Spring Based Analytical Models

A structural system can be modeled using a network of springs with pre-defined spring characteristics. For linear analysis, the stiffness of the spring system is considered constant, while for nonlinear analysis, the system changes as a function of the displacement (strain, curvature, rotation or deformation). Over the years, one of my major goals has been to develop practical and rational nonlinear spring models for realistic assessment of structural systems. Some of the major spring models developed and being developed are:

Models for nonlinear assessment of RC structural members
Models for nonlinear assessment of RC beam-column joints
Models for nonlinear assessment of anchorages in concrete
Models for assessment of RC joints strengthened with FFHRS
Models for nonlinear assessment of RC structures
Models for nonlinear assessment of RC structural members under fire
Models for nonlinear assessment of masonry infilled RC frames

Models for nonlinear assessment of RC structural members

The nonlinear behavior of RC structural members can be modeled through distributed plasticity (e.g. 3D Finite element) approach or through lumped plasticity (e.g. beam elements) approach. The primary nonlinearities in a frame member are flexure (defined by moment-rotation characteristics), shear (defined by shear force-shear deformation characteristics), axial (defined by axial force-axial deformation characteristics), torsion (defined by torsional moment-torsional rotation characteristics) and bond (defined by bond strength-slip characteristics). In lumped plasticity approach, the bond behavior is best considered indirectly in moment-rotation characteristics.

As a pre-requisite to performing nonlinear static pushover analysis, I developed models for nonlinear assessment of RC structural members using lumped plasticity approach. Note that these models were based on formulations available in literature but the most suitable models were identified through validation against a relatively large and versatile database. The formulations used for the generation of nonlinear characteristics are given in details in my PhD thesis as well as in various publications. To optimize the efforts, small VB based programs were written and implemented as macros in MS Excel.

Multi-spring model to simulate nonlinear behavior of beam-column joints

The seismic performance of non-seismically designed reinforced concrete frame structures is often governed by their beam-column joints. The high joint shear forces resulting due to a high rate of change of bending moment over relatively short length might result in joint shear failure characterized by diagonal cracks in the joint core. Such joint shear failures have been one of the major reasons for the failure of several RC frame structures during past earthquakes. Therefore, in order to realistically simulate the seismic performance of non-seismically designed structures, modeling the seismic behavior of the joints of such structures is essential. The nonlinear joint behavior can be captured either through detailed 3D finite element modeling or by modeling the joint through a network of nonlinear springs.

While 3D finite element modeling of joints are able to capture the realistic behavior of the joints, provided good modeling techniques and constitutive laws are used, they are not practical to model the complete structures and generate their capacity curve due to computational efforts and costs involved. Multi-spring models within the framework of lumped plasticity approach offer a good solution to perform nonlinear seismic analysis of RC frame structures considering nonlinear joint behavior. One such model was developed within the framework of my PhD Thesis which uses one rotational spring and two shear springs to simulate the contribution of the joint shear distortion to the global drift, in case of an exterior joint. The model uses critical principal tensile stress as the failure criterion to identify first joint shear cracking and ultimate failure of the joint, in case of joints with beam bars bent in. The model is reliable, fast, accurate and is based on the realistic deformation behavior of beam-column joints under seismic loads. The model is easily implementable in any software capable of performing nonlinear analysis within the framework of lumped plasticity approach. Further details on the joint model can be obtained from the following publication:

Paper Joint Model

Models for nonlinear assessment of anchorages in concrete

The design of anchorages (fastenings) in concrete is traditionally performed using the force-based methods such as concrete capacity design (CCD) method. The design load for an anchorage is calculated corresponding to different failure modes and the least of these values defines the allowable load that can be applied for the anchorages. While calculating the failure loads, it is assumed that all the anchors in an anchor group have equal stiffness and they are all loaded through an anchor plate which is stiff. This approach does not allow or account for any redistribution of the forces in the anchorage due to possibly ductile behavior that can be achieved in case of anchor steel failure or reinforcement failure. This might lead to a non-uniform level of safety in different configurations of anchorages. Furthermore, specifically with respect to the anchorages used to connect the elements required for strengthening (esp. seismic strengthening) of existing structures, the force based design methods often lead to no solutions or unreliable solutions.

A possible solution to improve the reliability and scope of the design of anchorages is by using nonlinear spring models. In this approach, the behavior of anchors is modeled through uniaxial springs associated with idealized spring characteristics. I used this approach first time to consider the nonlinear behavior of the anchorages used for connecting the haunch elements with the concrete frame (see next article below). Later, this approach was also used to model the structure-anchor-piping interaction. My research group is working on extending this approach to develop generalized methods for design of anchorages, which are not covered by the existing design methods.

Model for assessment of RC joints strengthened with FFHRS

Beam-column joints often form the weak links in a reinforced concrete frame structure subjected to seismic actions. To ensure the safety of structures against earthquakes, strengthening of beam-column joints may be frequently needed. One of the best solutions for strengthening of beam-column joints is the fully fastened haunch retrofit solution (FFHRS), in which the haunch elements are connected to the frame structure using post-installed anchors. To assess the behavior of joints retrofitted using FFHRS, within the framework of my PhD, I developed a spring-based model that gives due consideration to the nonlinear behavior of frame members, joint panel as well as the anchorage (fastening) used to connect haunch elements and the frame members. The model is developed within the framework of lumped plasticity approach, is practical and implementable in the commercial software capable of performing nonlinear structural analysis (e.g. Pushover). Furthermore, by associating a suitable hysteretic rule with the model, it can be used for nonlinear dynamic analysis of structures with joints retrofitted with FFHRS. The details of the model can be obtained from the following reference:

Paper Modeling of FFHRS

Models for nonlinear seismic assessment of RC structures

The various models to simulate the nonlinear behavior of different components of the must interact with each other in order to simulate the nonlinear behavior of the structure. All the models developed and discussed above were combined in the structural model and verified against the experimental results. The models are shown to be able to capture the nonlinear behavior of the RC structures under static loads (Pushover analysis), quasi-static cyclic loads (cyclic analysis) as well as dynamic loads (dynamic analysis). The extended pivot hysteresis model was associated as the hysteretic rule for defining the rule for loading, unloading and reloading. With the practical yet reliable models, the performance of the RC structures can be assessed with high accuracy through a reasonable modeling and computational effort.