The paper describes a study of incremental-iterative solution techniques for geometrically non-linear analyses. The solution methods documented are based on a modified Newton-Raphson approach, meaning that the tangent stiffness matrix is computed at the commencement of each load step but is then held constant throughout the equilibrium iterations. A consistent mathematical notation is employed in the description of the iterative and load incrementation strategies, enabling the simple inclusion of several solution options in a computer program. The iterative strategies investigated are iteration at constant load, iteration at constant displacement, iteration at constant ‘arc-length’, iteration at constant external work, iteration at minimum unbalanced displacement norm, iteration at minimum unbalanced force norm and iteration at constant ‘weighted response’. The load incrementation schemes investigated include strategies based on the number of iterations required to achieve convergence in the previous load step, strategies based on the ‘current stiffness parameter’ and a strategy based on a parabolic approximation to the load-deflection response. Criteria for detecting when the applied external load increment should reverse sign are described.

A challenging example of a circular arch exhibiting snap-through (load limit point) behaviour and snap-back (displacement limit point) behaviour is solved using several different iterative and load incrementation strategies. The performance of the solution schemes is evaluated and conclusions are drawn.

A study of incremental-iterative strategies for non-linear analyses

Nonlinear analysis of free-form shells by flat finite elements

A distributed plasticity model for concrete-filled steel tube beam-columns with interlayer slip

SUMMARY Geometrically or physically non-linear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to ineciency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is dened on a subplane of the degrees-of-freedom (dof’s) hyperspace or a hyperspace constructed from specic functions of the degrees-of-freedom. This unied approach includes many existing methods which have been proposed by various authors. The entire computational process is driven from only one control function which is either a function of a number of degrees-of-freedom (local subplane method) or a single automatically weighted function that incorporates all dof’s directly or indirectly (weighted subplane method). The control function is generally computed in many points of the structure, which can be related to the nite element discretization. Each point corresponds to one subplane. In the local subplane method, the subplane with the control function that drives the load adaptation is selected automatically during the deformation process. Part I of this two-part series of papers fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms. The comparison is carried out by means of a number of geometrically and physically non-linear examples. General conclusions are drawn with respect to the eciency and applicability of the subplane solution control method for the numerical analysis of engineering problems. Copyright ? 1999 John Wiley & Sons Ltd.

Enhanced solution control for physically and geometrically non-linear problems. part i|the subplane control approach

Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.

Arc-length technique for nonlinear finite element analysis

A compatibility method of analysis is developed for the study of the inelastic in-plane behavior of multistory steel rigid frames subject to sidesway. The method provides for the following conditions to be satisfied throughout the frame simultaneously: (1) The internal forces are in equilibrium with the external forces on the deflected frame; (2) the curvature of each portion of each member satisfies the appropriate nonlinear moment-thrust-curvature relationship; and (3) the distortions of all the parts satisfy the condition of continuity of the frame. The analysis is obtained by successive approximations. A special technique is developed to facilitate the analysis at the condition of ultimate load, and in the early stages of instability of the frame. Good agreement is obtained with tests conducted by others on three full-scale frames.

Analysis of Unbraced Multistory Steel Rigid Frames

A procedure for obtaining minimum weight of plastic designed, rigid jointed steel frames, is presented and beam-column interaction is considered. Zonal linearization of the nonlinear relationship between unit weight and plastic-moment capacity of standard rolled shapes allows linear programming. Convergence to a minimum-weight design by considering a limited number of feasible solutions is discussed.

Design of Steel Frames for Minimum Weight

An approximate method of determining the ultimate strength of wide-flange steel columns under biaxially eccentric load is developed. Eccentricity of load is the same at both ends. Equilibrium is enforced only at midlength, and displacements u and v of the center of gravity and the angle of twist ∖a/ are assumed to vary sinusoidally. Results are presented in the form of six sets of interaction curves which cover a wide range of eccentricities. Results of tests by others on 78 specimens with yield stresses ranging from 30 to 60 ksi are compared with the predictions of the interaction curves. The agreement is very good; 51% of the test values are within ± 5% of the predicted values, 80% are within ± 10% and 94% are within ± 15%. A simple procedure for using the interaction curves in the general case where eccentricities at one end differ from those at the other is suggested. Predicted strengths are in good agreement with tests on four specimens by others.

Strength of Steel Columns with Biaxially Eccentric Load

"Based on a thesis submitted ... for the degree of doctor of philosophy in civil engineering".

Edwin H. Gaylord

Papers

Analysis of Unbraced Multistory Steel Rigid Frames

Design of Steel Frames for Minimum Weight

Strength of Steel Columns with Biaxially Eccentric Load

Minimum weight of plastically designed steel frames

Discussion of “Strength of Plate Girders in Shear”