pro. ” Final Reportj International Association for Bridge and Structural Engineering. such as occurs in beams. E. the zero for tensile stress would be lowered. although the concrete has cracked..300fi for cube strength or cylinder strength. Abeles. The results are not complicated advance in experimentation for this purpose.. or puts it into a relative compression. assistirlg in resisting bending through the steel between the cracks. ‘(hyperstrength” account for tlhe so-called This means that.1308 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part 2 Dec. “Nature of Bond in Pre-Tensioned Prestressed Concrete. and between cracks lowers the stress in the steel. The usual assumption of perfect bond does not occur. . 10. = 52. Cambridge. W. before and after cracking. I use a simple formula for the initial modulus. By A. V. V. International Association for Bridge and Strut. Jack 1054. University of London. “The [Jse of High-Strength Steel in Ordinary Reinforced and Prestressed Concrete Beams. Close to the support the steel may actually be in compression. 2. the compressed length is quite short compared with a beam. it is still. P. ” ~retimi?~ary Publication. R. that in the authors’ test piece. Fourth Congress..000~ or 58. 1953. ” ACI Janney. Fourth Congress. Abeles. Abeles. Cox. pp. if this were so. June 1942. Cambridge. partly if not wholly. 4. 1952. discussion of “Tests of Reinforced Concrete Beams with Recommendations for Attaining Balanced 1)esign. and which lies between Jensen’s and the proposed formula in Fig. J. 8(a). 50. It is altogether an excellent and stimulating paper.” by Kenneth C. P. Ltr. L. Im~>erial Colleze of Science and Teclxnology. . L. Institution of Civil Engineers (London).ULTIMATE STRENGTH DESIGN 1309 By A. we have car~ied out a large number of beam tests and obtained results which agree closely with those shown in Fig. and Its Application to Ultimate Load Design. University of I. maximum strength occurs. the steel strain is assumed to be the total strain minus the strain due to prestress.6 GUfor concretes weaker than about 4000 psi and 0.ondon. The value of F varies from about 0. the latter being obtained from the measured total steel tension and the applied bending moment. mainly of procedure. 6. In bonded prestressed beams. bulb it has been obt~ined. formations will have comparatively low values. In the usual beams the actual creep deappreciable creep in the concrete.4 as a safe value of [62. The majority of reinforced concrete beams are under-reinforced and in those cases the difference in calculated ultimate bending strength according to conventional calculations and to ultimate strength design is negligible. of compression is determined from the lever arm value. 1953. We have also found that 0. This will not *Pro femor of Concrete Technology. “Further Research in Reinforced Concrete. except at wide cracks when the ratio of the steel strain to the virtual concrete strain equals F. such as columns and some types of beams with high quality steel. as shown in Fig. Since c:ylinder strength is about 0.5 c. with the assumption of a linear distribution of strain. We have adopted 0. because a lowering of the neutral axis will decrease the concrete stresses to a high degree. which might be of general interest. London. academic interest. Part HI. Using fitrain gages to determine the position of the neutral axis at failure.. L. Paper No. we have cletcrmined the average compressive stress in terms of the cube strength Cu and found that safe values are 0. In over-reinforced cross sections. 2. ” Proceedings.8 to 1 according to bond conditions. during the last 10 years. these values agree well with the . ~ Precise determination of the shape of the compressive stress distribution is a slow It has become. and the moment of resistance of a beam is then easily obtained from the above safe values.002 is a safe minimum value of the concrete strain at which. we agreej too. 58Y4. for concretes stronger than about 4000 psi. 6. BAKER* At Imperial College. Aug.8 cube strength. Malmo. By ULF BJUGGRENI This excellent paper gives rise to some reflections. sincx safe values of the average compressive stress and the position of the center of compression obtained from many beam tests proThe position of the center vides sufficient information for design purposes. tSkfmska Cementaktiebokwet. Sweden. England. The strain values at failure determine the position of the neutral axis. however. the compressive area takes up a rather large Sustained loads can in that case give rise to part of the whole crosti section. Department of Civil Engineering. L. A. Some further tests have been carried out on the bending simulation machine referred to in the paper and agree well with the beam tests. ~Baker.kIkj values of Fig. . 109=+ 1A U. when the distance to the neutral axis is small. quite a lot on the effect of transverse strains.’Q~=2.i . should the creep deformations be included in the ultimate concrete strain or not? If we take an extreme case of a cross section having a sustained highest compressive stress of about 2000 psi. I have found with a 12 in. A similar effect can be obtained if the compressive area of the cross section is sustained by concrete with low stresses. { w [8.= n / 4-- yl’ .. happen The average compressive stress at compression failure depends.7310 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part 2 Dec. which is 40 percent of the value of the ultimate strain recommended by the authors..”’ /-.-1. 7956 .:.* G /“- /’ . the corresponding elastic strain is of the order 0.0012. . ‘+ 4’ ““6 ..“.75 —-- c/W t =oqe mdoys Fig.1. high cross section (cube strength 9000 psi) that the height of the . as the authors have pointed out. . This can especially be the fact. uz4+L4p A -.””X /’4 G / . A—Initial tnodulus of elasticity in flexure The question is: to the same extent in over-reinforced cross sections./ . Appropriately placed stirrups thus can to a certain degree increase the ultimate compressive fcmce. / .5 .0004 and the creep deformation might be of the order 0. . c have pro*Professor of Architectural Science. In the past it has been necessary to assume a shape for the concrete stress distribution.43 log t -b 0. g These considerations lead to the statistical concept of a concrete specimen made up of a number of elements with strengths distributed over a specified range. deformation c~f concrete at moderate loads. Williams and . There may be more favorable results obtained by such small beams as joists. COWAN* The paper is an outstanding contribution toward the design of reinforced and prestressed concrete for ultimate strength. Reinforced. As seen from Fig. cent of the ultimate load has in fact been observed by Berg. Sydney. pp. and thereafter ceasing to contribute to the load resisting capacity of the model.Kloot. This may be represented in a mechanical model by a number of springs rupturing progressively with increasing strain. Neither can be reconciled with mathematical theories of the inelastic deformation of concrete which have been published. A. By HENRY J. V. meaLsured from Fig. University of Sydney. extensive experimental data of the shape of the stress-strain diagram itself. As a result. Finally I would like to mention the obtained values of the initial modulus of elasticity. 4 using an ultrasonic method. their presence has also been inferred from tensile tests by Blakey and Beresford. The presence of minute cracks at about 60 percracks above a certain limit.z has suggested that the departure of the concrete stress-strain diagram from visco-elastic behavior is due to breakdown of the adhesion beThis presumes the formation of microtween the cement and the aggregate. namely F (t. On the other hand. Glanville’s experimental in 1930 established the existence of inelastic strains which were proportional to the applied stress. w/c) == (0. and to prove the accuracy c}f the assumption by comparing the predictions of the resulting We have now for the first time theory with the failing loads of test beams. as a function of the 10th logarithm of the age of the concrete and of the cement-water ratio. 10. where t = age of concrete in days. Australia.75 c/w) 10’. The most significant features of results presented are the relatively high ultimate strains and the descending portion of the stress-strain diagram. 21. and “by Jones. the dimensions of the test specimens used by the authors correspond to the compressive area of an usual beam with a depth of about 60 in.ULTIMATE STRENGTH DESIGN 1311 compressive stressin the concrete was higher than the cube strength. I have arranged the values in flexure. 3 using an optical method. the result seems to be significant as a first approximation. and Prestressed JotJrnal. that in this case these values ought to depend on one quality factor and one time factor. D. ” . Freudenthal. working on the behavior of timber specimens in tension. a number of theological models for the deformation of concrete were proposed~ consisting of viscous dashpots and While these models satisfactorily reproduce the long-term elastic springs. tThe mod@ p~oposed by A. they fail to explain the high ultimate strains and the descending portion of the stress-strain diagram. ROSS ~’Creep and Shrinkage in Plain.ss an example. Starting from the point of view. Institution of CmiI Engineers (London).24 + 1. 1943. Concrete. 33-57] may serve . s evidently satisfactory. the failing. When this duced by the mechanical model shown in I r n-for the elastic and inelastic deformation of L4- L Fig.. but due to the blocking of the second dashpot some of the inelastic deformation is not recovered. rapidly to produce negligible viscous and it should be practicable to dedeformation termine parameters to fit various types of experimental curves. When the model is loaded sufficiently rapidly to produce negligible deformation clf the dashpots. C has therefore been constructed to reproduce all the known features of the inelastic def orrnation of concrete. It consists of two series of viscous dashpots and elastic springs in parallel. “ ‘The mechanical model shown in Fig. INSTITUTE Part Q Dec. and it is necessary in practice to stipulate a limiting ultimate strain. it deforms elastically until the fracturing load . B—Load-deformation curve with the experimental data presented Fig. 1956 duced a mathematical theory for a model of this type. B. loads of these brittle springs being distributed over a range. the dashpot of one of the units being blocked by nonreturn valves. C when it is deformed sufficiently in this paper i. and a progressive inelastic deformation at a decreasing rate. When the model is unloaded the procedure is reversed. C—Comprehensive mechanical model concrete model is deformed by a moderate load. With this reservation. a stress-strain cwrve obtained from this theory is shown in The theoretical curve obFig. so that the brittle springs are not damaged. it exhibits an immediate elastic deformation proportional to the applied load. Chosing suitable parameters.1319 JOURNAL OF THE AMERICAN CONCRETE -! : : . the total inelastic deformation over an infinite period being proportional to the applied load. pro- Fig. however.. viously proceeds to an infinite strain at a very small stress.4 .a o 0 1 2 3 4 5 ST.. since in a statistical theory an occasional element may remain undamaged. and a series of brittle springs in parallel which behave elastically up to failure. the agreement O@ 06 . 1952.. and Beresford.2-2. at loads above the working loads but below the failing loads under rapid loading) present a problem that must be investigated before ultimate strength methods can be applied to reinforced concrete structures with perfect confidence. It has similarly been observed that micro-crack formation.. Y. II. Wash. H. 70. A.” British Journal of Applied Physics. C2. W. A. New York. 12. ~fTen~ile Strains in Cloncret. accepted that the declining portion of the stress-strain diagram is due to irreversible micro-crack formation. R. increases the load bearing capacity of the concrete structure. Stationery Officej London.d bearing capacity of the concrete structure loaded continuously to destruction.. It may be seen that the repeated loading of reinforced concrete structures at high loads (i. ‘(. The Inelaetic Behavior oj Structrwes. Williams.c-Part No. * When the model is deformed sufficiently slowly to allow for appreciable deformation of the dashpots and the load is carried at the same time above the fracturing load of the weaker brittle springs..dley . and the subsequent inelastic deformation is irreversible. SSSR. Seattle.Jownal of Applied Science (Melbourne). concrete cylinder as the means of demonstrating the Instead of the immediate plasticity of concrete in response to overload. V. pp.. if d?icient time is . V. V. Commonwealth Scientific and Industrial Research Organization. The outstanding success of the authors in developing an experimental technique for determining the complete stress-strain diagram under rapid loading holds out hope for the solution of the problem of repeated loading. John Wiley and Sons. 2. 165. It has been observed that inelastic deformation due to creep. Glanville. Berg.nd Plasticity of Concrete. 3. REFERENCES 1. by ironing out the peak moments in a redundant structure over long peric~ds of time. J. London. healing.. “Stress-StrainRelationship: A Mathematical Model.& Hadley. HADLEYt The series of tests reported in this paper is notable in its abandonment of the standard 61x 12-in.” Australian . Division of Building Research. and Kloot.. 1955. ) 4. Blakey. ~~The Nauk. If it is. D. (Translated as Communication No.. 6. 3. Road Research Library. problem of the Strength e. N.. H.M. E. M. Akad. 1950. pp.. F. F. however.ULTIMATE STRENGTH DESIGN 1313 of the weakest brittle spring is reached.~s allowed. as compared with a structure made from a similar elastic material failing in a brittle manner at the maximum stress. pp. 617-620. Jones.” Do~l. O. H. 1952.Melbourne. then the increase in strength may not hold for repeated high loads. ” Repd 5. Freudenthal.e . 3. 19130. 1-13.1950. +H. increases the loa. Consulting Engineers.A Method of Studying the Formation of Cracks in Materials Under Stress. 229-232. [(The fJreeP or F]ow of concrete Under Loadt” Building Research Technicat Paper No. then both types of inelastic deformation are reproduced. By HOMER M. *ln practice some of the micro-cracks would probably close up due to autogeno. Department of Scientific and Industrial Research. by extending the stress-strain diagram and increasing the ultimate strain. ” has a familiar look. simultaneously subjected to direct stress and bending.n Fig. concrete cylinders—U. The weakness and “come-apartness” of concrete so strained in a brief period of time has not been lost by changing the form of the specimen. nor a definite strength after a definite time of applying load at a definite rate. not much space is left for plasticity. and others. 9. It is to be hoped that the authors will state their recommendations respecting the outer limits of the plastic range.010 without word of the concrete’s coming apart. S. This permits a more gradual application of load and more gradual adjustment to load on the part of frame than would be possible in a briefer time. 8 of Laboratory Rej~ort No.0035 boundaries of the elastic range. 9 days. 49.0025 in. ” These designations now definitely identify the curves with those bearing the same descriptions in Fig. SP-12. Oct. nor a definite state of cracking and shattering. Its physical basis is vague and uncertain and indefinite. Hognestad’s paper. per in.” . where they appear in Mr. S. 1952. Not the Ieatit unsatisfactory feature of this ultimate strength theory is the fact that there is no definite and decisive physical boundary to limit its supposed plastic range.009 and 0.00~3 in.003 to 0. 2. Bureau of Reclamation. except for those of lowest strength. Bureau of Reclamation. The elastic theory has the maximum cylinder strength which is something positive and specific and determinable. 1956 cylinder there is substituted the channel-shaped frame. There is a marked similarity in the stress-strain curves of two types of specimen. 13 days. T-his theory has neither a definite strain. 3. U. none reach far beyond strains of 0. 7.” The chief difference between these two figures is that the curves are now designated. The identity of those frames shown i. the curves extending to strains of 0.002 strains of the elastic range and the 0. that once graced and adorned the outer reaches of (‘the plastic range”? As a matter of fact. Proc. titled “Tests of 3 x 6-in. It is noted that about 15 min was given to the testing of each frame. between the 0. “Inelastic Behavior in Tests of Eccentrically . would have abruptly terminated both the curve and the frame at that point. Mar. V. 1947. 122. per in. p. and a new foundation for ultimate strength design is presented. These then are the cylinders whose stress-strain relationships cylinder. per in. top to bottom: ‘[41 days. On comparison the curves are found to be identical with those of Fig. wherein we learn that “each curve represents the loading of a single 3 x 6-in. Fig. “Where is the rose of yesteryear?”: those strains of 0.010 in. 5 is not disclosed but the shattered condition they are in was no sudden development preceded by a relatively lengthy period of plastic integrity.Loaded Short Reinforced Concrete Columns. The results of tests of these frames and of companion cylinders is presented in Fig.009 and 0. ACI JOURNAL. We have been offered the Saliger speculation. The writer is of the opinion that the sudden application of a small increment of load to any of these frames that had attained a strain of 0.1314 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE ParI 2 Dec. Perhaps the most significant feature of the curves is their limited extent. square by 1~ in. The principal weakness of this experimental method is the unknown extent to which the introduction of the steel plates locally changes the stress conditions in the concrete. It was tested on a 130-cm span with equal. the force. at the middle.. completely out of contact with its neighbors. ” It is far from clear to the writer that the introduction of steel plates in Brice’s beam did actually change the stress condition in the concrete appreciably. cm wide. of full beam width. approximately third-point loads. These plates were placed and the concrete was cast against their ends and against the little forms which excluded it from the central open galp portion beneath glass shaped.ULTIMATE STRENGTI-I DESIGN 1315 for concrete in compression Professor Hognestad believed applicable to flexure and which he then took as the basis of “the present analysis. after all the labored efforts that have pattern? In any event it is probably well been made to lhave it a “plastic” that this beam be described in some detail so that “its principal weakness” can be more widely evaluated. Brice’s test beam is a novel and most interesting one. there is created a concrete beam with reinforcement continuous from end to end. . Brice’s experimental efforts” to determine flexural stress-distribution in a beam receive the somewhat hurried brush-off that they do in this paper II . were not in contact with one another but were separated by spaces of 0. reacts against the concrete but it is the pressure. that produce the deformations in the steel and these steel deformations are not so large as to involve any question as to their in- . P. It is only the concrete. at their ends narrowing to a section 1 cm. 1 cm thick.” i. long. thick having about ~c in. why of all things did the stress pattern so perversely revert to one with straight-line variation. that of 1952. At midspan a stack of eight steel plates. to a central section % in. of course. At midspan there is a short length from which concrete is completely excluded. as described. the gap being crossed by the reinforcing bar at the bottom and at the top by a Bet of steel plates about ~ in.e. The eight plates.5 Each plate had two small projecting prongs at each end for anchoring into the concrete and holding it in position. pressing Such a beam appears innocent enough. which reduced section extended for a length of 4. Its dimensions are slight: 7 x 16 x 150 cm. against the steel plates on their end surfaces that produces the recorded deformations in the steel. a straight-line stress-distribution was found. But if it did. . concrete but each plate reduces. with a midspan zone of constant moment and zero shear. not the strains and deformations of the ccmcrete. replaced that length of concrete in the upper two-thirds of the section and an open gap or hole. 14 cm long. to repeat. exerted by the concrete. The steel. was created in the concrete beneath them. across which opening the reinforcing bar extended. In plan view these plates were hour7 cm.41:3 cm. it Thus. gap The ends of these plates bear against the full width of the between them. them. about 8 cm long. Under the circumstances prevailing it is scarcely to be wondered at that L. even at high loads. if one can entertain the thought of a flexural unit stress greater than the compressive the 12 in. high cylinder be expected say. Therefore despite the addition of a new section on ultimate strength design in the ACI Building Code and other “party line” efforts. Now it is true that the plates show a straight-line stress distribution pattern “even at high loads. cannot the wide prevalence of that belief is not to be questioned. direct. then the straight-line theory The straight-line works admirably and agrees with the beam performance.. ” but even so that fact is not. based on the straight-line theory. should should. It is to be hoped. as stated. high cylinder by a Slater-Lyse amount. ‘1 956 terpretatiou: they are purely elastic. way the beam’s theory flexural and capacity be explained Consequently. showed a straight -line variation of stress “even at high loads. can be explained value of stress in flexure is to be no more than that of the providing a substitute theory straight-line from the region in the vicinity consideration. does not convincingly demonstrate that they have “locally changed stress conditions in the concrete.” All they have done is conflict flagrantly with the theory which our authors seek to establish. in a fashion not dissimilar to the tones of a trombone strength or even approximate under varying measure degrees of extension. ” Would it not. quite independent of concrete deformations and the interpretation thereof. both by the authors and by others. Nearly 30 years ago Gonnerman’s tests showed that the strength of concrete in 6 in. cylinders with 2000-psi concrete. than to scrap altogether and to adopt a new one founded on the wonder tale of concrete’s instant plasticity. 50 percent higher wifh 3000-psi and stronger concrete. and most ingenious method of determining what the stresses in concrete actually are. therefore. discreditable to them. Another possible view is that Brice has developed a simple. per se. it may be advisable to hold back a bit pending a proper investigation rather than an arbitrary rejection of Brice’s findings. Some 25 years ago Slater and Lyse found the flexural strength of concrete. it seems much simpler the straight-line theory and better of f. that this promising field of investigation will be thoroughly and open-mindedly explored. which. at this stage of things. high cylinder of some other height assumed the only the of cylinder.1316 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part Q De. strength Just of its but if the is by that why the compressive to be the exact strength maximum cylinder. The installation of a set of Brice plates in the midsection of a concrete channel frame might present its difficulties but with the ordinary beam there is little or no trouble to be foreseen. in sufficiently reinforced concrete beams to be 67 percent higher than that of 6 x 12-in. or the of the flexural rationally concrete rather than that the strength of a 6 in. theory being the one with which tc) change the value everyone is familiar. be a most amazing development if additional Brice-beams should reveal naught but straight-line stress variation? . of the neutral axis which the straight-line unit strength of excludes On the other hand. scrapping activates theory of the 12 in. diameter cylinders varied inversely with the height of the cylinder. which have design. ) . (3) may be rewritten in the form M. This means for a rectangular section that the substituting rectangular stress area has a depth of 2kzc. difference in the following We substitute a rectangular stress block for the actual one. the largest specimen had a section of 80 x 160 cm (about 32 x 64 in. (a).. In the USSR. determine simply the value of ~JZ. and different testing arrangements applied.(b) Both Eq. 1). 477 that “the tests cover only one type of agg~gate. 1 Other investigations have been carried out in several European countries. (a) according to is significant. for instance. (a) and (b) yield the same result. Czechoslovakia Technical .. instead of Eq. (a).k. we consider it in practical computations strength. and one size of specimen without compression reinforcemerit. Prague. the sections of which had areas twice to 64 times smaller. of these European countries.67.ULTIMATE STRENGTH DESIGN 1317 By KONRAD HRUBAN* The investigation by the authors forms a conclusive contribution to the ultimate strength theory. 9.~ for University. eccentrically loaded columns of rectangular and I-shape sections have been tested up to considerable dimensions. the assumed uniform stress in the pressure zone (the so-called flexrwal strength of concrete) we obtain. Denoting with fl ~. I would like to make a few comments to the authors’ remark on p.fc’ =~ffiez’ ”””””’””””’”’”” “’””’”””” ...fV d ( I–2P. if h% k.). To obtain the correct moment M~zt giveq by Eq. The value of k2/klk3 depends on the concrete Table 2 it varied between 0. Comparative tests were carried out on columns. = A. Let us now compare the quantitative test data arrived at in the paper.. (c) The standard specifications adopted the ultimate strength *Professor of Concrete Technology. I think. different types of aggregates and cements have been used. the authors’ investigation gives qualitative information of fairly general validity. both the actual and the substituting stress block must have the same volume and give the same line of action of the total compression C (see Fig. therefore. All the accumulated evidence leads incontestably to the conclusion that the stress-strain relationship for concrete in flexure is nonlinear and has to be represented by curves similar to those shown in Fig. one maximum aggregate size.48 and 0. the expression . The In Europe.. ” 1 compared the test data given in the paper with results obtained elsewhere. Eq.. which has been achieved by means of an original and ingenious method. way. I _. 1948 (in Russian). Hamman.1318 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part Q Dec. pp.61 to the values of kzjklk~ corresponding 0.z which are 5 to 7 percent higher than those calculated by means of Eq. Scotland. This has but an insignificant influence on the resulting ultimate moment given by Eq. The small variations can easily be explained by local influences. we have to take account that 1 psi = 0. Moscow.b~ is the 20-cm compare the test results.’.. Aberdeen. specifications) have been determined which was suggested by the discusser with regard to previous test evidence: f’.. 2. for instance. 126-135 (in Russian).z (given in the 1955 according to the following formula. CSN 732001.05~. given by the authors. The results 0. M.. 1490+f’cubs ““””””””’””””’””””””” in kg per sq cm. Russian standard specifications of 1948 give values of the strength f’fl. we obtain the 2000 0. f’.. “Design of Reinforced Concrete Structures. (b). the difference against Eq. V. (a) does not reach 3 percent. Evidently. — 4000 0.. By J. ‘[New Investigations of Eccentrically Loaded Elements. (d). the greatest difference between both respective numbers is 3 to 4 percent of the authord values. strength of concrete. Here. 3.64 6000 0. Bulgakov. . the standard values of f’fl. 1956 each category of concrete so that it corresponds to the test value of k2/klk3 The category of concrete is defined by the 20-cm cube according to Ilq.. 1952. j: psi Table 2 Eq.65 Middle European tests.0703 kg per sq cm and that f’. (c).” Prague. Lee for it avoids the complications introduced by the presence of the rein*Lect. Marischal College. In my opinion the conclusions arrived at in the paper may be considered as a definitive answer to the question of the concrete stress distribution. University of Aberdeen.Moscow. (c) Using these relations and substituting into Eq. PRENTIS* On p.69 8000 0.b. In this country.67 are given in the third column of Table A.~b~ is regarded as kl[kzks according to being equal to 1.” Problems of the Present Reinforced Concrete Construction. into of concrete (d) has been derived. S. 462 the authors point out that their method has advantages over the and methods of analyzing beam test data proposed by Prentis. TABLE REFERENCES 1. 1955 (in Czech). A = 1435f’.54 ::). GOST 4286-48 (USSR standard specifications) and NiTU-3-48 (Russian norms and regulations). (c).rer in Engineering. It can therefore be stated that the findings of the authors are also quantitatively in acecmd with European investigations. with (d) To those cube from strength which Eq. lboth groups of tests lead to practically the same conclusions. It would be possible to test identical specimens at different rates of loading. Sept. V. The writer would like to congratulate the authors on the ingenious way in which they have done this. and from the complications due to reinforcement. hizving isolated the problem of determining the concrete stress concrete tensiie stresses. J. it is difficult to estimate the extent to which finite spacing of cracks upset distribution of stress above neutral axis. which. With each beam it was possible to measure strains at three sections along Due to irregular cracking the compressive the length of the central span. Proc.” ACI JOURNAL. inherent difficulties in measuring accurately the concrete strains in a beam.or 8-in. is more important than the two already mentioned. If time effects are negligible then the resultant stress-strain curves from such tests would be the same whatever the rate of straining. A number of beams were tested under fourpoint loading to give a central span subjected to a uniform bending moment.) and the position of the neutral axis. M. the section of failure G is somewhat greater than the mean value and the depth of the compression zone is less than that indicated by the strain gages.m&!& of Inelastic Bending Stress in Concrete Beams. interest here. The validity of this is questionable but it appears that the authors could use their test methods to verify it. due to the finite and often irregular spacing and form of the tensile cracks. Hamman. . However. In deliberately ensuring that no tensile stresses are generated the authors avoid these difficulties. These tests are referred to in a paper by the writer~ but the relevant points may be mentioned here. or alternatively. for the difficulty of measuring the steel strain can be overcome* and the tensile stresses are generally insignificant in their effect. 30:317. however. twice the strain at the other and so strained twice as fast. gage length to determine the maximum concrete strain (e. 53$P&. “A. Beam tests by the writer have provided evidence substantiating this assumption. and Lee are well founded. say. crack spacing there appears to be a logical difficulty in applying the results to normal reinforced concrete beams where all these factors are present. A comparison of the results from tests on pairs of identical specimens in which one of a pair is tested without cracks while its mate is tested with cracks would demonstrate the effect to which the crack spacing and presence The writer must confess to a personal of tensile stresses upsets the analysis. One of the basic assumptions of all the methods is that concrete stress is a function of strain only.ULTIMATE STRENGTH DESIGN 1319 There is a further advantage forcement andl the concrete tensile stresses. The writer would like to suggest that it is desirable that further tests be carried out in which tensile cracks are allowed to form. time and stress history effects being negligible. Thus. single specimens could be tested so that the strain at one edge was held at. It is necessary to measure average strain readings over a 6. Furthermore. in some respects. strain in the concrete and the position of the neutral axis varied between *Prentis. 1956.. There are. such tests would indicate the extent to which the beam analyses of Prentis. . it is simple to show that 6 (. it will be assumed that concrete may be represented as a certain combination of Maxwell and Kelvin bodies. extrapolation to i = ~ from direct compression tests performed at different rates should give a linear relation between stress and strain. . namely. where it is zero. . Had time effects not been negligible have resulted in a diflerent stress-strain curve for each section. . differed for each stress-strain curves were found to be the same. . .London. . . pronounced variation in strain rate should certainly not be neglected. 1956 sections. . . . .. . that the proposed curved distribution of stresses in concrete leads to fairly accurate prediction of flexural capacity in prismatic members of rectangular cross section. . The methods used for computing stress-strain curves in the prisms are objectionable principally because of the assumption that concrete stress is a function of strain only and does not depend on the history or rate of loading. Again. Nevertheless. . . Now. ROSENBLUETH* stress disthe strain section the for testing this would The paper constitutes an important contribution to the theory of ultimate strength design. . .1320 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part Q Dec. . Institute of Engineering and Geo~hysics.(A3) c *Research Professor. f as well m several other investigators. .8 j. Mexico. . . “The Creep or Flow of Concrete Under Load..) f= —-----x . . Although distribution. . 1930. . . . . Using the writer’s method of analyzing the data the tribution was drawn for each of the three sections. . . . the stress-strain relation can be put in the form f= ~(e) . the behavior of each element of concrete will be idealized as that of a system of linear springs and viscous dampers. It is well established that stress-strain relations in concrete are strongly time sensitive. . it does warn against unwarranted extrapolation and neglect of time effects. E. c (Al) The effects of such a where the dot denotes derivative with respeet to time. . . 1!?. University of Mexico. Under these conditions. . Mexico City. . . at least up to stresses equal to 0. Admitting the validity of the proposed idealization in the range O S f S 0. one may write 6= . . . assuming linear strain distribution in the prisms.. and hence the rate of straining. ” Building Department of Scientific and Industrial Research. To gain some idea as to effects of the variable strain rate. . . That is. . has shown conclusively that this is nearly the case.’. . By E. The first concerns stress distribution in prisms loaded similarly to those re~orted in the paper. . . . . . .8 t~. . W. . The time taken each beam was 4 to 5 hr. H. tGlanville. . . . Admitting a linear variation of c. . . This discussion should not detract from the main conclusion of the paper. . . . .(A2) where o depends on the strain rate and strain history. the strain rate in the prisms varies approximately in proportion to the distance from the neutral axis. Glanville. Research Technical Paper No. Conclusions derived in the paper may be grouped in two categories. . the general shape of the curves is quite similar. . . . Since j. .(A5) t.. . The 10 percent difference found in initial moduli in the tests can best be explained on the basis of differences in the method of casting cylinders and prisms... .. Similarly. . . with respect to e. in first approximation. . . . for a constant strain rate i = v 4=w ( l–e–~ “) . (17) . (A4) and (A5) give. . . (A4) and (A5) is ... (A7) where -y and E represent the damper and spring constants.8 fC’. . hand. .. that the same order as experimental errors. = 0. That is. 9. (16) and (17) is less than the maximum observed difference in the tests and of It must be concluded. . .. . . For example.. D/jC is quite small in practice. .*GJ and from Eq. which is representative of the curves in Fig.. . each concrete element idealized as a simple spring and dashpot in series. .. . . for strain-flexure curves. . Eq. wd.) ++’(=o)c. therefore.8 fi. . = 0. . ] . . Clearly. where the prime denotes derivative [d(d+. for uniform compression at the rate v = v..8 j. .. .(A6) It is true that D = O only when ~/eC is constant.’. in the range covered by the tests one may take.ULTIMATE STRENGTH DESIGN 1391 and the stress distribution (A3) is linear at all times up to j. . .=.83 Ee. D=@(G) computed -@(ec)cc] from Eq. The difference between values off.. . computed from Eq. fO= : ~: fc dz = .. values from both conditions of equilibrium does not disprove the contention that stress-strain curves are time dependent. . From Eq. .. ... .. . (A4) to (A6) and assuming vC-y = 2.[%(~. .6 EG at j.6 EeC at fc = 0.. .. again with VJY = 2. was found from the average of values deduced from Eq. ....025. . (16) j. .=.(A4) from Eq. On the other and. .?7 Ee. fc = 0. Then. .. = 0. (16) and (17). . . . . which corresponds to no internal darnping. retipectively. the maximum difference between values of ~. . Eq. (A2) and (A7) give f. However. Substituting in Eq. one obtains D/fC = 0.. The initial tangent moduli are theoretically both equal to E. . the near equality of f. stress-strain compression beams presence role This of rapidly cylinders In have and there the only. 9). The in the quality theory is that kz was not found only plausible fibers explanation of the lies prisms. paper. the strain sections distribution distribution containing of is a discontinuity compression there and to would of plane at every strain assumption is adhered compression a discontinuity was not the only operative this phenomenon statistical whereas selected effects. an extrapolating neglected statistical reinforced invalidates imply infinite trapolation. cracks. Lincoln. SMITH and L. University of Nebraska. it was is justified practically the to conclude up curve tests. different fc = () (Fig.’. zone still at be of kz. Hanson. . of ultimatebeams. and McHenry are to be congratulated on the ingenious and thorough method of test used in determining the shape of the stress block as associated with ultimate strength design. 8 show experimental and actual difference to simpler strength determined prediction differ prisms that reinforced stress indeed a systematic likely lead experimentally distribution. a slight systematic curvature in the strain The curvature is concave downward (Fig. from those as load values in and characteristic constant increases. = 0. M.8 jf. Time each If Data such been of standarci concrete The important crack. Hognestad. Professors *Asw&~~t of Engineering Mechanics. constants. fail at the weakest section in tension prisms those tested will more sections concrete beams weakest developed compression. reinforced from situation also. exaggerate From stress that the the computed values discussion prisms it resembled that in the play example. more accurate It seems corrected a trapezoidal as Jensen’s. fiber. the latter effects. To really appreciate the contribution of this paper. whic~ tends to distribution. E. axis. By G. concrete differences of At higher stresses there is. since is undoubtedly in axis reinforced shifts favorable. YOUNG* curve Messrs. besides. the in upward u is no longer z decreases objections should monotonically are in such valid. The differ neutral on prisms. for concrete and for of the prisms. the all fibers foregoing constants Fig. with EWwould deduced of the behavior of concrete in flexure than the stress-strain from standard compression tests. the strains factors tension a strict be plane. one has only to review the literature of ultimate theories which have evolved from numerous assumed shapes of the stress Knowing the actual shape of the stress block block in the compression zone.13QQ JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part 2 Dec. 1956 An apparent exactly equal in slight discrepancy to 1/3 at with the proposed (1). Neb. there which would tension to for shearing in the tests tended fail which to at the have are other in the strictly cracks. strains if Even at the that zone exfor in the zone neutral of cracks cannot that Lhe actual and loaded foregoing levels than to effecti For distribution at all stress rather in the plane to ~. in the compression zone will provide a more fundamental understanding of failure and possibly provide new avenues of approach to ultimate design. As section. su~mmary block for from loading the beams that subject which moment nature end for of the moment is not the to could No of without the paper. true members the category of pure at the attempted. *Reaetwch Student. 9 and the plotted points for kt in Fig. (Fig. under in the stress actually axial. . The condition of dead loading of a compression member such as a cylinder would not permit a reduction in the load and therefore a reduction in stress could not occur even if the stiffness of the testing machine was infinite. after force Pz was initially increased with PI until a maximum which Pz decreased quite rapidly near failure. 12 indicate that within the degree of predictability of concrete that the ultimate stresses might be identical and k3 = 1. as described by parameter ii3. The difference between the two cases is that in the beam bending about the axis GH. but about EN which for low loads lies at the centroid This axis EF is the “neutral axis” about which bending takes section. Australia. University of Sydney. A comparison of the stress-strain curves in flexure and direct compression in Fig. 466 and 467 indicate that the was reached. but of few axial load and not is test moment. In other words. reducing eccentricity e. J. Civil Engineering Department. loads case. to some force P at an represent the actual conditions of a concrete authors have introduced a system of loading P and e such that zero strain occurred in definition. It is interesting to note that the previous wide deviation between ultimate stress in flexure and in direct compression. occurs about of the place. analogous concrete beam would be GH (Fig. combination occur load. this dots which the by the conditions. Sydney. has been narrowed. D) in fact the represents 8 x 6 x 16-in. problem has been in a beam. that solved It is implied shape at failure This indicated. the whereby it was possible to vary the fibre CD. termined and not come appear stated of the methods ABCD conditions for the pure to be solution different flexural ildeally. do loads near failure in a testing machine simulate loads under the force of gravity? The description of test results on pp. There is some question as to whether similar conditions exists for specimens subjected to dead loads as in a structure. that this was the neutral axis. In an attempt to beam failing in compression.ULTIMATE STRENGTH DESIGN 1323 This paper and supporting evidence verify the dip in the compression stress-strain curve of concrete after ultimate stress is reached when under test in a testing machine. in practice. TAYLOR” The authors must certainly be congratulated on a really convincing attempt to solve the problem of the compressive stress distribution at ultimate loads But it must be realized that they have defor members subject to flexure. but in the authors’ tests bending does not occur the axis DC. By A. the loading some doubt. hence deducing by The neutral axis position for an E). block under test. It would be of interest to know if the sum of PI and Pg ever decreased as the specimen approached failure. values of kg reached and exceeded a value of 1/2. ‘2 ‘1– (P. 6) are also not applicable to concrete beams subject to flexure for another reason. D FQ. According to Eq. ” AUTHORS’ CLOSURE The authcms greatly appreciate the contribution made by the discussers in clarifying the fundamentals of ultimate strength design. not analogous. (12) and (13) klka=— P. In three out of five tests in which graphs were given for k~ against concrete With kz strain (Fig. 6). the depth of the compressive zone varies due to increased cracking and the introduction of plasticity. In an actual beam test. cylinders. against strain (Fig. bcf[ P. therefore. E The two cases are. equal to 1/2. +P. there was a striking similarity between flexural stress-strain relations and those obtained in concentric compression tests on 6 x 1Z-in. The sets (of curves given for ?cZ and klk.13Q4 JCXJRNAL OF THE AMERICAN CONCRETE INSTITUTE Part $! Dec. In effect. since although the actual neutral axes are effectively in the same position. + P. al + Pz a. each point on the lcz and iilks curves represents values which could be obtained for particular loads on different beams with different effective depths. For this particular condition we thus have an axially loaded specimen. to find that even though moments do exist: “.)C where c is equal to the depth of the compressive zone in a beam but is represented by the constant total depth of the specimens tested by the authors. therefore. It is not surprising. the bending neutral axes are in different positions but have been disguised by the axial load in the authors’ tests. . 1956 A B 1 f P L Fig. In any one of these beams the depth of the compressive zone would vary and not remain constant as in the tests. the line of action of the compressive force must be at the centroid of the section. Studies of in- . . in spite of differences in materials and testing techniques. While the authors’ investigation was in progress. the radical change in aggregate type to lightweight materials caused only a minor change in the stress distribution properties. and the eccentricity corresponding to a position of the neutral axis at an edge of the cross section was determined. Typical eccentricities were O. . the authors have recently studied the ultimate strength design specifications for reinforced concrete structures issued in that country in 1949 (N and TU-49) and in 1955 (123-55 MSPMChP) .0 times the kern distance. By comparison with test results obtained in the USSR and in several European countries Hrubam concludes that the authors’ findings are applicable beyond the type of aggregate.rectly and quantitatively The authors’ applicable to ultimate strength design of structural concrete. Hanson at the PCA Laboratories. an excellent agreement exists between the German and the American test results for sand and gravel concretes. F that. and it is gratifying that this discussion aff orals an opportunity for comparison of independent findings in different parts of the world. similar work was being carried out independently at the Munich Institute of Technology in West Germany under the direction of Rusch. which for most cases utilize an equivalent rectangular stress block. 0. For all specimens within one group. l?. These German findings are compared to the PCA results in Fig. 0. cement.ULTIMATE STRENGTH DESIGN 13!25 elastic concrete stress distribution have recently been intensified throughout the world. “ Recently. and specimen used. may be helpful in considering design values suitable for practical ultimate strength design. By applying suitable statistical methods. The German tests were made with sand and gravel aggregates. A. the authors have had occasion . .6. strain measured at an outside face at ultimate strength was plotted versus applied eccentricity. strong consideration must also be given to tests of reinforced concrete beams and columns .1-3 It was found that these specifications. ” and ‘(To make definite design recommendations. Test results have been added for three new sand and gravel concretes and for 15 lightweight aggregate concretes tested recently by J. position on this question was stated in the paper as “The numerical constants obtained . are basecl on fundamental concepts regarding inelastic concrete stress distribution that are strikingly similar to those presented in the paper.4 Groups of identical specimens were tested with a different and constant eccentricity for each test.2. Furthermore. Applicability to structural concrete Several of the contributions to this discussion considered the extent to which the test results presented in the paper are d. It is seen in Fig. however. Though they have not had an opportunity to examine the pertinent test reports published in the USSR. Comparison witlh other data Baker reports that the results of a large number of beam tests carried out at Imperial Colllege agree closely with the results reported in the paper. information regarding the stress distribution similar to that developed by the authors was arrived at. . . and 1. 001 0 10100 A . psi strength properties of stress distribution .I?ijgch’s Tests G .000 g >0 ‘L& . . 0 G G ~---E(22 )2) A &go-o* o ~’L’ A o – PCA Sand “rai. F—Ultimate Prism strength. A ““’”’a’ ‘“ .1396 JOIJRNAL OF THE AMERICAN CONCRETE INSTITUTE Part 2 Dec. 1956 Average Stress = k.003 .= ~ Gravel ~ A 9020 10... ka f: 1 “./’ .004L $ ~ 3 x OJ @3_ —.PCA Lightweight 3000 4020 5000 6000 71XKI KKO 2m0 Concrete Fig. f~. k a) =--<:~ . The variation of P.)/ be ja’. regarding flexural cracking. effects. that development of new special test methods may be helpful in clarifying fundamental principles. and caienary effects. As shown in Fig. at It is further felt that in future studies of these least not at the present time. which he believes is also related to an increase in steel strength by virtue of the fact that the steel is embedded in concrete. He also reported observation of a similar restraining effect of concrete below the neutral axis.nce. Abeles suggested a design equation accounting for such contribution. on the other hand. it is advisable to carry out reinforced concrete experiments under entirely realistic conditions. h’evertheless. models may aid in discovering relationships between properties of different materials. fully developed to reflect actual properties of materials. The authors feel that most hyperstrength effects are of an unreliable and they should therefore not be considered in ultimate strength design. Mechanical models Mechrmical models such as those discussed by Cowan and Rosenblueth When they are careare valuable tc]ols in fundamental materials research. It is possible. pre-tensioned beams. such as a change from sand and gravel to lightweight aggregates. even in cases when ultimate strength was ccmtrolled by compression. and Rosenblueth pointed out that concrete in tension between flexural cracks may contribute to ultimate beam strength. it is felt that the principal purpose of the test method developed should be to clarify principles and to study effects of variables in the composition of concretes. it is evident that consideration must be given principally to tests of reinforced members. = (Pl + P. for insta. 6. and beams of T-section. Bjuggren pointed out the restraining effects of stirrups that may increase the strength of the compression zone in beams. and it is therefore believed that the same values would have been reached in a similar test under gravity ldads. reentrant corners in the concrete compression nature. particularly when the distance to the I’urther sources of hyperstrength effects are strain neutral axis is small.ULTIMATE STRENGTH DESIGN 13Q7 to compare ultimate strength computed by the numerical constants given in the paper to measured ultimate strength of a variety of structural concrete members. or relationships between different properties of the same material. Agreement within a few percent has been found for columns. Rosenblueth used a simple model consisting of one spring and one dashpot in series to in- . Prentis. The ultimate strength design factors given in Fig. + Pz during loading is reported in the paper in terms of klk. hardening of the reinforcement. zone. Hyperstrengih effects In studies of the ‘(hyperstrength” effects mentioned in several discussions. Smith and Young considered the effect of gravity loads. Ashdown. a slight decrease in kllcz took place before final collapse of the specimen for low concrete strengths. 8 and 12 correspond to the maximum value of PI + P1 for each test. although the quantitative estimate may be questioned. The significant question in using the assumption that all “fibers” follow one and the ~samestress-strain curve was in the authors’ view not one of time effects.. (23).2 percent would then be reduced to 0. should be taken as the full yield point stress of the steel. but rather one of a possible effect of the stress gradient on the stressstrain relationship. however. Abeles requested an opinion regarding the applicability of Eq. This is certainly true. this assumption is entirely reasonable in developing ultimate ~strength equations for reinforced concrete members. 1956 vestigate the stress-strain relationship f or concrete stresses below 0. that compression reinforcement is yielding at ultimate strength. b In ultimate strength design. Prentis reported that no effect of strain rate was found in his beam tests.’. However. for reinforcement with yield points not exceeding 60. On the other hand. that this should be done in design only after such beneficial effects have been demonstrated by tests of reinforced members. Thus. In some cases this implies a consideration of beneficial If compression reinforcement of alloy steel quality with a creep effects. it is often assumed. It is felt.1328 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE Part 9 Dec.003 to 0. numerous tests of reinforced concrete beams and columns have shown that linear distribution of strain closely approximates reality if strains are measured over a gage length greater than the average crack spacing. Such application was discussed in a recent paper. the assumption of linear strain distribution is not warranted. it seems logical to consider the beneficial effects of creep that lead to a “pre-compression” of the steel.8 j.8 . All studies of creep in concrete with which the authors are familiar indicate that for the test conditions discussed. (23) to the balanced reinforcement of prestressed beams. Strains at a crack over a short gage length probably exceed these values. the creep strain would be of the order of 1 percent of the elastic strain.005 reported in recent American literature were in most cases measured by 6-in. His conclusion that the stress is less than the product of strain and initial tangent modulus is certainly correct. yield point of about 90.000 psi is used with high-strength concrete. The strength of eccentrically loaded columns subject to high sustained loads has been estimated on that basis.000 psi. The balanced percentage computed by Abeles as 2. As far as further developments of ultimate strength design are concerned. to study local conditions at cracks. it is felt that the effects of creep and repeated loading may better be investigated by tests of reinforced members. but j. CVshould be taken as the yield strain minus the effective prestress strain. o To obtain an approximate value of pb by Eq. Bjuggren raised the question regarding the extent to which creep deformaFor research purtions should be included in the ultimate concrete strain. Strains Rosenblueth suggests that the distribution of strain across a reinforced beam section cannot be linear at flexural cracks. The ultimate concrete strains of 0. electric gages. poses it is certainly convenient to do so. Modulus of elasticity The relationship between modulus of elasticity. 3. Modifications of that sort are indeed possible.Z7O computed by the authors’ ultimate strength factors. which is in reasonable agreement with the values of 0. The authors believe this is due primarily tothe low factor of 2/3 to 0. he prefers to retain the straightline method.77 to 1.ienti$. . State Publishing Office.ULTIMATE STRENGTH DESIGN 1399 percent. Tal.08 percent he quoted from European tests. and c/w ratio suggested by Bjuggren is indeed an interesting one. Hadley’s discussion indicates that he finds it impossible to accept any evi- dence of “immediate plasticity” in concrete. but they lead to complicated design criteria which are basically irrational. 846 pp. 1956. the R values quoted would have been 0. Had a conversion factor of 0. 2. K. 1. (in Russian). (in Russian). He also reported that European tests of beams with high-strength wire reinforcement have indicated higher values of R = lVf. The authors feel that the important similarities between the two cases concerned are the position of the neutral axis defined as the axis of zero strain and the fact that the same gradient of stress and strain is present in both cases. modified in such a way as to give about the same answer with respect to measured ultimate strength as is reached by inelastic analysis. “Ultimate Strength Design of Reinforced Concrete Structure. E.” All-Union S. 19.90 been used. REFERENCES Y 1 VEB Verlag Teehrvik Berlin. It seems reasonable to expect that the constants in his equation may depend on the conditions of specimen storage. The volume of test data involved has during the past two decades become so comprehensive and so conclusive that another paragraph or two in this closure. Moscow. Ministry of Industrial Construction USSR. as has been shown by others. W.300 which is in reasonable agreement with the value O.7 used by Abeles to convert the compressive strength of cubes to cylinder strengths. Structural Engineering Society. ” could scarcely serve a useful purpose. Moscow. devoted perhaps to the “roses of yesteryear.ZJbd2j.241 pp. time. Straight-line theory Many discussions regarding adaptation of ultimate strength design concepts to everyday practice in a design office have been recorded in the literature of reinforced concrete between 1930 and the present time. 83 pp. 123-55-MSPMChF” (1955 design code). (German translation of Russian textbook). Taylor suggested that the authors’ test method does not represent the conditions in the compression zone of a beam.245 to 0.’ than those corresponding to the factors given in Table 2. “Instructions for the Design of Sections of Reinforced Concrete Structural Elements. Sachnowski... K. 1955. “Stahlbeton-Konstruktionen. On the basis that bending takes place about different axes. If the authors interpret Hadley’s discussion correctly.56. 1330 JC)URNAL OF THE AMERICAN CONCRETE INSTITUTE Part $! Dec.)) ACI JOURNAL. Proc. V. 1. J. 727-755. D.. . E. 94 pp. 1955. 52. E. Deutscher Ausschuss fur Stmhlbeton. Jomney. 601-620. and McHenry.. Elstner. pp. 1956.” ACI JOURNAL.. “Ultimate Flexural Strength of Prestressed and Conventionally Reinforced Concrete Beams. Hognestad. Mar. “Versuche zur Festigkeit der Biegedruckzone. Riisch. ‘Sustained Load Strength of 13ccentrically Loaded Columns. V.. M. H.. 1956. Berlin. 1956 4. C. 5. Feb. Pp. Pro. 52.. ” Heft 120... R. R. 6. Viest. and Hognestad.