It is conventional among civil engineers to design structures starting from estimating the load it should resist all the way to calculating the corresponding material stresses and then strains. However, this load-based approach does not guarantee a valid solution all the time, an example being a value of stress in a section can sometimes have two or no corresponding strain values based on its material properties. An alternative approach is to look at the same problem from the other end, setting a strain value as a failure point based on a set of criteria and calculating the capacity at the section, member and structural level. This approach gives a complete idea of the capacity and behavior of structure at each level and hence has become a vital part in the design of structures.
Everything is Non-Linear: Sources of Non-Linearity and Stiffness in Structures
Starting from the material level itself, the stress-strain relationship for reinforced concrete is nonlinear. Even if the materials the structural elements are made of have a linear stress-strain relationship, phenomenon like cracking of the section starts imparting non-linearity at the section, member and structure level. In addition to this, the geometry of the member, boundary conditions and other time dependent parameters can further add nonlinearity to a structure. So, the conventional notion of analysing a structure based on the simple set of equations, [F] = [k][u], do not hold most of the time in a structure and there are no single set of value of stiffness that satisfy these equations except for the case where the structures behave completely linearly. The values of this stiffness, k, vary non-linearly in material in terms of modulus of elasticity and in the sections, members and finally the structures. The nonlinearity in structure therefore should be properly addressed in the design and analysis of a structure.
Structural Mechanics: A Four Player Game
The complete behavior of structures can be understood by four key concepts, viz., actions, stresses, strains and deformations, and the relationships between them. The equilibrium relationships between actions and stresses, the constitutive relationships between stresses and strains, the compatibility/kinematic relationships between strains and deformations, and the stiffness relationships between actions and deformations are the four key relationships between the four players. These relationships can be used to determine the capacity of a structure from a given deformation or the deformations from a given set of loading. The design of structures essentially boils down to the tweaking of these relationships to achieve a desired set of behaviors and hence these relationships become the key players in the game of designing a structure.
Is an Inclined Member a Beam or a Column?
One may consider horizontal elements as beams and vertical elements as columns, but this notion starts to break when we consider an inclined member – is it a beam or is it a column? The orientation of the member whether vertical, horizontal or inclined should not be used to decide whether the members should be treated as columns or as beams, a distinction made to choose the action sets for design. The presence of high axial loads decreases the flexural capacity of a member and hence this should be considered in a design of members which are called columns. Additionally, the presence of P-delta effects in the member are also distinguishing characteristics between columns and beams wherein columns are designed considering P-delta effects while the latter is not.
The Moment Curvature Curve
As a structural engineer, it is important to understand the complete behavior of a structure subjected to a wide range of loading. For the case of axial-flexural members, we are concerned with the sectional response of these members. So, to understand their complete response, we must be able to simultaneously visualize the stress strain behavior of materials, the composite action of concrete and reinforcement, the effect of confinement reinforcements and the equivalent actions of different loadings on the section. All of this information is effectively packed into one curve, the moment-curvature curve, allowing us to completely understand the sections subject to axial-flexural loading from cracking of the section to yielding and all the way to ultimate failure of the section.
Effects of Different Parameters on Moment Curvature and Ductility of Reinforced Concrete Section
Moment curvature curves and its variations with changes in materials and geometry of the section give us a lot of ideas about how we can make a section more ductile and also allows us to see the effectiveness of our preliminary designed structure in terms of its ductility. Larger amount of compressive longitudinal reinforcement, smaller amount of tensile longitudinal reinforcement, lower yield strength, smaller rebar diameter and effective lateral confinement all contribute to a higher ductility of a section. In addition to these properties, the axial load on the section plays a vital role as it has the effect of reducing the moment capacity as well as significantly reducing the ductility of a section.
Insights from Maximum Curvature Surface
What may appear to be an eye candy on the surface, these maximum curvature surfaces are an extension to the normal moment curvature curves and provide a deep insight into the change in ductility of a section when subjected to different levels of axial loading as well as when subjected to biaxial bending. It is evident from this surface that the presence of axial loads significantly decreases the ductility of a section. Similarly, significant effects of biaxial moments are seen in the sections subjected to lower levels of axial loading where circular sections show a lower decrease in their ductility when subjected to biaxial loading when compared to rectangular sections. This allows one to effectively provide ductility through detailing in a section that is to be subjected to biaxial bending.
How do Different Parameters Affect the Capacity of a Section Subjected to Axial-Flexural Loading?
The capacity of a section subjected to Axial-Flexural loading can be visualized with the help of load-moment-moment plot of the capacity interaction surface providing a great deal of information about its different types of capacities – pure axial capacity, pure bending capacity, biaxial bending capacity and a combination capacity of these. Variations like increasing the depth of the section or the strength of the concrete in a section lead to an increase in its capacity against compression-controlled failure. Similarly, increasing the yield strength of the rebar can lead to an overall higher capacity of the section. This brings up an important point that the nature of this interaction surface is only affected by the different cross-sectional properties and is independent of the applied actions.
