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INVESTIGATION OF THE EFFECTS OF DIMENSIONAL AND MATERIAL PROPERTIES ON THE MOMENT-CURVATUER REALTIONSHIP OF REINFORCED CONCRETEWritten Report byROBERTO GABRIEL CORNADO P. CAPULONGSubmitted toMARNIE GIDUQUIOIn partial fulfillment of the requirements inMCE 614 – REINFORCED CONRETE DESIGNMASTERS OF CIVIL ENGINEERINGMAJOR IN STRUCTURAL ENGINEERINGUNIVERSITY OF SAN CARLOSJANUARY 2018?ABSTRACTReinforced concrete is a highly resilient composite material used in the construction of structures that takes advantage of the contrasting mechanical properties of steel and concrete. Due to its prevalent use, it is important to understand the behavior of reinforced concrete under different configurations and conditions and what affects its performance as a material.The Moment-Curvature graph is a method of graphically describing the behavior of reinforced concrete beam sections by plotting the curvature of the section at against its flexural load at a given instance. This document aims to explore the relationship between the curvature of a concrete section and the moment. This is accomplished by investigating the curvature-moment capacity diagram of a given section to set up a control, and then proceed to manipulate different parameters of the concrete section such as the overall beam dimensions, number of tensile and compression reinforcement and strength of steel and concrete. The values of the depth of compression block, the flexural capacity and the curvature at the points before the onset of initial cracking, immediately after the onset of initial cracking, at the yielding of the tensile steel and finally at the crushing of concrete. These points define the different phases of the behavior of concrete under flexural loading.Figure 1 graphically illustrates the results of the first seven cases, using the Hognestad Model for stress-strain I concrete and using the Priestley et al model for Grade 60 steel to generate the moment-curvature graph. Figure 1. Summary of Moment-Curvature Graphs?INTRODUCTIONReinforced concrete is a highly resilient composite material used in the construction of structures, that takes advantage of the contrasting mechanical properties of steel and concrete, whose interplay of properties allows for the construction of highly complex structures that are both structurally sound and economical. Concrete as a material takes advantage of its high compressive strength, low coefficient of thermal expansion, resistance to weather and fire, its workability in its fluid state and its relatively low cost is paired with steel’s properties of high tensile stress, ductility and toughness to compensate for concrete’s low tensile strength and brittle nature.Due to the non-homogenous nature of concrete, the behavior of concrete is typically described in phase or stages, each phase defined by a separate function, rather than defining its behavior using a single function. Therefore, a considerable amount of effort has been exerted to define the behavior of its individual components and how they behave under stress as well as the relationships between the different properties of steel and concrete to simulate and understand the behavior of reinforced concrete as a seemingly homogenous material.Concrete is mainly regarded for its compressive strength, however, it has a very small resistance to tensile stress which significantly affects how reinforce concrete behaves. Thus, it is important to also study the stress-strain relationship of concrete under tension as well as that in compression.Figure 2: Stress Strain Curve for unreinforced ConcreteThe behavior of steel is different from that of concrete and is evident in the stress –strain curve of steel. The curve is defined by three regions, all of which occur at all grades of steel. The three regions or phases of steel are the elastic phase which is the region defined by a linear function, the yielding phase defined by the plateau, hardening phase defined by the positive parabolic arc and necking, defined by the negative parabolic arc. Figure 3. Illustrates these phases against the stress-strain curve for reinforcing steel.The primary focus of the discussions will be the behavior of concrete under flexural loading. As such, concessions and basic assumptions need to be made in order to set the working parameters for the analysis. The basic assumptions of flexure area. Plane sections remain plain before and after bendingb. The strain in reinforcement and the concrete is directly proportional to its distance from the neutral axis.c. Using a given stress-strain curve model for steel and concrete, the stresses for each material for a given strain can be computed.Reinforced concrete has three principal points of interest that will be investigated. The firs is the point on onset of initial cracking determined by the rupture of concrete under tension.?Initial Onset of CrackingCracking occurs when the stresses in the bottom most fiber of the concrete reaches the modulus of rupture of concrete, fr, That is, the strain due to tension is equal to the tension rupture strain of concrete. Having obtained this value, the critical moment of rupture can be computed using the equation: From this, we can then proceed to calculate the curvature at cracking moment as defined by the equation: This gives us the first critical point in our moment-curvature model, the cracking point.Yielding StageAfter the onset of cracks, the concrete loses its ability to resist tension and the tension force it was resisting prior to rupture will now be loaded unto the tension steel as the crack propagates throughout the section.  As this occurs, the section undergoes increased strain without significant increase in the flexural load capacity. Once the steel takes on the tension demand from the concrete, moment-curvature graph once again follows a liner behavior similar to that prior to concrete cracking, until the tension steel reaches the yielding point. At the point of yielding, the strain at the tensile reinforcement is equivalent to: The depth of the compression block can be computed using the equilibrium of internal forces, resulting in the equation: Having obtained the depth of the compression block, c, we can then proceed to compute the strain at various depth of the concrete by proportioning the depth with the strain at the depth under investigation. Obtaining the strain at the level of the tensile reinforcement, compression reinforcement and at the center of application of the compressive force, we can then compute the compressive force in the concrete compression block, the moment capacity of the section at flexural yielding and the curvature at flexure yielding.Given these, the second critical point of the moment curvature model, the yielding point, may now be obtained. In actual practice, this is the point practitioners are most concerned of as structures are typically designed in a manner that the steel yields first under tension before the concrete yields under compression or crushing. This is what commonly referred to as under-reinforced design.Ultimate Strength PhaseAt the onset of yielding, the section takes on significantly less flexural load in proportion to the increase in strain, if there is any increase in flexural capacity at all. The strain in the section gradually increases until it reaches the strain value of 0.003, where the concrete reaches its maximum stress, fc’.Significance of the ResearchDue to the prevalent use of reinforced concrete, wherein the key criteria is always the life and safety of its occupants, it is important to understand how concrete behaves in order to properly design our structures. An investigation into the moment curvature diagram of the reinforced concrete section gives allows us to understand the relationship between the two factors that drive the design of structures, the load demands, in this case flexural demand, and the serviceability requirements of the structure, which is described by the deflection of the members. Knowledge and an in depth understanding of the relationship between these two demands allows for optimization of the structure under design based on the serviceability requirement and load demands without sacrificing one or the other. Understanding the effects of various parameters on the moment capacity and curvature of the concrete is also of importance, in order to properly solve design problems in a strategic manner.This paper aims to answer or to produce the following at the conclusion of this report:a. To be able to produce calculations to determine the moment capacity and the curvature of the section at the different stages of failureb. To be able to produce a moment-curvature diagram of reinforced concrete and to define the relationships between the twoc. To be able to produce illustrate and describe the effects of manipulating the different parameters of on the moment curvature diagram of reinforced concrete?REVIEW OF RELATED LITERATUREThe moment curvature diagram of reinforced of concrete is dependent on the ability to determine the stress and strain relation of concrete and steel as individual materials.  Countless studies have been conducted to try and define this relationship. The succeeding computations will rely on three different stress strain models.Hognestad Stress- Strain Model for Concrete Figure 4. Hognestad Stress – Strain Model for ConcreteThe Hognestad model graphically illustrates the relationship of the strain against the compressive strength of concrete. In this model, the ascending branch is defined by the second degree equation:? The ascending function terminates when the stress in the concrete reaches the value of fc, and the strain in the concrete reaches the value 0.002. The post peak function of the stress strain curve is a linear function defined by the equation: This descending function terminates at a strain value of 0.0038 and a stress value of 0.85fc’. This descending function is marked by the shallow slope, designating an appreciative increase in strain while the compressive strength of the concrete gradually decreases until crushing.METHODOLOGYEight cases were observed using the Priestley and the Hognestad’s stress strains models. The eight cases were analyzed for their flexural behavior by graphing their moment curvature models. In all case, the critical points for each were defined and investigated, namely the initial cracking stage, the flexural yielding stage and lastly the ultimate stage. A control section was set up and was investigated first to determine a baseline set of data, while the other cases were derived from the control section with one property altered for every case.Calculations were completed using Mathcad while graphical representations were made using Microsoft Excel.Table 1 presents the dimensional and material properties of the different sections under investigation. It also presents the variations made for each case, as shown by the parameters highlighted.?Table 1. Summary of Geometrical and Material Properties of Each CaseParameter Description Case 0 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7b, mm Section width 300 300 300 300 300 400 300 300h, mm Section depth 450 450 450 450 450 450 600 450d’, mm Concrete Cover 40 40 40 40 40 40 40 75As’, mm2 Area of Compression Steel Reinforcement 982 982 0 982 982 982 982 982As, mm2 Area of Tension Steel Reinforcement 1473 982 1473 1473 1473 1473 1473 1473fc’, MPa Compressive strength of Concrete 35 35 35 35 42 35 35 35fy, MPa Yield Strength of Steel 420 420 420 275 420 420 420 420 ?Table 2 summarizes the stress strain models used to construct the moment –curvature relationship graph for the control section. ?ANALYSES OF RESULTSUsing the methodology described previously, the moment curvature diagram for reinforced concrete was obtained from eight different concrete sections. The effects of the various parameters will be discussed and their corresponding effects on concrete will be extracted from the data acquired.Presented below in table 3 are the numerical values of the depth of compression block, moment capacity and curvature of each case at the instant before and after cracking, at the point of yielding and at the point of ultimate stress and strain.Table 3. Summary of Calculation ResultsCASE 0 1 2 3 4 5 6 7Before Cracking c 238.81 234.75 248.96 238.81 238.56 238.17 317.99 237.88 Mcr 44.50 42.11 42.52 44.50 48.21 56.37 79.98 37.99 ? 6.60E-07 6.48E-07 6.93E-07 6.60E-07 6.81E-07 6.58E-07 4.94E-07 6.57E-07After Cracking c 238.81 234.75 248.96 238.81 238.56 238.17 317.99 237.88 Mcr 44.50 42.11 42.52 44.50 48.21 56.37 79.98 37.99 ? 3.66E-06 5.11E-06 3.76E-06 3.66E-06 3.96E-06 4.65E-06 3.21E-06 4.56E-06Yielding c 133.80 110.79 147.19 146.11 129.73 119.39 159.83 133.00 My 199.13 135.41 198.49 94.31 200.10 202.14 283.22 177.31 ? 7.88E-06 7.23E-06 8.32E-06 4.14E-06 7.76E-06 7.46E-06 5.30E-06 9.11E-06Ultimate c 71.90 59.63 80.62 71.90 66.72 62.52 73.85 87.58 Mn 218.43 150.26 219.02 135.69 221.32 223.10 309.46 200.15 ? 4.17E-05 5.03E-05 3.72E-05 4.17E-05 4.50E-05 4.80E-05 4.06E-05 3.43E-05?Table 4 presents a summary of the increase and decreases in terms of percentages in the dependent parameters against the results of the dependent parameters of the control section.Table 4. Increase in Dependent Parameters in PercentageCASE 0 1 2 3 4 5 6 7Before Cracking c 0.0% -1.7% 4.2% 0.0% -0.1% -0.3% 33.2% -0.4% Mcr 0.0% -5.4% -4.5% 0.0% 8.3% 26.7% 79.7% -14.6% ? 0.0% -1.9% 5.0% 0.0% 3.1% -0.3% -25.1% -0.4%After Cracking c 0.0% -1.7% 4.2% 0.0% -0.1% -0.3% 33.2% -0.4% Mcr 0.0% -5.4% -4.5% 0.0% 8.3% 26.7% 79.7% -14.6% ? 0.0% 39.6% 2.7% 0.0% 8.3% 27.1% -12.4% 24.7%Yielding c 0.0% -17.2% 10.0% 9.2% -3.0% -10.8% 19.5% -0.6% My 0.0% -32.0% -0.3% -52.6% 0.5% 1.5% 42.2% -11.0% ? 0.0% -8.3% 5.6% -47.4% -1.6% -5.4% -32.8% 15.6%Ultimate c 0.0% -17.1% 12.1% 0.0% -7.2% -13.0% 2.7% 21.8% Mn 0.0% -31.2% 0.3% -37.9% 1.3% 2.1% 41.7% -8.4% ? 0.0% 20.6% -10.8% 0.0% 7.8% 15.0% -2.6% -17.9%From table 3 and 4, the value of c, Mcr and the ? for Case1, where the tensile steel reinforcement was reduced by one bar, or by 33%, all dropped, although not significantly except for the moment at yielding and nominal moment capacity which saw a decrease of 30%, proportional to the decrease in tensile steel area. The curvature after cracking and at the ultimate stress point where they increased which denotes that the deflection of the member sharply increased after the point of cracking while at the same time significantly reducing the amount of additional flexural steel it can withstand. Reducing the tensile steel in a section forces the compressive depth block to become shallower in order to satisfy the equilibrium of internal forces. These contribute not only to a decrease in moment capacity but also an appreciable increase in member deflection.Comparing the results the control against case 2, where the compression steel was removed completely, it can be observed that no significant change in any of the dependent parameters occur. The largest increase in in Case 2 is the increase in the depth of the compressive stress block, a response to the loss of compression steel wherein the increased depth compensates for the lack of compression steel.Comparing the control with case 3, wherein the steel grade was changed from grade 60 to grade 40 reducing the specified yield strength from 420 MPa o 275 MPA, a difference of 35%, there is no appreciable difference before or after the initial crack onset. However, there is a significant decrease in the moment capacity at yielding at the ultimate nominal moment capacity. A significant decrease of 53% and 38% was experience by the steel due to the decrease in yield strength. To achieve equilibrium within the internal forces, the concrete compression block also decreased its depth by 48% to compensate for the reduced tensile strength of the steel.Case 4, where the compressive strength of concrete was increased by 20% shows no comparable difference to the control. However, there is an increase in the critical rupture moment of concrete by 8% as well as the critical strain at the onset of cracks, thereby delaying their occurrence. However, the increase in moment capacity post cracking is less significant due to the fact that much of the section moment capacity is now owed to the tensile steel rather than the concrete.Case 5 sought to explore the effects of increasing the section width by 33.33% in relation to the moment capacity and curvature of the section. An increase in section width increased the critical moment of rupture of the concrete section by 26.7% prior to cracking without significant changes to the deflection. However, upon the onset of cracking, the section experience an increase in deflection by 27%. This combined with a high critical rupture moment is an undesirable scenario wherein a structure has undergone considerable deformation, without the presence of any of the tell tales signs of damage which is cracking. Post cracking, the depth of compression block has decreased to compensate for the increase in section width. Notable also is the 15% increase in deflection at the nominal moment capacity of the section.Case 6 presents an increase in the overall section depth by 150mm or 33%. This increase in beam depth has significantly increased the moment capacity of the section at all stages while decreasing the deflection of the member. Increasing the beam depth effectively increases the moment arm of your tensile steel area, increasing the moment capacity of the section. To compensate, the depth of the compression block also increased to achieve equilibrium. The increase in moment capacity at all stages, while decreasing the deflection significantly makes this section modification ideal.Case 7 saw the concrete cover increase by 35mm, which is a common case for concrete structure poured against earth. Before and after the onset of cracking, no significant change in the depth of the compression block occurred, however, the increased concrete cover decrease the moment arm of the tensile steel area, which resulted in a decrease in the critical moment capacity of the section during cracking. The decrease in the depth of the tensile steel can also be attributed to the increase in deflection at the onset of cracking.From the data acquired from the investigation of the different sections, the following can be said for the relationship of the moment capacity of a concrete section, the deflection or curvature of the member and the various parameters in the design on flexural concrete members: a. Tensile Reinforcement – decreasing the tensile reinforcement decreases the depth of the compression block as well as the Moment capacity at yielding and nominal moment capacity. At the nominal moment capacity, the deflection is also significantly larger due to less tension members counteracting the deflection. The structure will deflect significantly before the concrete collapses under crushing.b. Compression steel – lowers the depth of the compression block. If removed, the deflection at nominal moment capacity is reduced. Onset of concrete failure occurs at the same time as control without appreciable deflection to foreshadow collapse.c. Steel grade/Specified Yield strength of Steel – reduction of steel grade results in earlier onset of yielding, at significantly lower deflection. Yielding occurs much faster at 50% of the yielding moment of the control. An increase in flexural load post yielding will lead to a proportionally significantly lager deflection with onset of concrete collapse occurring at a much smaller demand with the same deflection. This condition is ideally a desirable failure scenario.d. Increased concrete strength – no appreciable change from the control. Serves to marginally increase concrete tensile capacity.e. Increased beam width – increased concrete tensile strength and increases deflection at the point of concrete collapse at the same ultimate moment capacity as the control.f. Increased beam depth – increasing the beam depth directly changes several of the beam’s properties including the depth of the steel tensile reinforcement and the depth of the concrete compression block and tension block, thereby effectively increasing the moment capacity of the section at all stages, while decreasing the beam deflection at the point of yielding and at the point of concrete crushing. Despite this, the concrete section still allows for appreciable deflection between yielding and collapse.?CONCLUSIONThe relationship between the moment capacity of the concrete and its deflection at various stages was observed through numerical data and graphical representation. These results are then used to predict the behavior of concrete, particularly in relation to how the section will fail under certain conditions and what conditions will yield the desirable outcomes or modes of failure. As observed, concrete should not only be defined by the amount of load it can withstand but also by how much it can deflect.From the analyses made, it can be concluded that the optimal method of increase the concrete’s strength and increase its serviceability by increasing the section overall depth.The analyses also shows that the reinforced concrete section will have significantly less ductility if the tensile resistance of the section is reduced, either by decreasing the tensile steel area of by decreasing the specified yield strength of steel.  It then stands to reason that increasing these parameters would in turn significantly increase its capacity and resistance to deflection. However, the question of which material will fail first will come into play with the increase in the sections tensile steel strength. This condition is recommended to be observed in future studies.