Plastic-zone advanced analysis – Formulation including semi-rigid connection

Abstract

A new numerical formulation includes the behavior of the semi-rigid connection into plastic-zone finite element model for steel plane frames. This Bernoulli-Euler formulation is an expansion and generalization of the slice technique with updated Lagrangian co-rotational system, presented before. Only one spring is placed in the finite element end node to allow the study of plasticity inside the member. The connection semi-flexibility factor eta controls this spring. It is defined for the beam length and can classify the connection as rigid or flexible. The eta effects on the vertical displacement function, the generality of the proposed model and the corresponding stiffness matrices are shown. The usage of another eta based on the finite element length is also explained. The advantages of semi-rigid connection behavior are discussed. Several answers were obtained, considering the connection stiffness and the inelastic behavior. This technique is applied to solve three benchmarked examples, applying connections of constant stiffness and semi-rigid curves from an existing data bank. The semi-rigid connection economy on beams, its appearing improvement of columns capacity, and the reduction of portal frame limit load justify this research. These examples are minimum requirements to develop the advanced direct analysis, which power is explored in the last one.

Introduction

Today it is common the talk about semi-rigid connection and its influence on steel structures analysis. The first research is from 1917, while at late fifties many tests, papers and concepts (beam-line, slope-deflection, and wind-connections) were already known [1]. This subject is definitely on the minds of researchers, designer engineers, and in the current specifications [2], [3], [4].

This semi-rigidity arises from the changes of angle (for instance theta, θ) between the beam and column due to the transferred moment (M), as pictured in Fig. 1(a). In some cases, the axial and shear forces can participate in the analysis [5], but these effects are often neglected, due to lack of data or the simplicity of the adopted models. The connection behavior can be simply described by the moment-rotation graph, usually called M-θ (M-theta) curve, shown in Fig. 1(b). The two possible models to consider the connection in the past, like the pinned-end or the fully fixed, represent the axes of this graph.

There are many types of connections with different M-θ curves [6], as endorsed by Fig. 1(c), where some are approximated from SCDB (Steel Connection Data Bank) [7], with the following meaning: SP (single plate), HP (header plate), DWA (double web angles), FEP (flush end-plate), TSA (top- and seat-angle), TSDWA (TSA with DWA), and EEP (extended end-plate).

The majority of these curves are flexible (or simple), which mean can resist a little moment (nonzero), with big rotations. These are the preferred type because of being easier to fabricate, ship, assembly, and have the required moment capacity for some steps. The most rigid attain nearly 65% of the beam plastic moment (M_p), after a reasonable rotation. Stiffer connections lead to high costs, which is undesirable for construction [8]. The challenge is to determine the adequate connection, data and cost for a given usage in a building. This is a complex subject too, because even when the information provided by the current literature is helpful, still can be insufficient for design.

Looking at M-θ curves and models, many recently works can be cited, as listed: (1) several laboratories experiments were done all round the world [9]; (2) computer 3D models representing the scaled connections [10]; (3) curves defined by direct empirical equations were proposed [11]; (4) curve fitting by selected numerical parameters that are raised to the power of statistically treated exponents, for a series of connection data [12]; (5) springs in series models associated to components behavior [13]; (6) the column “joint behavior” (panel zone) separated from the connection rotation [14], etc. Other researchers were compiling all available technique [15].

In this work, as in general, the semi-rigid behavior is accessed by the M-θ curve only, which represents the own connection; as seen in Fig. 2(a). This implementation is a little inaccurate since the transition is at the column flange outer face. The next steps of research can involve: (1) the shear effect and eccentricity of Fig. 2(b), and (2) the joint rotation due to the panel zone deformation, indicated in Fig. 2(c) [14].

All this reveal that when is evaluated the plasticity together with second order effects, the semi-rigid connection must be included in the analysis due to its high nonlinear behavior. Still there are several ways to introduce the connection behavior into finite element (FE) method. In the past, this began with elastic analysis from Monforton and Wu [16]. Step by step, research was going to the inelastic models, like the refined plastic-hinge (PH) approach from Chen and his co-authors [7], [17]. Meanwhile, Chan and his team adopted a hybrid FE [18], [19]. In both cases, there is a pair of springs working together: one for plasticity evaluation and other for connection behavior. Ackroyd [20] introduced the connections in plastic zone (PZ) analysis, following the same strategy of [17], with the portal method. Foley [21] solved 3D models of multistory building, using supercomputers with parallel processing (for vectoring, sub-structuring and condensing the big structural matrices) to reduce the solution elapsed time.

Since 2010, many researches about advanced analysis, several with semi-rigid connections, were published worldwide. Some have different FE formulations, including 3D and dynamic analysis of seismic performance [22], [23]. Meanwhile, in Brazil, the basic slice technique was shown by Lavall [24], Silveira and his scholars [25] followed the cited researchers [17], [18], [19]. Ziemian and associates allowed the talks of these concepts at universities, using the Mastan2 program (3D inelastic analysis with two kinds of refined PH model), for teaching purposes [26].

Today, this world trend is also in Brazil, with PZ works [27], [28], [29]. Beyond the called “main aspects” studied before [28], that are the initial imperfect geometry and the residual stress (RS); now this PZ advanced analysis includes the semi-rigid connection formulation presented here.

However, this PZ technique is different from others, as it uses several FE by member. And although the displacements provide the rotations, the moments are defined by stress integration of the slices, which are subjected to plasticity and RS effects. Therefore, the point calculated by the analysis can be out from the given M-θ curve, what requires reduce the load increment and increase the number of iterations. In short, the connection curve brings new requirements to attend.

The main objective of this work is to develop the FE with the connection at only one side, to study plasticity inside the member. Even though many formulations were available, only few are suitable for the slice technique, among them, some follows a different order for the connection evaluation or have only one FE by member, etc. The beam example from Section 3.1 [30], among others tests, evidenced the need of several iterations, before reach the desired answers. One advantage is that the number of iterations grows slowly, insofar as the plasticity spreads.

The similar idea of placing a zero length FE with the spring stiffness inside appeared with the Stardyne package [31], which the author worked with as a program user (1978–80). The Stardyne also had sophisticated routines to condense and minimize the structural stiffness matrix. Nowadays, these routines can be installed wherever needed, but the software of this work has none. New packages (e.g. OpenSees [32]) also condense the structural matrix including the spring equations, with the same method shown before [7], [17], [18], [19]. However, with the present formulation, the stiffness matrices generation and assembly is simpler, since every FE has nonzero length and standard [6:6] matrix size. Moreover, the formulation shows the explicit interaction between the connection and the FE theory, as seen in the study of eta, of the Section 2.4. This idea was born in 2004, presented firstly at the Qualifying (2008), and is one of the major contributions of the Thesis [29].

In this paper, the basis of proposed formulation, the parameter for connection control η (eta), and the FE stiffness matrix are shown. Three numerical examples are detailed and discussed, including the linear M-θ connections, the SCDB standard semi-rigid ones and the advanced analysis too. The simplicity, reliability, and quality of this method as research tool are proved.