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 Cracking Cracking 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. Obtainign the strain ath e 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 ther 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 Research Due 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 understandin 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 reinfocred 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.The