Question

Give two examples of how ductility can be used to analyze a statically indeterminate structure at its strength limit state, as though it were statically determinate.

Answer 1

Good Morning Sir/Madam

The problem stated above to analyze Statically Indeterminate structure by using DUCTILITY concept is explained below:

Nowadays, reinforced concrete (RC) structures are largely used in engineering works from small constructions to tall buildings and bridges. The possibility to build any kind of geometry and format for the structural systems have given to reinforced concrete the status of general construction material. The costs associated to the construction of the buildings and other structures are still competitive when compared to several materials. Furthermore, RC as a structural material, especially when used for example in statically indeterminate structures, gives to the elements the capacity to behave according to the assumptions specified by the designer of the structural system, since the designing and execution of the structure respect the design hypothesis, limitations imposed by the codes and good procedures in situ. Regarding to the RC beams, the most part of these structural elements is designed to support bending moments and shear forces that come from the external loads. But, in general, the bending moment effects are more important than the shear ones, especially when the beams have higher values of the span-to-depth ratio, in which the shear strains can be neglected and the Euler-Bernoulli’s hypothesis governs the overall structural behavior and mainly the ultimate strength of those elements.

Among many aspects required in RC beams and slabs design, ductility has become mandatory by the standard codes (^{ABNT NBR 6118: 2014}; ^{EUROCODE 2: 2004}; ^{ACI 318: 2002}). In this context, ductility can be defined as the ability to support large plastic deformations before failure without significant resistance loss. The main reasons to consider ductility as a mandatory characteristic in the modern structural design are: ductility prevents brittle ruptures, which is a failure mode that must always be avoided; elements with ductile behavior have higher plastic rotation capacities when compared to brittle elements and contribute to large deformations/displacements before a physic rupture (^{Ko et al. 2001}); ductility of cross sections are essential to provide bending moment redistribution along the beam as longitudinal reinforcement steel yields ensuring the redundant behavior of hyperstatic structures (^{Kara and Ashour 2013}). Another important application in which the ductility is essential to guarantee safe behaviors of RC structural systems is related to dynamic loads generated by seismic tremors. In such cases, the ductility of the structural elements must be predicted and quantified in a detailed way to avoid severe damage and brittle failures of the buildings (^{Lopes et. al 2012}; ^{Arslan 2012}; ^{Demir et al. 2016}).

However, the prediction and the assessment of ductility as a single value to describe how ductile or brittle is an element or some cross section is not an easy task. There are several parameters interacting each other that influence the ductility at ultimate limit states. Moreover, it is impossible to dissociates ductility from rigidity of a RC element because they are function of almost the same parameters as: longitudinal reinforcement ratio, concrete strength, concrete softening branch in compression, crack pattern development, tension stiffening, bond-slip relationship (^{Lee and Pan 2003}; ^{Oehlers et al. 2009}; ^{Lopes et al. 2012}). The curvature depends on the material strain levels and their limits. On the other hand, these strain values are function of the neutral axis position, which influences the effective depth and longitudinal reinforcement ratio.

In order to at least guarantee a ductile behavior for RC structural elements, the design codes impose some restrictions to the relative neutral axis position (β_x) on the cross section determinations.

^{EUROCODE 2 (2004}) and ABNT NBR 6118 (2014) recommend for ensure ductility in RC beams the following values: β_x ≤ 0,45 for f_ck ≤ 50 MPa and β_x≤ 0,35 for f_ck > 50 MPa at ultimate limit state. Therefore, when the codes adopt those recommendations for the neutral axis position, the balanced condition, in which β_x = 0,628 (limit between domains 3 and 4), is no longer accepted because it can lead to a brittle failure (^{Araújo 2009}). Although this form to solve the problem of ductility is simple and intuitive, it does not quantify the ductility of the designed RC cross sections. Moreover, several important mechanisms that interfere on the overall behavior of the RC beams are not take into account, such as the evolution of damage along time as the loading conditions change; confinement effect of the compressed concrete provided by stirrups and tension stiffening (^{Oehlers et al. 2013}).

Damage models (^{Mazars 1984}; ^{La Borderie 1991}; ^{Flores-Lopes 1993}; ^{Cervera et al. 1996}) have been adopted successfully to represent the rigidity loss of the structural elements as a function of the loading conditions. In the most of damage models, the damage is defined by a scalar parameter that quantifies the local state of degradation and penalizes the material stiffness at that point. To perform the damage assessment, there are several constitutive models with their own assumptions and hypothesis about the starting and evolution of the damage. But as a general approach, damage models depend on the stress/strain state at a point (integration points localized along the finite elements) and an evolution criterion for damage. By adopting damage model approaches, the limitation on the EI values and its variation along the structural elements is overcome.

In the context of the absence of explicit quantification methods for ductility, ^{Lee and Pan (2003}) proposed an algorithm based on simplified analytical equations to RC beams designing considering that ductility factor as an input data. In their approach, the authors considered the ^{Kent and Park’s (1971}) analytical model for concrete in compression to take into account the confinement effect caused by the transversal reinforcement. ^{Daniell et al. (2008}) studied the ductility in RC elements considering the plastic rotation capacity as a measure of ductility and associated it with ability to redistribute bending moments. ^{Arslan (2012}) carried out a parametric study varying some parameters listed in the design codes in order to ensure enough ductility for RC structures when subjected to earthquakes.

The concept of aanalysis of a Indeterminate structure by ductility showed influence of the confinement effect in RC beams ductility provided by shear reinforcement, considering a numerical approach via FEM. The confinement effect was incorporated to a unidimensional FEM mechanical model through a calibration procedure of the internal damage parameters adopted by the scalar Mazars’ damage model. The calibration was carried out by a Nonlinear Least Square Method based on the Gauss-Newton strategy solution. Therefore, the effect of the confinement was automatically incorporated to the constitutive concrete compression law after concrete reach the stress × strain curve peak. The paper’s main contribution is related to the use of a simplified numerical approach based on the calibration of the internal damage parameters taking into account the confinement effect. If the constitutive law of the material is known, the proposed procedure can be applied, which gives a general status to the model. A parametric analysis was carried out in a RC beam to study the effect of the confinement on the ductility, varying the neutral axis position, concrete strength and volumetric transversal reinforcement ratio.

These are some of the application methods for analyzing Statically Indeterminate structures using DUCTILITY parameters.

Thanking you Sir/Madam