Mechanics Research Communications 56 (2014) 123–129 Contents lists available at ScienceDirect Mechanics Research Communications journal h om epa ge: www.elsevier.com/locate/mechrescom On the effect of constraint parameters on the generalized displacement control method Sofie E. Leon a , Eduardo N. Lages b , Catarina N. de Araújo b , Glaucio H. Paulino a,∗ a Department of Civil Engineering, University of Illinois, Urbana, IL, USA b Center for Technology, Federal University of Alagoas, Maceió, Alagoas, Brazil a r t i c l e i n f o Article history: Received 2 March 2013 Received in revised form 6 October 2013 Accepted 24 December 2013 Available online 7 January 2014 Keywords: Nonlinear solution schemes Generalized displacement control method Arc-length control methods Planar truss Space truss a b s t r a c t In this work, we investigate the generalized displacement control method (GDCM) and provide a mod- ification (MGDCM) that results in an equivalent constraint equation as that of the linearized cylindrical arc-length control method (LCALCM). Through numerical examples, we illustrate that the MGDCM is more robust than the standard GDCM in capturing equilibrium paths in regions of high curvature. More- over, we also provide a geometric and physical interpretation of the method, which sheds light on the general class of path following methods in structural mechanics. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Arc-length control type algorithms are among the most popu- lar and widely used for solving nonlinear problems because they consider simultaneous iteration on both load and displacement variables. The arc-length control method (ALCM) has had measured success in its various forms, which include spherical, cylindrical, elliptical, and linearized versions (Wempner, 1971; Riks, 1972, 1979; Ramm, 1980; Crisfield, 1981, 1983). However, in some cases difficulties in recovering equilibrium paths with various versions of the ALCM have been reported (Carrera, 1994; Feng et al., 1996; Ritto-Correa and Camotim, 2008). To overcome issues of numerical stability near limit points associated with the ALCM, Yang and Shieh (1990) proposed the generalized displacement control method (GDCM). Since its intro- duction, the GDCM has been widely used for structural mechanics applications, including geometric nonlinear analysis of steel, con- crete and composite frames, and thin structures. Additionally, it has been used in the development of several new beam, plate, and shell elements, which are suited for large deformations. Rather than using the GDCM to solve a complex nonlinear problem, this work is focused on analysis of the nonlinear solution method itself. ∗ Corresponding author at: 205 N. Matthews Avenue, 3129 Newmark Lab, Urbana, IL 61801, USA. Tel.: +1 217 333 3817; fax: +1 217 265 8041. E-mail addresses: leon7@illinois.edu (S.E. Leon), enl@lccv.ufal.br (E.N. Lages), catarina@lccv.ufal.br (C.N. de Araújo), paulino@illinois.edu, paulino@uiuc.edu (G.H. Paulino). The success of the method in solving a nonlinear problem depends on the selection of an initial load factor. Typically, a small value is chosen to capture complex nonlinearities, but the method can yield poor or non-convergent results for slightly larger values of the load factor. In this work, we present a modification to the GDCM constraint equation that yields convergent results even at high values of the initial load factor. We will also show that this slight modification results in an equivalent formulation, namely in the constraint equation, as the linearized cylindrical arc-length control algorithm (LCALCM). Recently, the difference was briefly mentioned by Leon et al. (2011) but was not fully studied, hence, this paper investigates and elaborates on the methods in detail. In a related study, Cardoso and Fonseca (2007) also analyzed the GDCM and identified that it could be expressed as an orthogonal ALCM. The remainder of this paper is organized as follows: Section 2 outlines the incremental-iterative procedure and introduces the nomenclature used in this work. Next the GDCM is presented in Section 3. In Section 4, we discuss the motivation for the modifica- tion to the GDCM and show that the resulting constraint equation is equivalent to that of the LCALCM. We highlight the difference between the original GDCM and the MGDCM through numerical examples in Section 5. Finally, concluding remarks are made in Section 6. 2. Incremental-iterative procedure The discrete equilibrium equation is f(u) = p, where the inter- nal forces, f, are a function of the displacements, u, and the applied 0093-6413/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2013.12.009