Igor Planinc

A quadratically convergent algorithm for the computation of stability points: The application of the determinant of the tangent stiffness matrix

Abstract This paper presents a quadratically convergent algorithm for the computation of stability points (limit and bifurcation points) in the finite element formulation of the problems of the nonlinear structural mechanics. One such approach was given by Wriggers and coworkers [Comput. Methods Appl. Mech. Engrg. 70 (1988) 329–347 and Int. J. Numer. Methods Engrg. 30 (1990) 155–176]. Their approach employs the eigenvector equation K T ψ = 0 as the characterization of the stability point. In the present paper, an alternative approach is proposed, which uses the equation det K T = 0 as the stability point condition. In combination with Newtons iteration scheme, this condition has so far been considered unsuitable for the implementation in the finite element analysis because it has been believed that it requires the full assembly of the derivative of the global tangent stiffness matrix, which is a very costly operation. Objectives of our paper are: (i) to derive a new quadratically convergent algorithm for the computation of stability points, which uses the condition det K T = 0 for the characterization of the stability point; (ii) to show that the full assembly of the global tangent stiffness matrix derivative is not required, and that the linearization of the determinant of the global tangent stiffness matrix can be done in an element-by-element fashion; and (iii) to indicate that, in terms of the number of algebraic operations, memory requirements and the computer programming effort, our algorithm practically equals the one based on an eigenvector equation, but has the advantage of a larger radius of convergence. An additional benefit of the proposed algorithm is that it can also be used without any modification as a comprehensive path-following procedure to calculate regular and singular (stability) points. The effectiveness of our algorithm is illustrated by numerical examples. Its convergence characteristics are compared with those of the algorithm based on the eigenvector equation. We show stability analyses of elastic-plastic planar frames. The algorithm is, of course, valid generally and is not limited to this particular kind of structures.

Buckling of layered wood columns

Slip between layers, material properties of layers and geometric non-linearities largely dictate both the bearing capacity and the ductility of a composite beam. That is why the accuracy of their modelling in the numerical analysis of the composite beams is of utmost importance. In the present paper we present a new strain-based finite element model which considers these issues in a highly effective manner. Each layer of the beam is modelled by geometrically exact Reissners beam model. The layers are assumed to stay in the contact during deformation but the relative tangential displacement (slip) is possible. The non-linear load-slip law of the interface is considered. The formulation is found to be accurate, reliable and computationally time-effective. The further objective of the paper is an analysis of the buckling force of axially compressed layered wood columns, being simply supported, fixed-pinned or continuous. We compare the present numerical results with the analytical values of [Girhammar UA, Gopu VKA. Composite beam-columns with interlayer slip-exact analysis, J Struct Engng ASCE 1993;199(4):1265-82] and with the values, recommended by the European code for timber structures [Eurocode 5, Design of timber structures, Part 1-1: General rules and rules for buildings, 1993; ENV 1995-1-1]. The comparisons indicate that the European code for timber structures provides very conservative estimates for the buckling load.

Abrasion Resistance of Concrete in Hydraulic Structures

The phenomenon of hydraulic structure concrete abrasion, caused by the water-carried sediment abrasion process, is analyzed in this paper. There was detailed investigation of laboratory procedure adequacy for definition of concrete abrasion level in standard ASTM C1138 hydraulic structures. Assessment was based on comparison of laboratory results and natural condition measurements of concrete abrasion resistance by test plot performance in the Vrhovo hydro power plant (HPP) stilling basin. There was adequacy testing of the concrete built in the HPPs evacuation structures on the Lower Sava River. There was modification of basic concrete composition by pozzolanic or polymer additives, along with granular rubber, polypropylene fibers, steel fibers, and the primary binder. A qualitative similarity was shown in the analysis on concrete abrasion level between field measurements and laboratory simulations, as well as ASTM C1138 laboratory method suitability for concrete abrasion resistance assessment in the spillway of the lower Sava Rivers HPP chain. Quantitative results comparison showed a good correlation between natural environment measurements and laboratory measurements for concrete at 900 days, while this correlation was not confirmed for the concrete at 90 days.

THE LATERAL BUCKLING OF TIMBER ARCHES

This paper presents the stability analyses of glulam arches subjected to distributed vertical loading. The present analysis employs a strain-based formulation of a nonlinear geometrically exact three-dimensional beam theory. The influence of the relative height of the arch on the lateral buckling load is studied. The buckling load is determined by bisection method with observing the sign of the determinant of the tangent stiffness matrix. The post-critical load deflection path is traced by a modified arc–length method. Such influences are shown for arches with a constant cross-section or constant volume. After determining the most favorable height of the arch, the influence of the number and position of lateral supports is shown. We also compare the deflections, bending, and radial stresses at the lateral buckling states to the limit values which are recommended by European standards.

BUCKLING OF AN AXIALLY RESTRAINED STEEL COLUMN UNDER FIRE LOADING

Analytical procedure, based on the linearized stability analysis, is presented for the determination of the buckling load and the buckling temperature of a straight, geometrically perfect, axially loaded steel column subjected to an increasing temperature simulating fire conditions. The nonlinear kinematical equations and the nonlinearity of material are considered. The stress strain relation for steel at the elevated temperature and the rules for reduction of material parameters due to increased temperature are taken from European standard EC 3. Theoretical findings are applied in the parametric analysis of a series of Eulers columns subjected to two parametric fires. It is found how the slenderness of the column, the material nonlinearity, the temperature dependence of material parameters, and the stiffness of restraints at supports effect the critical temperature. While these parameters have major influence on the critical temperature, they have no effect on the shape of the buckling mode.

Analytical solution of linear elastic beams cracked in flexure and strengthened with side plates

A new mathematical model and its analytical solution for the analysis of the stress–strain state of a linear elastic beam cracked in flexure and strengthened with plates on its lateral sides is presented. Both the longitudinal and the transversal interactions at the side plate/beam interface are considered. Linear behaviour of the contact connection is assumed. The method is based upon the linearised planar beam theory of Reissner. The weakening of the beam induced by the flexural crack is modelled conventionally as a rotational spring. The suitability of the theory is demonstrated in a case presentation involving the comparison between analytical results of the present beam (one-dimensional) model, the experiments and the numerical results of a full three-dimensional solid model created in the LUSAS finite element analysis software. An excellent agreement between the results is observed and the proposed formulation is found to be accurate and reliable. Finally, the solution is employed in an engineering analysis, discussing the effects of the material and the geometric properties of selected characteristic cases of the observed beams on the static and kinematic quantities, including the boundary conditions of the side plates, the longitudinal and the transversal stiffness of the connection, the size of the cracks, the span of the beam, and the length and the stiffness of the side plates. For the cracked cantilever beam, a substantial effect of any of these parameters is found. In contrast, for the cracked two-span continuous beam, only the effect of the stiffness of the side plates and the effect of the length of the beam spans are noticeable.

Buckling of Slender Concrete-Filled Steel Tubes with Compliant Interfaces

THIS PAPER PRESENTS AN EXACT MODEL FOR STUDYING THE GLOBAL BUCK-LING OF CONCRETE-FILLED STEEL TUBULAR (CFST) COLUMNS WITH COMPLI-ANT INTERFACES BETWEEN THE CONCRETE CORE AND STEEL TUBE. THIS MOD-EL IS THEN USED TO EVALUATE EXACT CRITICAL BUCKLING LOADS AND MODES OF CFST COLUMNS. THE RESULTS PROVE THAT INTERFACE COMPLIANCE CAN CONSIDERABLY REDUCE THE CRITICAL BUCKLING LOADS OF CFST COLUMNS. A GOOD AGREEMENT BETWEEN ANALYTICAL AND EXPERIMENTAL BUCKLING LOADS IS OBTAINED IF AT LEAST ONE AMONG LONGITUDINAL AND RADIAL INTERFACIAL STIFFNESSES IS HIGH. THE PARAMETRIC STUDY REVEALS THAT BUCKLING LOADS OF CFST COLUMNS ARE VERY MUCH AFFECTED BY THE INTERFACIAL STIFFNESS AND BOUNDARY CONDITIONS.

Buckling Loads of Two-Layer Composite Columns with Interlayer Slip and Stochastic Material Properties

This paper presents an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The model is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure is investigated for a different number of random variables, sample points, and various degrees of response surfaces. The results show that the probability of failure is considerably affected by the type (deterministic or probabilistic) of the loading and its distribution.

Optimal size, shape, and control design in dynamics of planar frame structures under large displacements and rotations

Size, shape, and drive optimization procedures are combined with an energy-conserving time-integration scheme for the dynamic analysis of planar geometrically non-linear frame structures undergoing large overall motions. The solution method is based on the finite-element formulation, employing the classical displacement-based planar beam finite elements described in an inertial frame. Finite axial, bending, and shear strains are taken into account. If the system is conservative, the energy and momenta conservation in the discrete system during motion is guaranteed. Size, shape, and drive design variables are introduced into the model. Shape parameterization is achieved by the design element technique, using Bezier patches. The sensitivity analysis is performed by the discrete approach and the analytical direct differentiation method. A gradient-based optimization method, using an automatically adjustable convex approximation technique, is employed. The efficiency and the applicability of the approach are demonstrated via numerical examples. The shape and the driving function of a load-moving robot arm are optimized to reduce oscillations in its final position. The shape of a steel frame is optimized to reduce oscillations after an idealized ground motion jerk.

Three-dimensional bimetallic layered beams with interface compliance: Analytical solution

This paper focuses on the development of a new mathematical model and its analytical solution for the analysis of the mechanical behavior of geometrically and materially linear three-dimensional two-layer bimetallic beams with interface compliance. Consequently, the analytical solution of bending of elastic three-dimensional two-layer composite beams with interface compliance is derived for the first time. In the illustrative example, a three-dimensional two-layer cantilever composite beam made of 6061-T6 aluminum and C83400 red brass is analyzed. It is shown that interface compliance could have a significant influence on the mechanical behavior of such a structure. Finally, the results for different mechanical quantities are tabulated and as such the analytical solution presented can be used as a benchmark solution of three-dimensional bimetallic composite beams.