Second-order Geometric Effects Research Articles

Four-Element Pseudodynamic Hybrid Simulation of a Steel Frame with Cast Steel Yielding Connectors under Earthquake Excitations

Cast steel energy dissipative components are increasingly being incorporated in structural systems in recent years. Cast steel yielding connectors (YC) demonstrate a stable symmetric hysteretic response with increased postyield stiffness at large deformations due to second-order geometric effects. In the present study, the seismic performance of a four-story steel structure equipped with cast steel YCs is investigated through pseudodynamic hybrid simulations using the University of Toronto 10-Element Hybrid Simulation Platform (UT10), where the structure is subjected to three maximum considered event (MCE)-level earthquakes. Four YCs in the seismic force–resisting system are represented physically, whereas the rest of the structure is modeled numerically. These experiments are performed as part of an initiative to produce a database of high-fidelity system-level benchmark test results and to critically study and improve existing numerical models, their calibration, and the loading protocols used in their calibrations. The experimental results can be used to study these aspects in the numerical models for YCs or systems with similar hysteretic characteristics. The hybrid simulations are followed by cyclic tests on two of the YCs to evaluate the remaining low-cycle fatigue life of the YCs and to provide information on the cumulative plastic ductility capacity of YCs.

Hazard-Independent Stability Sensitivity Study of Steel and RC Frame Structures

Structural stability relates directly to the robustness of the system even against the abnormally large load or an unexpected event which might cause perturbation- changes from the normal state of the structural system- from significant damage. This study aims to examine the sensitivity of frame systems (primarily steel moment resisting frame systems) to the initial damage and second-order geometric effects, that may arise as a result of the design load and abnormally large load coming from the unexpected event. Incremental analysis is used to track the development of second-order effects. Planar Frame models are first examined to establish the patterns of the stiffness losses occurred with various cases of hazard-independent damages. The comparison of the anticipated behavior on Reinforced Concrete (RC) frame systems is investigated through buckling analysis of steel and RC frame systems. Observing the patterns, the study is extended to a 3D model, four-story moment frame structure, located in a coastal area and exposed to a design hurricane event, thereby addressing multi-hazard issues. The impact from the amount and location of the hazard-independent damage as well as the complexity of the frame system is studied for steel frame system which generates the overall idea of individual member perturbations and stability failure of the system, as a whole.

Deterministic Wind Load Dynamic Analysis of High Rise Steel Buildings Including P-Delta Effects

This study concerns with the investigation of the second-order geometric nonlinearity effects of P-Delta analysis on the dynamic response of high rise steel buildings due to deterministic wind load. Linear and nonlinear time history analyses were conducted to analyze different tall steel building models adopted in the study. Five steel building models ranging from 10 to 50 stories were numerically modeled and analyzed using finite element code ETABS (version 16.0.3). Deterministic dynamic wind load per ASCE 7-10 is applied to the buildings as a main lateral load. Comparative study between linear and nonlinear time history analyses reveals that nonlinear time history analysis including P-Delta effects displayed larger values of buildings lateral sway than those of linear time history analysis. Generally, including P-Delta effect in the nonlinear analysis increases the flexibility of the building structure, and thus increases response peak values and that peak values occur at a longer time periods indicating lesser response oscillations. The study recommends that P-Delta effect need to be addressed by any dynamic wind analysis for tall steel buildings with 20 story height or more.

Frequency Domain Analysis for Geometric Nonlinear Seismic Response of Tall Reinforced Concrete Buildings

This paper aims to study the second-order geometric nonlinearity effects of P-Delta on the dynamic response of tall reinforced concrete buildings due to a wide range of earthquake ground motion forces, including minor earthquake up to moderate and strong earthquakes. The frequency domain dynamic analysis procedure was used for response assessment. Reinforced concrete building models with different heights up to 50 stories were analyzed. The finite element software ETABS (version 16.0.3) was used to analyze reinforced concrete building models.
 The study reveals that the percentage increase in buildings' sway and drift due to P-Delta effects are nearly constant for specific building height irrespective of the seismic design category assigned to the building. Generally, increase in building lateral displacement and story drift due to P-Delta effects for all seismic design categories is less than 2% for 10 story buildings, whereas this increase for 20 stories or taller buildings is significant with a maximum value around 16% for 50 story building. As for column forces, the study shows that, generally, columns bending moment increases and shear force decreases when P-Delta effects accounted for. In conclusion, the study recommended that the effects of P-Delta need to be addressed for all SDCs allowed by ASCE7-10 and the most important factor to abandonment P-Delta effects is the building height limit.

Energy components in nonlinear dynamic response of SDOF systems

The equation of motion for the response of SDOF systems is presented in terms of energy for a general class of hysteretic relationships that include stiffness degrading inelastic and bilinear elastic behavior, for systems subjected to earthquake ground motions in the presence or absence of P-delta effects. The evaluation of expended energy is presented in a continuous form, along with the evaluation of input energy associated with second-order geometric effects. Examples illustrate the dissipation of energy through damping and hysteretic response and the increase in input energy associated with P-delta effects.

Nonlinear analysis of hexagon-based tensegrity ring: Effect of slackened and yielded cables

This paper addresses static analysis of tensegrity rings as the last generation of tensegrity systems. It discusses also combined geometric and material nonlinearities. A numerical iterative method based on updated Lagrangian formulation is used. The Lagrangian formulation is subjected to Crisfield’s spherical arc-length constraint. The resulting algorithm is new as it takes into account some slackening cables and yielding of others on the whole structure nonlinear behavior. Second-order geometric effects are included through higher order nonlinear stiffness matrices. Material nonlinearity cables-based yielding is modeled by means of the tangent stiffness modulus which is used not only to evaluate elastic rigidity matrix but second order elastic rigidity matrices as well. Results obtained from single or assembly of several hexagon-based ring cells submitted to different loading are discussed in detail.

Immobilizing 2-D Serial Chains in Form-Closure Grasps

The immobilization of nonrigid objects is a relatively unexplored area in grasp mechanics. In this paper, we consider the immobilization of freely moving serial chains of n hinged polygons using frictionless point fingers. We first set this problem in the context of classical grasping theory by showing that chain immobilization can only be achieved with equilibrium grasps. Then, we describe two immobilization approaches based on first- and second-order geometric effects. Based on curvature effects, chains of n ≠ 3 hinged polygons with nonparallel edges can be immobilized by n + 2 frictionless point fingers. Serial chains of three hinged polygons form an exception to this rule. Based on first-order geometric effects, we describe how to immobilize any chain of n hinged polygons with only one extra contact for the entire chain, using a total of n + 3 frictionless point fingers. Moreover, the immobilizing grasps are robust with respect to small contact placement errors. The results are illustrated with examples and described as readily implementable procedures.

A Linear Complementarity Approach for the Non-convex Seismic Frictional Interaction between Adjacent Structures under Instabilizing Effects

The paper deals with a numerical treatment of the dynamic hemivariational inequality problem concerning the elastoplastic-fracturing unilateral contact with friction between neighboring structures under second-order geometric effects during earthquakes. The numerical procedure is based on an incremental problem formulation and on a double discretization, in space by the finite element method and in time by the Houbolt method. The generally nonconvex constitutive contact laws are piece-wise linearized, and in each time-step a nonconvex linear complementarity problem is solved with a reduced number of unknowns.

Load and Resistance Factor Design and Analysis of Stepped Crane Columns in Industrial Buildings

Load and Resistance Factor Design (LRFD), generally known as Limit States Design (LSD) outside of the United States of America, is replacing working stress design (WSD) worldwide for good reason. Not only does it provide much more uniform reliability with load and resistance factors based on the statistical variation of the relevant variables but also, by implicitly eliminating unnecessary conservatism at the same time, it can effect economies as well. A major advantage of LRFD over working stress design is inherent in the fact that the ultimate limit states, those associated with failure of all or part of the structure, are checked against factored load combinations, which are chosen to have the required low probability of being exceeded. This means that both the second-order geometric effects, resulting from the deformation of the structure and the non-linear effects due to material behavior can be taken into account in a straightforward manner at the load levels associated with failure. The load deformation response can be determined to whatever load level is desired. Herein this level is taken as that approaching the formation of the first plastic hinge. In working stress design the response is assumed to be linearly elastic but is not established beyond the working load level. Second-order effects cannot be included directly. The differences in the two analyses are demonstrated by White and Hajjar who show responses for various types of analyses.

Finite-Element Model for Thin-Walled Space Frame Flexible Connection Behavior

A finite-element formulation for modeling flexible connection behavior in space frames composed of thin-walled members is presented. The finite-element model is based on a distributed linear spring representation over the physical contact area of the connection that is capable of representing translational, rotational, and warping deformations of the connection. Second-order geometric and nonlinear moment-rotation response of the connection is considered. An updated Lagrangian formulation is used to model second-order geometric nonlinear effects. A Ramberg-Osgood representation is used to model the nonlinear moment-rotation connection behavior. Details for transforming the resulting stiffness equations for connection-to-member eccentricities are included. Results for a cubic, portal space frame composed of I-section and C-section members are presented to demonstrate the versatility of the developed element.

Combined flexure and torsion of I-shaped steel beams

Design standards provide little information for the design of I-shaped steel beams not loaded through the shear centre and therefore subjected to combined flexure and torsion. In particular, methods for determining the ultimate capacity, as is required in limit states design standards, are not presented. The literature on elastic analysis is extensive, but only limited experimental and analytical work has been conducted in the inelastic region. No comprehensive design procedures, applicable to limit states design standards, have been developed.From four tests conducted on cantilever beams, with varying moment–torque ratios, it is established that the torsional behaviour has two distinct phases, with the second dominated by second-order geometric effects. This second phase is nonutilizable because the added torsional restraint developed is path dependent and, if deflections had been restricted, would not have been significant. Based on the first-phase behaviour, a normal and shearing stress distribution on the cross section is proposed. From this, a moment–torque ultimate strength interaction diagram is developed, applicable to a number of different end and loading conditions. This ultimate limit state interaction diagram and serviceability limit states, based on first yield and on distortion limitations, provide a comprehensive design approach for these members. Key words: beams, bending moment, flexure, inelastic, interaction diagram, I-shaped, limit states, serviceability, steel, torsion, torque, ultimate.

On Work-Hardening Adaptation of Discrete Structures Subjected to Dynamic Forces in the Presence of Second-Order Geometric Effects*

ABSTRACT The present paper deals with work-hardening adaptation of discrete rigid-plastic structures subjected to loadings which vary within a given domain according to an unknown history, with inertia forces, viscous forces, and second-order geometric effects also being included. Considering structural elements characterized by a piecewise linear yield surface, a piecewise linear work-hardening law, and a linear strain-rate sensitivity law, we give an adaptation criterion of “statical” type as well as a method of providing a priori bounds to deformation parameters such as displacement, plastic strain, and plastic strain intensity. These bounds can be rendered most stringent by solving a minimization problem of mathematical programming on an equivalent boundary value problem of finite plasticity. A simple application concludes the paper.

A formulation of the force method in the range of large displacements

Abstract The force method is formulated with regard to analysis problems of discrete elastic structures in the range of large displacements (but small strains). It is shown that the usual concepts — degree of redundancy, redundant force , release , etc. — which characterize the force method in the range of infinitesimal displacements are still valid in the present context (second-order geometric effects). The degree of redundancy is affected by the rank of the geometric stiffness matrix, while the (external) flexibility matrix of the primary structure proves to be the sum of an elastic flexibility matrix and a geometric one. A few comments on computational aspects and on the need of future developments conclude the paper.

Adaptation of rigid-work-hardening discrete structures subjected to load and temperature cycles and second-order geometric effects

W. Prager has recently introduced the notion of rigid-plastic shakedown. The present paper gives: (1) an extension of Prager's shakedown theorem to the general case of discrete structures and for a broader class of hardening rules, with temperature and geometric effects included; (2) two different methods of bounding maximum shakedown deflections. The results obtained turn out to be a nontrivial analogy of the elastoplastic shakedown theory. Essential differences occur because there is no direct relationship between plastic deformations and residual stresses in the case of rigid-plastic structures.

Mathematical programming methods for displacement bounds in elasto-plastic dynamics

In elastic-plastic structures subjected to dynamic external actions, if unbounded plastic deformations are developed, either local failure due to plastic fatigue (alternating plasticity) or gradual divergence of the deformed configuration (incremental collapse) will occur. Therefore, the boundedness in time of total plastic strains, and hence of total plastic work (usually referred to as adaptation or shakedown) is necessary for structural safety, in the sense that it rules out the occurrence of the above critical phenomena. Necessary and sufficient conditions for shakedown have been established by several authors. However, in many instances adaptation is not sufficient to ensure safety. In fact, even if plastic deformations can be proved to be finite, they can exceed some critical limit or exhaust the material ductility. In particular, for dynamic loading histories that cease after a certain time, a structure will certainly shakedown under any load amplitude, so that a safety criterion based on this event is clearly meaningless. Typical histories of this kind are earthquake or blast loadings. When the loading history is known, it is possible, in principle, to assess safety by following the actual plasto-dynamic evolution of the system, but this is often a laborious task and in several cases it provides far more information than is actually needed. On this remark rests the interest of methods capable to provide some essential information, such as upper bounds on maximum deflections or strains, through a moderate computational effort. In recent years several alternative techniques have been developed to bound from the above various quantities of interest. With the exception of very simple situations, the best bounds that can be obtained through any one of these methods involve the solution of constrained optimization problems. In this paper a study of several deformation bounding techniques is performed. The problem is formulated and the main previous results are outlined first with reference to general continua made of hardening materials. Then a class of discrete structural models (such as some finite element discretizations) is considered and, on this basis, two categories of deformation bounding techniques are described from the previous main results. All these techniques, some of which are new, permit the optimization of the upper bound by solving one or more mathematical programming problems of special forms. Some of the bounding procedures are shown to have merely theoretical interest, since they lead to cumbersome numerical procedures or to very coarse bounds. The formulations that appear to have practical application are compared from various standpoints (type of loading history, different hardening rules, influence of second order geometric effects, quantities to be bounded) and first assessment of their practical usefulness is attempted. Generalizations to second-order geometric and thermal effects and to situations in which the time history is not completely known are envisaged.

Inelastic Analysis of Reinforced Concrete Frames

The imposed rotation method for the inelastic analysis of frames rests on the interpretation of the actual bending moment distribution as the superposition of linear elastic moment responses to loads and to unknown plastic rotations regarded as imposed strains. The method given herein is a generalized formulation, covering second-order geometric effects and moment-axial force interaction. Finite element models of frames and piecewise linear moment-rotation laws for critical sections are assumed. Recent algorithms for solving quadratic programming and linear complementarity problems are shown to be efficiently applicable to the analysis of multistory frames. The safety factor with respect to local failure because of “brittle” flexural behavior turns out to be attainable by a suitably modified linear programming procedure.

Bounds on total dynamic deflections of elastoplastic structures allowing for second-order geometric effects

A word w is obtained by an arbitrary n-pattern interpretation of a word x if there are n homomorphisms h1,h2,…,hn and a positive integer k such that w=hi1(x)hi2(x)…hik(x) with 1⩽ij⩽n for all 1⩽j⩽k. Arbitrary multiple pattern interpretations of words are naturally extended to languages. We start with a short parallelism between pattern descriptions and arbitrary multiple pattern descriptions, and then investigate some closure properties of the families of languages obtained by arbitrary multiple pattern interpretations of finite, regular, and context-free languages, respectively. We show that the first of these families forms an infinite hierarchy and give a characterization of the arbitrary multiple pattern interpretations of finite languages. Two concepts of ambiguity and inherent ambiguity of arbitrary multiple pattern interpretations are defined. It is shown that both properties are decidable for arbitrary multiple pattern interpretations of finite languages but strong ambiguity is not decidable for arbitrary multiple pattern interpretations of context-free languages.

Upper Bounds on Deformations of Elastic-Workhardening Structures in the Presence of Dynamic and Second-Order Geometric Effects∗

Abstract Discrete models of elastoplastic structures are considered, Piecewise linear yield conditions and hardening rules are assumed. On this basis, a deformation bounding method resting on the use of fictitious loads as proposed first by Ponter [6, 7], is developed for situations in which: (a) the geometry changes affect the equilibrium equations but their effects may be expressed by bilinear terms in the pre-existing stresses and additional displacements (“second-order geometric effects”); (b) inertia and viscous damping forces play a significant role. Comparisons are made with different bounding methods previously established by the author [3,4], for the same classes of structures and mechanical situations.