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ACI JOURNAL TECHNICAL PAPER Title no. 83-3 Stress-Strain Relationship for Reinforced Concrete in Tension by Domingo J. Carreira and Kuang-Han Chu A stress-strain relationship to represent the overall behavior of rein- sian in Reference 1 is also proposed for the average forced concrete in tension, which includes the combined effects of stress-strain diagram o
  ACI JOURNAL TECHNICAL PAPER Title no. 83-3 Stress Strain Relationship for Reinforced Concrete in Tension by Domingo J. Carreira and Kuang-Han Chu A stress-strain relationship to represent the overall behavior of reinforced concrete in tension which includes the combined effects of cracking and slippage at cracks along the reinforcement is proposed. The serpentine curve previously used for the compression stress-strain relationship is also used in tension with parameters that are physically significant. These parameters can be determined experimentally from reinforced concrete prismatic specimens or estimated from proposed empirical relationships. The effects of the testing procedures gage length shrinkage reinforcement test specimen characteristics cracking and concrete strength and extensibility on the stress-strain diagrams for plain and reinforced concrete in tension are discussed. Keywords: cracking fracturing); extensibility; reinforced concrete; shrinkage; slippage; stress-strain relationships; tensile strength; tension; tension tests. The shape of the concrete stress-strain diagram in tension depends heavily on the testing procedure used. When plain concrete is tested in direct tension, using a testing machine in which the strain rate cannot be controlled, a linear diagram with a brittle failure is usually obtained. Experiments have shown that if the strain rate is controlled, the stress-strain diagram of plain concrete in tension is nonlinear and has well-defined ascending and descending branches. The diagram is also influenced by shrinkage, microcracking, and reinforcement. Testing of plain and reinforced concrete beams confirms the existence of the complete nonlinear stress-strain diagram in tension. A stress-strain relationship is proposed for reinforced concrete in tension which is similar to the relationship for compression proposed in Reference 1. The use of the same type of stress-strain relationship for plain concrete in compression and tension has been proposed in Reference 2. Nonlinear stress-strain diagrams in tension have been used in finite element analysis of slabs. Concrete tensile strength was needed to better fit the analytical results to the experimental data from reinforced concrete slabs. PROPOSED STRESS-STRAIN RELATIONSHIP IN TENSION The same general form of the serpentine curve used for the complete stress-strain relationship in compres- ACI JOURN L January-February 1986 sian in Reference 1 is also proposed for the average stress-strain diagram of reinforced concrete in tension fr = { (EIE;) fr {3-1+(EIE, )i3 1) fr = the stress corresponding to the strain E fr = the point of maximum stress, considered as the tensile strength t, = the strain corresponding to the maximum stress fr { = a parameter that depends on the shape of the stress-strain diagram The average diagram represents the overall or resultant behavior of concrete in tension restrained by the steel reinforcement. By a suitable choice of parameters in Eq. 1), the combined effects of cracking, slippage, and bond along the reinforcement may be included. The experimental determination of parameters fr , E/ and { presents more difficulties than determining the corresponding parameters in compression. Recommended design values for parameters fr , E/ and { are discussed in view of the available test data for concrete in tension. PLAIN CONCRETE IN TENSION, INFLUENCE OF TESTING PROCEDURE The term tensile strength and the stress-strain relationship in tension have no absolute meanings, but must be expressed in terms of the specific test proce dure used. 3 Three kinds of tests have been used for plain concrete testing: the direct tension test, the beam test, and the splitting test. Among the three methods of testing, the results from both the beam and splitting tests are based on the elastic theory. Therefore, only the direct tension test can provide the complete stress-strain diagram in tension beyond the elastic behavior. However, most of the testing machines cannot absorb the energy released when the load on the specimen begins Received June 7, 1984, and reviewed under Institute publication policies. opyright© 1986, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the November-December 1986 ACI JOURNAL if received by Aug. I, 1986. 21  A CI member Domingo J Carreira is an engineering specialist in the Structural Project Division of Sargent Lundy in Chicago, 1/1 He is involved in the design and construction of nuclear and fossil power plants. Dr. Carreira is a member and past chairman of CI Committee 209, Creep and Shrinkage in Concrete; a member o CI Committee 301 Specifications for Structural Concrete; and also a member of the Subcommittee of Nuclear Accreditation of the CI 3591 SME Boiler and Pressure Vessel Code. CI member Kuan-Han Chu is Professor Emeritus of Civil Engineering at the 1//inois Institute of Technology liT) in Chicago. He was employed with the firms of mmann Whitney and D. B. Steinman in New York, N.Y. before joining liT in 1956. Dr. Chu has published numerous articles in various technical journals. to decline. t was therefore concluded that concrete in tension behaves linearly almost to its tensile capacity without any significant descending branch or strainweakening portion of the stress-strain diagram. Nevertheless, the nonlinear behavior of plain concrete in tension has been observed experimentally. Hughes and Chapman 2 and Evans and Marathe 4 have used modified testing machines to obtain stress-strain diagrams in tension with ascending and descending branches similar to those obtained in compression Specimen failure in both the ordinary and the modified testing machines is caused by one crack propagating across the entire cross section and separating the specimens into two parts. The stress-strain diagrams from plain concrete specimens are strongly affected by the location of the crack with respect to the gage length. INFLUENCE OF SHRINK GE A long curing period by immersing the specimen in water prior to testing causes a more uniform strain distribution in the specimen because swelling offsets the effects of the autogenous and drying shrinkage. f test specimens are allowed to dry, however, the moisture migration to the surface causes transient nonlinear strain gradients across the thickness of the specimen. Differential shrinkage strains produce tensile stresses at the surfaces and compressive stresses toward the center. These tensile stresses cause microcracking. When tensile stress is applied to a dried specimen, shrinkage cracks propagate toward the interior of the specimen. This results in an average tensile strength lower than that of saturated specimens. The main factor affecting the test results when shrinkage is present is the location of the zero stress and strain reference point. Fulwood 5 showed the effect of drying and wetting on the modulus of rupture. By immersing 4 x 4 x 20 in. plain concrete beams in water for 1 hr before testing and by testing them when wet, the modulus of rupture increased from 5 to 54 percent more than that of similar specimens cured in water up to 7 days and allowed to dry until testing by third-point beam loading at 28 days. Ledbetter and Thompson 6 were able to measure the shrinkage-or swelling-induced stresses in the concrete caused by the presence of steel reinforcement. Their test specimens consisted of a thick-walled steel tube encased in a 6 x 24-in. cylinder of concrete. The steel tube served to internally restrain the shrinkage, measure strains, and apply the tensile load to the concrete. 22 t was observed trom the external 10a0 versus steel strain diagram that, after the initial crack, the reduced concrete section carried some load even at strains several times that of the one corresponding with the initial crack. When the load was removed to approximately one-half the initial crack load, the unloading line was parallel to the steel-tube load-elongation line. The hor izontal displacement of the unloading diagram was caused mainly by the restrained shrinkage. After further unloading, the diagram curved away from the line parallel to the steel-tube line, which resulted in additional residual strain. This additional strain is due to the crack not closing completely because: 1) broken particles are displaced on the fracture surface, 2) early shrinkage strain gradients are nonlinear and shrinkage stresses are relieved after cracking, and 3) residual creep strain is also present. TESTING OF REINFORCED CONCRETE N DIRECT TENSION Two types of specimens have been used to test reinforced concrete in tension: the direct tensile and the flexural test specimens. To define the concrete stress strain relationship in tension, the direct tensile test specimens are of special interest. Most reinforced concrete direct tension test specimens consist of one reinforcing bar centrally cast in a concrete prism. 7- 11 Specimens with multiple reinforcing bars have them symmetrically arranged along the concrete prism. 7ã  2 Specimens are stressed by applying the load at the ends of the protruding bars. The load applied to the bars is partially transferred to the concrete by bond along the development length at the ends of the prism. The elongation measured on the concrete prisms includes the reinforcement and concrete elongations as well as the slippage at the bar-concrete discontinuities. For uncracked specimens, discontinuities are located at both ends of the concrete prisms. For cracked specimens, discontinuities are also at both sides of each pri mary crack as defined by Broms 7 and described by Goto 8 and Illston and Stevens. 13 Elongations measured within the development length from the prism ends include the slippage caused by the stress transfer. Elongations measured at least one development length away from the prism ends can be used to determine the average stress-strain relationship of reinforced concrete. The minimum length of the concrete prism should be twice the development length 14 plus three to four times the minimum cross-sectional dimension 7 to include at least three primary crack systems within the gage length. Elongations measured from this gage length will represent the initially uncracked portion of the stress-strain diagram as well as the complete diagram while cracking progresses. Elongations measured within the development length from the specimen ends do not represent the ascending branch and the peak strength portion of the stressstrain diagram because there is slippage at the ends of the uncracked specimens included. ACI JOURN L January-February 1986  EFFECT OF INTERNAL CRACKS Broms 7 tested 37 reinforced concrete direct tension specimens and 10 flexural specimens and measured the crack widths at reinforcement stresses up to 69.7 and 66.6 ksi, respectively. He advanced the notion that cracks initiate close to the reinforcement and that crack lengths are governed by the spacing between two adja cent cracks. On this basis, the crack length will decrease linearly with decreasing crack spacing. Broms also defined the mechanisms of crack formation in which primary cracks and secondary cracks will develop as cracking progresses depending on the stress level and the reinforcement geometry). Goto 8 tested one deformed bar direct tensile specimen and reached the following conclusions: 1. Shortly after the formation of primary cracks, small internal cracks are formed around the deformed bars. 2. Secondary cracks do not appear at the concrete surface. 3. Internal cracks on both sides of the primary cracks form cones with their apexes near the bar lugs and with their bases generally directed toward the nearest pri mary cracks. 4. Internal cracks started at a stress level lower than 14 ksi shortly after primary cracks formed. 5. Internal cracks are influenced by the geometrical characteristics of the reinforcing bar deformations. 6. Along the portion of the bars on which internal cracks occur, the adhesion between steel and concrete is lost. Therefore, the bond mechanism depends on the bearing of the concrete cones against the lugs of the bar. 7. Complete relaxation of the external tensile load, after formation of internal cracks, does not return the stress in the reinforcement embedded in concrete to zero, even at primary crack locations. Illston and Stevens 13 used the research data reported in Reference 9 to confirm most of Goto's findings. They also found that concrete surrounding the reinforcing bar is often strained well beyond its tensile capacity and that a loss of bond adhesion may occur over a considerable length of the reinforcing bar, regardless of the type of steel used. Illston and Stevens did not agree with Goto's conclusion that the lugs are generally responsible for the initiation of cracking or that the cracks necessarily coincide with the lugs. They also pointed out that the pattern of internal cracking development is not affected to any considerable degree by the geometry of the lugs. These findings show the existence of a multiple crack pattern in reinforced concrete under tension, whereas one single crack causes the failure of plain concrete specimens. Therefore, the overall behavior of reinforced concrete in tension depends on the restraining action of the steel reinforcement, on crack propagation, and on the formation of a system of primary cracks and secondary internal cracks around and along the bars. ACI JOURNAL January February 1986 BOND·SLIP RELATIONSHIP ND TENSION STIFFENING Two approaches to crack and bond analysis have been used: 1) the bond-slip relationship and 2) tension stiffening by concrete. Analysis of data in References 11 and 12, in light of the discussion on testing reinforced concrete in tension, shows that both bondslip and tension stiffening by concrete are different aspects of the reinforced concrete response in tension. Bond slip relationship The bond-slip relationship for finite element analysis was introduced by Nilson  5 in 1968. To model the bond between steel and concrete, he used closely spaced discrete spring linkages connecting the steel elements with the adjacent concrete elements. The properties of these spring linkages were defined by the local bond-slip curves he derived. Also, the model included provisions for the changing internal topology caused by cracking for the concrete nonlinear behavior in compression and for the bond nonlinearity. In considering the calculated steel displacements versus the stress in the steel, Nilson arrived at the concept of tension stiffening by concrete some years before the first experimental results were reported. In 1971, Nilson 10 tested a steel bar centrally located along the 18- in. length of a 6 x 6 in. concrete block cross section. Bond-slip was calculated from the difference in the bar and concrete displacements at locations along their interface referenced to a transverse plane. Houde and Mirza   conducted an extensive testing program to study the bond-slip relationship. A total of 62 direct tension specimens and 6 beam-end specimens were tested. Specimens were reinforced with a single bar, size No.4, 6, and 8. Fig. 1 shows the applied stress versus total elongation of No. 4 bars concentrically embedded in 16 in. long concrete prisms tested by Houde and Mirza.   Tension stiffening y concrete When reinforcing bars embedded in concrete are stressed in tension, the stress-elongation diagram shows that the concrete assists the reinforcement in carrying the tensile force, even after severe cracking has developed. The concrete contribution is called tension stiffening by concrete or simply tension stiffening. Experimental data on tension stiffening are limited. References 11 and 12 report the only available data on tests performed on direct tensile specimens reinforced with commercial deformed bars. The curves in Fig. 2 show the stress applied to the reinforcement s versus the average strain.  2 AVERAGE TENSILE STRESS STRAIN DIAGRAMS FROM TEST DATA If the effect of slippage at specimen ends is excluded and an average strain compatibility is assumed along the gage length, the contribution of the concrete up to the yield of the bar is given by Eq. 4) in terms of 23  Total elongation on a 400-mm gage mm) 0.2 0.4 0.6 0.8 1.0 ·;;; ; ., .c = .. . .. . Q. Q. ' 70 60 50 40 30 20 Transformed member elongation 6 X Sin.) Series No. 14 No.4 bar 12.7mm) lcf = 6.81 ksi 46.9 MPa) 400 300 ., Q. E ., .c = 2 .. . 2 .. . Q. Q. ' ree bar elongation 100 10 15 20 25 30 35 40 Total elongation on 16-inch gage 10· 3 in.) Fig 1 Applied steel stress versus total elongation from axially loaded tensile specimens in Reference 11 E 500 400 300 ., .c .. = :¥ 200 c. c. ct 100 I I I I I I I I I 1 I_ 236 in. _ 1 1 6000 mm) I 1 1 pb 1 X X 1 pb II r 1 I + I ____ f 'm = : 1/1 I= 138 in. 3500 mm) 10 15 20 25 0.6 :: ;; .c .. 0.4 2 0.2 30 , . a. c. ct Fig 2 Tensi/e stress-strain diagrams for axially loaded test specimens, showing the tension stiffening by concrete, from Reference 12 the average tensile stress. For equilibrium and neglecting the effect of the concrete area displaced by the reinforcement 2) 24 For the average strain compatibility therefore where j 3) j (4) reinforcement and concrete strain, respectively average strain measured elongation divided by the gage length) reinforcement elastic modulus reinforcement and concrete cross-sectional area, respectively As/Ac the externally applied load to the reinforcement stress externally applied to the reinforcement average tensile stress in the concrete Eq. 4) assumes the direct tensile specimens are stressed by applying the load AJs to the reinforcement. For a given elongation Em, the concrete contribution in the average sense j may be found as the difference between the stress externally applied to the reinforcement and the average stress in the embedded portion of the reinforcement s - Em Es , multiplied by the steel ratio p. Oncef is known, the concrete stress-strain diagram can be plotted as in Fig. 3 and 4. When Eq. 4) is applied to the data in Reference 11, the stress-strain relationship obtained is not unique. For example, the peak stress and initial slope differ for each curve because the 16 in. long concrete prism used barely ACI JOURN L I January-February 1986
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