ABSTRACT This paper provides an overview on how a reinforced concrete beam behaves under flexural loading through the moment-curvature diagram. It also describes the three distinct stages a flexural member goes through before collapsing. This paper highlights the effects of varying configuration, dimension, and property of materials used in the capacity of the members by comparing moment-curvatures at critical stages.
For comparison eight rectangular cross-sections, two stress-strain models for concrete (Hognestad (1951), and C35-Kent and Park (1971) Models), three stress-strain models for steel (Priestley et al. (1996) for Grade 60 steel, A1035 – ACI ITG 6 (2010), and SD685 – Wang et al. (2009) Models), two compressive strengths of concrete (35 and 42 MPa), and two grades of steel (Grades 40 and 60) were used in the experiment. The results were then analyzed and the reason for their variation is discussed.
INTRODUCTION Ductility, by definition, is the ability of a member to undergo deformations without a substantial reduction in its flexural capacity (Baji et al, 2011). For reinforced structures where resistance to brittle failure during flexural loading is required, ductility ensures structural integrity. It is known that concrete works well in compression, but not in tension, due its relatively low tensile strength and ductility. This is addressed by the inclusion of reinforcement having higher tensile strength or ductility. It then becomes a composite material known as reinforced concrete. The most common reinforcing material is the deformed steel bar due to its many advantages. Its reinforcing schemes are designed to limit, control and/or resist tensile stresses in particular regions of the concrete which might cause unacceptable cracking and/or structural failure. Behavior of concrete member is important in understanding the possible modes of failure.
Ultimate failure leading to collapse can be due to crushing of the concrete, which occurs when compressive stresses exceeds its strength, yielding or failure of the rebar, which occurs when bending or shear stresses exceeds the strength of the reinforcement, and/or bond failure between the concrete and the rebar. Such behavior can be understood in the generation of moment-curvature diagrams of sections. SIGNIFICANCE OF THE STUDY This paper studies the behavior of reinforced concrete beam specimens with varied dimensions, configuration, and material properties under flexural loading. The moment-curvature diagrams were evaluated and the mode of failure determined. Results of this study will provide future designers an overview of the behavior of concrete beam under flexural loading, a discussion on how different parameters affect the ductility of the beam section, and a guide on what to alter in the design if a certain capacity or mode of failure is desired.
OBJECTIVES This paper must be able to: 1. produce calculations of the moment and curvature of reinforced concrete sections at cracking stage, yielding stage, and ultimate stage; 2. generate moment-curvature diagrams using actual stress-strain diagrams; and3. discuss the effects of different parameters towards the ductility of reinforced concrete beam.
REVIEW OF LITERATURE Basic Assumptions in Flexure Theory In the analysis of flexural members, the following assumptions are made:? Plane sections before bending remain plane after bending. This means that strains above and below the neutral axis are proportional to the distance from the neutral axis as stated in Bernoulli’s principle. ? Strain in reinforcing bars is equal to concrete at the same level, provided that no slippage can occur between the two materials and their bond is sufficient to keep them acting together under different load stages.? The stress-strain diagrams for concrete and steel are known. The stresses in concrete and its reinforcement can be computed from these diagrams.? The tensile strength of concrete may be neglected. ? The maximum strain of the extreme compression fiber of concrete at ultimate strength is assumed to be 0.
003 based on the guidelines of ACI-318. Moment Curvature If a gradually increasing moment is applied to a relatively long (shear will not have a large effect on its behavior) flexural reinforced concrete member until it fails, the member goes through three distinct stages before collapsing. First is the uncracked concrete stage, second is the concrete cracked–elastic stresses stage, and third is the ultimate-strength stage.Uncracked Concrete Stage. At this stage, tensile stresses are less than the modulus of rupture (the bending tensile stress at which concrete begins to crack).
The entire cross section of the beam resists bending, with compression on one side and tension on the other. Figure 1. Stress and strain diagrams at uncracked concrete stage Concrete Cracked–Elastic Stresses Stage. At this stage, cracks begin to develop at the bottom of the beam as the modulus of rupture of the concrete has already been exceeded. The moment at which these cracks begin to form is referred to as the cracking moment, Mcr. Mcr is when the tensile stress at the bottom of the beam is equal to the modulus of rupture, fr. As the loads are increased further, cracks quickly spread up to the vicinity of the neutral axis, and the neutral axis begins to move upward. After the bottom has cracked, the concrete in the cracked zone cannot resist tensile stresses thus the steel resists is alone.
Figure 2. Stress and strain diagrams at cracked concrete stage Beam Failure—Ultimate-Strength Stage. At this stage it is assumed that the reinforcing bars have already yielded. Concrete compression stresses begin to change appreciably and the tensile cracks, as does the neutral axis, move farther upward. Figure 3. Stress and strain diagrams at ultimate strength stage ? is defined as the angle change of the beam section over a certain length and is equal to in which ? is the strain of a beam fiber at a distance y from the neutral axis. The three stages of beam behavior are better illustrated in a moment-curvature diagram. The first stage of the diagram is for uncracked concrete stage.
In this range, the strains are small, and the diagram is nearly vertical and very close to a straight line. When the moment is increased beyond the cracking moment, the slope of the curve will decrease a little because the beam is not quite as stiff as it was before the concrete cracked. The diagram will follow almost a straight line from Mcr to the point where the reinforcing is stressed to its yield point. Until the steel yields, a fairly large additional load is required to appreciably increase the beam’s deflection. After the steel yields, the beam has very little additional moment capacity, and only a small additional load is required to substantially increase rotations as well as deflections. The slope of the diagram is now very flat.
Figure 4. Moment-Curvature Curve Stress-Strain Models To understand the behavior of any material under stress, it is necessary to establish its stress-strain characteristic. However, both concrete and steel have nonlinear actual material behavior. Thus, it is best described by idealized stress-strain models. Hognestad (1951) Model for concrete Figure 5.
Hognestad (1951) Model Hognestad proposed a stress-strain equation for unconfined concrete. In his model, the ascending branch is represented by the equation where ?o = 1.8fc’/Ec. The post-peak branch is represented by a line where stress at 0.
0038 strain is 0.85fc’. C35-Kent and Park (1971) Model for concrete Figure 6. Kent and Park (1971) Model Kent and Park (1971) proposed a stress-strain equation for both unconfined and confined concrete. In their model, the ascending branch is represented by a modification of the Hognestad second degree parabola by replacing ?o by 0.002.
The post-peak branch is represented by a straight line whose slope is defined as a function of concrete strength, where . ?50u is the strain corresponding to the stress equal to the 50% of the concrete strength for the unconfined concrete equal to for fc’ in MPa. Priestley et al. (1996) Model for Grade 60 steel The stress-strain relationship for this model is presented by:fs = 29,000?s (ksi) for 0 ? ?s ? 0.00207fs = 60 (ksi) for 0.
00207 ? ?s ? 0.008(ksi) for 0.008 ? ?s ? 0.
12 A1035 steel – ACI ITG 6 (2010) Model The stress-strain relationship for this model is presented by:fs = 29,000?s (ksi) for 0 ? ?s ? 0.0024(ksi) for 0.0024 ? ?s ? 0.02f = 150 (ksi) for 0.02 ? ?s ? 0.06 SD685 steel – Wang et al. (2009) Model The stress-strain relationship for this model is presented by:fs = 29,000?s (ksi) for 0 ? ?s ? 0.00345fs = 60 (ksi) for 0.
00345 ? ?s ? 0.01(ksi) for 0.01 ? ?s ? 0.
0.097 METHODOLOGY This study has two experimental setups. Both were directed towards determining the moment-curvatures relationship of rectangular reinforced concrete flexural members. Experimental Setup 1A reinforced concrete section with a width of 300 mm, height of 450 mm, concrete strength of 35 MPa, specified steel yield stress of 420 MPa, concrete cover of 40 mm, 2 compression reinforcements and 3 tensile reinforcements with a diameter of 25 mm, and transverse reinforcements with a diameter of 10 mm, was taken as the control specimen. The depth of compression block, curvature, and positive nominal flexural capacity were then calculated just before flexural cracking, right after flexural cracking, flexural yielding, and ultimate stage using the Hognestad (1951) Model for concrete and Priestley et al. (1996) Model for Grade 60 steel neglecting the effect of confinement with the aid of MathCad PTC Prime. The process was repeated with the experimental specimens enumerated below keeping all other configurations and properties the same with the control section.
Case 1: Tensile reinforcement was reduced to 2-25 mm Ø.Case 2: Compression reinforcement was removed.Case 3: Grade of steel was changed to Grade 40.Case 4: Specified concrete strength was increased to 42 MPa.Case 5: Beam width was increased to 400 mm.Case 6: Beam depth was increased to 600 mm.Case 7: Cover concrete was increased to 75 mm. Results were then compared and analyzed.
Experimental Setup 2A reinforced concrete section with a width of 300 mm, height of 450 mm, concrete strength of 35 MPa, specified steel yield stress of 420 MPa, concrete cover of 40 mm, 2 compression reinforcements and 3 tensile reinforcements with a diameter of 25 mm, and transverse reinforcements with a diameter of 10 mm was used. Moment-curvature interaction diagrams of the section were then constructed using C35-Kent and Park (1971) Model for concrete and the steel models enumerated below. Case 1: Grade 60-Priestley et al. (1996) ModelCase 2: A1035 – ACI ITG 6 (2010) ModelCase 3: SD685 – Wang et al. (2009) ModelCase 4: Experimental Data 1 (Grade 60)Case 5: Experimental Data 2 (A1035)Case 6: Experimental Data 3 (SD685) The moment-curvature diagrams were generated by creating a general program in Mathcad PTC Prime for each case and taking different concrete strains at the extreme compression fiber of the section to come up with moment and curvature values that corresponds to one point of the curve. Plotting of all the points gave a moment-curvature curve for the section.
RESULTS AND DISCUSSIONS RESULTS Experimental Setup 1 CASE01234567JustBeforeCrackingc238.814234.748248.
425E-05Table 1. Summary of results Figure 7. Moment-curvature relationship comparison of resultsExperimental Setup 2 Figure 8. Moment-curvature relationship comparison of cases with theoretical properties (Cases 1, 2, and 3) Figure 9. Moment-curvature relationship comparison of reinforced concrete with Grade 60 steel Figure 10. Moment-curvature relationship comparison of reinforced concrete with A1035 steel Figure 11. Moment-curvature relationship comparison of reinforced concrete with SD685 steelANALYSES OF RESULTS Experimental Setup 1 Figure 12. Moment-curvature relationship comparison of Case 0 and Case 1 Case 1 exhibits the effect of the area of the tensile reinforcement.
As area of tensile reinforcement is decreased, depth of the compression block is also reduced resulting to a reduced moment capacity and increased curvature. Figure 13. Moment-curvature relationship comparison of Case 0 and Case 2 Case 2 exhibits the effect of the compression reinforcement. Compression reinforcement helps the compression block. In the summation of forces, it adds a compressing force which could help resist the effect of loads and minimize curvature. Figure 14.
Moment-curvature relationship comparison of Case 0 and Case 3 Case 3 exhibits the effect of the grade of steel. By decreasing the yield strength of steel, depth of compression block stays almost the same but in the summation of forces, tensile force will be lesser resulting to a reduced moment capacity.It is evident that the moment capacity of the beam will increase as yield strength of steel is increased.
However, this does not indicate that the higher the steel grade, the safer the design. There is a risk of over-reinforcing the section. This means that the concrete will crush prior to yielding of tensile reinforcement . The beam will fail in a brittle manner, meaning failure without warning, which should be avoid. Figure 15. Moment-curvature relationship comparison of Case 0 and Case 4 Case 4 exhibits the effect of the specified concrete strength.
If the concrete strength is increased, depth of compression block is also increased. This results to a larger ultimate curvature. Figure 16. Moment-curvature relationship comparison of Case 0 and Case 5 Case 5 exhibits the effect of the beam width. Beam width affects the magnifies the compression block and eliminates the need for compression reinforcement. Figure 17.
Moment-curvature relationship comparison of Case 0 and Case 6 Case 6 exhibits the effect of the beam depth. If beam depth is increased, c is increased resulting to a larger moment capacity. Figure 18. Moment-curvature relationship comparison of Case 0 and Case 7 Case 7 exhibits the effect of the concrete cover.
If concrete cover is increased, the strain for steel is decreased, thus, fs is decreased resulting to a decrease in moment capacity. Experimental Setup 2 A1035 and SD685 are both high strength steels. As seen in their equations, both Priestley and SD685 models have constant values of fs at the middle range strains while the A1035 model did not. In the computations, is was discovered that for A1035 model, given fc’is 35 MPa, the section was over reinforced. Crushing of extreme compression concrete fiber happened first before tension steel yielded. CONCLUSION From the analysis carried out from varying parameters, there are significant differences in the ductility and moment-curvature relationship of the reinforced concrete.
The main observations are:1. As area of tensile reinforcement is decreased, moment capacity is reduced and curvature is increased;2. Compression reinforcement helps the compression block resist the effect of loads and minimize curvature;3. As the yield strength of steel decreased, moment capacity is reduced;4.
As the concrete strength is increased, depth of compression block is also increased resulting to a larger ultimate curvature;5. Beam width affects the magnitude of the compression block;6. As beam depth is increased, c is increased resulting to a larger moment capacity; and7. As concrete cover is increased, moment capacity is reduced. REFERENCES Reddiar, M., 2009, “Stress-Strain Model of Unconfined and Confined Concrete and Stress Block Parameters” Marnie et al, 2015, “High-Strength Flexural Reinforcement in Reinforced Concrete Flexural Members under Monotonic Loading”, pp.
794 Baji, H., 2011, “Investigation of Ductility of RC Beams Designed Based on AS3600” James K. Wight, and James G. MacGregor, “Reinforced Concrete Mechanics and Design 6th Edition” Jack C. McCormac, and Russell H.
Brown, “Design of Reinforced Concrete 9th Edition” Mander, J., Priestley, M. J. N., and Park, R.
(1988). “Theoretical Stress-Strain Model for Confined Concrete.’ Journal of Structural Engineering, 114, 1804. ACI (2011). “Building Code Requirements for Structural Concrete and Commentary.” Report ACI 318-11, Americal Concrete Institute, USA.
Kent, DC., & Park, R. (1971). Flexural Members with Confined Concrete. ASCE Journal of the Structural Division, 97(7) 1969-1990.